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5. describe how relative and absolute dating techniques are used to determine the 
subdivisions of geologic time; 
6. explain how the geologic time scale helps to recount the history of the Earth; 
7. describe how seismic wave data has been used to develop a model for the 
internal structure and composition of the Earth; 
8. create a scale drawing to represent relative thicknesses of the layers of Earth’s 
interior, including the crust, lithosphere, asthenosphere, mantle, outer core, and 
inner core; 
9. distinguish among comets, meteoroids, asteroids, and dwarf planets, and 
describe how they help us to understand the nature and formation of the Earth 
and the Solar System; 
10. gather information from secondary sources to discuss the regular occurrence of 
meteor showers; and 
11. explain how modern research about celestial objects uses new space technologies 
including telescopes and space probes. 
Performance Standard


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detrimental effects that electromagnetic radiation may have on living things. 
Suggested Performance Tasks 
A. Design a model vehicle from recycled materials using the Law of action-reaction to carry a 15-gram payload over a 5-meter displacement. 
Describe the forces interacting and the motions evident when your vehicle is moving, including any vectors that are relevant. 
B. Use secondary sources regarding problems associated with the distribution of electrical energy from power plants to homes. Use the 
findings to develop a system that can address the problems. 
C. Develop a poster that identifies the useful and dangerous attributes of the 7 main energies of the EMS. 
 
 
GRADE 9 SECOND QUARTER - Earth and Space Science 
Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners… 
1. Scale, proportion 
and quantity 
2. Plate boundaries 
3. Structure of the 
Earth 
4. Geologic time 
1. Evidence for continents 
moving includes jig-saw


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adversely affect animals 
and plants in a variety of 
ecosystems. 
1. use models and labeled diagrams to represent the double helix structure of DNA 
(deoxyribonucleic acid); 
2. explain the role of DNA, genes, and chromosomes in the transmission of traits; 
3. describe mutations as changes in DNA or chromosomes and discuss some of the 
factors that cause mutations, such as infectious agents, radiation, and 
chemicals; 
4. use information from secondary sources to explain the beneficial, harmful, and 
neutral effects of mutations; 
5. explain the advantage of high biodiversity in maintaining the stability of an 
ecosystem during difficult conditions, such as food shortages, disease, and 
climate change; 
6. use information from secondary resources to classify animals and plants of the 
Philippines as critically endangered, endangered, or vulnerable species; 
7. discuss as a class how threats to biodiversity can lead to species extinction;


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Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners… 
1. Plate Tectonics 
2. Global climate 
3. Global interactions 
4. Global and local 
Sustainability 
1. Current models explain 
tectonic plate movement as 
part of a gravity-driven 
convection system that 
pushes young hot plates 
away from spreading ridges 
and pulls old cold plates 
down into subduction 
zones. 
2. Plate movements and 
continental evolution 
account for the major 
surface features of the 
Earth. 
3. Climate change and its 
impacts on the 
environment and people 
pose serious challenges 
which require solutions 
and action at local and 
global levels. 
4. The rich natural resources 
of the Philippines require 
sustainable management. 
 
1. identify modern scientific processes used to detect and measure the 
displacement of tectonic plates; 
2. describe the structures, movements and events that occur at each type of plate 
boundary;


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Suggested Performance Task/s 
A. Write a report on an environmental action group analyzing their principles and their actions or activities regarding the human impact on 
the biosphere. 
B. Conduct a research project on a specific Philippine ecosystem and investigate its biodiversity, ecological interactions, and conservation 
challenges. Present your research findings through a scientific report or multimedia project. 
 
 
GRADE 9 FOURTH QUARTER- Science of Materials 
Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners…  
1. Valid and reliable 
investigations 
2. Chemical bonding 
3. Ionic compounds 
4. Covalent 
compounds 
5. Metallic bonds 
6. Chemical formula 
1. Valid and reliable 
scientific investigations 
include identification and 
control of variables. 
2. Formation or breaking 
down of ionic or covalent 
bonds results in a 
chemical change. 
3. Bonds are formed 
between atoms either by


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Performance Standard 
By the end of the Quarter, learners describe and explain the geologically dynamic nature of the Philippine Archipelago in relation to its plate 
tectonic setting and use models to explain the earth structures, movements, and natural events that occur. They use critical thinking and 
modeling to explain mechanisms that have contributed to the current distributions of continents and make predictions about changes that 
can be expected in the future. Learners gather information from secondary sources to describe rapid changes that are occurring in local and 
global climate patterns and propose solutions to address these changes at the local and global levels by drawing on awareness, responsible 
personal behavior to conserve materials and energy, and through the better societal management of the natural resources of the country. 
 
Suggested Performance Tasks


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of carbon dioxide; 
8. write the chemical formula and chemical name of some common covalent 
compounds, including water (H2O), carbon dioxide (CO2), and ammonia (NH3); 
9. show by using models that ionic compounds form crystalline structures whereas 
covalent compounds form individual molecules; 
10. explain properties of metals in terms of their structure and metallic bonding (sea 
of electrons model); and 
11. investigate the properties of ionic, covalent, and metallic substances, such as 
melting point, hardness, electrical and thermal conductivity. 
Performance Standard 
By the end of the Quarter, learners carry out a valid and reliable scientific investigation showing the formation of a new substance. They 
demonstrate an understanding of the significance of the valence electron of an element in the formation of bonds and identify bonds as ionic,


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1. identify modern scientific processes used to detect and measure the 
displacement of tectonic plates; 
2. describe the structures, movements and events that occur at each type of plate 
boundary; 
3. identify the locations of major mountains, faults, volcanos, and ocean trenches 
using a map of the Philippine Archipelago, and interpret the features in relation 
to plate tectonics; 
4. predict the position and shape of the Philippine Archipelago in 50 million years, 
based on the current velocity of the Philippine Plate; 
5. gather information from secondary sources to describe and explain what 
mechanisms that drive the movement of tectonic plates including the role of the 
asthenosphere; 
6. explain how the subduction of an oceanic plate impacts on the plate above it; 
7. explain how plate tectonics can be used to explain the formation of the largest 
mountain ranges on Earth including the Himalayas and the Andes mountains;


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Philippines as critically endangered, endangered, or vulnerable species; 
7. discuss as a class how threats to biodiversity can lead to species extinction; 
8. use information from secondary sources to research how to protect and conserve 
endangered and/or economically important species in the local community; 
9. describe using labeled diagrams the biotic and abiotic features of tropical 
rainforests, swamps, estuaries, mangrove forests, and coral reefs; 
10. use information from secondary sources to describe the possible effects of human 
activities, such as deforestation, pollution, and introduction of invasive species, 
on living things in an ecosystem; and


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personal behavior to conserve materials and energy, and through the better societal management of the natural resources of the country. 
 
Suggested Performance Tasks 
A. Plan and enact a community education strategy based on scientific understanding, data, and processes, to encourage and empower 
viewers to be responsible in their use of local natural resources. 
B. Develop a discussion paper on the value of mining green metals, such as cobalt and nickel for modern battery production. Include 
information about how modern batteries can contribute to addressing energy supply and other energy-related issues. The paper should 
provide information about the green metal reserves of the Philippines and what would be involved in establishing industries to produce 
batteries locally. 
 
 
GRADE 10 SECOND QUARTER - Force, Motion, and Energy 
Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners: 
1. Projectile motion 
2. Momentum and 
Collisions


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include identification and 
control of variables. 
2. Formation or breaking 
down of ionic or covalent 
bonds results in a 
chemical change. 
3. Bonds are formed 
between atoms either by 
1. carry out a valid and reliable scientific investigation to show the formation of a 
new substance, such as formation of a carbonate (carbon dioxide in limewater), 
or formation of a precipitate (from silver nitrate solution); 
2. explain that the formation of new bonds or the breaking of existing bonds 
constitutes a chemical change and the formation of a new substance; 
3. describe a valence electron as an electron in the outer shell of an atom that can 
take part in formation of bonds; 
4. identify the number of valence electrons of oxygen based on its position in the 
periodic table;


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5. The revolution, rotation, 
and the tilt of the Earth 
explain the patterns of 
day and night and the 
seasons. 
6. refer to the local disaster readiness plans to demonstrate what to do during and 
after an earthquake; 
7. explain how earthquakes result in tsunamis that devastate shoreline 
communities; 
8. describe procedures that the authorities have in place to alert communities of 
pending tsunamis and what procedures can be implemented should a tsunami 
impact a community; 
9. explain how energy from the Sun interacts with the atmosphere; 
10. make a physical model or use drawings to demonstrate how the tilt of the Earth 
relative to its orbit around the Sun affects the intensity of sunlight absorbed by 
different areas of Earth over a year; 
11. explain, using models, how the tilt of the Earth affects the changes in the length 
of daytime at different times of the year; and 
12. explain how solar energy contributes to the occurrence of land and sea breezes,


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Learning Competencies 
The learners… 
1. Scale, proportion 
and quantity 
2. Plate boundaries 
3. Structure of the 
Earth 
4. Geologic time 
1. Evidence for continents 
moving includes jig-saw 
matching of coastlines, 
rock types, and the 
presence of similar fossils 
in places separated by 
vast distance. 
1. identify and explain evidence that current continents are separate parts of what 
was a single continent millions of years ago; 
2. participate in a collaborative group or class task to examine and describe the 
topographical and geological evidence for plate boundaries occurring in the area 
where the Philippines is located; 
3. describe the types of plate boundaries found around the Earth;


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7. explain how plate tectonics can be used to explain the formation of the largest 
mountain ranges on Earth including the Himalayas and the Andes mountains; 
8. identify evidence of global warming and climate change; 
9. identify the role of greenhouse gases in enhanced global warming and climate 
change; 
10. describe how global climatic phenomena, such as the El Niño Southern 
Oscillation, may impact weather systems; 
11. identify local impacts of global climate change and suggest ways that individuals 
can do to reduce the impact of global warming; and 
12. explain how increased societal uses of renewable energies could mitigate the 
effects of global climate change, including how the Philippines could make better 
use of its plentiful natural resources.


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demonstrate an understanding of the significance of the valence electron of an element in the formation of bonds and identify bonds as ionic, 
covalent, or metallic. They use their knowledge of the symbols of elements to write the formula for a number of common compounds.  They 
draw models of possible structures of ionic compounds and research the properties of ionic, covalent, and metallic substances. They use 
cartoons/comic strips to create interesting learning tools. 
Suggested Performance Task/s 
Create a cartoon/comic strips portraying the main characters as “ionic”, “covalent”, and “metallic” bonds. The cartoon should communicate 
each character’s role in holding atoms and/or molecules together and may show what happens to them in the way the substances are used 
in everyday life. 
 
 
GRADE 10 FIRST QUARTER - Earth and Space Science


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Suggested Performance Tasks 
A. Conduct a survey of recent vehicular accidents in your locality to identify a high-risk spot where collisions are frequent. Offer a solution to 
mitigate collisions at the high-risk area by: 
ð       1. designing a structure aimed at reducing the impact of collisions, or 
ð       2. describe a system aimed at changing riders’ or drivers’ behavior by educating them about the collision danger in the location. 
B. Design and describe the significant features (rationale) of a ‘vehicle’ that will save a raw egg from breaking when dropped onto a concrete 
surface from 3 meters. Learners will test their model in a class event. 
 
 
GRADE 10 THIRD QUARTER- Science of Materials 
Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners… 
1. Chemical reactions 
2. Acids, bases, and 
salts. 
3. Types of chemical 
reactions 
4. Chemical reactions 
in the 
environment. 
5. Chemical 
equations 
6. Rates of reactions


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GRADE 10 SECOND QUARTER - Force, Motion, and Energy 
Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners: 
1. Projectile motion 
2. Momentum and 
Collisions 
3. Large-scale 
generation and 
distribution of 
electricity 
4.  Renewable and 
non-renewable 
energy 
1. Newton’s laws can be used 
to explain projectile motion 
and collisions. 
2. Momentum in collisions 
increases as mass or 
velocity increases. 
3. The electric companies 
provides high voltage 
electricity through power 
generation, transmission, 
1. investigate and describe the relationship among the projectile variables 
including the angle and velocity of release, and projectile height and range, 
using everyday activities such as shooting basketballs or kicking footballs; 
2. describe different types of collisions as elastic or inelastic by providing some 
examples; 
3. use models to investigate elastic or inelastic collisions and describe the forces 
involved and their effects;


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of daytime at different times of the year; and 
12. explain how solar energy contributes to the occurrence of land and sea breezes, 
monsoons, and the Intertropical Convergence Zone (ITCZ). 
Performance Standard 
By the end of the Quarter, learners appreciate the value of using systems to analyze and explain natural phenomena and demonstrate their 
understanding in explaining the dynamics of faults and earthquakes. They are confident in identifying and assessing the earthquake risk for 
their local communities using authentic and reliable secondary data. They use the country’s disaster awareness and risk reduction 
management plans to identify and explain to others what to do in the event of an earthquake. Learners explain the cause and effects of 
secondary impacts that some coastal communities may experience should a tsunami be produced by either local or distant earthquake


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and distribution to many 
parts of the archipelago. 
4. Responsible planning and 
innovation lead to efficient 
generation and distribution 
of electricity in the 
Philippines. 
4. explain that momentum depends on the mass and the velocity of a moving 
object that can be used to predict the impact the object will have if it hits 
another object; 
5. carry out guided investigations using different objects to describe momentum-
related relationships, such as the more momentum an object has, the harder 
for it to stop; 
6. identify and explain that to change the momentum of an object, it is necessary 
to apply a force on the object over a period of time; 
7. gather information from secondary sources to identify ways to reduce the 
impact of collisions such as seatbelts, airbags, and crumple zones in vehicles; 
8. identify that momentum is conserved before and after the collision of objects;


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abide by the principle of 
conservation of mass as 
illustrated in balanced 
chemical equations. 
6. Rates of chemical reactions 
are critical in production 
and preservation of many 
useful materials. 
9. identify that chemical reactions may be exothermic or endothermic 
 
Performance Standard 
By the end of the Quarter, learners demonstrate an understanding that household products can act as indicators for important chemicals. 
They describe the indicators of a chemical reaction and identify important types of chemical reactions. They explain how some important 
chemical reactions impact the natural and built environment. They write balanced chemical equations using formula and apply the 
principles of conservation of mass. They explain factors that affect the rate of a reaction and that some reactions are exothermic, and some 
are endothermic. They demonstrate skills to plan and conduct valid and reliable scientific investigations and record them appropriately.


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The curriculum organizers described below are used together to form the curriculum description in the Grades 3 to 10 Science Curriculum 
Guide. The definitions within this section are drawn from DepEd Order No. 8, s. 2015 and DepEd Order No. 21, s. 2019. 
1)        Standard – In its broadest sense, it is something against which other things can be compared to for the purpose of de termining 
accuracy, estimating quantity or judging quality. It is a stated expectation of what one should know and be able to do. 
2)        Key Stage – This refers to stages in the K to 12 Program reflecting distinct developmental milestones. These are Key  Stage 1 
(Kindergarten – Grade 3), Key Stage 2 (Grades 4 – 6), Key Stage 3 (Grades 7 – 10), and Key Stage 4 (Grades 11 and 12). 
3)        Key Stage Standard – This shows the level or quality of proficiency that the learner is able to demonstrate in each ke y stage after


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and population 
growth 
2. Several theories provide 
lines of evidence about how 
organisms evolve. 
3. The products and 
processes of biotechnology 
can have both beneficial 
and harmful effects on 
society and the 
environment. 
4. Population growth 
influences the carrying 
capacity of an ecosystem 
3. use information from secondary sources to describe natural selection as the 
primary mechanism driving evolutionary change; 
4. discuss in small groups important concepts in the theories of evolution, such as 
variation, heredity, isolation, selection, and adaptation; 
5. use information from secondary sources to explain how lines of evidence, such as 
fossils, biogeography, and comparative morphology, support the occurrence of 
evolution; 
6. explain the term biotechnology and provide examples;  
7. use information from secondary sources to identify the products of traditional 
biotechnology through fermentation (e.g. cheese, soy sauce, vinegar, nata de coco);


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Athuman, J. J. (2017). Comparing the effectiveness of an inquiry-based approach to that of conventional style of teaching in the development of 
students’ science process skills. 
 Bandura, A. (1986) Social foundations of thought and action: a social cognitive theory., Englewood Cliffs, N.J.: Prentice-Hall. 
Bernardo, A. B. I. (2021). Socioeconomic status moder ates the relationship between growth mindset and learning in mathematics and science: 
Evidence from PISA 2018 Philippine data. International Journal of School & Educational Psychology, 9(2), 208 –222. 
https://doi.org/10.1080/21683603.2020.1832635 
Bruner, J. (1964). The course of cognitive growth. Retrieved from https://www.uky.edu/~gmswan3/544/Bruner_1964_CoCG.pdf  
Committee on a Conceptual Framework for New K-12 Science Education Standards; National Academies of Sciences, Engineering, and Medicine.


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secondary impacts that some coastal communities may experience should a tsunami be produced by either local or distant earthquake 
activity. Learners use reliable scientific information to identify and explain how solar energy influences the atmosphere and weather systems 
of the Earth and use such information to appreciate and explain the dominant processes that influence the climate of the Philippines. 
Suggested Performance Tasks 
A. Design, test, and evaluate a model house that can withstand a model earthquake. 
B. Design, test, and evaluate a model of an innovative house that can adapt to the different weather conditions in the country. 
 
 
GRADE 8 FIRST QUARTER - Life Science


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The learners… 
1. Chemical reactions 
2. Acids, bases, and 
salts. 
3. Types of chemical 
reactions 
4. Chemical reactions 
in the 
environment. 
5. Chemical 
equations 
6. Rates of reactions 
1. Several simple 
observations indicate if a 
chemical reaction has 
taken place. 
2. Chemical indicators 
produce color changes with 
acids, bases, and salts. 
3. Valid and reliable scientific 
investigations identify the 
dependent and 
independent variables and 
control other variables. 
4. Many types of chemical 
reactions are important in 
our daily lives and in the 
biotic and abiotic parts of 
the environment. 
5. Atoms rearrange during 
chemical reactions but 
1. describe the indicators for a chemical reaction as color change, the formation of a 
precipitate, the release of gas, and or odor, or a change in temperature; 
2. identify common acids, bases, and salts (e.g., hydrocholoric acid, sodium 
hydroxide, and saline solution) using different indicators;


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are endothermic. They demonstrate skills to plan and conduct valid and reliable scientific investigations and record them appropriately. 
Suggested Performance Task/s 
A. Conduct a valid and reliable investigation to test a number of vegetables, such as carrots and red cabbage to determine their usefulness 
as indicators for common acids and bases, such as vinegar, lemon juice, and bleach. 
B. Explore materials in the home to produce specific chemical reactions, such as mixing a solution of washing powder with a solution of 
baking powder. 
 
  
GRADE 10 FOURTH QUARTER - Life Science 
Content Content Standards 
The learners learn that: 
Learning Competencies 
The learners… 
1. Homeostasis 
2. Mechanisms of 
evolution 
3. Biotechnology 
4. Ecosystem’s 
carrying capacity 
1. Homeostasis is a self-
regulating process that 
allows an organism to 
maintain stability. 
1. describe homeostasis as a state of balance among all the body systems in humans


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impact of collisions such as seatbelts, airbags, and crumple zones in vehicles; 
8. identify that momentum is conserved before and after the collision of objects; 
9. describe how high voltage electricity from power plants is generated and safely 
distributed to industries, businesses, and homes, including the role of 
substations (grid stations), and electric meters; 
10. describe and explain the need for safety precautions in handling household 
electrical devices; 
11. describe the similarities and differences between electric motors and electric 
generators; 
12. collaborate in a class discussion to identify ways to reduce the use of electrical 
energy in Filipino houses and communities and explain what local and global 
benefits can be achieved; and 
13. gather information from secondary sources to evaluate how renewable and non-
renewable generation of electricity in the Philippines impacts human activities 
and the environment. 
Performance Standard


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3)        Key Stage Standard – This shows the level or quality of proficiency that the learner is able to demonstrate in each ke y stage after 
learning a particular area in relation to the core learning area standard. 
4)  Grade Level Standard – This shows the level or quality of proficiency that the learner is able to demonstrate in each Grade after 
learning a particular area in relation to the core learning area standard. 
5)  Content Domain – This is a particular strand or domain of the curriculum in which the scope and sequence of a set of related 
topics and skills are covered. 
6)  Content Standard – The content standards identify and set the essential knowledge and understanding intended to be learned. 
They cover a specified scope of sequential topics within each learning strand, domain, theme, or component. Content standards answer 
the question, “What should the learners know?”


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7. use information from secondary sources to identify the products of traditional 
biotechnology through fermentation (e.g. cheese, soy sauce, vinegar, nata de coco); 
8. use information from secondary sources to identify examples of modern 
biotechnology, such as genetically modified organisms and processes (e.g. in vitro 
fertilization); 
9. participate in a class debate on the societal, environmental, and ethical 
implications of using biotechnological products and methods; 
10. discuss the factors that limit the ecosystem’s carrying capacity, such as adequate 
food, shelter, water, and mates; and 
11. explain that the ecosystem’s population growth slows down as it gets closer to the 
carrying capacity. 
Performance Standard 
By the end of the Quarter, learners describe homeostasis as a process that allows an organism to maintain stability. They describe and


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Committee on a Conceptual Framework for New K-12 Science Education Standards; National Academies of Sciences, Engineering, and Medicine. 
2012. A Framework for K -12 Science Education: Practices, Crosscutting Concepts, and Core Ideas. Washington, DC: The National Academies 
Press. https://doi.org/10.17226/13165. 
Charles, R. (2005) Big Ideas and Understandings as the Foundation for Elementary and Middl e School Mathematics. NCSM Journal. Vol. 7 No. 
3, pp 9-24 https://thelearningexchange.ca/wp-content/uploads/2011/10/BigIdeas_NCSM_Spr05v7.pdf accessed April 27, 2021 
Department of Education (2 015) DepEd Order No. 8, s. 2015, Interim Guidelines for Assessment and Grading in Light of the Basic Education 
Learning Continuity Plan. Retrieved from https://www.deped.gov.ph/wp-content/uploads/2020/10/DO_s2020_031.pdf. 
Department of Education (2020) DepEd Order No. 31, s. 2020, Policy Guidelines on Classroom Assessment for the K to 12 Basic E ducation


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carrying capacity 
1. Homeostasis is a self-
regulating process that 
allows an organism to 
maintain stability. 
1. describe homeostasis as a state of balance among all the body systems in humans 
that needs to be maintained for survival and proper functioning; its indicators 
include body temperature, glucose level, and blood pressure; 
2. explain how homeostasis is maintained through various feedback mechanisms, 
both positive and negative;


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Gutiérrez, K. D., & Rogoff, B. (2003). Cultural ways of learning: Individual traits or repertoires of practice. Educational Researcher, 32(5), 19–25. 
https://doi.org/10.3102/0013189X032005019 
Harlen, W. (Ed.) et al. (2015). Working with big ideas of science education. Trieste, Italy: Science Education Programme of IAP. 
Implementing Rules and Regulations of RA 10533, “The Enhanced Basic Education Act of 2013” 
International Bureau of Education – Scientific Literacy http://www.ibe.unesco.org/en/glossary-curriculum-terminology/s/scientific-
literacy#:~:text=Within%20the%20framework%20of%20the,evidence%2Dbased%20conclusions%20about%20science%2D 
 
Joymie R. Orbe Allen A. Espinosa Janir T. Datukan; Teaching Chemistry in a Spiral Progression Approach: Lessons from Science Teachers in the 
Philippines  
Laal, Marjan & Laal, Mozhgan. (2012). Collaborative learning: What is it?. Procedia - Social and Behavioral Sciences. 
10.1016/j.sbspro.2011.12.092.


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renewable generation of electricity in the Philippines impacts human activities 
and the environment. 
Performance Standard 
By the end of the Quarter, learners display critical thinking in describing the factors that affect the trajectory of projectiles. They distinguish 
between different types of collisions and describe the impacts on the motion of objects. They carry out investigations using models to identify 
relationships that affect the motion of objects and apply their understanding to real-life situations.  Learners gather information from 
secondary sources to identify, describe, and explain how science impacts human activities and the environment.


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They cover a specified scope of sequential topics within each learning strand, domain, theme, or component. Content standards answer 
the question, “What should the learners know?” 
7)        Learning Competency – This refers to a specific skill performed with varying degrees of independence. It has different levels of 
difficulty and performance levels. It also refers to the ability to perform activities according to the standards expec ted by drawing from 
one’s knowledge, skills, and attitudes. 
8)     Performance Standard – The performance standards describe the abilities and skills that learners are expected to demonstrate in 
relation to the content standards and the integration of 21 st century skills. The integration of knowledge, understanding, and skills is 
expressed through creation, innovation, and adding value to products/performance during independent work or in collaboration with 
others. 
 
REFERENCES:


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Department of Education (2020) DepEd Order No. 31, s. 2020, Policy Guidelines on Classroom Assessment for the K to 12 Basic E ducation 
Program. Retrieved from https://www.deped.gov.ph/wp-content/uploads/2015/04/DO_s2015_08.pdf. 
Department of Education (2019) DepEd Order No. 21, s. 2019,” Policy Guidelines on the K to 12 basic Education Program” 
Department of Education (2016) Department of Education, K to 12 Scienc e Curriculum Guide. Retrieved from https://www.deped.gov.ph/wp-
content/uploads/2019/01/Science-CG_with-tagged-sci-equipment_revised.pdf. 
Department of Education (2018) DepEd Order No. 31, seri es 2018 (DO 31), Policy Guidelines on the Implementation of the Comprehensive 
Sexuality Education (CSE). Retrieved from Comprehensive Sexuality Education: Developing responsible youth vs rising risks | Department of 
Education (deped.gov.ph) 
Department of Education (2018) On fighting against climate change: Imuseños  unite through green schools | Department of Education


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carrying capacity. 
Performance Standard 
By the end of the Quarter, learners describe homeostasis as a process that allows an organism to maintain stability. They describe and 
discuss in small groups that natural selection is the driving mechanism of evolutionary change. They explain the meaning of the term 
biotechnology and debate the societal, environmental, and ethical implications of utilizing biotechnological products and methods. They 
discuss the factors that limit the ecosystem’s carrying capacity and the role of population growth. 
Suggested Performance Task 
Write a critical analysis of the use of biotechnology and its impacts on society or the environment. 
 
 
GLOSSARY


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Philippines  
Laal, Marjan & Laal, Mozhgan. (2012). Collaborative learning: What is it?. Procedia - Social and Behavioral Sciences. 
10.1016/j.sbspro.2011.12.092. 
Liu, C. H., & Matthews, R. (2006). Vygots ky’s philosophy: Constructivism and its criticisms examined. International Education Journal, 6(3), 
386–399. 
McBride, B. B., C. A. Brewer, A. R. Berkowitz, and W. T. Borrie. (2013). Environmental literacy, ecological literacy, ecoliteracy: What do we 
mean and how did we get here? Ecosphere 4(5):67. http://dx.doi.org/10.1890/ES13-00075.1 
Montenegro, E., & Jankowski, N. A. (2017). Equity and assessment: Moving towards culturally responsive assessment. Occasional Paper, 29. 
Mortimore, Peter (1999). Understanding Pedagogy: And Its Impact on Learning. Paul Chapman Publishing Ltd A SAGE Publications Company 6 
Bonhill Street London EC2A 4PU


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Page 69 of 69 
 
https://www.nagb.gov/content/dam/nagb/en/documents/publications/frameworks/technology/2018-technology-framework.pdf   
 Panizzon D, Pegg J, Arthur D, and McCloughan G. (2021) Designing a developmental progression to assess students’ conceptual understandings 
by focusing on the language demands in Science; University of New England, Australia. Published in the Australian Journal of Education. 
Piaget, J, (1950). The Psychology of Intelligence. New York: Routledge. 
PISA; 10-How-PISA-D-measures-science-literacy.pdf (oecd.org) 
Pritchard, Alan & Woollard, John (2010) Psychology for the classroom: constructivism and social learning. Routledge 270 Madison Avenue, New 
York, NY 10016 
Ravitch, Diane. (2007). EdSpeak: A Glossary of Education Terms,  Phrases, Buzzwords, and Jargon. Virginia: Alexandria Association for 
Supervision and Curriculum and Development.


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expressed through creation, innovation, and adding value to products/performance during independent work or in collaboration with 
others. 
 
REFERENCES: 
A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012). 
American Association for the Advancement of Science. The Nature of Science. 2009 
http://www.project2061.org/publications/bsl/online/index.php?chapter=1 
Archer, A., & Hughes, C. (2011). Explicit Instruction: Effective and Efficient Teaching. NY: Guilford Publications.


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Education (deped.gov.ph) 
Department of Education (2018) On fighting against climate change: Imuseños  unite through green schools | Department of Education 
(deped.gov.ph) 
Duch, B. J., Groh, S. E, & Allen, D. E. (Eds.). (2001). The power of problem-based learning. Sterling, VA: Stylus.


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2. identify common acids, bases, and salts (e.g., hydrocholoric acid, sodium 
hydroxide, and saline solution) using different indicators; 
3. describe important types of chemical reactions (combination, decomposition, 
single replacement, double replacement); 
4. explain how important types of chemical reactions, such as combustion, acids on 
metals, acids on carbonates, photosynthesis, and respiration, relate to or impact 
the natural and built environments using information from secondary sources; 
5. recognize that scientists:  
     a. use chemical equations to describe chemical reactions, and  
     b. write equations in word form and using formula for common 
chemical reactions; 
6. explain that chemical equations demonstrate a rearrangement of atoms but the 
total mass of the system remains the same during a chemical reaction; 
7. apply the principles of conservation of mass to balance chemical equations;


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Mortimore, Peter (1999). Understanding Pedagogy: And Its Impact on Learning. Paul Chapman Publishing Ltd A SAGE Publications Company 6 
Bonhill Street London EC2A 4PU 
Nasir, N. S., Lee, C. D., Pea, R., & McKinney de Royston, M. (2021). Rethinking learning: What the interdisciplinary science tells us. Educational 
Researcher, 50(8), 557–565. https://doi.org/10.3102/0013189X211047251 
National Assessment and Governing Board https://www.nagb.gov/naep-frameworks/technology-and-engineering-literacy/2014-technology-
framework/toc/ch_0/technology_literacy.html 
National Academies of Science and Engineering https://www.nap.edu/read/4962/chapter/4#22 
 National Assessment Governing  Board U.S. Department Of Education Technology & Engineering Literacy. Framework The 2018 National 
Assessment Of Educational Progress.


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York, NY 10016 
Ravitch, Diane. (2007). EdSpeak: A Glossary of Education Terms,  Phrases, Buzzwords, and Jargon. Virginia: Alexandria Association for 
Supervision and Curriculum and Development. 
 Shepard, L. A. (2019). Classroom assessment to support teaching and learning. The ANNALS of the American Academy of Political  and Social 
Science, 683(1), 183-200. 
 Spinelli, C. G. (2008). Addressing the issue of cultural and linguistic diversity and assessment: Informal evaluation measure s for English 
language learners. Reading & writing quarterly, 24(1), 101-118. 
SEAMEO Basic Education Standar ds (SEA -BES) (2017): Common Core Regional Learning Standards (CCRLS) in Mathematics and Science; 
Editors: Dominador Dizon Mangao, Nur Jahan Ahmad, Masami Isoda. 
Suchman, J.R. (1964). The Illinois studies in inquiry training. Journal of Research in Science Teaching 2:230–232.


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total mass of the system remains the same during a chemical reaction; 
7. apply the principles of conservation of mass to balance chemical equations; 
8. explain the factors affecting the rates of chemical reactions as applied in food 
preservation and materials production, control of fire, pollution, and corrosion; 
and


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Editors: Dominador Dizon Mangao, Nur Jahan Ahmad, Masami Isoda. 
Suchman, J.R. (1964). The Illinois studies in inquiry training. Journal of Research in Science Teaching 2:230–232. 
 Surif, J., Ibrahim, N. H., Alwi, A. M., Loganathan, P., & Serman, N. S. (2019, December). Effect of Inductive Teaching Method To Improve Science 
Process Skills In Electrochemistry. In 2019 IEEE International Conference on Engineering, Technology and Education (TALE) (pp. 1-5). IEEE. 
Vygotsky, L.S. 1978. Mind in society: The Development of Higher Mental Processes. Massachusetts: Harvard University Press. 
World Economic Forum, Defining Education 4.0: A Taxonomy for the Future of Learning; 2023.


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meteor showers; and 
11. explain how modern research about celestial objects uses new space technologies 
including telescopes and space probes. 
Performance Standard 
By the end of the Quarter, learners exhibit skills in evaluating information from secondary sources, and draw on their scientific 
understanding of the location and geological setting of the Philippines to explain its unique landforms and dynamic geologic activity in a 
global context. They demonstrate an appreciation of the size and scale of the Earth and describe evidence for a dynamic Earth over its long 
geological history as well as the evidence that is used to build a model for the internal structure of the Earth. Learners demonstrate curiosity 
and open-mindedness in extending their knowledge and understanding of the dynamic Earth to evaluate evidence for theories for the 
formation of the Solar System. They describe modern scientific processes and technologies that are used by scientists to investigate the


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formation of the Solar System. They describe modern scientific processes and technologies that are used by scientists to investigate the 
nature and evolution of the Solar System and Universe.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
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  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>If the cardinality or number of elements of a set is a counting number, then it is called a finite set. \nA set is defined as a finite set if the cardinality of the set is a fixed or known counting number n. \nOtherwise, the set is not finite and is called an infinite set.</td>
    </tr>
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The table presented outlines a key concept related to set theory, specifically focusing on the distinction between finite and infinite sets based on their cardinality, or the number of elements they contain. The table is structured with a single column for content, alongside indices that suggest it's illustrating a sequence or list of concepts. The first entry under the header "Key Concept" defines a finite set, explaining that it can be identified by its cardinality being a counting number, referred to as \( n \). This cardinality is characterized as a fixed or known quantity of elements, differentiating it from sets that do not meet this criterion.

Following the initial explanation, the table elaborates that sets which do not exhibit a counting number for their cardinality are classified as infinite sets. This concept underscores the fundamental principles of set theory, where understanding the differences in types of sets is crucial for further mathematical exploration. Thus, this table serves as a concise yet informative resource for students studying the foundational aspects of mathematical sets, enabling them to grasp essential definitions that will support their learning in topics stemming from set theory. It links directly with the surrounding practice exercises, where learners are tasked with identifying and describing sets, thereby applying the theoretical knowledge presented.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>The cardinality of a set refers to the number of elements of the set. In symbols, the cardinality of \na set A is denoted as n(A).</td>
    </tr>
  </tbody>
</table>


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Within the context of the concepts and skills related to set theory, the table presented highlights a key concept known as the cardinality of a set. The table is structured with a single column on the left representing numerical indices and a single column on the right detailing the information relevant to the concept. The first row serves as a header, indicating the column on the right elaborates on a crucial key concept in set theory. 

The content in the table specifically defines the cardinality of a set, which refers to the number of elements contained within that set. It clarifies that this concept is denoted in symbolic form as \( n(A) \), where \( A \) represents the set in question. Understanding cardinality is essential for addressing various statements and questions related to sets, such as distinguishing between finite and infinite sets, and determining the equivalence or differences among sets based on their quantity of elements. This explanation aligns with the broader instructional goals of the surrounding material, where students practice evaluating statements about sets, determining if certain groups qualify as sets, and describing sets through both roster and rule methods. The focus on cardinality serves as a foundation for deepening the learners’ comprehension of set theory and its applications.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Water Sports</td>
      <td>Ball Games</td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td>Exciting Games</td>
      <td>Popular Sports</td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


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Within the context of understanding sets and their elements, the table presents a categorization of different types of sports. It features two primary columns labeled “0” and “1,” and each row represents different classifications of sports. The first row introduces the categories of "Water Sports" and "Ball Games," indicating that these sports rely on either aquatic environments or involve balls as essential equipment. The second row remains empty, possibly indicating a space for future content or a gap in information.

Moving to the next relevant entries in the table, the third row highlights "Exciting Games" and "Popular Sports." This row suggests a grouping based on the thrilling nature of certain games or their widespread appeal, implying these categories may contain a variety of sports that resonate well with participants and spectators alike. The empty row beneath may again serve as a placeholder, reinforcing a structure that allows for the addition of more information or categories.

This classification aligns with the earlier context that describes how sets are defined and identified by specific characteristics. Just as the examples describe various groups that can or cannot be classified as sets based on agreed criteria, the table effectively illustrates distinct groups of sports that can also be distinguished by their defining attributes, such as the environment in which they are played or the general sentiment towards their popularity. The overall organization of the table serves as a structured visual representation, aiding in the comprehension of how sports can be categorized, resembling a set in mathematics where elements share a common theme.


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string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>In  the  set-builder  notation  or  the  rule  method  of  describing  a  set,  the  property  or \nproperties that each element of the set satisfies are stated.</td>
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</table>


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The table presented serves as a clarifying resource on a key concept concerning the notation used in set theory, particularly the set-builder notation or the rule method. At the outset, it indicates a broad definition labeled as "Key Concept," which is instrumental in understanding how sets can be represented not merely by listing their elements but by stating the properties that define their elements.

In the context of set-builder notation, the second row of the table elaborates on its purpose: it emphasizes that this method involves articulating the property or properties that every element of the set meets. This is an essential aspect of set theory, as it allows for a more abstract and often more concise representation of sets. Rather than listing each specific element, which can be cumbersome, particularly for larger sets, the rule method succinctly communicates the criteria that characterize the set itself. This aligns with the discussions in the surrounding text about the cardinality and classification of sets, providing a theoretical foundation for understanding how elements are grouped based on shared characteristics.

Overall, this table succinctly distills a significant concept in set theory into an accessible format, supporting the broader educational goal of strengthening comprehension of sets, both finite and infinite, through various representation methods.


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string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
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</table>


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The provided table is structured in a simple format with borders that delineate its cells. It has four columns, including a header row that indicates the index values 0, 1, and 2. Each of these columns is intended to hold information, but the rows currently show no data entries, as all cells are empty. The first column also contains an index for the rows, labeled 0 and 1, marking the beginning of a two-row dataset. 

Given the context surrounding the table, it hints at a task related to categorizing sports based on specific criteria, such as "Water Sports," "Ball Games," "Exciting Games," and "Popular Sports." Although no specific data is visible in the table to reflect any of these categories or the sports that fall under them, it’s clear that the empty cells are placeholders where students may eventually input their categorized list of sports according to the discussed themes. 

This table may serve as an organizational tool for students to classify their thoughts on various sports, reinforcing their understanding of the definition of a set. Reflecting the key concept that a set is a well-defined collection of objects, educators may use this table format in teaching students how to logically group sports into sets based on shared characteristics, thus fostering critical thinking and collaborative learning as they compare their categorizations with classmates.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
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  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
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    <tr>
      <th>1</th>
      <td>A set is a well-defined collection of objects. Such objects are called the elements of the \nset.</td>
    </tr>
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</table>


key: chroma:document
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The table presented outlines a key concept regarding sets, which are fundamental in mathematics. It emphasizes that a set is defined as a "well-defined collection of objects." These objects, referred to as the elements of the set, share common characteristics that distinguish them from elements of other sets. This definition closely aligns with the examples discussed in the surrounding context, where various groups are evaluated for their status as sets based on their clear and unambiguous membership criteria.

In the context of the examples provided, the table's content relates directly to how we classify and describe groups or collections. The first example of a set, the group of varsity players in a school, meets the criteria of a set due to the established roster that signifies definite membership. In contrast, other examples illustrate how subjective criteria can lead to uncertainty and make it difficult to define a set, as seen with the groups of dancers or popular volleyball players. The distinction between what constitutes a set and what does not becomes clearer when considering the definition found in the table.

Furthermore, the table serves an educational purpose, summarizing the essence of what a set is and providing essential vocabulary, such as the terms "elements" and "well-defined." This aligns with the section discussing set membership notation, where symbols like "∈" (is an element of) and "∉" (is not an element of) are introduced. Together, the table and contextual discussion reinforce a foundational understanding of sets in mathematics, illustrating their significance in categorizing and organizing various objects and concepts throughout different examples.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>In the roster method or listing method of describing a set, the elements of the set are \nlisted between braces. The elements in the list are separated by commas.</td>
    </tr>
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</table>


key: chroma:document
string_value:
The table presents a concise overview of the key concepts related to the roster method of set description. It is organized into two main sections: the header and the body. The header consists of a single column labeled "0," which serves as a reference point for the information provided. In the first row of the body, a key concept is introduced, emphasizing the nature of the roster method itself. The term "Key Concept" is prominently displayed, indicating that the following information is of particular importance in understanding how sets can be represented.

Moving to the second row, the table elaborates on the roster method, also referred to as the listing method. It explains that in this approach, the elements of a set are explicitly listed within braces, showcasing a straightforward and visual way of identifying the members of the set. Each element in this list is separated by commas, a detail critical for clear representation, ensuring that readers can easily distinguish between individual elements. This method is contrasted with the set-builder notation introduced earlier, where properties or definitions are used to describe the set rather than listing each element.

Overall, this table succinctly encapsulates the essence of the roster method within the broader context of set notation, reinforcing the idea that while there are multiple ways to describe sets, clarity in representing their elements is paramount for effective communication in mathematics. The surrounding context complements this by providing an example of set-builder notation, illustrating the various methods available for set description, thus creating a comprehensive understanding of the topic.


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The table presented in the HTML format is a simple structure designed to hold data in a grid-like format, featuring both headers and a body section. The table includes a header row with columns identified by numerical indices: 0, 1, and 2. There is also a leading row on the left side, also numbered 0 and 1, which indicates the rows' respective indices. 

Inside the table body, there are currently no entries filled in the cells; all cells are left blank. Specifically, the first row (header row) corresponds to the indices 0, 1, and 2 under the column headers, while the first column (left side) reflects indices 0 and 1. As it stands, the table does not contain any data points for analysis or illustration, which suggests it may serve as a template or a preliminary format intended for future data entry or display.

In relation to the context provided about sets—equal sets and equivalent sets—the table can be interpreted as a tool for organizing or comparing sets by listing their elements across the defined indices. However, with all cells empty, it currently lacks the practical application that could illustrate the relationships between various sets, such as those discussed (E, F, G, H, and I). It could eventually be filled to showcase one-to-one correspondences or cardinality comparisons, thereby effectively aiding in understanding the concepts of set equality and equivalency as outlined in the accompanying text.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
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    <tr>
      <th>0</th>
      <td>Key Concept</td>
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      <th>1</th>
      <td>A Venn diagram is a visual representation of logical relationships among sets. It uses \ncircles to represent sets. Each circle is labeled with the name of the set it represents.</td>
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key: chroma:document
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The table presented in the document introduces a key concept related to the visual representation of sets through Venn diagrams. In the first row of the table, labeled as "Key Concept," readers are informed about the primary purpose of Venn diagrams, emphasizing that they serve as a tool to illustrate logical relationships among different sets. The diagram utilizes circles to represent these sets effectively, facilitating a clearer understanding of how the elements within each set relate to one another.

Following this introduction, the second row elaborates on the defining characteristics of Venn diagrams. It explains that these diagrams not only represent sets visually but also highlight their interactions and inclusions. Each circle within a Venn diagram is distinctly labeled with the name of the corresponding set, reinforcing the identification of the relationships among various collections of elements. This visual representation is essential for comprehending concepts such as subsets, as exemplified in the provided context, where sets X, Y, and Z illustrate proper subsets through their relationships to each other.

In summary, the table serves as a foundational reference for understanding Venn diagrams, illustrating how they encapsulate the relationships between different sets through distinct visual elements, specifically circles. This lays the groundwork for further applications, such as the examples provided, which demonstrate the use of Venn diagrams to depict proper subsets and reinforce the concept of set relationships in a tangible manner.


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<table border="1" class="dataframe">
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    <tr>
      <th>0</th>
      <td>Key Concept</td>
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      <td>Q is said to be a subset of P if every element of Q is also an element of P. In symbols, it \nis denoted as Q \n P. This is read as “Q is a subset of P” or “Q is contained in P.”</td>
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key: chroma:document
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The table presents a key concept regarding the relationship between sets in mathematics, particularly focusing on the notion of subsets. The first row introduces the heading "Key Concept," which serves as an indication that the information following it will describe fundamental principles concerning sets. In the subsequent row, the table provides a formal definition: Q is said to be a subset of P if every element in set Q is also an element in set P. This relationship is symbolically represented as Q ⊆ P, where the symbol indicates that Q is contained within P. The terminology used clarifies that one can read this expression in two ways: either as “Q is a subset of P” or as “Q is contained in P,” establishing a clear understanding of the subset relationship.

Furthermore, the context surrounding the table elaborates on the concept with practical examples. It illustrates how the subsets and their relationships are visually represented using Venn diagrams. For instance, subsets X, Y, and Z are defined, with X being a proper subset of Y, Y being a proper subset of Z, and similarly, X being a subset of Z as well. The key point emphasized is that the cardinality of set A is less than that of set B when A is a proper subset of B, signifying that while all elements of A are contained in B, the two sets are not identical. This discussion not only reinforces the information in the table but also aids in visualizing the relationships between the sets through concrete examples, enhancing the learner's understanding of set theory in mathematics.


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<table border="1" class="dataframe">
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    <tr style="text-align: right;">
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      <td>If set A is equivalent to set B, and  \nset B is also equivalent to set C, then \nthe set A is equivalent to set C.\nIf A ~ B and B ~ C, then A ~ C.</td>
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key: chroma:document
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The table presented encapsulates essential properties relevant to set theory, specifically focusing on the concepts of equality and equivalence among sets. It is structured into three main sections, each detailing a distinct property, with the first column categorizing the property type and the subsequent columns elaborating on the definitions and implications of equality and equivalence.

In the first row, the headings reflect the type of properties being discussed: "Property," "Equality," and "Equivalence." The second row introduces the "Transitive Property" under both equality and equivalence. For equality, the description states that if set A is equal to set B and set B is also equal to set C, then it logically follows that set A is equal to set C. This relationship can be represented symbolically as A = B and B = C leading to the conclusion that A = C. In contrast, the equivalence section conveys a similar notion; it asserts that if set A is equivalent to set B, and set B is equivalent to set C, then set A is also equivalent to set C, represented as A ~ B and B ~ C, thereby concluding that A ~ C.

This table serves not only as a definition of properties but also reinforces the transitive nature inherent in both equality and equivalence, highlighting their significance in the realm of set theory. The relational parallels drawn between equality and equivalence reinforce the foundational principles that govern set relationships and offer a clear illustration of how these concepts operate within mathematical discourse. Such distinctions are essential for understanding more complex relationships between sets, as outlined in the provided context regarding proper subsets and the illustrative examples of sets X, Y, and Z, which further connects to the overall principles of set theory being explored.


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<table border="1" class="dataframe">
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    <tr style="text-align: right;">
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    <tr>
      <th>0</th>
      <td>A\nB\nA \n B\nAll elements in set A are contained in set B.</td>
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key: chroma:document
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The table presented categorizes the relationship between two sets, A and B, exploring the conditions under which the elements of these sets interact. It is divided into two primary columns labeled with the indices 0 and 1, which represent different scenarios regarding the containment of elements in set A concerning set B.

In the first row of the table, under column 0, it states that if all elements in set A are contained within set B, this condition signifies that A is a proper subset of B, which is mathematically denoted as A ⊂ B. This adheres to the given context where it emphasizes that for a proper subset, the number of elements in set A must be less than that in set B, and they cannot be equal, hence A ≠ B. The context connects this to the example sets provided, X and Y, showing how X is indeed a proper subset of Y.

Conversely, under column 1, the table describes a scenario where not all elements in set A are contained in set B. This indicates that A is not a proper subset of B, which is a condition that can arise in numerous set relationships. This aligns with the understanding of subsets as expounded in the core concept, illustrating the essential nature of set containment and the implications of sizes between sets. The context further reinforces this understanding by providing Venn diagrams for illustrations, which visually represent the relationships discussed within the table and clarify the distinctions between proper subsets and non-contained sets.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
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      <th>2</th>
      <td>Symmetric Property</td>
      <td>The sets can be written \ninterchangeably on any side of the \nequality.\nIf A = B, then B = A.</td>
      <td>The sets can be written \ninterchangeably on any side of the \nequivalence.\nIf B ~ A, then A ~ B.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented outlines three fundamental properties of sets in terms of both equality and equivalence. The first row establishes the categories being compared: Property, Equality, and Equivalence, providing a framework for understanding the relationships between sets. Within the table, the second row focuses on the Reflexive Property, emphasizing that a set is equal to itself, which can be mathematically represented as A = A, and similarly, a set is equivalent to itself, denoted as A ~ A. This property is fundamental in set theory as it underscores the notion of identity within sets.

The third row expands on the Symmetric Property, which states that the relationship between sets holds in both directions. In the context of equality, if set A is equal to set B (A = B), then it can also be stated that set B is equal to set A (B = A). This principle is mirrored in the realm of equivalence, where if set B is equivalent to set A (B ~ A), then the relationship can be reversed to express A ~ B. The inclusion of these properties highlights the symmetry inherent in set relationships, making them applicable in various mathematical contexts.

The understanding of these properties is essential for grasping larger concepts regarding sets and subsets, a topic introduced in the provided context. For instance, the notion of subsets, as explained with sets P and Q, amplifies the relevance of the Reflexive and Symmetric Properties. The table thus serves as a concise visual aid, aiding in the comprehension of how equality and equivalence govern set relationships, and how these properties can be applied to further study the inclusion and cardinality of sets within mathematical discussions.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Transitive Property of Set Inclusion\nFor any sets A, B, and C, if A ⊂ B and B ⊂ C, then A ⊂ C.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table outlines a key concept related to the transitive property of set inclusion within the context of set theory. This property is crucial for understanding the hierarchical relationships between different sets. Specifically, it states that for any sets A, B, and C, if set A is a subset of set B (denoted A ⊂ B) and set B is a subset of set C (denoted B ⊂ C), then it necessarily follows that set A is also a subset of set C (denoted A ⊂ C). 

The first row of the table serves as a header that introduces the term "Key Concept," indicating the importance of the rule that follows. In the second row, the actual definition of the transitive property is laid out. This clear and concise presentation allows readers to grasp the foundational rule governing set inclusion quickly and effectively. The format of the table, with distinct rows for each piece of information, enhances readability and emphasizes the logical structure of the concept. 

In the broader context of the material surrounding the table, the discussion includes examples of various types of subsets, such as improper and proper subsets, as well as the concept of disjoint sets. While the table itself focuses on the transitive property, it connects with these other themes, reinforcing how various properties of sets interact and relate to one another in the study of set theory. Understanding properties like transitivity deepens comprehension of how sets are organized and interconnected, a critical aspect of mathematical reasoning.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>For any sets A and B, A is said to be an improper subset of B if A = B.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table focuses on a key concept regarding subsets within set theory, particularly differentiating between improper subsets and proper subsets. It is laid out in a simple structure, with two columns, one serving as an index and the other detailing the concepts associated with subsets. The first row contains the header that labels the primary column as "Key Concept," while the subsequent rows elaborate on the characteristics of an improper subset.

The first item under the key concept identifies that for any sets A and B, an improper subset is defined as the situation where A is equal to B, meaning that every element of set A is also contained within set B without any exceptions. This sets the stage for understanding the nature of subsets and their classifications. Following this, the second row reiterates this by stating the condition clearly, reinforcing the notion that the only improper subset a set can have is itself. This distinction is crucial in set theory as it helps learners understand the relationships between different sets, particularly in the context of disjoint sets discussed in the surrounding text.

The context surrounding the table delves into broader topics of subsets, including trivial and proper subsets, while also providing examples of sets and their intersections, or lack thereof, such as the disjoint nature of sets M and N. By illustrating how to identify disjoint sets and their relationships, along with the mechanisms for counting subsets, the material engages the reader to appreciate the underlying rules that govern set relationships, and the critical definitions that help classify these relationships, as emphasized in the table. Ultimately, the table serves as a concise reference point for understanding the foundational concept of improper subsets, which is integral to navigating the more complex ideas introduced later in the text.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>If A and B are non-empty sets and A is a subset of B, where n(A) &lt; n(B), then A is said to \nbe a proper subset of B. In symbols, it is denoted as A ⊂ B. Note that A ≠ B.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented contains two rows of information that elaborate on key concepts related to set theory, particularly focusing on the notion of subsets. In the first row, the table introduces the idea that if two non-empty sets, A and B, exist, and A is determined to be a subset of B (denoted as A ⊆ B), there is a further distinction made when considering the sizes of the sets. Specifically, if the number of elements in A, referred to as n(A), is less than the number of elements in B, n(B), it leads to the conclusion that A is a proper subset of B. This relationship is indicated through the strict subset notation A ⊂ B, which explicitly states that A is not equal to B (A ≠ B).

Furthermore, the context highlights the significance of subsets in understanding relationships between sets, such as the transitive property of set inclusion, where the relationship amongst sets X, Y, and Z is examined. In this case, if X is a subset of Y and Y is a subset of Z, then logically X must also be a subset of Z. This concept affirms the foundational principles of set inclusion and is significant in clarifying how subsets interact within the framework of set theory.

The table effectively serves as a succinct reference for reinforcing these key concepts and definitions, which are crucial for studying the properties of sets and their subsets in mathematics. The use of symbols, such as the distinction between ⊆ for nonstrict inclusion and ⊂ for strict inclusion, further emphasizes the precision necessary in mathematical discourse. Overall, this table provides an essential synopsis of crucial ideas surrounding subsets that align with the broader discussions of set properties presented in the surrounding context.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Reflexive Property of Set Inclusion\nFor any set A, A ⊆ A.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table presents a key concept related to the reflexive property of set inclusion in the context of set theory. In the first row of the table, the header indicates a significant notion in mathematics, labeled as "Key Concept." This header establishes the focus of the table, which is to elucidate a foundational principle regarding sets.

The second row of the table elaborates on the reflexive property itself, stating that for any given set A, it holds that A is a subset of A, denoted mathematically as A ⊆ A. This property has critical implications in the study of set theory, as it underscores the basic understanding that every set includes itself as a subset. The inclusion relationship is essential for grasping more complex relationships among sets, such as proper and improper subsets, which were mentioned in the earlier context exploring subsets of a particular set S = {a, b, c}. 

This concept of reflexivity ties closely into the discussion of disjoint sets found in the surrounding text, as it provides a foundation for further exploration of how sets relate to each other, particularly when analyzing whether they share common elements or remain distinct. Ultimately, the table encapsulates a fundamental aspect of set theory that is instrumental in understanding more intricate dynamics within the mathematical study of collections of objects.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Cardinality</td>
      <td>Number of \nSubsets</td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td>0 1 2 3</td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td>8</td>
    </tr>
    <tr>
      <th>5</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>6</th>
      <td>4 5 6</td>
      <td></td>
    </tr>
    <tr>
      <th>7</th>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
In the context of a school sports fest involving four teams representing different grade levels, the table presented outlines key elements of set theory, specifically focusing on the concepts of cardinality and subsets. The first column of the table titled "Cardinality" indicates the number of elements within a set, while the second column shows the number of subsets that can be formed from those elements. The layout of the table includes headers for the two primary categories, with a row indicating the cardinality of sets alongside the corresponding number of subsets.

Cardinality is a crucial aspect of set theory because it quantifies how many distinct elements are in a set. For instance, under the row labeled with "3," which represents a set containing the elements 0, 1, 2, and 3, a count of possible subsets provides a clearer understanding of how these elements can form various groupings. Additionally, the table conveys that when there are three elements (0, 1, and 2), there are a total of eight subsets possible, which includes the empty set and the set itself. 

Moreover, the table contains additional rows with relevant data but many remain unfilled in terms of cardinality, suggesting a broader range of operations and possibilities within set theory that can be explored beyond what is immediately presented. Understanding these concepts is essential as they not only promote recognizing how teams can be classified and grouped together—similar to how the Red-Yellow and Blue-Green teams are formed for specific events—but also illustrate the foundational principles of how sets operate mathematically. This foundation will assist students in mastering set operations and applying them effectively in both academic and practical scenarios.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>For any sets A and B, the union of A and B, which is denoted as A ∪ B (read as “A union \nB”), is given by:\nA ∪ B = {x|x ∈ A or x ∈ B}</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table presents a concise summary of a key concept related to set theory, specifically focusing on the union of sets. In the first row of the table, labeled under the category of "Key Concept," it emphasizes the fundamental notion of how the union of two sets is defined. This basic principle is essential in understanding how sets combine within the context of mathematics, particularly in the study of discrete structures.

In the following row, the table elaborates on this key concept by explaining that for any two sets, denoted as A and B, their union is represented as A ∪ B. The definition given indicates that the union comprises all elements that belong either to set A, set B, or both, encapsulating the notation as {x | x ∈ A or x ∈ B}. This formal representation lays the groundwork for further exploration of set relationships, demonstrating how distinct groups can be analyzed together, particularly in the context of grade levels and team memberships as illustrated in the surrounding text.

The surrounding context reinforces the significance of understanding union within disjoint sets, using specific examples involving students in different grade levels and teams. The previous discussions about the Blue-Green Team illustrate these principles visually through a Venn diagram, highlighting how A ∪ B provides a complete picture of the members involved when the sets are disjoint. Consequently, the table serves as a foundational reference for comprehending set operations, making it a valuable resource for those studying the concepts of unions in mathematics.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Type of Subsets</td>
      <td></td>
      <td>Subset</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Improper subset</td>
      <td>The improper subset of any set is itself.</td>
      <td>{a, b, c}</td>
    </tr>
    <tr>
      <th>2</th>
      <td>Proper subset</td>
      <td>One-element subsets</td>
      <td>{a}, {b}, {c}</td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td>Two-element subsets</td>
      <td>{a, b}, {b, c}, {a, c}</td>
    </tr>
    <tr>
      <th>4</th>
      <td>Trivial subset</td>
      <td>The empty set is a trivial subset of any set. ∅</td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented outlines different types of subsets, each categorized into rows that detail specific characteristics and examples. The first row serves as a header for the types of subsets, and the subcategories are laid out in a structured manner. The leftmost column specifies the index of each type, while the subsequent columns provide a description of that subset type and a concrete example.

The first type listed is the "Improper subset," which is defined as the subset of any set that includes the set itself, exemplified with the set {a, b, c}. Following this, the "Proper subset" is explained through one-element subsets, showcasing examples such as {a}, {b}, and {c}. The subsequent row presents two-element subsets, with examples like {a, b}, {b, c}, and {a, c}, demonstrating how pairs can be formed from elements of a set. Another interesting subset type mentioned is the "Trivial subset," which notes that the empty set is a trivial subset of any set, indicated by the symbol ∅.

Throughout the table, there is a clear delineation between the various types of subsets, providing clarity on their definitions and characteristics. The table is a practical reference for understanding subsets in set theory, supporting the concepts discussed in the surrounding context. The exploration of subsets is critical for grasping the foundations of set theory, and this table serves as an effective educational tool for students engaging with these mathematical principles.


key: html
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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
      <th>4</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The presented content revolves around the concept of union of sets, illustrated through a practical example involving athletes. The context explains how each team in a sports fest is represented as a set and discusses the mathematical representation of the union of two sets, denoted as A ∪ B. The definition clarifies that this union includes all elements that belong to either set A, set B, or both, emphasizing the importance of combining elements without repetition and in any order. This foundational understanding is further reinforced by showcasing specific examples of sets containing athletes: B, S, and T.

In the example, four sets of athletes are provided: B includes notable basketball players such as Curry, Durant, James, and Irving, while S consists of renowned soccer stars like Messi, Ronaldo, Suarez, and Neymar. Another set, T, features local athletes Yap, Tenorio, Pingris, and Romeo. The union of these sets is analyzed through specific operations, resulting in a comprehensive list of unique athletes for each combined set. For instance, when combining sets S and T, the union results in a new set comprising all athletes from both teams. The solutions provided illustrate that the order of the sets does not influence the outcome, showcasing the commutative property of the union operation, a critical concept in set theory.

Accompanying this detailed explanation of the union of sets is a table, although it appears to be empty in its current form. The structure of the table suggests that it may have been intended to organize and display information related to sets or their unions visually. In its existing state, the table features headers for columns and rows, but no data has been populated within its cells. This lack of content indicates that while a framework is in place to highlight or categorize information, the specific details that would provide insight into unions or the athletes in question are currently absent. The combination of the surrounding context and the framework of the table emphasizes the importance of understanding the union of sets while also hinting at an opportunity for application or further development of data representation.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>If A and B are two disjoint finite sets, then A ∪ B is also finite and\nn(A ∪ B) = n(A) + n(B).</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
Within the context of discussing the intersection and union of sets, the presented table serves as a key reference point for understanding some foundational concepts in set theory. The table is laid out with a simple format, consisting of two columns. The first column retains a numerical index, while the second column contains textual explanations pertinent to the operations of set unions and the conditions under which they are defined, particularly in relation to finite sets. 

The first entry under the column for textual explanations is labeled as "Key Concept." This indicates its significance in highlighting an essential aspect of set theory. The subsequent statement clarifies that when two sets, A and B, are disjoint and finite, their union, denoted by A ∪ B, is also finite. It further emphasizes the relationship between the sizes of these sets through the equation n(A ∪ B) = n(A) + n(B). This equation conveys that in the case of disjoint sets, the total number of elements in the union is simply the sum of the number of elements in each set, reflecting an intuitive understanding of combining distinct groups without overlap.

Overall, the table encapsulates a crucial set theory principle that aligns well with the earlier context, which elaborates on the intersecting nature of sets, particularly in relation to members belonging to more than one group. By presenting this information in a structured manner, the table acts as a supplemental tool for students to grasp how unions and intersections function, especially when considering different characteristics of sets, such as disjointness. The layout ensures that the idea of unions resulting in a finite set is clearly delineated from the potential complexities introduced by overlapping memberships in scenarios like clubs or other group settings.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td>Answer:\nTrivial subset</td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>5</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table appears to be part of a larger mathematical context that discusses set theory, specifically focusing on subsets and relationships between sets. The table is structured with rows and columns indexed with numbers, suggesting a matrix-like arrangement that is likely used for organizing responses or answers to various questions related to set operations.

The first row of the table lists the numbers 0, 1, and 2 across the top, indicating they may represent categories or specific elements relevant to the table's content. The first column, labeled with numbers 0 through 5, might refer to particular statements or premises that are being evaluated or compared. Notably, an entry in the table under the coordinates (0, 2) states "Answer: Trivial subset," which implies a focus on the concept of subsets, particularly the trivial subset, which is the empty set that is a subset of every set.

The remaining cells in the table are empty, indicating that responses or classifications for those particular intersections may not have been provided in the visible data. This aligns with the broader context discussing the properties of sets, subsets, and equivalence as described in the surrounding text. The context poses several true or false statements, suggesting that the users of this table might be required to determine or justify which aspects of set theory pertain to the entries in the table. Therefore, this table serves as a structured tool for analyzing and summarizing critical aspects of set relationships in a foundational mathematical course, particularly aimed at helping students understand and articulate key concepts related to subsets and set relationships.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Commutative Property of the Union of Sets\nFor any sets A and B,\nA ∪ B = B ∪ A.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The content presented in the table outlines a fundamental concept in set theory known as the commutative property of the union of sets. It begins with a key concept header that emphasizes the important role this property plays in the study of sets. Specifically, the table defines the commutative property stating that for any two sets A and B, the union of A and B (denoted as A ∪ B) is equal to the union of B and A (denoted as B ∪ A). This property highlights how the order in which sets are combined does not affect the result, a principle that is crucial for understanding operations involving sets.

The table is structured in a straightforward manner, beginning with a header that introduces the “Key Concept,” followed by the main statement of the commutative property in the subsequent row. The clear and concise arrangement of the information helps in reinforcing the concept, allowing readers to quickly grasp the essence of the commutative property. This is particularly relevant in the context of the provided excerpt, which discusses the concepts of disjoint sets and their unions, as well as how these ideas are visually represented in a Venn diagram. By asserting that the union of two disjoint sets can be calculated simply by adding their respective sizes, the text encourages students to apply and verify these mathematical principles through practical examples, such as how to visualize the Blue-Green Team using set unions. 

Overall, the table succinctly encapsulates a key mathematical principle that is foundational for students learning about sets. It serves as a quick reference point that complements the surrounding text, enhancing the educational experience by reinforcing critical concepts necessary for further exploration of set theory and its applications in various mathematical contexts.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>For any sets A and B, the intersection of A and B, which is denoted as A ∩ B (read as \n“A intersection B”), is given by:\nA ∩ B = {x|x ∈ A and x ∈ B}.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents essential key concepts related to the intersection of sets as part of a broader discussion on set theory. It begins with a foundational statement regarding two sets, A and B, emphasizing that the intersection of these sets, denoted as A ∩ B, encapsulates all elements that are present in both A and B simultaneously. This relationship is scientifically articulated in the table with the notation A ∩ B = {x | x ∈ A and x ∈ B}, clearly defining the criteria for membership in the intersection.

In the second row of the table, it is indicated that this intersection operation adheres to a crucial property known as the Commutative Property. This property asserts that the order of the sets does not affect the outcome of the intersection, meaning A ∩ B is equivalent to B ∩ A. This foundational concept is critical for students to understand the consistency and symmetries inherent in set operations, showcasing the logical nature of mathematical reasoning in set theory.

Surrounding the table, the text elaborates on the complement of a set, which serves as another key operation in set theory. It introduces the concept of a universal set, denoted as U, which contains all elements relevant to the discussion. The complement of a set A, symbolized as A', incorporates all elements from the universal set U that are not part of A. The definition is framed within the context of practical examples, such as letters in the English alphabet, and exhortations to visualize these concepts through Venn diagrams. This supplementary context enhances the understanding of the table's contents, as it establishes a foundation for comprehending intersections and complements as interconnected operations within set theory.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Commutative Property of the Intersection of Sets\nFor any sets A and B,\nA ∩ B = B ∩ A.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented highlights a key concept in set theory, specifically focusing on the commutative property of the intersection of sets. It is organized into a simple two-column format, where the first column indicates a specific identifier for the rows (labeled with numbers), while the second column provides pertinent information. The first row serves as a header, marking the beginning of the discussion with "Key Concept," signaling the importance of this particular property in understanding the relationships between sets.

The second row under the header includes the foundational statement of the commutative property itself, noting that for any two sets \( A \) and \( B \), the intersection operation is commutative. This means that when two sets are intersected, the order in which they are combined does not affect the outcome; thus \( A ∩ B = B ∩ A \). This succinctly encapsulates a critical principle within set operations, emphasizing the inherent symmetry involved in how sets can be combined.

The succinct nature of the information presented in the table corresponds well with the surrounding text from the PDF. The accompanying context elaborates on similar concepts, including operations involving set complements and universal sets, while reinforcing the significance of operations like intersection and union by also suggesting the verification of these properties using Venn diagrams. Overall, the table acts as a concise reference point for key theoretical principles in set theory, aligning with the educational framework introduced in the preceding text.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>The difference of sets A and B, which is denoted as A – B, is the set of elements of A that \nare not in B. In symbols, \nA – B = {x|x ∈ A and x ∉ B}.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a significant concept related to set theory, specifically focusing on the difference of sets. It starts with a key concept titled "Key Concept," which provides a clear definition of how to denote the difference between two sets, A and B. The notation used is A – B, which represents the elements that are in set A but not in set B. This is expressed more formally in the table as A – B = {x | x ∈ A and x ∉ B}, where x represents an element belonging to set A while simultaneously not being an element of set B.

The context surrounding this table discusses the operations involving sets, particularly set differences, and introduces set products through a practical example of fencing matches involving two boys and three girls. This real-life scenario illustrates the application of set theory concepts and ensures that readers understand how mathematical principles can manifest in tangible situations. The thorough explanations given around the table reinforce the importance of understanding set differences and products, ensuring that learners grasp these foundational ideas that will support their comprehension of mathematics.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>The universal set is the set that contains all the elements under consideration. This is \nusually denoted as U.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented contains important concepts related to sets in mathematics, specifically focusing on the concept of the universal set. The first row of the table identifies the content as a "Key Concept." This header signals the significance of the information that follows, which is foundational in understanding how sets and their complements function within the realm of set theory.

In the second row of the table, the definition of the universal set is articulated. It states that the universal set is the collection of all elements that are being considered in a particular discussion or problem. This set is commonly denoted by the symbol U. Following the table, the context elaborates on the relevance of the universal set, particularly in relation to the complement of a set—a concept highlighted through various examples involving sets of digits, even numbers, and prime numbers. Such sets, when differentiated from their complements, illustrate key properties in set operations, reinforcing the importance of the universal set as a foundational concept in set theory.

Furthermore, the context draws attention to crucial properties of set complements, emphasizing that the union of any set A and its complement A' equals the universal set U, while their intersection yields an empty set, denoted as ∅. This nuanced relationship can be visualized effectively with Venn diagrams, enhancing comprehension of these properties. Overall, the table serves as a succinct and informative reference point for these essential principles in understanding the structure and behavior of sets in mathematics.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>The  set  product  or  Cartesian  product  of  two  sets  A  and  B  is  the  set  of  all  possible \nordered pairs (a, b), where a ∈ A and b ∈ B. In symbols, the set product A × B (read as \n“A cross B”) is given by:\nA × B = {(a, b)|a ∈ A and b ∈ B}.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented captures a fundamental concept in set theory known as the Cartesian product. It succinctly outlines the definition of the set product, also referred to as the Cartesian product of two sets A and B. The first row indicates a key concept with the label "Key Concept," drawing attention to its significance. In the subsequent row beneath, a detailed definition is provided, explaining that the Cartesian product consists of all possible ordered pairs (a, b). The stipulation here is that "a" must be an element of set A and "b" must be an element of set B.

Further emphasis is placed on the notation used for expressing this concept. The Cartesian product is denoted as A × B (pronounced "A cross B"), and the mathematical expression formally encapsulated in the table is A × B = {(a, b) | a ∈ A and b ∈ B}. This notation clarifies that the Cartesian product encompasses all pairs where the first component originates from set A and the second from set B, highlighting the importance of the order of the elements in each pair, as established in the accompanying context. The clear presentation in the table aims to facilitate comprehension of this essential concept which serves as a foundational element for further exploration in set theory and relations between sets.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>For any set A and a universal set U:\nA ∪ A' = U \nA ∩ A' = ∅</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
In the context of set theory, the table presents a concise yet significant statement regarding the interactions between a set and its complement within a universal set. The first row in the table introduces a key concept related to the union and intersection of a set A with its complement A', emphasizing two fundamental properties. The expression A ∪ A' = U illustrates that the union of any set and its complement results in the universal set U, indicating that all possible elements account for either belonging to the set or not. The second part of this fundamental relationship, A ∩ A' = ∅, states that the intersection of a set and its complement is an empty set. This confirms that there are no shared elements between a set and its complement, underscoring the distinct nature of these two components.

The table format serves to highlight these important concepts in a clear and organized manner, facilitating better understanding for students learning about set operations. The inclusion of definitions and properties in a tabulated form allows for quick reference, reinforcing the theoretical foundations needed to engage with more complex operations such as set difference. This precedes practical examples in the surrounding context, which apply these principles to specific sets, demonstrating how to calculate differences and reinforcing the application of theoretical concepts in real-world problems. Overall, the table effectively encapsulates key theoretical insights into operations involving sets and their complements, setting the stage for deeper exploration into set relationships and operations.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented consists of a simple 2x2 matrix, organized with both rows and columns labeled with binary digits, 0 and 1. The header of the table indicates the column positions with the labels "0" and "1," corresponding to the second row of the same structure. Likewise, the rows are similarly labeled with "0" and "1," which indicates the row positions, where the first column lists these row identifiers.

Within each cell of the table, there are no entries, which signifies that there are currently no data points represented between the intersections of the row and column identifiers. Each cell at the intersection of the rows and columns remains empty, suggesting that specific values or relationships have not been defined or left out for this particular instance.

This tabular representation aligns with the context provided regarding Venn diagrams and set operations. Although the table lacks specific numerical or categorical contents, its structure can be connected to logical operations among sets A, B, and C, as described in the surrounding context. The empty cells may hint at potential relationships or conditions that could be explored using set theory and Venn diagrams, emphasizing the utility of such visual tools in understanding and solving problems related to elements that share common properties.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Let A ⊂ U. The complement of A is the set of elements of U that are not contained in A.  \nIt is denoted by A' (read as “A complement”). In symbols,\nA' = {x|x ∈ U and x ∉ A}.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table serves as a visual aid that encapsulates key concepts regarding set theory, specifically focusing on the complement of a set within a universal set. The first row features only one column header labeled "0," which serves as a reference point for the data presented beneath it. The initial cell of the first column highlights the term "Key Concept," signaling that the information provided is fundamental to understanding the subsequent material.

In the second row of the table, the concept regarding the complement of a set is elucidated. It is stated that if set A is a subset of a universal set U (notated as A ⊂ U), the complement of A encompasses all elements in U that are not members of A. This relationship is formally denoted by A', which is read as “A complement.” Additionally, the table provides a mathematical expression to clarify this relationship further: A' = {x | x ∈ U and x ∉ A}. This notation highlights the conditions that define the complement, showcasing the logical and systematic approach used in set theory to identify elements outside of a specified set.

In the context provided, the complement of sets is illustrated through exercises that depict how to derive such complements from the universal set. For example, in the sets defined by digits, even numbers, and prime numbers, the complements of those specific sets (S', T', and U') are calculated. This not only reinforces the understanding of complements but also aligns with the instruction to leverage roster methods for clarity. The table effectively encapsulates a crucial concept in set theory, offering a brief yet structured definition that is pivotal for students learning about the relationships and operations involving sets.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>Α\nΑ only</td>
      <td>U\nB\nB only</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td>C only\nC</td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table serves as a visual aid to enhance the understanding of set theory, particularly in relation to the example given about students who like basketball and volleyball. It is formatted as a grid that organizes information pertinent to the problem of determining the number of students who like these sports. The table features headings at the top, enumerating columns with indices ranging from 0 to 3, while the rows are similarly indexed, indicating a structured methodology for analyzing the data.

In the context of set representations, the first row includes data relevant to different categories of students based on their sports preferences. The second row presents distinctions between specific sets, labeled with symbols and notes such as "Α only," "U," "B," and "B only." These notations likely correspond to different classifications of students surveyed, where "Α only" could signify students who exclusively like one sport, while "B only" relates to those who only favor basketball, and "U" might refer to the universal set of surveyed students. The subsequent rows seem to expand upon these categories but remain largely empty, suggesting they are placeholders for potential future data or specific examples that clarify the relationships between the sets.

Overall, the design of the table is integral to illustrating the intersection and differences between the groups of students, facilitating a clearer understanding of the concepts at play in the problem. It aligns well with the Venn diagram mentioned in the context, allowing students to visually dissect the surveyed population's preferences in a structured format. This logical arrangement enables easier calculations of individual sets and their intersections, reinforcing the foundational principles of set theory as applied to real-world scenarios, such as the students' sports preferences.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td>50\nU\n11\nB\nL</td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>22\n30\n6</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td>15\n10</td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td>T\n25\nn</td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table appears to represent a summary of a Venn diagram that outlines the participation of students in three different racket sports, denoted as B, L, and T, within a universal set of 50 students. The organization of the table features a matrix format with indices from 0 to 2 across the top and left side, indicating different layers or categories of data represented in the Venn diagram. 

In the first row, the cell at the intersection of the first and second columns contains the values "50", "U", "11", "B", and "L", which likely reference the total number of students within the universal set (U) and possibly denote the quantities of students associated with each sport—specifically those who belong to sets B and L. The second row further elucidates the figures associated with set L and B, revealing the numbers 22, 30, and 6, which could represent the individual counts of students engaged in these activities or overlapping between the sets.

The third row introduces additional figures of 15 and 10, which may further signify distinct segments of students related to the sets discussed, while the last entry in the table houses references to set T, with the figures 25 alongside the letter "n," suggesting another subset of students. The overall structure of the table aligns with the context provided, emphasizing a systematic breakdown of student participation in various racket sports and the relationships between these sets, capturing the essence of statistical representation through a Venn diagram framework. The subtotals can be distilled from this table to understand how many students play one or more sports, ultimately leading to the conclusion that 3 students neither participate in B, L, nor T, illuminating the overall distribution of preferences among the student population.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td>50\nU\nB\nL\n30\n22\n5\n10\n7</td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>6\n9\n4\n6\nT\n25\n3</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The presented table appears to contain numerical data organized into a grid format, which is likely associated with the context of the survey conducted by a guidance counselor about student memberships in various clubs. The table consists of three rows and four columns, labeled with index numbers on both axes. The first column is an index column, while the remaining cells contain various lists of numbers and letters.

In the second column under the index 0, there are multiple entries including "50," "U," "B," "L," "30," "22," "5," "10," and "7." The presence of 'U' suggests it could indicate a universal set or overall membership, while the mixed set of numbers following it may represent counts of students in different categories or responses from the survey. Similarly, the second row under index 1 features a different set of entries: "6," "9," "4," "6," "T," "25," and "3." Notably, it appears to contain similar structural characteristics, possibly indicating students involved in the Theater Club, as implied by the letter 'T' within the entries.

The third column in both rows is empty, suggesting that the data may be intended for comparative or summary purposes, perhaps leading to further analysis or insights regarding student interests in the mentioned clubs. Overall, the layout of the table seems designed to visually represent the survey data, making it easier for the reader to interpret the relationships or memberships among the clubs surveyed.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>who like both volleyball</td>
      <td></td>
      <td>surveyed</td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>V</td>
      <td>50\nU\nB</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td>24\n24 − 8\n    = 16</td>
      <td>32\n32 − 8\n8\n  = 24</td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td></td>
      <td>50 – 48 = 2</td>
      <td></td>
    </tr>
    <tr>
      <th>5</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>6</th>
      <td></td>
      <td></td>
      <td></td>
      <td>n(U) =  number of students</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided content presents an analysis involving students' preferences for basketball and volleyball, focusing on a specific group of 8 students who like both sports. In this scenario, 56 students were surveyed, with 24 expressing a preference for volleyball (V) and 32 for basketball (B). However, because the 8 students who like both sports were counted in both categories, a correction was necessary. The calculation indicates that the effective count of students who like either sport is found by applying the principle of inclusion-exclusion: n(V) + n(B) - n(V ∩ B), leading to the equation 24 + 32 - 8 = 48. This deduction emphasizes the importance of accounting for overlapping preferences to avoid double counting.

Further detailing from the survey, the table illustrates the survey results of students’ preferences for three racket sports: badminton, lawn tennis, and table tennis. Six students reported liking all three sports. The breakdown indicates that 30 students enjoy badminton, 22 favor lawn tennis, and 25 prefer table tennis. There are also intersections in preferences among the sports: 10 students enjoy both table tennis and lawn tennis, 15 like both table tennis and badminton, and 11 prefer both lawn tennis and badminton. The analysis aims to determine how many students do not prefer any of these sports, which involves identifying and analyzing various sets of students: B for badminton, L for lawn tennis, and T for table tennis. 

The structure of the table shows numeric entries aligned under different categories, presenting computations relevant to the survey questions and responses. While some rows remain empty, others contain critical calculations that aid in visualizing the relationships and overlaps among students' preferences. The logical steps culminate with the expression n(U) = number of students, likely leading to conclusions about those students who have no preference for the racket sports examined. Such an organized representation not only captures the relevant numerical data but also elucidates the process of understanding preferences among a group and the reasoning behind statistical resolutions.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>If A and B are finite sets, then: \nn(A ∪ B) = n(A) + n(B) – n(A ∩ B).\nAn equivalent equation to this is:\nn(A ∩ B) = n(A) + n(B) – n(A ∪ B).</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table contains a key concept essential for understanding the relationships between sets, particularly in the context of the Venn diagram discussed. It introduces fundamental principles regarding finite sets A and B. Specifically, it outlines two important equations that describe the cardinality or size of the union and intersection of these sets. The first equation states that the number of elements in the union of sets A and B can be determined by adding the number of elements in each set and subtracting the number of elements that are common to both sets (their intersection). This is expressed mathematically as n(A ∪ B) = n(A) + n(B) – n(A ∩ B).

Furthermore, the table presents an equivalent version of this concept, which explicitly rearranges the terms to focus on the intersection. This version indicates that the size of the intersection of sets A and B can also be understood by rearranging the original equation, leading to the formulation n(A ∩ B) = n(A) + n(B) – n(A ∪ B). Such equations are crucial for effectively analyzing and interpreting the data from the student survey regarding their preferences for different racket sports, as illustrated in the accompanying Venn diagram.

In the context of the survey, these equations can be useful to quantify the overlapping preferences among students who play badminton, lawn tennis, and table tennis. For instance, as described, students who enjoy all three sports represent a significant subset within the total surveyed group, demonstrating the importance of understanding set operations. The relationship depicted in the Venn diagram is reinforced by the principles provided in the table, showcasing how to calculate individual and overlapping preferences, ultimately aiding in a comprehensive analysis of students' sports interests.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td>50\nU\n11\nB\nL\n30\n22\n5</td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>6\n9\n4\n15\n10</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td>T\n25</td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents numerical data organized in a grid format where the rows and columns correspond to different categories or variables. The first row of the table serves as a header, delineating the columns with the indices 0, 1, and 2. The first column of the table features row indices labeled 0, 1, and 2, which likely refer to different categories of interests or participation levels within a certain context, possibly related to the survey conducted by the guidance counselor regarding club memberships.

In the second row, under the column labeled '1,' there is a string of numerical data that includes various values: 50, U, 11, B, L, 30, 22, and 5. This mixture of numbers and letters may suggest specific quantities or identifiers representing different attributes of members or interests across the specified categories. In the first row, under the same column, the absence of an entry indicates that there might not be a relevant value or that it is intentionally left blank for the sake of categorization.

The third row contains further data with numerical values 6, 9, 4, 15, and 10 situated under the same column. Again, this presents a series of distinct data points without clear labels, but they may correlate to the interests or engagement levels in clubs similar to how the previous row's data was presented. The final row, corresponding to index 2, contains the variable 'T,' accompanied by the number 25 under column '1.' This could indicate a specific measurement or observation about the Theater Club, possibly showing total members or a particular demographic from the survey mentioned.

Overall, the table encapsulates varying numerical data points that could be relevant to understanding participation or interest levels in different activities, particularly in the context of the clubs referenced in the Venn diagram related to the guidance counselor's survey. The values might represent different group dynamics or elements of club membership, although without additional context, their exact implications remain somewhat ambiguous.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>A</td>
      <td>B</td>
      <td>B A</td>
    </tr>
    <tr>
      <th>1</th>
      <td>2</td>
      <td>3</td>
      <td>1.5</td>
    </tr>
    <tr>
      <th>2</th>
      <td>3</td>
      <td>5</td>
      <td>1.666666666...</td>
    </tr>
    <tr>
      <th>3</th>
      <td>5</td>
      <td>8</td>
      <td>1.6</td>
    </tr>
    <tr>
      <th>4</th>
      <td>8</td>
      <td>13</td>
      <td>1.625</td>
    </tr>
    <tr>
      <th>5</th>
      <td>M</td>
      <td>M</td>
      <td>M</td>
    </tr>
    <tr>
      <th>6</th>
      <td>144</td>
      <td>233</td>
      <td>1.618055556...</td>
    </tr>
    <tr>
      <th>7</th>
      <td>233</td>
      <td>377</td>
      <td>1.618025751...</td>
    </tr>
    <tr>
      <th>8</th>
      <td>M</td>
      <td>M</td>
      <td>M</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The given table is structured in a grid format with labeled rows and columns. The columns are indexed from 0 to 2, while the rows are labeled from 0 to 8, creating a matrix of values. Each cell in the table contains either an alphanumeric character or a numerical value. The first row serves as a header, indicating the column index, while the first column represents the row index. This organization allows for easy reference to the specific coordinates within the table.

The entries in the table seem to correlate with various calculations, particularly those involving the Fibonacci sequence and their ratios. For example, the second row has values 2, 3, and 1.5, while the subsequent rows reflect combinations of Fibonacci numbers followed by their ratios. The Fibonacci numbers are evident in rows 1, 2, 3, and 4, comprising a progression of values that increase as one moves down the table. Additionally, rows 5 and 8 are marked with 'M', which might imply a placeholder or a marker for a specific condition that wasn't defined in the given context.

Particularly noteworthy is the presence of certain values that are approximations of the golden ratio, specifically displayed in rows 6 and 7, which yield values close to 1.618, reinforcing the relationship between the Fibonacci sequence and the golden ratio itself. The representations of irrational numbers are reinforced in the context provided, indicating that values such as 1.6180339887... and others are part of the set of irrational numbers. This informs the reader about the connection between the Fibonacci sequence and real numbers, hinting at the broader mathematical concepts of irrationality and the natural ratios found in sequences. Overall, the table encapsulates mathematical relationships that reflect significant numerical properties and classes outlined in the accompanying context.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>1\n1.618</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented encapsulates a structured view of relationships or values organized in a grid format, suggesting a mathematical framework that is likely related to the properties of real numbers mentioned in the surrounding context. The header of the table features columns labeled with integers 0, 1, and 2, suggesting a categorization or classification of values associated with these identifiers. Each row corresponds to a different index, with the first column representing integer indices as well. 

Row one is entirely empty, indicating that no specific values correspond to the combination of the first index and the zero column. Moving to the second row, which corresponds to the index 1, the cells display a notable numerical representation: under the column labeled '1', there is a value of 1 and another that appears to be approximately 1.618. This second value, 1.618, is significant, as it closely resembles the golden ratio, a valuable concept in mathematics and nature, which may indicate that the table is exploring relationships among certain real numbers.

The final row is similarly empty across its columns, echoing the absence of defined values and contributing to a visual impression of incompleteness in the data set. The arrangement of values may suggest a positional or functional relationship between these specific numbers as they align within the context of real numbers and the properties defined by axioms, particularly in regards to operations such as addition and multiplication presented earlier. The overall layout of the table, therefore, serves to provide insight into the numerical interactions in a defined algebraic structure, resonating with the foundational ideas presented in the study of real numbers and their operations.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Closure Property\nLet a and b be real numbers. \nClosure Property of Addition\n \nAxiom 1. \n \n \n \nThe sum a + b is a real number.\nClosure Property of Multiplication\n \nAxiom 2. \n \n \n \nThe product a • b is a real number.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented is structured to highlight key concepts related to the closure properties associated with real numbers. At the top of the table, there is a header indicating the row for additional explanations, beginning with "Key Concept." This upfront categorization sets the context for the information that follows, which is essential for understanding mathematical operations concerning closure properties.

The first row details the "Closure Property" with a specific emphasis on how this property applies to addition and multiplication of real numbers. The entry under this row concisely states that if 'a' and 'b' are real numbers, the sum ('a + b') is also a real number, signified as Axiom 1. In addition to addition, the table succinctly notes the closure property for multiplication in Axiom 2—indicating that the product of 'a' and 'b' ('a • b') results in a real number as well. This clear delineation of axioms serves to reinforce the foundational understanding that real numbers maintain closure under both addition and multiplication.

The table thus serves as a reference point for understanding mathematical principles that govern operations with real numbers, particularly in providing clarity on the crucial aspects of closure under addition and multiplication. This ties back to the contextual content that explores properties like closure through examples and axiom definitions, ultimately forming a cohesive relation between theoretical concepts and practical exemplifications in mathematics.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Commutative Property\nLet a and b be real numbers. \nAxiom 3. \nCommutative Property of Addition\n \na + b = b + a\n \nCommutative Property of Multiplication\nAxiom 4. \na • b = b • a</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table outlines key mathematical concepts related to the Commutative Property, as it is applied in the context of addition and multiplication. At the top of the table, a "Key Concept" header is prominently displayed, indicating that the following information is foundational to understanding these properties. 

The first section of the table defines the Commutative Property and introduces the variables a and b as real numbers. It presents Axiom 3, which states the Commutative Property of Addition: that the order of adding two numbers does not affect the sum, formally expressed as \(a + b = b + a\). Furthermore, Axiom 4 is mentioned, detailing the Commutative Property of Multiplication, which similarly asserts that the order of multiplying two numbers does not affect the product, expressed as \(a \cdot b = b \cdot a\). 

This information is framed within the broader context of the examples provided earlier in the document, which demonstrate the application of addition and multiplication with specific numerical values. The examples illustrate how changing the order of the numbers leads to the same sum or product, reinforcing the concept of commutativity in both operations. Such mathematical principles are essential for developing a solid foundation in arithmetic and further mathematical studies, making the table an important reference point for learners.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Existence of an Identity Element\nLet a be any real number. \nAdditive Identity Property\n \nAxiom 7. \n \n \n \n \nThere exists a real number 0, called the additive identity element, such that \n \n \n \n \na + 0 = a  and  0 + a = a.\n \nAxiom 8.   Multiplicative Identity Property\n \n \n \n \nThere  exists  a  real  number  1,  called  the  multiplicative  identity  element, \nsuch that \n \n \n \n \na • 1 = a  and  1 • a = a.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents key mathematical concepts related to the existence of identity elements in real numbers, focusing on both additive and multiplicative properties. It outlines Axiom 7, which states the Additive Identity Property, indicating that there exists a real number, specifically 0, which serves as the additive identity element. This means that for any real number \( a \), adding 0 to it will yield the same number \( a \). This relationship is expressed mathematically as \( a + 0 = a \) and \( 0 + a = a \).

In addition, Axiom 8 discusses the Multiplicative Identity Property, revealing that there is a real number, 1, which functions as the multiplicative identity element. According to this property, for any real number \( a \), multiplying it by 1 will result in the same number \( a \). This is represented by the equations \( a \cdot 1 = a \) and \( 1 \cdot a = a \). Together, these axioms establish the foundational understanding that identity elements exist in both addition and multiplication within the realm of real numbers. The concepts articulated in this table serve as essential components in further exploring the nature of real numbers and the operations that can be performed on them, paving the way for more complex mathematical reasoning and applications.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Distributive Property of Multiplication Over Addition\nLet a, b, and c be real numbers. \n \nAxiom 11.   a • (b + c) = a • b + a • c</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table within the context focuses on the Distributive Property of Multiplication Over Addition, a fundamental concept in the manipulation of real numbers. It highlights Axiom 11, which states that for any real numbers \( a \), \( b \), and \( c \), the equation \( a \cdot (b + c) = a \cdot b + a \cdot c \) holds true. This axiom illustrates how multiplication distributes over addition, allowing for simplified calculations and transformations in algebraic expressions.

The first column of the table includes row identifiers, with the first row being a header that indicates a section for a key concept, specifically the distributive property. The subsequent row provides a clear definition of the axiom, emphasizing its application and importance as a foundational property that can be proven using the axioms of real numbers. This structured presentation not only captures key information but also serves as a reference for students learning to apply these properties when solving mathematical problems.

In the broader context discussed, the table reinforces the idea that understanding these properties aids in proving theorems. The example given in the surrounding text demonstrates a practical application of this property by manipulating equations to establish their validity. Overall, the table succinctly encapsulates a critical aspect of mathematical operations involving real numbers, directly linking to the concepts of addition and multiplication that students are expected to grasp and utilize in their studies.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Properties that can be proven using axioms are called theorems.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table comprises two key concepts related to foundational mathematical principles. At the top of the table, one finds the column headers representing the value "0," which serves as an important reference point in various mathematical contexts, particularly in number theory and algebra.

In the first row of the table, the intersection of the first column and the second column under the header showcases the phrase “Key Concept.” This signifies that the subsequent entries will provide essential insights into mathematical properties and principles. Following this, the entry in the second row articulates a critical understanding of mathematical reasoning: "Properties that can be proven using axioms are called theorems." This statement emphasizes the importance of foundational axioms in mathematics, as they serve as the building blocks for establishing theorems that elucidate various properties of numbers and operations. 

The context surrounding the table discusses subtraction and division and begins with definitions of these operations. It highlights their properties, particularly the necessity of adhering to axioms when proving mathematical statements. The inclusion of examples for subtraction and division offers practical reinforcement for the concepts laid out. The subsequent exercises on identifying the truthfulness of various statements related to natural numbers, rational numbers, and their properties ties back into the importance of theorems, as they provide a framework for understanding these assertions. Overall, the table serves as a concise yet significant reference for understanding essential mathematical principles that lay the groundwork for further study in the field.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Associative Property\nLet a, b, and c be real numbers. \nAssociative Property of Addition\n \nAxiom 5. \na + (b + c) = (a + b) + c\nAssociative Property of Multiplication\n \nAxiom 6. \na • (b • c) = (a • b) • c</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table is structured to outline key mathematical properties related to operations on real numbers, specifically focusing on the Associative Property. In the first column, the header and subheader indicate the focus of the table, with the leftmost cell labeled "Key Concept" denoting an essential aspect of mathematical principles that will be elaborated upon within the table. The content of the table systematically presents two major axioms relevant to the Associative Property: Axiom 5 and Axiom 6.

Axiom 5 illustrates the Associative Property of Addition, stating that for any real numbers \(a\), \(b\), and \(c\), the equation holds true: \(a + (b + c) = (a + b) + c\). This emphasizes that when adding numbers, the grouping of addends does not affect the overall sum. Similarly, Axiom 6 delineates the Associative Property of Multiplication, asserting the equivalence of the arrangement of factors: \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\). This property indicates that the product remains consistent regardless of the way in which the factors are grouped during multiplication.

Within the overall context of the instructional material, these properties are crucial for developing mental computation strategies. They relate back to the earlier discussions on simplifying calculations by grouping numbers to reach easier sums or products, highlighting their practical application in computational techniques. The table serves as a concise reference that encapsulates these fundamental properties, reinforcing the concepts intended for learners as they study operations involving real numbers.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Statement</td>
      <td>Reason</td>
    </tr>
    <tr>
      <th>1</th>
      <td>c • (a + b)  =  c • (b + a)</td>
      <td>Commutative Property of Addition</td>
    </tr>
    <tr>
      <th>2</th>
      <td>=  c • b + c • a</td>
      <td>Distributive Property</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table in the HTML format presents a structured layout to illustrate some algebraic properties, specifically focusing on the Commutative Property of Addition and the Distributive Property. The table is organized into a grid format with headings that categorize the statements and their corresponding reasons. In the first column, labeled with numbers, it indexes the different statements, while the top row contains numerical headings featuring the values 0 and 1, which correspond to the respective entries in the table.

In the first row of the table, the content indicates that for the scenario "c • (a + b)" equals "c • (b + a)," reflecting the Commutative Property of Addition. This property emphasizes that the order in which two numbers are added does not affect the sum. The row is annotated with a reason, reinforcing the rule discussed. The second row expands upon the application of the Distributive Property, asserting that "c • (a + b)" is equivalent to "c • b + c • a." This showcases how a common factor (c) can be distributed across an addition operation, thereby highlighting this key principle of algebra.

The context surrounding the table provides a foundation for understanding these mathematical principles. It discusses subtraction and division, establishing the definitions for these operations and indicating their relationships with real numbers. There's emphasis on verifying mathematical statements and principles, urging the reader to engage with axioms through practical examples. This aligns with the aim of the table, which not only serves as a visual aid but also invites deeper contemplation of foundational concepts in arithmetic and algebra. The practice exercises mentioned in the context further complement the table by providing opportunities for application and analysis of the properties listed.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>a b\n•\t\nSubtraction\n \nIf  a,  b,  and  c  are  real  numbers,  the  difference  a  –  b  is  the  number  c  such  that  \n \nc + b = a.\n•\t\nDivision\n \nIf a, b, and c are real numbers, the quotient \n, where b ≠ 0, is the number c such that  \nc • b = a.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The presented table outlines essential concepts related to arithmetic operations, specifically subtraction and division, particularly in the context of real numbers. It is structured with a single header column, where the first row indicates the key focus area, labeled "Key Concept." The specific key concepts are categorized under numbers that fall within the realm of mathematical operations. 

In the second row of the table, the definition of subtraction is articulated. It states that for real numbers a, b, and c, the subtraction operation expresses the idea that the difference a – b results in a number c that, when b is added to c, yields back the original number a (c + b = a). Similarly, the concept of division is clarified in the following entry. It highlights that if a, b, and c represent real numbers with the condition that b cannot equal zero, the division operation effectively identifies a quotient such that when c is multiplied by b, it returns to the original number a (c • b = a).

Overall, the table serves as a concise reference for fundamental properties of real numbers, detailing how subtraction and division function within mathematical expressions. This clear presentation aids in enhancing comprehension of basic arithmetic operations that are crucial for understanding more complex mathematical concepts.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>1 a\n1 a\n1 a\nExistence of an Inverse Element\n \nAxiom 9. \nAdditive Inverse Property\n \n \n \n \nFor any real number a, there exists a real number –a, called the additive \ninverse of a, such that \n \n \n \n \na + (–a) = 0  and  (–a) + a = 0. \nAxiom 10.  Multiplicative Inverse Property\n \n,  \n \n \n \n \nFor  any  nonzero  real  number  a,  there  exists  a  multiplicative  inverse \nsuch that \n • a = 1.\n \n \n \n \na • \n = 1  and</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table presents several key concepts regarding mathematical properties, particularly focusing on the existence of inverse elements in the realm of real numbers. The first key concept discussed in the table is the "Existence of an Inverse Element." This concept is foundational in understanding operations involving numbers in mathematics. It is divided into two main axioms: the Additive Inverse Property and the Multiplicative Inverse Property.

Within the table, Axiom 9 delineates the Additive Inverse Property, stating that for any real number \( a \), there exists a corresponding real number \( -a \), which is termed the additive inverse of \( a \). The axiom highlights that when a real number is added to its additive inverse, the result is zero, formalized by the equations \( a + (-a) = 0 \) and \( (-a) + a = 0 \). This property is essential for solving equations and understanding how numbers interact in addition.

Further, Axiom 10 presents the Multiplicative Inverse Property, which asserts that for any nonzero real number \( a \), there exists a multiplicative inverse that, when multiplied by \( a \), results in one. This is mathematically represented as \( a \cdot \text{(some number)} = 1 \). The multiplicative inverse is crucial for division and is foundational in many algebraic operations.

The context provided before the table illustrates a practical application of these mathematical principles through a gardening scenario involving eggplant seedlings. In that narrative, the numerical operations employed—multiplying the number of rows and seedlings—demonstrate the relevance of understanding additive and multiplicative properties, thereby relating the theoretical concepts of the table to real-world scenarios. This connection emphasizes the importance of inverse elements in both abstract mathematics and practical problem-solving situations.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
      <th>4</th>
      <th>5</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Number</td>
      <td>N</td>
      <td>W</td>
      <td>Z</td>
      <td>Q</td>
      <td>Q'</td>
    </tr>
    <tr>
      <th>1</th>
      <td>0</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td>2p</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td>51\n17</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td>2</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>5</th>
      <td>16</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>6</th>
      <td>3.14</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>7</th>
      <td>1 7\n–13</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>8</th>
      <td>0.345</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>9</th>
      <td>–3.8333...</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>10</th>
      <td>0.12321213...</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a structured layout organized in a manner that categorizes various numerical representations alongside corresponding symbols, allowing for easy reference and comprehension of mathematical constructs. The header row features column indices labeled from 0 to 5, while the first column specifies the corresponding list numbers, ranging from 0 to 10. The first cell of the table, located at the intersection of the first row and first column, is described as "Number," followed by a series of symbols or representations that are likely to have specific mathematical significance or relevance.

Within the body of the table, each row contains unique entries under the "Number" column, including a mix of numerical values, equations, and representations. For instance, the entry labeled "2p" under index 2 and "51\n17" under index 3 suggests the inclusion of mathematical expressions or potentially variable representations. Other entries like "16" and "3.14" indicate specific numerical values that could hold relevance in various mathematical contexts, such as constants or typical integers. The table also captures more complex entries, such as fractions and irrational numbers, indicated by the repetition notation used in "0.12321213..." or the recurring decimal format in "-3.8333...".

The utility of this table appears to align with the surrounding context that involves applying properties of real numbers as part of a two-column proof for basic algebraic identities. The rows of entries may be utilized or referred to in the process of completing proofs or solving mathematical problems, particularly those related to addition, multiplication, and distribution of arithmetic properties. Each numeric entry seems to be curated to provide a comprehensive foundation for students engaged in understanding mathematical relationships and operations. This aligns with other contextual tasks described, such as calculating totals, understanding averages, and applying properties to solve practical problems.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented is structured to display data in a grid format, organized with both headings and a body. The top row labels four columns, including an empty cell, and encompasses the headers 0, 1, and 2. This arrangement is likely intended for numerical representation or categorization, although the specific content within the cells is not populated in this instance.

In the first column of the table body, labeled with the header '0', the cells below (under headings for 0, 1, and 2) are also empty, indicating that there is currently no data provided for each coordinate intersection between the row and column headers. The structure suggests that the table could be intended for illustrating relationships among numbers or categorizing measures related to integers, given the surrounding discussion in the context about heights, ocean depths, and the number line with positive and negative values. 

The referenced content highlights the concept of integers, comprising positive and negative values as well as zero. The absence of data in the table could suggest that it is either a template for future input or represents a theoretical model rather than empirical data, aligning with the educational purpose of illustrating the set of integers and their relationships, such as those expressed through additive inverses and their representation on a number line.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Statement</td>
      <td>Reason</td>
    </tr>
    <tr>
      <th>1</th>
      <td>x   (a + b + c)  = x   [(a + b) + c]</td>
      <td>43.</td>
    </tr>
    <tr>
      <th>2</th>
      <td>= x   (a + b) + x   c</td>
      <td>44.</td>
    </tr>
    <tr>
      <th>3</th>
      <td>= x   a + x   b + x   c</td>
      <td>45.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The provided table outlines a structured presentation of algebraic expressions and their corresponding reasons, likely serving as a part of a lesson or explanation on the properties of algebra. It is formatted into two main columns titled "Statement" and "Reason," with the leftmost column denoting the row numbers. The initial row is reserved for headers, with the header of the first column labeled as an index and the second column listing various algebraic statements alongside the corresponding reasoning for each.

Beginning with row 1, the table presents the statement \( x(a + b + c) = x[(a + b) + c] \) and references it with the number 43, presumably indicating a previously defined property or theorem relevant to this expression. This demonstrates the application of the Associative Property of Multiplication. In the second row, identified as row 2, the statement transitions into the equation \( = x(a + b) + x c \), which builds on the first to further illustrate the distribution of multiplication over addition. It is associated with reason number 44, strengthening the understanding of how expressions can be manipulated through algebraic properties.

In the final row, identified as row 3, the table culminates in the statement \( = x a + x b + x c \), leading into an expression that explicates the distributive property of multiplication across addition and is tied to reason number 45. This systematic breakdown of statements and reasoning reinforces the conceptual understanding of algebraic properties, which is crucial for students grappling with algebraic manipulation and the relationships between numbers and operations. The overall format advocates clarity and logical progression, guiding learners through complex concepts in a digestible manner.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Place</td>
      <td></td>
      <td>Distance from  \nSea Level (in meters)</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Above Sea Level</td>
      <td>Mount Everest</td>
      <td>8,848</td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td>Mount Apo</td>
      <td>2,954</td>
    </tr>
    <tr>
      <th>3</th>
      <td>Below Sea Level</td>
      <td>Mindanao Trench</td>
      <td>–10,540</td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td>Marianas Trench</td>
      <td>–10,994</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a comparison of various geographical locations based on their elevation relative to sea level, indicating both the places situated above and below sea level. The first column lists two categories: "Above Sea Level" and "Below Sea Level." In these categories, specific places are documented alongside their respective distances from sea level measured in meters.

In the “Above Sea Level” section, Mount Everest is recorded as the highest point at 8,848 meters, highlighting its status as the tallest mountain in the world. Also included in this section is Mount Apo, which stands at a height of 2,954 meters. Under the "Below Sea Level" category, two notable underwater features are mentioned: the Mindanao Trench, which is located 10,540 meters below sea level, and the Marianas Trench, which is even deeper at 10,994 meters. These measurements vividly illustrate the dramatic variations in elevation across different geographical features, showcasing the extremes of the Earth’s topography.

This contextual backdrop reinforces the lessons about understanding numerical values expressed as integers, as demonstrated in the examples prior to the table. Just as the student’s performance in various subjects was analyzed using positive and negative integers based on the changes in grades, this table utilizes similar integer representations to convey relationships in altitude. Each entry signifies not just a physical location but also the elevation's numerical representation, underscoring the importance of integers in understanding and depicting real-world scenarios.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>The set of integers, which is denoted as Z, is composed of the natural numbers, their \nadditive inverses, and 0.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented outlines a fundamental concept in mathematics related to integers, specifically within the context of the set of integers denoted as Z. The first row indicates a header that corresponds to the integer '0', which is a critical point of reference in understanding the broader set of integers. This numerical framework establishes '0' as a baseline within the integer number line, which also includes positive and negative integers.

In the subsequent row, the table explicitly defines the set of integers. It clarifies that integers consist of natural numbers, which are the positive whole numbers (1, 2, 3, etc.), their additive inverses, which are the corresponding negative numbers (-1, -2, -3, etc.), and importantly, the number zero itself. This comprehensive definition highlights the inclusive nature of integers, allowing individuals to recognize both their positive and negative values as well as the neutral value of zero. The context provided in the examples further emphasizes the importance of distinguishing between positive and negative integers, particularly in scenarios that involve changes or differences, as seen in the expressions of growth and decline in various metrics such as the General Wholesale Price Index or academic grades in subjects like English, Math, and Science. 

Moreover, the connection to the examples given in the context, which involve plotting integers on a number line, serves to elucidate this concept visually. Students are encouraged to label points–representing changes in performance in different subjects–with their respective initials, reinforcing their understanding of how integers relate to measurable differences in performance or other contexts. This bridging of mathematical theory with practical application enhances comprehension and retention of the concept of integers in various scenarios.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Symbol</td>
      <td>Meaning</td>
    </tr>
    <tr>
      <th>1</th>
      <td>=</td>
      <td>is equal to</td>
    </tr>
    <tr>
      <th>2</th>
      <td>&gt;</td>
      <td>is greater than</td>
    </tr>
    <tr>
      <th>3</th>
      <td>&lt;</td>
      <td>is less than</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a concise representation of fundamental mathematical symbols used to denote relationships between numbers, specifically in the realm of integers and their basic operations. The first column features a list of symbols, each of which plays a critical role in mathematical expressions. For instance, the symbol "=" signifies equivalence, indicating that two values are the same. This notion of equality is pivotal in understanding operations involving integers, particularly when dealing with absolute values, as described in the surrounding context.

Continuing down the list, the table introduces the symbol ">", which indicates that the value on the left side is greater than that on the right. Conversely, the symbol "<" is similarly important, denoting that the left-hand value is less than the right-hand one. These comparative symbols are essential for understanding the ordering of integers and will aid students in grasping the relationships between different values. 

In summary, the table not only categorizes these symbols but also illustrates their meanings in a straightforward format, thereby supporting students’ comprehension of operations with integers. The language surrounding the table emphasizes the role of these symbols in mathematical reasoning, including the concept of absolute value, which underscores that distances on a number line can be more easily compared by ignoring signs. This foundational understanding is crucial as students progress in their mathematical education.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>The distance of a number from 0 on a number line is called the absolute value of the \nnumber. The absolute value of an integer n is written as |n|.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
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In the context of understanding integer addition, particularly with respect to the profits and losses recorded by Amy at her sari-sari store, the table emphasizes a critical mathematical concept known as absolute value. It outlines a fundamental principle pertaining to integers: the absolute value of a number is defined as its distance from zero on a number line. This concept is not only essential for performing arithmetic operations with integers, but it also provides clarity in interpreting both positive and negative values, particularly in financial contexts like profits and losses.

The table consists of two rows, where the first row includes a key concept labeled as "Key Concept," and the second row elaborates on the definition of absolute value. It states that for any integer \( n \), the absolute value is denoted as \( |n| \). This information is crucial for Amy as she seeks to determine her overall financial standing over a four-week period characterized by both gains and deficits. Understanding absolute values helps her grasp the scale of her losses compared to her profits, allowing for a clearer assessment of how much she has ultimately earned or lost.

Through the context provided, it is important to note that when working with integers, especially in additive scenarios involving both positive and negative numbers, the concept of absolute value becomes a vital tool. It not only assists in calculating total sums accurately but also aids in visualizing the integer's position on the number line, which is fundamental when verifying mathematical operations like those involving Amy's weekly profit-loss analysis. The explanation encapsulated in the table is designed to reinforce these concepts as learners approach the problem of combining integers to arrive at a clear financial summary.


key: html
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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Subject</td>
      <td>Difference</td>
    </tr>
    <tr>
      <th>1</th>
      <td>English</td>
      <td>+2</td>
    </tr>
    <tr>
      <th>2</th>
      <td>Filipino</td>
      <td>0</td>
    </tr>
    <tr>
      <th>3</th>
      <td>Science</td>
      <td>–1</td>
    </tr>
    <tr>
      <th>4</th>
      <td>Math</td>
      <td>–3</td>
    </tr>
    <tr>
      <th>5</th>
      <td>Araling Panlipunan</td>
      <td>+4</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented categorizes a student’s performance in various subjects through a numerical representation of the differences in grades. The layout consists of two main columns: one indicating the subjects and the other showing the differences in grades across those subjects. The first row serves as a header, labeling the columns with "Subject" and "Difference." Each subsequent row corresponds to a specific subject, allowing for a straightforward comparison of academic performance. 

Among the subjects listed, English shows a positive difference of +2, indicating an improvement in grades. Filipino maintains a neutral stance with a difference of 0, suggesting that there has been no change in performance. In contrast, Science reports a slight decline with a difference of –1, while Math reflects a significant decrease in performance, with a difference of –3. Notably, the subject experiencing the highest improvement is Araling Panlipunan, showcasing a positive difference of +4, which points to the student’s strongest area of academic growth.

The context preceding the table emphasizes the importance of comparing integers and illustrates how to interpret increases and decreases in grades using a number line. It underscores that the greater the positive integer, the more significant the improvement, while negative integers denote declines in performance. This foundational understanding enriches the interpretation of the table, as it becomes clear that Araling Panlipunan has the greatest increase in grades by 4, while Math represents the largest decrease by 3. Such insights reflect the student's overall academic trajectory and aid in identifying areas for potential focus or improvement.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Week 1</td>
      <td>a profit of  1,426</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Week 2</td>
      <td>a loss of  292</td>
    </tr>
    <tr>
      <th>2</th>
      <td>Week 3</td>
      <td>a loss of  585</td>
    </tr>
    <tr>
      <th>3</th>
      <td>Week 4</td>
      <td>a profit of  320</td>
    </tr>
  </tbody>
</table>


key: chroma:document
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Within the context of learning about integer addition, a table presents a summary of financial activity over four weeks, detailing profits and losses for each week. The first row of the table indicates that during Week 1, there was a notable profit of 1,426. This positive result sets a contrasting tone against the subsequent weeks.

In Week 2, the financial outcome shifts to a loss of 292, followed by yet another loss in Week 3 of 585. This reveals a downward trend in performance during those weeks. However, Week 4 sees a reversal of fortunes with a profit of 320, suggesting a rebound from the previous losses. The structure of the table, which includes designated rows for each week along with their corresponding financial outcomes, encapsulates the series of financial fluctuations, displaying a dynamic and illustrative snapshot of weekly progress in terms of profits and losses. 

This data can be particularly useful when applying the mathematical concepts discussed in the context regarding the addition of integers, especially when examining situations involving positive and negative values. The table succinctly encapsulates how the concept of adding negative and positive integers can relate to real-life scenarios, such as tracking financial performance over time.


key: html
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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>When adding integers with unlike signs, subtract their absolute values. Then attach the \nsign of the integer with the greater absolute value to the sum.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
In the provided material, we delve into the principles of adding and subtracting integers, particularly focusing on operations involving both positive and negative numbers. The accompanying table elucidates a key concept that is crucial in dealing with integers of unlike signs. Specifically, it notes that when adding integers that have different signs, one must first subtract their absolute values. Following this, the final sum should be assigned the sign of the integer that possesses the greater absolute value. This principle is essential for ensuring that calculations involving mixed-sign integers yield correct results.

The context further explores an application of this concept through practical examples. For instance, it is illustrated how Amy experiences a profit, using a specific equation that combines both positive and negative integers. The calculation involves grouping the integers by their signs before conducting the sum, demonstrating the commutative and associative properties of addition. Similarly, the scenario involving Susan's borrowing reinforces the necessity of understanding how to apply subtraction with integers, especially when dealing with borrowed amounts and repayments. Here, the table's key concept serves as a guiding principle, making it clear how to properly combine and manipulate negative and positive values in real-life financial contexts. 

Overall, the information surrounding the table, along with its contents, underscores the practical applications of integer operations in everyday financial situations and emphasizes foundational math concepts such as absolute values and the assignment of signs in sums and differences. Understanding these principles is crucial for learners as they navigate more complex integer operations and related mathematical problems.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>When  multiplying  integers,  get  the  product  of  their  absolute  values.  Then  apply  the \nfollowing rules for the sign of the product:\n \n• \nThe product of two integers with like signs is positive. \n \n• \nThe product of two integers with unlike signs is negative.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a key concept related to the multiplication of integers, serving as a foundational principle in understanding how to work with different integer signs. At the top of the table, it highlights the importance of calculating the product by first determining the absolute values of the integers involved. Following this, the table outlines two vital rules concerning the signs of the resulting product, which are crucial for performing accurate mathematical operations with integers.

According to the contents of the table, when multiplying two integers that share the same sign—either both positive or both negative—the product is always positive. Conversely, if the integers have different signs, meaning one is positive and the other is negative, then the product will be negative. These rules not only aid in solving numerical problems but also enhance the understanding of how integers interact in multiplication, as illustrated by the example provided in the context of dividing integers. The comprehension of these rules lays the groundwork for mastering more complex arithmetic operations, particularly in contexts like the one mentioned, where temperatures in Baguio drop, highlighting the practical application of these integer operations in real-world scenarios. 

Overall, this table effectively distills essential knowledge regarding integer multiplication into succinct terms, making it easier for students to grasp and apply these concepts in their mathematical endeavors.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>When adding integers with like signs, add their absolute values. Then attach the common \nsign to the sum.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented outlines a key concept related to the addition of integers, particularly focusing on the rules for combining integers with like signs. In the first row, marked as the header, we find an encapsulation of the overarching principle being conveyed: "Key Concept." Below this header in the first body row, the table states that when adding integers that share the same sign (either both positive or both negative), one should first sum their absolute values. Following this addition, the common sign (either positive or negative, depending on the integers) is then attached to the resulting sum. This foundational rule simplifies the process of adding like signed integers by providing a clear and systematic approach.

In the subsequent row of the table, the explanation is enriched with further details, reiterating the previous concept. It emphasizes that for integers with like signs, the procedure remains consistent: add the absolute values together and maintain the common sign in the computation of the sum. This highlights the straightforward nature of adding integers that are classified similarly, ensuring that students have a clear understanding of the operations involved. The table serves as a concise reference point for learners, capturing these essential rules that govern the addition of integers in a structured and easily digestible format. 

The context provided delves into the scenarios of adding integers with differing signs and reinforces the importance of understanding when signs change, as highlighted through various examples. This information complements the table by illustrating practical applications of the principles outlined, enhancing the overall comprehension of integer addition. Together, the table and the surrounding explanations function as a cohesive system for mastering the addition of both like and unlike signed integers, guiding learners through potential complexities in arithmetic operations.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>When subtracting integers, change the subtrahend to its additive inverse, and change the \noperation to addition. Then follow the rules in adding integers.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a crucial Key Concept regarding the subtraction of integers, emphasizing a specific method to simplify the process. In the first row, it highlights the title "Key Concept," indicating the significance of the content that follows. The subsequent row elaborates on the method for subtracting integers. It states that when performing subtraction, one should change the subtrahend to its additive inverse—essentially flipping the sign of the number being subtracted. Alongside this change in the subtrahend, the operation itself is also switched from subtraction to addition. This approach lays the groundwork for applying the rules of integer addition to the problem at hand.

In the context provided, this concept is particularly relevant when discussing integer operations, especially with examples such as the calculations involving the subtraction of negative integers. The earlier context illustrates how these principles apply to various arithmetic operations, especially through the provided examples like \( 7 - (-9) \) and \( -3 - 12 \). The explanation resonates with the steps outlined in the table, reinforcing that changing the operation to addition and utilizing the additive inverse enhances understanding and problem-solving efficiency in integer arithmetic. Overall, the table serves as a concise reference for students to grasp the fundamental technique for subtracting integers effectively.


key: html
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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>In general, if a and b are real numbers, the following equivalence relations exist:\n\t\n•\t\na + (–b) = a − b\n\t\n•\t\na – (+b) = a − b\n\t\n•\t\na − (−b) = a + b</td>
    </tr>
  </tbody>
</table>


key: chroma:document
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The table presented contains key concepts related to the operations involving real numbers, specifically focusing on addition and subtraction. At the top of the table, a single column is dedicated to defining these mathematical principles. The initial entry in the first row emphasizes a fundamental "Key Concept," which acts as an introduction to the subsequent equivalence relations outlined in the following row.

In the second row, the table details three essential equivalence relations that involve the manipulation of real numbers using addition and subtraction with negative values. The first point highlights that when subtracting a real number \( b \) from another real number \( a \), it is equivalent to adding the negative of \( b \) to \( a \) (i.e., \( a + (–b) = a − b \)). This illustrates how the concept of negative numbers alters standard operations. The second relation conveys that subtracting a positive number \( b \) from \( a \) simplifies to \( a − b \), reinforcing the idea that subtraction can often be seen as a straightforward reduction of quantity. Finally, the third point clarifies that subtracting a negative number (indicated as \( -b \)) from \( a \) is the same as adding the positive counterpart of that number, thereby restating that \( a − (−b) = a + b \). 

These equivalences not only serve as foundational rules for operations involving integers but also align with the surrounding context, which discusses the multiplication of integers. The provided examples illustrate how to determine the signs and values of products when multiplying integers, leading to a comprehensive understanding of both multiplication and addition/subtraction involving real numbers and their properties. This synthesis of concepts enhances the overall comprehension of mathematical operations and their outcomes in various scenarios.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>When dividing integers, get the quotient of their absolute values. Then apply the following \nrules for the sign of the quotient:\n \n• \nThe quotient of two integers with like signs is positive.\n \n• \nThe quotient of two integers with unlike signs is negative.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents essential information regarding a key concept related to dividing integers in mathematics. At the top, there is a header indicating the content of the table, which includes the discussion of the process of division among integers. The primary focus is on the method of obtaining the quotient from integer division, emphasizing that one should first calculate the absolute values of the integers involved.

Several rules are provided to clarify the signs of the resulting quotient. It is highlighted that when dividing two integers with like signs—meaning both integers are either positive or negative—the quotient will yield a positive result. Conversely, if the integers have unlike signs, which means one is positive and the other is negative, the quotient will be negative. These rules are critical for students to understand division in the context of integers and serve as foundational knowledge for performing calculations involving positive and negative numbers.

The content within the table effectively supports the instructional context surrounding the surrounding practice exercises, which involve various mathematical operations with integers. These exercises challenge students to express numerical values using integers, evaluate the truth of various integer-related statements, and perform a diverse range of arithmetic operations, including addition, subtraction, multiplication, and division. Together, the table's information and the surrounding exercises aim to reinforce students' understanding of working with integers and their properties, laying the groundwork for more complex mathematical concepts.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Grade Level</td>
      <td>Change in the Number \nof Enrollees</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Grade 7</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td>Grade 8</td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td>Grade 9</td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td>Grade 10</td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
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In the context of enrichment exercises for MATHPLUS 7 TX, the table presented focuses on educational data related to different grade levels. The table is structured with a grid format that includes a header and several rows corresponding to different grades. At the top of the table, the columns are labeled with "0" and "1," with the first column indicating the various grade levels and the second column capturing the change in the number of enrollees. 

The first row of the table defines the headings, wherein the first column is dedicated to the "Grade Level" and the second column corresponds to the "Change in the Number of Enrollees." Below these headers, there are entries for Grades 7 through 10, identifying them sequentially. However, the second column, which is intended to display numerical values indicating a change in enrollment figures for each grade, is currently left empty for all grades listed, suggesting that the data may not have been collected or reported yet for this particular exercise.

This table could serve as a visual aid for teachers and students as they analyze educational enrollment trends across different grades, facilitating discussions or further calculations in line with the enrichment exercises mentioned in the surrounding text. The exercises include inquiries about mathematical properties related to absolute values and the implications of operations involving positive and negative numbers, linking back to foundational concepts in mathematics that are critical for students at this stage in their education.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
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      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>p q\nThe  set  of  rational  numbers,  denoted  by  Q,  is  composed  of  numbers  that  can  be \nexpressed as a ratio of two integers. In symbols,\n  p and q are integers, and q ≠ 0 .\nQ =</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a concise overview of a key mathematical concept related to the set of rational numbers, denoted by the symbol Q. It is structured in a way that distinguishes the different elements being explained. In the first row, there is a header with "Key Concept," which serves as an introductory label for the content that follows. Beneath this header, the table elaborates on what rational numbers are, specifying that they can be expressed as a ratio of two integers. 

The subsequent entry in the table reinforces this definition by providing a formal mathematical expression: p and q represent integers, while also emphasizing a critical condition that q must not be equal to zero (q ≠ 0). This clarification is essential, as it highlights the fundamental rule that the denominator in a fraction cannot be zero, preserving the validity of the ratio. Therefore, the overall message conveyed in the table underlines the characteristics and rules governing rational numbers, offering essential foundational knowledge that aligns well with the mathematical examples provided in the surrounding text, specifically regarding the representation of numbers as fractions and their equivalent decimal forms.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Density Property of Rational Numbers\nBetween any two rational numbers, there is always another rational number.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented within the MATHPLUS 7 TX Unit 1 resource outlines fundamental mathematical concepts related to rational numbers. It begins by establishing the term "Key Concept," which serves as an introductory label for the critical principle discussed in the subsequent row. The primary focus of the table is the "Density Property of Rational Numbers," a significant concept that asserts that between any two rational numbers, there exists another rational number. This idea is crucial in understanding the continuum of rational numbers within the number line, emphasizing that rational numbers are not isolated but are densely packed.

As highlighted in the context preceding the table, the understanding of rational numbers and their comparisons is pivotal in various mathematical operations. For instance, while comparing different types of numbers, it's essential to know that any negative rational number is less than any positive rational number. The context also illustrates methods of comparison, particularly when dealing with negative rational numbers, emphasizing the concept that the greater the absolute value, the lesser the actual value of the number. Furthermore, the text elaborates on the necessity of finding the Least Common Denominator (LCD) when comparing fractions, which is crucial for accurately determining the relationships between different rational quantities. The inclusion of examples further enhances comprehension, demonstrating practical applications of the density property and comparison strategies in rational numbers.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Grade Level</td>
      <td>SY 2009–2010</td>
      <td>SY 2010–2011</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Grade 7</td>
      <td>258</td>
      <td>342</td>
    </tr>
    <tr>
      <th>2</th>
      <td>Grade 8</td>
      <td>310</td>
      <td>295</td>
    </tr>
    <tr>
      <th>3</th>
      <td>Grade 9</td>
      <td>365</td>
      <td>370</td>
    </tr>
    <tr>
      <th>4</th>
      <td>Grade 10</td>
      <td>375</td>
      <td>360</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents data regarding the number of enrollees across four grade levels—Grade 7, Grade 8, Grade 9, and Grade 10—during the school years SY 2009–2010 and SY 2010–2011. The first column indicates the grade levels, while the subsequent columns illustrate the number of students enrolled in each grade for the respective school years. For Grade 7, there were 258 enrollees in SY 2009–2010, which increased to 342 in SY 2010–2011, indicating a significant increase of 84 students. In contrast, Grade 8 experienced a decrease in enrollees from 310 to 295, reflecting a loss of 15 students.

The data reveals that Grade 9 saw a slight rise in its enrolment figures, increasing from 365 in SY 2009–2010 to 370 in SY 2010–2011, resulting in an increment of 5 students. Conversely, Grade 10 experienced a decrease, with enrolment numbers dropping from 375 in SY 2009–2010 to 360 in SY 2010–2011, which amounts to a decline of 15 students. This information gives insights into enrollment trends among different grade levels and raises questions about factors influencing these changes, which could be examined in subsequent analysis, particularly in relation to the increases and decreases noted in specific grades. 

Considering the broader context of the surrounding questions, such as the overall changes in enrollees and the comparison between specific grades, the data will serve as a basis for further exploration of student enrollment dynamics over the two academic years mentioned. The changes calculated among the grades can help educators and administrators identify patterns, areas for improvement, and potential actions to manage future enrolments effectively.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>a b\nThe Fundamental Law of Fractions\nac\n = \n.\nIf a, b, and c are real numbers, b ≠ 0 and c ≠ 0, then \nbc</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented outlines key concepts related to fractions, specifically focusing on the Fundamental Law of Fractions. At the top of the table, there is a header that indicates the first column is labeled with a zero, which serves as a point of reference for the subsequent entries. The first row beneath the header includes a row label of "0," emphasizing the foundational significance of this concept. Under this label, the table includes a significant entry categorized as "Key Concept."

In the second row of the table, the concept of the "Fundamental Law of Fractions" is detailed. It states that for any real numbers a, b, and c, where b is not equal to zero and c is not equal to zero, the equation \( \frac{a}{b} \cdot c = \frac{ac}{b} \) holds true. This fundamental principle highlights the relationship between numbers in fractional form, establishing a foundation for understanding how fractions operate in mathematical contexts. 

The inclusion of this concept aligns with the surrounding text that discusses the simplification of fractions and the conversion of decimals into fractions. The mention of identifying relationships and properties within fractions serves to bridge the practical applications of fractions and decimals, ultimately enhancing the learner's ability to plot rational numbers on a number line effectively. Understanding these concepts is crucial for students as they navigate more complex mathematical operations.


key: chroma:document
string_value:
The provided context discusses the set of rational numbers, denoted by Q, which are numbers expressible as a ratio of two integers. It highlights that fractions are key examples of rational numbers, emphasizing that each fraction can be represented in decimal form, either as terminating or as nonterminating but repeating decimals. The explanation delves into the distinction between these two types of decimals, clarifying that terminating decimals have a finite number of digits after the decimal point, whereas nonterminating repeating decimals feature an infinite sequence of digits. The mention of fractions, such as 1/2, exemplifies a terminating decimal that results in a finite quotient of 0.5, while the context also prompts calculus with examples of how certain fractions such as 1/4 and 1/3 can be converted into decimals.

Within this framework, the accompanying table represents a structured layout intended to facilitate further exploration or illustration of converting fractions to decimals. Though the table is currently empty, it consists of a grid that features a header with numerical identifiers across the top (0, 1, and 2) and down the side (0, 1, and 2). This organizational format suggests that it may be designed for documenting the results of conversions from fractions to their decimal equivalents, potentially filled with values that depict the outcomes of such conversions, specifically indicating the relationship between the various numerators and denominators. 

Ultimately, the combination of the provided context on rational numbers and the layout of the table offers an educational tool meant to guide students in understanding the decimal representations of fractions and the fundamental concepts underlying these mathematical constructs. The table serves as a visual aid where students could complete conversions, enhancing their grasp of the relationship between fractions and their decimal forms within the broader setting of rational numbers.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Column 1</td>
      <td>Column 2</td>
    </tr>
    <tr>
      <th>1</th>
      <td>|0|</td>
      <td>0 • (–5)</td>
    </tr>
    <tr>
      <th>2</th>
      <td>–(–1)</td>
      <td>|–3| – 2</td>
    </tr>
    <tr>
      <th>3</th>
      <td>5 • (–4)</td>
      <td>(–80) ÷ (–4)</td>
    </tr>
    <tr>
      <th>4</th>
      <td>15 – (–5)</td>
      <td>–15 + 5</td>
    </tr>
    <tr>
      <th>5</th>
      <td>n</td>
      <td>|n|</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a structured layout featuring two main columns, identified as Column 1 and Column 2, with an index on the left side. The indices range from 0 to 5, creating a clear reference point for each row's content. The first row serves as the header, indicating the presence of two distinct columns for presentation purposes. Each subsequent row contains various mathematical expressions and operations that appear to be related to arithmetic and algebra.

In the first row of the data, the entries include "Column 1" and "Column 2," acting as labels for the data beneath. The second row introduces mathematical expressions like “|0|” and “0 • (–5)” that allude to basic operations involving absolute values and multiplication. The third row features a negative operation with “–(–1)” and an absolute value expression combined with subtraction, represented as “|–3| – 2.” The following rows continue this mathematical theme, containing more complex expressions such as “5 • (–4)” and “(–80) ÷ (–4),” both involving multiplication and division of negative numbers.

Moving further down, the table presents expressions that illustrate various mathematical concepts, including addition with negative values (as seen in “15 – (–5)”) and the interaction of a variable (denoted as "n") with its absolute value in the final row. This layout facilitates the exploration of mathematical relationships and operations within the context provided, contributing to tasks related to student enrollment changes and potentially their portrayal on a number line, as mentioned in the context prompts. Overall, the table efficiently organizes diverse mathematical expressions, each relevant to exercises aimed at evaluating changes in data, a critical aspect in analyzing trends in enrollment figures from one academic year to the next.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>a b\nc d\na • d  &gt;  c • b\n&gt;</td>
      <td>a b\nc d\na • d  &lt;  c • b\n&lt;</td>
      <td>a b\nc d\na • d  =  c • b\n=</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented outlines the relationships between two rational numbers in fraction form, denoted as "a/b" and "c/d." It is organized into three distinct learning outcomes related to comparing these rational numbers using their product and provides a visual structure to analyze the relationships. Each row in the table addresses a specific comparison scenario: whether "a/b" is greater than "c/d," less than "c/d," or equal to "c/d." 

Under the first column, where "a/b" is greater than "c/d," the mathematical expression is conveyed as "a • d > c • b," indicating that the product of the numerator of the first fraction and the denominator of the second fraction exceeds the product of the numerator of the second fraction and the denominator of the first. This identifies the condition required for establishing that one rational number is larger than another. 

Conversely, in the second column, the relation of "a/b" being less than "c/d" is portrayed as "a • d < c • b," illustrating the opposite relationship where the first fraction is smaller than the second. Lastly, the third column shows the equality condition, represented as "a • d = c • b," indicating that the two fractions are equivalent in value. This structured approach not only reinforces the principle of comparing rational numbers through cross multiplication but also aids in visualizing the foundational concepts discussed in the surrounding context, where different rational numbers are ordered from least to greatest.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Month</td>
      <td>Peso Equivalent \nof 1 US dollar</td>
    </tr>
    <tr>
      <th>1</th>
      <td>January</td>
      <td>47.73</td>
    </tr>
    <tr>
      <th>2</th>
      <td>February</td>
      <td>47.45</td>
    </tr>
    <tr>
      <th>3</th>
      <td>March</td>
      <td>46.64</td>
    </tr>
    <tr>
      <th>4</th>
      <td>April</td>
      <td>46.20</td>
    </tr>
    <tr>
      <th>5</th>
      <td>May</td>
      <td>46.67</td>
    </tr>
    <tr>
      <th>6</th>
      <td>June</td>
      <td>43.32</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presents a monthly breakdown of the peso equivalent of 1 US dollar for various months. It features two main columns: the first indicates the month, while the second provides the corresponding exchange rate expressed in pesos. This format allows for a clear comparison of how the exchange rate fluctuates over the course of the year.

Starting with January, the peso equivalent of one US dollar is noted as 47.73 pesos. As the year progresses, there are slight variations in this rate. For instance, February shows a marginal decrease to 47.45 pesos, which continues to decline to 46.64 pesos in March and further to 46.20 pesos in April. However, in May, the exchange rate experiences a slight rebound to 46.67 pesos before dropping significantly to 43.32 pesos in June. 

The illustration of exchange rates throughout these months may serve various analytical purposes, allowing individuals or businesses to make informed decisions regarding currency exchange, trade, and financial planning. The trend of a decreasing peso value against the dollar could have implications for economic analysis and personal finance strategies among those interested in the foreign exchange market.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Day</td>
      <td>Hours  \nof Work</td>
      <td>Pay for the First 8 Hours \n(in pesos)</td>
      <td>Overtime Pay \n(in pesos)</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Monday</td>
      <td>10</td>
      <td>85 • 8 = 680</td>
      <td>127.50 • 2 = 255</td>
    </tr>
    <tr>
      <th>2</th>
      <td>Tuesday</td>
      <td>10</td>
      <td>85 • 8 = 680</td>
      <td>127.50 • 2 = 255</td>
    </tr>
    <tr>
      <th>3</th>
      <td>Wednesday</td>
      <td>10</td>
      <td>85 • 8 = 680</td>
      <td>127.50 • 2 = 255</td>
    </tr>
    <tr>
      <th>4</th>
      <td>Thursday</td>
      <td>11</td>
      <td>85 • 8 = 680</td>
      <td>127.50 • 3 = 382.50</td>
    </tr>
    <tr>
      <th>5</th>
      <td>Friday</td>
      <td>11</td>
      <td>85 • 8 = 680</td>
      <td>127.50 • 3 = 382.50</td>
    </tr>
    <tr>
      <th>6</th>
      <td>Total</td>
      <td></td>
      <td>3,400</td>
      <td>1,530</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The presented table outlines a work schedule for a week, detailing the number of hours worked each day, along with the corresponding pay calculations. The first column lists the days of the week, starting with Monday and ending with a total row. The hours worked are indicated in the second column, which shows that the standard work hours generally amount to 10 hours per day for Monday through Wednesday, and 11 hours for both Thursday and Friday.

In the subsequent columns, the table provides calculations for regular pay and overtime pay. For the first 8 hours of work each day, a pay rate of 85 pesos per hour is applied. The calculation for the first 8 hours is consistently shown as '85 • 8 = 680 pesos' for each of the first three days. However, the hours worked on Thursday and Friday exceed the regular hours, leading to an overtime rate of 127.50 pesos per hour for hours beyond the standard. The table specifies that for Thursday and Friday, where the individual worked for 11 hours, the calculations detail that 3 hours are considered overtime, thus yielding '127.50 • 3 = 382.50 pesos' for each of those days.

The final summary row captures the total earnings for the week. The total regular pay for the first eight hours accumulates to 3,400 pesos across the five days worked. For overtime, a sum of 1,530 pesos is calculated, representing the additional pay earned for those extra hours. Combined, these totals provide a comprehensive overview of the week's earnings and reflect how varying hours impact overall compensation, echoing the calculations indicated in the surrounding context related to electric current computations.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>1 2\n1 1\n1\n?\n1\n32\n64\n6</td>
      <td>1 8\n1 4</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented contains data organized in a matrix format, consisting of three columns and two rows, including headers. The first column lists index values while the top row categorizes numerical values in two separate columns labeled with index numbers 0 and 1. The first index row is further divided into specific entries, where it appears to represent the values or relationships that pertain to the analysis or calculations provided in the surrounding context.

Delving into the content of the cells, we see that the first cell under the header for column 0 includes a sequence of mixed numbers: "1 2", "1 1", "1", "?", "1", "32", "64", and "6." The inclusion of a question mark suggests that there might be an unknown value or a variable that requires resolution or interpretation. This arrangement could indicate relationships between numerical values or representations of fractions and their respective whole number counterparts. Under the header for column 1, we have "1 8" and "1 4," which denote specific fractions, potentially relevant to any calculations or analyses involving ratios or proportional reasoning, as illustrated in the preceding examples. 

This structure effectively showcases numerical relationships, potentially connecting to the aforementioned contextual examples dealing with length conversion and simplification of complex fractions. The values in the table might serve not only as standalone data points but also as tools toward further mathematical operations, echoing the analytical process demonstrated in the context, where numerical precision is critical in deriving accurate answers.


key: chroma:document
string_value:
The table presented reflects an organized layout designed to track the relationship between the number of dots along each side of a square and the total number of dots contained within that square. This concept directly correlates with the discussion about perfect squares and their square roots, as outlined in the surrounding text. The context emphasizes that the total number of dots represents a perfect square, with the number of dots along each side corresponding to the square root of that perfect square. The reference to numbers from 1 to 10 illustrates the foundational principles of square numbers and their subsequent roots, which is essential for grasping the set of irrational numbers and their properties.

Each row in the table is indexed, indicating a numerical sequence where the first column serves as a reference point for the number of dots along the sides of the squares. However, within the table, the cells adjacent to these indices are currently empty, suggesting that this is an exercise aimed at encouraging learners to fill in the appropriate values. For instance, while the first entry indicates that with 1 dot along each side, there is a total of 1 dot, the pattern continues with each subsequent perfect square—2 dots yield a total of 4, 3 dots lead to 9, and so on, culminating in the realization that the total number of dots equals the square of the count of dots along each side. This activity not only solidifies understanding of square numbers but also prepares students to explore more complex concepts relating to irrational numbers.

Furthermore, this exercise serves as a visual and calculative approach to mathematics, fulfilling the objective of demonstrating how square roots and perfect squares are linked. The educator's goal is to stimulate critical thinking by making connections between geometric representations and numerical patterns, inviting learners to delve deeper into the nature of numbers—rational as well as irrational.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td>Solution:\nDay\nMonday\nTuesday\nWednesday\nThursday\nFriday</td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>5</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>6</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>7</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>8</th>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table features a structure with multiple rows and columns, indicating a systematic approach to displaying information, although specific data points are not filled in for most of the cells. At the topmost row, the first column is left empty, while the subsequent columns are labeled with numerical indices: 0, 1, and 2. The first column on the left side of the table includes index numbers ranging from 0 to 8, demonstrating a sequential arrangement.

In the ninth cell of the first column, under the index 0, there appears to be a fragment of text or a prompt indicating a "Solution" followed by the labels for the days of the week—Monday through Friday—under the third column. This suggests that the table might be meant to record or illustrate a series of solutions or data points associated with those days, though the specific entries related to those days are absent from the current view. The consecutive rows, numbered from 1 to 8 in the first column, are empty, indicating that there may be no additional solutions provided yet or that the table is a template awaiting data entry.

The context surrounding the table explains a word problem that involves calculating electric current using a known formula. This example demonstrates how to substitute values into a formula to derive a solution, specifically for a light bulb’s electric current given its resistance and voltage. Although the connection between the word problem and the provided table is not explicitly clear, it may indicate that this table is designed for students to work out solutions to similar problems or record results from exercises they may perform throughout the week. The absence of data in the table elements could imply a preparatory or instructional phase where students will fill in their findings as they work through problems related to mathematics and electrical calculations involving decimals.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>p q\n, \nThe  set  of  irrational  numbers  is  the  set  of  numbers  that  cannot  be  expressed  as \nwhere p and q are integers, and q ≠ 0.</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented in the HTML format contains a concise representation of a key concept related to numbers, particularly focusing on the distinction between rational and irrational numbers. The first row identifies the content with a header labeled "Key Concept," and the following rows define a critical aspect of the number system. Specifically, it articulates the definition of irrational numbers, which are described as numbers that cannot be expressed as a fraction where both the numerator (p) and the denominator (q) are integers, with the stipulation that the denominator must not equal zero.

In the second row of the table, the definition is laid out clearly, emphasizing that irrational numbers are not expressible in the form of p/q. This is significant in the context of square roots, as the surrounding text discusses both principal and negative square roots in relation to perfect squares and their properties. The distinction emphasized in the context about numbers like -4 being the negative square root, but not the principal square root, aligns seamlessly with the broader discussion of rational versus irrational numbers. Thus, the table serves as a reference point for understanding the foundational concepts of number classification, which is pivotal for manipulating and solving problems involving square roots and areas, as highlighted in the examples provided in the surrounding text.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>1 4\n6</td>
      <td></td>
      <td>1 4\n11</td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td>1 2\n7</td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td>3 4\n8</td>
    </tr>
  </tbody>
</table>


key: chroma:document
string_value:
The table presented provides a structured representation of certain numeric relationships, likely aimed at facilitating students' understanding of number classifications, including rational and irrational numbers. The table is organized into four columns, with the first column indicating the row number and the subsequent columns (labeled 0, 1, and 2) containing various numerical values or expressions.

In the first row, corresponding to 0, the table lists "1 4\n6" in the first column. This notation suggests a fractional representation, possibly indicating a combination of whole numbers and fractions. In the second column, under the "1" header for the first row, there is no value presented. The third column contains "1 4\n11," which could signify another fractional or compound number. Moving on to the next row, identified as "1," the first column is left empty, while the second column includes "1 2\n7," indicating another numeric representation. The third column remains blank. Finally, in the last row labeled "2," the first and second columns contain no values, while in column three, the entry "3 4\n8" suggests a similar format as seen previously.

The absence of certain values, especially in specific rows or columns, might highlight certain numeric intervals or show how different numbers can be positioned on a number line, especially in relation to irrational numbers discussed in the surrounding context. The context elaborates on the concept of irrational numbers, particularly pi, emphasizing the importance of understanding these differences when locating various types of numbers, including how they may be represented on a number line. This table likely serves to further the learning objectives by providing practical examples that students can work with, helping them define and locate both rational and irrational numbers effectively.


key: chroma:document
string_value:
The table presented in the HTML format appears to be a simple two-row by two-column grid. Each cell within the grid is currently empty, including the headers, which are labeled "0" and "1" for both columns and rows. The empty nature of the cells may suggest that this table is intended to be used for illustrative or computational purposes rather than presenting finalized data.

In relation to the surrounding context, which focuses on approximating square roots and identifying the intervals between perfect squares, this table may serve as a visual aid. For instance, in the examples provided, the discussion revolves around determining whole number approximations by using perfect squares. The empty cells could potentially be filled with numbers or values representing squares, whole number approximations, or specific intervals, thereby allowing for a clearer demonstration of the concepts being discussed. 

The structure of the table might facilitate exercises or problems in which students can input values based on their understanding of how to approximate numbers like 70 and 8 through associated perfect squares. This ties back to the provided context about finding square roots and helps clarify where values fall within those ranges, possibly reinforcing the learning of students as they engage with such exercises.


key: html
string_value:
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
      <th>4</th>
      <th>5</th>
      <th>6</th>
      <th>7</th>
      <th>8</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>1</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>2</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>3</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>4</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>5</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>6</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>7</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>8</th>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>


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The presented table is structured with an 8x9 grid, comprising both header and row labels. The table is devoid of any content in the cells, indicating it is currently empty. The header row spans from column 0 to column 8, while the first column on the left indicates the row numbers from 0 to 8. Each cell in the table is bordered, suggesting a format designed for clarity and organization, which facilitates the input of data or calculations.

In the context of the mathematical concepts discussed in the surrounding text, the table likely serves a purpose related to numerical estimations or the approximation of square roots. The method detailed in the context involves steps to compute an approximation for the square root of a radicand, using various calculations such as division, averaging, and comparing squared results with the original radicand. Although the table does not contain any specific numbers at present, it may be intended for recording intermediary results or final outcomes of the estimation process outlined. 

As students engage with the exercise of estimating square roots, they may utilize this table to comprehensively track their calculations through each iterative step. With the guidance provided in the context, the table could effectively manage numerical values to validate the convergence toward the correct estimate, as shown in the example of approximating the square root of 8. Therefore, while presently unfilled, the table holds strategic importance for the process of mathematical estimation and verification.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
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  <tbody>
    <tr>
      <th>0</th>
      <td>Number of Dots along \nEach Side of the Square</td>
      <td>Total Number  \nof Dots</td>
    </tr>
    <tr>
      <th>1</th>
      <td>1</td>
      <td>1</td>
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    <tr>
      <th>2</th>
      <td>2</td>
      <td>4</td>
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    <tr>
      <th>3</th>
      <td>3</td>
      <td>9</td>
    </tr>
    <tr>
      <th>4</th>
      <td>4</td>
      <td>16</td>
    </tr>
    <tr>
      <th>5</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>6</th>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <th>7</th>
      <td>5 6 7 8 9</td>
      <td></td>
    </tr>
    <tr>
      <th>8</th>
      <td></td>
      <td></td>
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    <tr>
      <th>9</th>
      <td></td>
      <td></td>
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      <th>10</th>
      <td>10</td>
      <td></td>
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The table presents a clear relationship between the number of dots along each side of a square and the total number of dots comprising the square. The first column, labeled "Number of Dots along Each Side of the Square," indicates the length of one side of the square measured in dots. This column is complemented by the second column titled "Total Number of Dots," which represents the total number of dots that can be found within that square configuration.

Starting from one dot to ten dots, the table illustrates that with one dot on each side, the total becomes one dot, forming a 1x1 square. As the number of dots along each side increases, the total number of dots follows a pattern corresponding to the formula for the area of a square, \( A = s^2 \). For example, with two dots along each side, the overall total is four dots, while three dots yield a total of nine dots. The correlation continues up to four dots, which results in sixteen total dots. Interestingly, the table leaves some rows blank after the fourth entry but provides an additional row under the fifth, showing that five dots on each side would not only extend the pattern but also demonstrate the total as twenty-five, illustrating the consistent progression of perfect squares.

This relationship implies a fundamental concept about square numbers and their roots, aligning with the context that discusses square roots and the calculation of areas, such as determining the dimensions of the room tiled with square pieces. Specifically, the previously discussed example outlines how, using the area of 400 square feet derived from tiling, one can apply the square root function to find the length of each side of a square room, reinforcing the learning concept represented in the table. Overall, the table serves as an educational visual aid to assist in understanding basic geometric principles and square numbers, directly linking to the broader topic of square roots.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
      <th>1</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Day</td>
      <td>Time (in seconds)</td>
    </tr>
    <tr>
      <th>1</th>
      <td>Monday</td>
      <td>10.43</td>
    </tr>
    <tr>
      <th>2</th>
      <td>Tuesday</td>
      <td>10.39</td>
    </tr>
    <tr>
      <th>3</th>
      <td>Wednesday</td>
      <td>10.56</td>
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    <tr>
      <th>4</th>
      <td>Thursday</td>
      <td>10.45</td>
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      <th>5</th>
      <td>Friday</td>
      <td>10.46</td>
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The displayed table presents data concerning the time taken on various days of the week, specifically measured in seconds. The structure of the table includes headings that denote the first column as "Day" and the second column as "Time (in seconds)," providing clarity on the nature of the data presented. Each row corresponds to a different day, with the first row serving as a label, indicating that the subsequent entries will follow the sequence of the week from Monday to Friday.

For each day, the table lists the specific time recorded, with Monday showing a time of 10.43 seconds. The subsequent days exhibit closely related times, with Tuesday at 10.39 seconds, a slight decrease from Monday's figure. Wednesday marks a noticeable increase, reaching 10.56 seconds, whereas Thursday returns to a lower time of 10.45 seconds. Finally, on Friday, the recorded time is 10.46 seconds, indicating a minor rise from Thursday. Overall, the table provides a concise overview of the time variations across the week, with values that exhibit slight fluctuations within a tightly grouped range, thereby offering insight into the performance metrics captured during these days.


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<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>Key Concept</td>
    </tr>
    <tr>
      <th>1</th>
      <td>If a and b are nonnegative numbers and b2 = a, then b is a square root of a. In symbols, \nit can be written as:\n \na  = b.</td>
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</table>


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The table presents a key concept regarding square roots and their properties in relation to nonnegative numbers. It consists of two main entries. The first header, titled "Key Concept," introduces the primary focus of the information contained in the table. Below this header is another important note indicating a fundamental relationship between two nonnegative numbers, \( a \) and \( b \). The table clearly states that if \( b^2 = a \), it means that \( b \) is a square root of \( a \). This mathematical relationship is succinctly expressed in symbolic form, represented as \( a = b^2 \).

The context surrounding this table provides an illustrative example regarding the extraction of square roots, specifically in the framework of rational and irrational numbers. The narrative begins by discussing the measurements of a square table and the implications of these measurements on the design of a cover, leading to a calculation of areas. The discussion highlights that the square root of a perfect square yields a rational number, while in the case of non-perfect squares—such as when determining the side length of a smaller square with an area of \( 2 \) square feet—the outcome is an irrational number. This serves to reinforce the information in the table, as it elaborates on the notion that the square root might not always yield a rational result, and it emphasizes the significant difference between rational numbers and irrational numbers, especially when exploring square roots of non-perfect squares. 

In summary, the table succinctly encapsulates a mathematical principle that provides clarity on square roots while the surrounding context illustrates its practical implications through real-world examples, clarifying the distinction between different types of numbers that can arise from such mathematical operations.


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vi
Linear Equations and Inequalities  
in One Variable
CHAPTER
4
Lesson 4-1 Mathematical Phrases and Sentences ............................................ 186
Lesson 4-2 Introduction to Equations .............................................................. 195
Lesson 4-3 Solutions to Linear Equations ........................................................ 202
Lesson 4-4 Applications of Linear Equations .................................................... 217
Lesson 4-5 More Applications of Linear Equations ........................................... 239
Lesson 4-6 Linear Inequalities in One Variable and Their Applications ............. 263
Lesson 4-7 Solutions to Absolute Value Equations and Inequalities ................. 276
Chapter Assessment ........................................................................................ 287
Measurement
CHAPTER
5
Lesson 5-1 Development of Measurement ........................................................ 292


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C & E
Publishing, Inc.
C & E Publishing, Inc.  
was established in 1993,  
and is a member of  ABAP, PBAI, 
NBDB, and PEPA.
Math PLUS (Practical Life Uses and Solutions) 7
Published by C & E Publishing, Inc. 
839 EDSA, South Triangle, Quezon City
Tel. No.: (02) 8929-5088
E-mail: info@cebookshop.com
Copyright © 2020 by C & E Publishing, Inc.,
Fidel R. Nemenzo, and Allan M. Canonigo  
All rights reserved. No part of this publication may be  
reproduced, stored in a retrieval system, or transmitted 
in any form or by any means—electronic, mechanical,  
photocopying, recording, or otherwise—without the prior  
written permission of the publisher.
Editing: Jose Dante V. Dela Merced
 Mark Anthony S. Espisua
 Jaymie M. Guillermo
Cover Design: Ruth Anne D. Ellorin
 
Illustrations: Apolonio Y. Capones Jr.
  Ruth Anne D. Ellorin
Layout: Maricar L. Sanchez
  Jolly Monique B. Dela Peña
  Darwin M. Tolentino
ISBN:  978-971-98-1293-7


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ix
Preface
Math PLUS (Practical Life Uses and Solutions) 7 is part of a junior high school 
math textbook series that is designed to cater to the needs of 21st-century learners. 
Each book progressively guides the students in mastering mathematics concepts 
and prepares them for higher-level mathematics learning. 
Each chapter begins with a real-life illustration of the topics involved. Each lesson 
in a chapter also starts with a real-life illustration of the lesson topic that is related 
to the chapter’s theme. The chapter and lesson openers motivate the students to 
learn about the importance of mathematics in their lives. 
Each lesson is composed of the following components:
  Probe and Learn provides discussions of the topics using illustrations, 
examples with step-by-step solutions, and key concepts. This component has 
the following subcomponents:
   Key Concept highlights important terms, laws, formulas, properties,


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1
Introduction to Sets
CHAPTER
1
A teacher conducts a class survey to find out which of two sports most of his 
students would want to engage in. Of the 50 students surveyed, 17 like archery, 
while 38 like fencing. If all of the students chose at least one sport, how many 
students like both archery and fencing? Problems like this can be solved by applying 
the concept of sets.
Every day, you group objects to make them handier. For instance, you put all 
your clothes in a closet or put together money of the same denomination. These 
activities also use the concept of sets, which is fundamental to modern mathematics.
Lesson 1-1. Basic Concepts About Sets
Lesson 1-2. Set Relations
Lesson 1-3. Set Operations
Lesson 1-4. Problem Solving Involving Sets


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v
Introduction to Sets
CHAPTER
1
Lesson 1-1 Basic Concepts About Sets ............................................................. 2
Lesson 1-2 Set Relations .................................................................................. 11
Lesson 1-3 Set Operations ............................................................................... 22
Lesson 1-4 Problem Solving Involving Sets ....................................................... 34
Chapter Assessment ........................................................................................ 46
The Real Number System
CHAPTER
2
Lesson 2-1 Real Numbers ................................................................................ 52
Lesson 2-2 Integers ......................................................................................... 66
Lesson 2-3 Rational Numbers .......................................................................... 83


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2
Probe and Learn
Look at the following list of sports:
archery gymnastics swimming
athletics handball table tennis
badminton hockey taekwondo
basketball judo tennis
boxing pentathlon triathlon
cycling rowing volleyball
equestrian sailing water polo
fencing shooting weightlifting
football softball wrestling
Have you heard or played any of these sports? What do they 
have in common?
Basic Concepts About Sets
Learning 
Objectives
At the end of the 
lesson, you should be 
able to:
•	 define	sets
•	 describe	sets	using	
the	roster	method	
and	the	rule	
method
•	 identify	the	
elements	of	 a 	 set
•	 find	the	cardinality	
of a set
•	 classify	sets	
according	to	size
LESSON
1-1


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3
These are the sports played in the Summer Olympic Games. You might have 
played some of them or watched them being played. Look again at the list of sports. 
Try to group the sports based on the following categories:
Water Sports Ball Games
Exciting Games Popular Sports
Now compare your list with that of your classmate. Did you write the same sports 
under each category? In which categories did you list different sports? Why did you 
have different lists in those categories?
Definition of a Set 
In the previous activity, the games you listed under each category form a group. 
Take note, however, that not all groups are sets. Consider the sports that you 
categorized as water sports. If you are familiar with all the sports in the given list, 
you could identify the water sports as follows:
rowing, sailing, swimming, and water polo.
This group of sports is an example of a set. It is a set because the common


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examples with step-by-step solutions, and key concepts. This component has 
the following subcomponents:
   Key Concept highlights important terms, laws, formulas, properties, 
postulates, or theorems that the students should look into to better 
understand the lesson.
   Do You Know gives interesting information or trivia about a specific term or 
concept mentioned in the lesson. 
  Practice provides assessment items in varying levels of difficulty to allow the 
students to master and apply what they have learned. This component is 
divided into three parts:
   Concepts and Skills is composed of knowledge-based items about the topics 
discussed.
   Applications is composed of problem-solving items that focus on real-life 
applications of the topics. 
   Enrichment Exercises is composed of items that go beyond the topics discussed 
to enhance the students’ critical thinking skills. 
Each chapter concludes with a Chapter Assessment to measure the acquired


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4
examPLe 1
Tell whether or not each group is a set. Explain.
a. group of varsity players in your school 
b. group of students who participated in a basketball competition
c. group of good dancers in your class
d. group of popular volleyball players
 Answers:
 a.  It is a set. Your school has a definite roster of varsity players.
 b.   It is a set. You can identify the students who participated in the competition. 
 c.   It is not a set. People may have different standards in recognizing good 
dancers.
 d.   It is not a set. Popularity is based on opinions, which may vary from one 
person to another.
Naming a Set and Its Elements
A set is conventionally denoted by a capital 
letter. You may use a letter that can remind 
you of the common characteristic or rule that 
describes the elements the set. For example, 
you could name the set of colors of the rings 
in the Olympic symbol as set O. The elements 
of set O are blue, yellow, black, green, and 
red.


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Measurement
CHAPTER
5
Lesson 5-1 Development of Measurement ........................................................ 292
Lesson 5-2 Conversion of Measurement Units .................................................. 306
Lesson 5-3 Temperature, Time, and Angle Measures ........................................ 319
Lesson 5-4 Numbers in Measurement .............................................................. 334
Chapter Assessment ........................................................................................ 343
Algebraic Expressions
CHAPTER
3
Lesson 3-1 Powers and Exponents ................................................................... 120
Lesson 3-2 Introduction to Algebraic Expressions ............................................ 130
Lesson 3-3 Introduction to Polynomials ........................................................... 140
Lesson 3-4 Operations on Polynomials ............................................................. 150


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5
examPLe 2
Suppose that P is the set of Philippine Basketball Association (PBA) players in 
2017. Which of the following is true?
a. Tim Yap ∉ P
b. Tim Cone ∈ P
c. Allan Caidic ∉ P
d. J ordan Clarkson ∈ P
 Answers:
 a. True; Tim Yap is not a PBA player.
 b. False; Tim Cone is a PBA coach, not a player. Thus, Tim Cone ∉ P.
 c.   True; Allan Caidic is a PBA player who played during the 1990s, not in 
2017.
 d.  False; Jordan Clarkson is an NBA player, not a PBA player. Hence, Jordan 
Clarkson ∉ P.
Methods to Describe Sets
The following are the two commonly used methods to describe sets:  
(1) the roster method or listing method; and (2) the rule method or set-builder notation.
In the roster method or listing method of describing a set, the elements of the set are 
listed between braces. The elements in the list are separated by commas.
Key Concept
For example, suppose that set V is the set that contains the vowels of the English


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Lesson 2-3 Rational Numbers .......................................................................... 83
Lesson 2-4 Irrational Numbers ........................................................................ 105
Chapter Assessment ........................................................................................ 115
Table of Contents
Preface  ...................................................................................................... ix


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vii
Statistics
CHAPTER
7
Lesson 7-1 Introduction to Statistics ............................................................... 486
Lesson 7-2 Presentation of Data ...................................................................... 503
Lesson 7-3 Measures of Central Tendency ....................................................... 532
Lesson 7-4 Measures of Variability .................................................................. 556
Chapter Assessment ........................................................................................ 574
Geometry
CHAPTER
6
Lesson 6-1 Basic Concepts of Geometry ........................................................... 348
Lesson 6-2 Angles ............................................................................................ 361
Lesson 6-3 Relationships Between Angles ........................................................ 373
Lesson 6-4 Relationships Between Lines .......................................................... 388


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you could identify the water sports as follows:
rowing, sailing, swimming, and water polo.
This group of sports is an example of a set. It is a set because the common 
characteristic of the identified sports is clear and definite. All of them are played in 
the water. On the other hand, the sports that are listed as “exciting games” may vary. 
What is exciting for you may not be exciting for others. Hence, the group of sports 
under “exciting games” is not a set. Study the following definition of a set:
A set is a well-defined collection of objects. Such objects are called the elements of the 
set.
Key Concept
Remember that the elements of a set should be described in a way that each 
element that belongs to the set can be easily identified. This may be done by giving a 
clear rule that every element of the set satisfies. Study the examples.


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6
In the set-builder notation or the rule method of describing a set, the property or 
properties that each element of the set satisfies are stated.
Key Concept
When using the set-builder notation, the rule should be written clearly so as to 
include only those objects that satisfy the described property. In set-builder notation, 
the general form a set can be written as:
A = {x|x is P },
where P is the defining property. This statement is read as “set A is the set of all x’s 
such that x satisfies P.”
Consider again the two sets, C and V, that you previously described using the 
roster method. In set-builder notation, you can write the sets as follows:
   V = {y|y is a vowel in the English alphabet}; and
   C = {z|z is a counting number from 1 to 1,000}.
You may use either the roster method or the rule method, whichever is more 
practical, to describe a given set.
examPLe 3
Describe each set using the rule method.
a. W = {0, 1, 2, 3, …}
b. B = {11, 12, 13, 14, 15, 16, 17, 18, 19}


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Lesson 3-4 Operations on Polynomials ............................................................. 150
Lesson 3-5 Special Products ............................................................................ 169
Chapter Assessment ........................................................................................ 180


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describes the elements the set. For example, 
you could name the set of colors of the rings 
in the Olympic symbol as set O. The elements 
of set O are blue, yellow, black, green, and 
red.
Set membership is denoted by the symbol 
∈, which is the lowercase Greek letter epsilon. 
Suppose that set B is the set of countries 
whose representatives won in at least one of 
the QubicaAMF Bowling World Cup (BWC) 
tournaments from 1965 to 2018. The 
Philippines belongs to that set of countries. 
Hence, you can write:
Philippines ∈ B,
which is read as “the Philippines is an element of set B.”
To denote that an element is not a member of a set, the symbol ∉ is used. For 
instance, Indonesia did not win in any BWC tournament from 1965 to 2018. So you 
can write:
Indonesia ∉ B,
which is read as “Indonesia is not an element of set B.” 
The Olympic rings symbol was designed 
by Pierre de Coubertin in 1913 for the 
Paris Congress of the Olympic Movement. 
It is made up of five interlocking rings—


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to enhance the students’ critical thinking skills. 
Each chapter concludes with a Chapter Assessment to measure the acquired 
knowledge and skills of the students.


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listed between braces. The elements in the list are separated by commas.
Key Concept
For example, suppose that set V is the set that contains the vowels of the English 
alphabet. Using the roster method, you can describe this set as follows:
V = {a, e, i, o, u}.
You can see that the elements of V are the letters a, e, i, o, and u. Since there are 
only five elements in V, it is easy to enumerate them all. Note that when writing the 
elements of a set, the order of the elements does not matter. So you can also write 
set V as {o, u, a, e, i} or {e, i, o, u, a}. Also, remember that the elements in a set should 
be distinct; that is, the elements should not be repeated in the list.
In cases when a set has many elements, an ellipsis (...) is used to indicate that 
the list of elements continues up to the last given element. For instance, suppose 
that set C is the set of the first 1,000 counting numbers. You may describe set C as 
follows:
C = {1, 2, 3, ..., 1,000}.


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Lesson 6-4 Relationships Between Lines .......................................................... 388
Lesson 6-5 Geometric Constructions ............................................................... 404
Lesson 6-6 Polygons ........................................................................................ 427
Lesson 6-7 Circles ........................................................................................... 463
Chapter Assessment ........................................................................................ 478
About the Author


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7
Kinds of Sets
Sets may be classified based on the number of their elements. Study the following 
definition:
The cardinality of a set refers to the number of elements of the set. In symbols, the cardinality of 
a set A is denoted as n(A).
Key Concept
Consider the following set:
P = {x|x is a planet in the solar system}.
Using the roster method, set P may be described as follows:
  P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}.
You can say that the cardinality of set P is 8. In symbols, n(P) = 8.
Now consider the following set:
  A = {x|x is a planet in the solar system whose name starts with the letter A}.
Is there a planet whose name starts with the letter A? Since there is no such 
planet, then A has no elements. Thus, n (A) = 0. 
A set that has no elements is called an empty set or a null set.  It is denoted as ∅ or { }.
Key Concept
Since n(A) = 0, you can say that A is an empty set.


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8
Practice 1-1
Concepts and Skills
Write True if the statement is true; otherwise, write False.
 1.  The empty set ∅ is the same as {0}.
 2. The set of fractions is an infinite set.
 3. The set of proper fractions is a finite set.
 4. The size of an empty set cannot be described.
 5. The set {1, 2, 3, 4, 5} is the same as the set {5, 4, 3, 2, 1}.
 6. The elements of a set may be written more than once in a set.
 7. It is impossible to enumerate all the elements of an infinite set.
 8.   Any set could be described using both the roster method and the rule method.
 9. In the rule method, the elements of a set are listed.
 10.  In the roster method, a set is described by specifying the properties its elements 
should satisfy.
Tell if each group is a set or not. Explain.
 11. collection of large numbers
 12. group of sea mammals
 13. group of distinct letters in your first name
 14. list of names of girls with long hair in your class 
 15. collection of fiction books in your school library


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11
Learning 
Objectives
At the end of the 
lesson, you should be 
able to:
•	 identify	the	relation	
among	sets
•	 illustrates	relations	
among	sets	using	
Venn	diagrams
Set Relations
Probe and Learn
Could you imagine yourself playing five different sports 
within a 12-hour period? You might be amazed at how athletes 
who play the modern pentathlon sustain their stamina. Modern 
pentathlon is an athletic competition that is composed of five 
different events that test one’s endurance and athletic versatility. 
This includes shooting, fencing, swimming, horseback riding, 
and running.
LESSON
1-2


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 25. P =  {Aguinaldo, Quezon, Laurel, Osmeña, Roxas, Quirino, Magsaysay, Garcia, 
Macapagal, Marcos, Aquino, Ramos, Estrada, Arroyo, Aquino, Duterte}
For items 26 to 30, find the cardinality of the sets described in 16 to 20.
 26. n(P)
 27. n(F)
 28. n(L)
 29. n(T)
 30. n(O)
For items 31 to 35, classify the sets described in 21 to 25 as finite or infinite.
 31. U
 32. A
 33. M
 34. C
 35. P
For items 36 to 40, suppose that M is the set of the sums of any two consecutive 
whole numbers between 10 and 50. 
 36. Describe set M using the roster method.
 37. Describe set M using the rule method.
 38. Classify M as finite or infinite.
 39. Find the cardinality of M.
 40. Tell whether or not 21 ∈ M. Explain.
Applications
Answer each item completely.
 41. Suppose that you will make a model of the solar system. Name all the objects 
that will be represented in your model. Describe the objects in your model 
using the roster method.


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10
Enrichment Exercises
Answer each item completely.
 46. Explain why an empty set is a finite set.
 47.  Give an example of an empty set. Use the rule method to describe the set.
 48.  Give an example of a finite set with a cardinality of 6. Use the rule method to 
describe the set.
 49.  Give an example of a set that can be best described using the rule method.
 50.  Give an example of a set that can be best described using the roster method.


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The Olympic rings symbol was designed 
by Pierre de Coubertin in 1913 for the 
Paris Congress of the Olympic Movement. 
It is made up of five interlocking rings—
blue, yellow, black, green, and red—
representing the five major regions of the 
world.


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12
Equal Sets and Equivalent Sets
Suppose that you have the following sets:
 P = {x|x is an event in modern pentathlon}; and
 E = {shooting, fencing, swimming, horseback riding, running}.
Note that all events in E are the sports described in P. Since sets E and P have the 
same elements, E and P are said to be equal sets. In symbols, you can write E = P.
Sets A and B are equal if they contain exactly the same elements. In symbols, A = Β.
Key Concept
When two sets, say C and D, are not equal sets, you can write the relation as  
C ≠ D.
Look at the following sets:
 E = {a, b, c, d, e}  F = {a, e, b, c, d}  G = {a, b, c, e, f}
You can verify the following relations:
 E = F    E ≠ G    F ≠ G
Another way to relate sets is to consider the number of their elements. Look at 
the following sets:
 H = {1, 4, 9, 16, 25, 36}  I = {x|x is a whole number less than 6}
Using the roster method, I = {0, 1, 2, 3, 4, 5}. You can see that H ≠ I. However, if


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practical, to describe a given set.
examPLe 3
Describe each set using the rule method.
a. W = {0, 1, 2, 3, …}
b. B = {11, 12, 13, 14, 15, 16, 17, 18, 19}
c. D =  {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
d. C =  {monitor, central processing unit, keyboard, mouse, speaker, printer, 
scanner}
 Answers:
 a. W = {x|x is a whole number}
 b. B = {y|y is a counting number greater than 10 but less than 20}
 c. D = {z|z is a day of the week}
 d. C = {v|v is a part of a computer system}


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examPLe 1
Consider the following sets:
 L = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}
 M = {x|x is a multiple of 10 from 10 to 100} 
 N = {x|x is a number from 5 to 50 that is divisible by 5}
Tell whether each relation is true or false. Explain.
a. L = M
b. L ~ M
c. L = N
d. N ~ L
e. M ≠ N
 Answers:
  To compare the given sets easily, describe first sets M and N using the roster 
method.
  L = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}
  M = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}
  N = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}
  Based on the given sets, you will have the following:
 a. True; L and M have exactly the same elements.
 b. True; n(L) = n(M) = 10.
 c. False; L and N have only some elements in common.
 d. True; n(N) = n(L) = 10.
 e. True; M and N do not have exactly the same elements.
Consider three sets A, B, and C . Suppose that A ~ B and B ~ C. Does it follow 
that A ~ C ? Look at the given sets in the previous example. Note that L ~ M and


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If A and B are non-empty sets and A is a subset of B, where n(A) < n(B), then A is said to 
be a proper subset of B. In symbols, it is denoted as A  ⊂ B. Note that A ≠ B.
Key Concept
examPLe 2
Consider the following sets:
 X = {1, 3, 5, 7, 9}
 Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 
 Z = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}
Illustrate each relation using a Venn diagram.
a. X ⊂ Y b. Y ⊂ Z c. X ⊂ Z
 Answers: 
 a.  Since all the elements of X are found in Y,  
and n(X) < n(Y), then X ⊂ Y is illustrated in  
the diagram on the right.
 b.  Since all the elements of Y are found in Z,  
and n(Y) < n(Z), then Y ⊂ Z is illustrated in  
the diagram on the right.
 c.  Since all the elements of X are contained in Z,  
and n(X) < n(Z), then X ⊂ Z is illustrated in  
the diagram on the right.
X1
0
2
4 6
83 5
7 9
Y
Y
3
0 1
2
5
9 8
6
Z
4
7
X
1
3
5 7
9
Z


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12. group of sea mammals
 13. group of distinct letters in your first name
 14. list of names of girls with long hair in your class 
 15. collection of fiction books in your school library
Describe each set using the roster method.
 16. set P containing perfect square numbers
 17. set F containing the colors in the Philippine flag
 18. set L containing the leap years in the 21st century
 19. set T containing three-digit palindromes 
 20. set O containing the siblings of an only child in a family
Describe each set using the rule method.
 21. U = 
 22. A = {2, 3, 5, 7, 11, 13, 17, 19, …, 97}
 23. M = {April, June, September, November}
 24. C = {5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49}


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the following sets:
 H = {1, 4, 9, 16, 25, 36}  I = {x|x is a whole number less than 6}
Using the roster method, I = {0, 1, 2, 3, 4, 5}. You can see that H ≠ I. However, if 
you will get the cardinality of each set, you will see that n(H) = n(I ). In such case, the 
two sets are said to be equivalent.
Sets A and B are equivalent if they have the same cardinality. In symbols, A ~ B.
Key Concept
Note that there is a one-to-one correspondence between the elements of equivalent 
sets; that is, each element of one set can be paired with exactly one element of the 
other set, and vice versa. In the previously given equivalent sets, H and I, you can 
denote the one-to-one correspondence as H ↔ I. In the following illustration, notice 
how each element of H can be paired with exactly one element of I.
H = {1, 4, 9, 16, 25, 36}
I = {0, 1, 2, 3, 4, 5}


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14
Property Equality Equivalence
Transitive Property
If set A is equal to set B, and set B 
is also equal to set C, then set A is 
equal to set C.
If A = B and B = C, then A = C.
If set A is equivalent to set B, and  
set B is also equivalent to set C, then 
the set A is equivalent to set C.
If A ~ B and B ~ C, then A ~ C.
Sets and Subsets
Another relationship involving sets is set inclusion, which means that a set may 
be contained in another set. Consider sets P and Q as follows:
 P = {x|x is a student in your class}
 Q = {y|y is a male student in your class}
Based on the description of sets P and Q, if a student in your class is male, then 
he is an element of Q and, at the same time, of P. In such case, Q is a subset of P.
Q is said to be a subset of P if every element of Q is also an element of P. In symbols, it 
is denoted as Q  P. This is read as “Q is a subset of P” or “Q is contained in P.”
Key Concept


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41. Suppose that you will make a model of the solar system. Name all the objects 
that will be represented in your model. Describe the objects in your model 
using the roster method.
 42. Condiments are food seasonings that improve the flavor of food. You want to 
enhance the flavor of the food you are eating. What set of condiments are you 
going to choose? Describe it as a set containing five elements using the roster 
method.
 43. Suppose that you will grow some herbal plants in your school backyard. Give 
a set of ten plants that you would consider planting. Use the roster method.
 44. Maria plans to establish a pet shop. Using the rule method, describe a set of 
animals that Maria could sell in her shop.
 45. Suppose that the sports coordinator in your school organizes a sports fest. 
Your school has no facilities for outdoor games; so only indoor games can be 
played. Describe a set of games you could suggest to the sports coordinator. 
Use the roster method.


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In example 2, note that X ⊂ Y and Y ⊂ Z. This implies that X ⊂ Z, which is verified 
in example 2c. Thus, the relation among the three sets is illustrated as follows:
Y
X
1 7
3
5
9
Z
0
2
4 6 8
This shows the transitive property of set inclusion.
Transitive Property of Set Inclusion
For any sets A, B, and C, if A ⊂ B and B ⊂ C, then A ⊂ C.
Key Concept
Consider another set W, which is the set of whole numbers. How do you compare 
Z and W? Which has more elements? In this case, Z = W and n(Z) = n(W). Hence,  
Z is said to be an improper subset of W. 
For any sets A and B, A is said to be an improper subset of B if A = B.
Key Concept
Since any set is equal to itself, you can also conclude that A ⊆ Α and B ⊆ Β.  
This property is referred to as the reflexive property of set inclusion. 
Reflexive Property of Set Inclusion
For any set A, A ⊆ A.
Key Concept
The symbol ⊆ is nonstrict, while the symbol ⊂ is strict. Note that if A ⊆ B, then 
either A ⊂ B or A = B.


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Consider three sets A, B, and C . Suppose that A ~ B and B ~ C. Does it follow 
that A ~ C ? Look at the given sets in the previous example. Note that L ~ M and  
M ~ N. Could you conclude that L ~ N ? If yes, is this property true for equality of sets? 
Study the following table that summarizes the properties of equality and equivalence 
of sets.
Property Equality Equivalence
Reflexive Property
A set is equal to itself.
A = A
A set is equivalent to itself.
A ~ A
Symmetric Property
The sets can be written 
interchangeably on any side of the 
equality.
If A = B, then B = A.
The sets can be written 
interchangeably on any side of the 
equivalence.
If B ~ A, then A ~ B.


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examPLe 3
Let S = {a, b, c}. How many subsets does S have? What are these subsets?
 Answer:
 Look for the improper, proper, and trivial subsets of set S.
Type of Subsets Subset
Improper subset The improper subset of any set is itself. {a, b, c}
Proper subset 
One-element subsets {a}, {b}, {c}
Two-element subsets {a, b}, {b, c}, {a, c}
Trivial subset The empty set is a trivial subset of any set. ∅
 Thus, S has 8 subsets, namely: {a, b, c}, ∅, {a}, {b}, {c}, {a, b}, {b, c}, and {a, c}.
How many subsets does a set with 4 elements have? Try answering the question 
using the set T = {a, b, c, d}. Use an organized way of finding all the subsets so you 
will not miss out any subset.
Disjoint Sets
Look at the following sets: 
 M = {1, 3, 5, 7, 9}  N = {2, 4, 6, 8, 10}  O = {1, 2, 3, 4, 5}
In what way can you show the relations among sets M, N, and O?
Notice that sets M and N have no common elements. Thus, M and N are disjoint 
sets.


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19
Refer to the following sets to answer items 11 to 20: 
 A  =  {x|x is an even digit} 
 B
 
=
 
{x|x is a multiple of 2 from 0 to 8} 
 C
 
=
 
{x|x is a prime number from 2 to 11} 
 
D
 
=
 
{x|x is an even prime number} 
 E
 
=
 
{x|x is a counting number between 2 and 11} 
 F
 
=
 
{x|x is a whole number less than 10 that is a multiple of 10} 
Write True if the statement is true; otherwise, write False.
 11. A ~ B
12. B ≠ C
 13. F = ∅
 14. D ⊄ B
 15. C ⊂ E
 16. C and A are disjoint sets.
From the six previously given sets, choose the sets that are being described in each 
item.
 17. two sets that are both equal and equivalent
 18. two sets that are equivalent but not equal
 19. three sets that are subsets of B
 20. a subset of A that is not a subset of E
For items 21 to 28, refer to the following diagram: 
C
A B
D
Write True if the relation is true; otherwise, write False.
 21. A ⊂ B
 22. A ⊂ D
 23. B ⊂ C
 24. D ⊂ C
 25. A ⊂ C ⊂ D
 26. B ⊂ A ⊂ C
 27. B ⊂ C ⊂ D
 28. D ⊃ C ⊃ B


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Reflexive Property of Set Inclusion
For any set A, A ⊆ A.
Key Concept
The symbol ⊆ is nonstrict, while the symbol ⊂ is strict. Note that if A ⊆ B, then 
either A ⊂ B or A = B.
Another subset of any set A is the empty set; that is, ∅ ⊂ A. This means that ∅ is 
a trivial subset of any set. Is ∅ ⊆ ∅? 
Remember that any set has at least one of these types of subsets—improper, 
proper, or trivial subset. Now you will study the subsets that a given set could have.


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20
For items 29 to 37, consider the set V = {0, 10, {100}}. Write True if the statement is 
true; otherwise, write False and modify the relation symbol to make it correct.
 29. 0 ∈ V
 30. ∅ ∈ V
 31. 10 ⊂ V
 32. {10} ⊂ V
 33. {100} ∈ V 
 34. {10} ⊂ {100}
 35. {0} ∈ V
 36. {{100}} = {100}
 37. {0} = ∅
For items 38 to 41, let S be the set of distinct letters in the word “smiles.” Identify all 
the subsets of S that satisfy each description.
 38. subsets that have no element
 39. subsets that contain only one vowel
 40. subsets that do not contain a vowel
 41. subsets that contain exactly three elements
Applications
Answer each item completely.
 42.  Your school is offering the following PE subjects: basketball, volleyball, table 
tennis, badminton, and baseball. Is the set of PE subjects offered in your school 
a subset of the set of sports played in the Summer Olympic Games? Explain.
 43.  Liza can choose three from the five elective subjects offered during a term.


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examPLe 4
Consider the following sets:
 O = {o|o is an odd number}
 E = {e|e is an even number}
 C = {0, 1, 2, 3, 4, 5}
Tell whether or not each pair of sets are disjoint. If the two sets are not disjoint, 
identify their common element(s).
a. O and C b. O and E c. E and C
 Answers:
 a. O and C are not disjoint sets. They both contain 1, 3, and 5. 
 b. O and E are disjoint sets.
 c.  E and C are not disjoint sets. Their common elements are 0, 2, and 4. 
Practice 1-2
Concepts and Skills
Write True if the statement is true; otherwise, write False.
 1. An empty set is a subset of itself.
 2. Any two equivalent sets are not disjoint sets.
 3. An empty set has no proper subset.
 4. If A ⊂ B, then A and B are not disjoint.
 5. An element of a set can be the same as another set.
 6. The improper subset of any set is equal to the empty set.
 7. A subset of B is a subset of C only if B is a subset of C.
 8. If A and B are both equivalent to C, then A and B are equal.


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Q is said to be a subset of P if every element of Q is also an element of P. In symbols, it 
is denoted as Q  P. This is read as “Q is a subset of P” or “Q is contained in P.”
Key Concept
 From Q  P, you also say that “P is a superset of Q.” It can be written as P  Q. If 
at least one element of Q is not contained in P, then Q is not a subset of P. In symbols, 
Q  P. What can you say about the cardinality of a subset and of its superset?
Set relations can be visualized using a Venn diagram. 
A Venn diagram is a visual representation of logical relationships among sets. It uses 
circles to represent sets. Each circle is labeled with the name of the set it represents.
Key Concept
The relation that set A is a subset (or not a subset) of another set B can be 
illustrated in the following Venn diagrams:
A B
A  B
All elements in set A are contained in set B. 
A B
A  B
Not all elements in set A are contained in set B.


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A set that has no elements is called an empty set or a null set.  It is denoted as ∅ or { }.
Key Concept
Since n(A) = 0, you can say that A is an empty set.
If the cardinality or number of elements of a set is a counting number, then it is called a finite set. 
A set is defined as a finite set if the cardinality of the set is a fixed or known counting number n. 
Otherwise, the set is not finite and is called an infinite set. 
Key Concept
The elements of a finite set are limited; thus, such elements can be completely 
listed down. For example, V = {a, e, i, o, u} is a finite set since n(V) = 5. Also, although 
an empty set has no element (or zero element), and 0 is not a counting number, an 
empty set is also considered as a finite set. An example of an infinite set is the set of 
counting numbers, C = {1, 2, 3, ...}. It is an infinite set because there are infinitely 
many counting numbers.
Is the set of the stars in the sky finite or infinite? You may think that since there


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22
Probe and Learn
In a school sports fest, four teams will participate. Each team 
will be composed of students in each year level. The grade 7 
students will be the Green Team. The grade 8 students will be 
the Yellow Team. The grade 9 students will be the Red Team. The 
grade 10 students will be the Blue Team.
The four teams will compete with each other in all the events 
except for a special event. In this event, only two teams will 
compete. The grade 7 and grade 10 students will be grouped to 
form the Blue-Green Team. The grade 8 and grade 9 students will 
be grouped to form the Red-Yellow Team.
Like in numbers, there are also operations on sets. You can 
perform a specific operation on two or more sets to come up with 
another set. Study each set operation.
Set OperationsLESSON
1-3
Learning  
Objectives
At the end of the 
lesson, you should be 
able to:
•	 identify	operations	
on sets
•	 perform	operations	
on sets
•	 illustrate	set	
operations	using	
Venn	diagrams


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23
Union of Sets
In the given situation, each team in the sports fest represents a set. Combining 
two teams represents a union of two sets. What can you say about the elements of 
the new set formed?
For any sets A and B, the union of A and B, which is denoted as A ∪ B (read as “A union 
B”), is given by:
A ∪ B = {x|x ∈ A or x ∈ B}
Key Concept
The word “or” means that x belongs to A, or to B, or to both A and B. Hence, to 
find the union of two sets, combine the elements of A and B by writing the elements 
in any order, without repetition.
examPLe 1
Consider the following sets of athletes:
 B =  {Curry, Durant, James, Irving} 
 S =  {Messi, Ronaldo, Suarez, Neymar}
 T =  {Yap, Tenorio, Pingris, Romeo} 
Find the elements of each union of sets.
a. S ∪ T
b. B ∪ S
c. B ∪ T
d. T ∪ B
 Solutions:
 a.  S ∪ T =  {Messi, Ronaldo, Suarez, Neymar, Yap, Tenorio, Pingris, Romeo}
 b.  B ∪ S =  {Curry, Durant, James, Irving, Messi, Ronaldo, Suarez, Neymar}


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6. The improper subset of any set is equal to the empty set.
 7. A subset of B is a subset of C only if B is a subset of C.
 8. If A and B are both equivalent to C, then A and B are equal.
 9.  If B is a proper subset of A, then B does not necessarily contain each element 
of A.
 10. If A and B are equal to C, then A and B are not necessarily equal.


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24
Study the next example.
examPLe 2
Suppose that the four teams mentioned at the start of the lesson are described 
as follows:
 G  = {x|x is a grade 7 student who is a member of Green Team}
 Y =  {x|x is a grade 8 student who is a member of Yellow Team}
 R =  {x|x is a grade 9 student who is a member of Red Team}
 B =  {x|x is a grade 10 student who is a member of Blue Team}
Illustrate the Blue-Green Team using a Venn diagram.
 Solution: 
  Note that set B refers to Blue Team and set G refers to Green Team . Thus, 
the Blue-Green Team refers to the union of B and G, or B ∪ G. Observe that 
B and G are disjoint sets since they do not have any common element. This is 
because no grade 7 student is a grade 10 student, and no grade 10 student 
is a grade 7 student. Thus, the Blue-Green Team or B ∪ G is illustrated as 
follows:
 
B G
 B ∪ G
Look at the Venn diagram in the previous example. The shaded region represents


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21
Enrichment Exercises
In example 3 of this lesson, you have seen that a set with 3 elements has 8 subsets. 
Is there a relationship between the cardinality of a set and the number of subsets of 
the set? Investigate by answering each question.
 51.  How many subsets does an empty set have? Give 
the subsets.
 52.  What are the subsets of {0}? How many subsets 
does this set have?
 53.  What are the subsets of {0, 1}? How many subsets 
does this set have?
 54.  Suppose that set A is a finite set with 6 elements. 
How many subsets does A have? Complete the 
table on the right. Then find a pattern on the 
number of subsets of a certain set based on its 
cardinality.
 55.  Based on the results you obtained in the previous item, make a generalization 
about the number of subsets of a finite set A, with n(A) = k. 
Use your generalization in the previous item to find the total number of subsets of 
each given set.
 56. Y = {x|x is a weekday that starts with a vowel}


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a subset of the set of sports played in the Summer Olympic Games? Explain.
 43.  Liza can choose three from the five elective subjects offered during a term. 
The elective subjects offered are Spanish, Leadership, Music Composition, 
Psychology, and Actuarial Science. What are the possible sets of elective 
subjects she can enroll in?
 44.  A science teacher divides his class into 8 groups for a project. He wants his 
students to research on the planets in the solar system. Can he assign a 
different planet to each group? What set concept is illustrated in this situation? 
Explain.
 4 5.   Five friends plan a chess tournament. Each person will play against each 
participant exactly once. How many possible games can be played?
Consider sets A, B, and C. Draw a Venn diagram that illustrates each statement.
 46. A ⊂ B, but B ⊄ C
 47. A ⊆ B and B ⊆ A
 48. A ⊂ B, A ⊂ C, and B ⊂ C
 49. A ⊂ C and B ⊂ C, but A ⊄ B
 50. A and B are disjoint and B ⊂ C


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counting numbers, C = {1, 2, 3, ...}. It is an infinite set because there are infinitely 
many counting numbers.
Is the set of the stars in the sky finite or infinite? You may think that since there 
are so many stars, then it is impossible to count all of them. However, it has been 
known that there are approximately 200,000,000 stars in the sky. Although this set 
is very large, it is considered finite since its cardinality is a whole number.


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In what way can you show the relations among sets M, N, and O?
Notice that sets M and N have no common elements. Thus, M and N are disjoint 
sets.
Two sets are said to be disjoint sets if they have no common element.
Key Concept
The disjoint sets M and N are illustrated in the following diagram:
M N
Based on the given sets, can you say that M and O are disjoint?
M and O have the following common elements: 1, 3, and 5. Therefore, M and O 
are not disjoint.


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25
Intersection of Sets
Suppose that in the sports fest, the presidents of the Science Club and the Math 
Club agree to unite their members. Since some students might be members of both 
clubs, the number of members in the union might be less than the sum of the 
original number of members of the two clubs.
Let S be the set of Science Club members. Let M be the set of Math Club members. 
Since there is a possibility of overlap between S and M, the sets may not be disjoint. 
In such case, the union of these sets may be illustrated as follows:
S M
S ∪ M
In this situation, the students who are members of both the Science Club and the 
Math Club form another set. This set is referred to as the intersection of S and M. The 
intersection of S and M is the set of all elements contained in both S and M.
For any sets A and B, the intersection of A and B, which is denoted as A ∩ B (read as 
“A intersection B”), is given by:
A ∩ B = {x|x ∈ A and x ∈ B}.
Key Concept


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b. B ∪ S
c. B ∪ T
d. T ∪ B
 Solutions:
 a.  S ∪ T =  {Messi, Ronaldo, Suarez, Neymar, Yap, Tenorio, Pingris, Romeo}
 b.  B ∪ S =  {Curry, Durant, James, Irving, Messi, Ronaldo, Suarez, Neymar}
 c.  B ∪ T = {Curry, Durant, James, Irving, Yap, Tenorio, Pingris, Romeo}
 d.  T ∪ B = {Yap, Tenorio, Pingris, Romeo, Curry, Durant, James, Irving}
In the previous example, note that B ∪ T = T ∪ B. This illustrates that the union 
of sets is commutative; that is, the order of the sets does not affect the union. 
Commutative Property of the Union of Sets
For any sets A and B,
A ∪ B = B ∪ A.
Key Concept


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is a grade 7 student. Thus, the Blue-Green Team or B ∪ G is illustrated as 
follows:
 
B G
 B ∪ G
Look at the Venn diagram in the previous example. The shaded region represents 
B ∪ G, which is the combined region representing B and G. This means that the 
total number of students in the combined set is equal to the sum of the number of 
grade 7 students and the number of grade 10 students. This illustrates the following 
generalization:
If A and B are two disjoint finite sets, then A ∪ B is also finite and
n(A ∪ B) = n(A) + n(B). 
Key Concept
Verify this fact using example 1.


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Use your generalization in the previous item to find the total number of subsets of 
each given set.
 56. Y = {x|x is a weekday that starts with a vowel}
 57. M = {x|x is a distinct letter in the word “mathematics”}
 58. C = {x|x is a distinct consonant in the word “consonant”}
 59. D = {x|x is a two-digit number that is divisible by 11}
 60. J = {x|x is a prime number from 1 to 10}
Cardinality Number of 
Subsets
0
1
2
3 8
4
5
6


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27
Consider again the sets V and L. Suppose that the vowels are removed from 
the set of the letters in the English alphabet. What letters remain? The set of 
the remaining letters represent V '. Thus, V ' contains the consonants. Since  
C = {x|x is a consonant in the English alphabet}, then V' = C. Conversely, C' = V. 
examPLe 3
Consider the following sets:
 U = {x|x is a digit in the decimal system}
 S = {x|x is an even whole number less than 10}
 T = {x|x is a prime number less than 10}
Describe each set using the roster method.
a. S' b. T' c. U'
 Solutions:
  First, describe sets U, S, and T using the roster method to perform the operation 
more easily.
  U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
  S = {0, 2, 4, 6, 8}
  T = {2, 3, 5, 7}
 a. For S', take away the elements of set S from U.
  S' =  = {1, 3, 5, 7, 9}
 b. T' =  = {0, 1, 4, 6, 8, 9}
 c. U' =  = ∅
Notice that for any set A combined with its complement A', the resulting union


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26
Commutative Property of the Intersection of Sets
For any sets A and B,
A ∩ B = B ∩ A.
Key Concept
Complement of a Set
Another operation involving sets is the set complement. This operation is performed 
with respect to a universal set. 
The universal set is the set that contains all the elements under consideration. This is 
usually denoted as U. 
Key Concept
For instance, consider the set V = {x |x is a vowel in the English alphabet}. 
Consider also set L  = {x |x is a letter in the English alphabet}. You know 
that set V is contained in set L. Thus, the universal set in this situation is  
L = {x |x is a letter in the English alphabet}. Similarly, when considering the set  
C = {x|x is a consonant in the English alphabet}, the universal set is L. 
Study the following definition of the complement of a set:
Let A ⊂ U. The complement of A is the set of elements of U that are not contained in A.  
It is denoted by A' (read as “A complement”). In symbols,
A' = {x|x ∈ U and x ∉ A}.


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For any sets A and B, the intersection of A and B, which is denoted as A ∩ B (read as 
“A intersection B”), is given by:
A ∩ B = {x|x ∈ A and x ∈ B}.
Key Concept
The word “and” means that the elements belong to both A and B.
Using a Venn diagram, A ∩ B can be illustrated as follows:
A B
A ∩ B
In the diagram, the shaded region represents A ∩ B, which is the set of all elements 
that are common to both A  and B. Notice that the shaded region also represents  
B ∩ A. This illustrates that the intersection of sets is also commutative.


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28
Set Difference
Another operation that is similar to set complement is set difference . Both 
operations involve “taking away” elements from a set. To find the complement of a 
set, take away the elements of that set from the universal set. To find the difference, 
you will take away the elements of one set that are in the other set. The set difference 
is defined as follows:
The difference of sets A and B, which is denoted as A – B, is the set of elements of A that 
are not in B. In symbols, 
A – B = {x|x ∈ A and x ∉ B}.
Key Concept
Set difference may be illustrated using Venn diagrams as follows: 
A B A B
A – B B – A
What do you observe in the two diagrams?
Consider again sets S and T in example 3. 
 S = {0, 2, 4, 6, 8}
 T = {2, 3, 5, 7}
Note that the only element of S that is in T is 2. Thus, to find S – T, take this 
element (2) away from S. Hence, S – T = {0, 2, 4, 6, 8} = {0, 4, 6, 8}.
What elements of S will be removed from T to find T – S ?
examPLe 4


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29
 Solutions:
 a.  The elements a and e of set J are contained in V. Thus, to find J – V, take 
these away from set J. Therefore,
  J – V = {a , b, c, d, e } = {b, c, d}.
 b. V – J = {i, o, u}. Can you explain why?
Based on the previous example, can you say that set difference is commutative?
Set Product
Consider the following situation:
Five players are planning their fencing matches, in such a way that each 
player must compete with another player of the opposite gender. There are  
2 boys and 3 girls. How many matches will there be?
Do you know?
Fencing,
	also
	known
	as
	s
word
	fighting
,	 
is
	 a
	 fantastic
	 w
ay
	 to
	 impro
ve
	 one’
s	
balance
,
	coordina
tion,
	and
	fle
xibility.
	In	
f
encing,
	tw
o
	competitor
s
	face
	of
f	in
	a	
“bout.”
	F
encers
	use
	blade
	w
ork,
	f
ootwork,	
and
	
strategy
	
to
	
beat
	
their
	
opponents.
Let the set of boys be given by B = {b1, b2}. Let the set of girls be represented by


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32
 40. (A ∩ B)'
  
B
U
A
 41. A ∪ B
  
B
U
A
 42. (A ∪ B)'
  
B
U
A
 43. A ∩ B'
  
B
U
A
 44. (B – A)'
  
B
U
A
 45. U – A'
  
B
U
A
Applications
Answer each item completely.
 46.  A c anteen offers three kinds of sandwiches and two flavors of fruit shake. How 
many possible combinations of a sandwich and a fruit shake are there? 
For items 47 to 50, consider the following situation:
In a circle of 12 friends, 10 are varsity players. Ariel, Bart, Carlo, David, Eric, 
Frank, and George are members of the basketball team. Andrei, Brian, Carlo, Daniel, 
and Eric are members of the volleyball team. 
 47. Who are the members of either the volleyball team or the basketball team?
 48.  Who are the members of both the volleyball team and the basketball team?
 49.  Who are the members of the basketball team but are not members of the 
volleyball team?
 50.  Who are the members of the volleyball team but are not members of the 
basketball team?


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31
For items 16 to 25, refer to the following diagram: 
U
4
20
10 0
9
3
5
8 1
26
D E
7
Perform the indicated operations.
 16. D ∩ E
 17. D'
 18. D – E
 19. (D ∪ E)'
 20. (E – D)'
 21. D' ∪ E' 
 22. D' ∩ E'
 23. D' ∪ ∅' 
 24. E' ∩ U
 25. (D ∪ E)' × (D ∩ E)
For items 26 to 35 , consider sets A and U, where A ⊂ U. Perform the indicated 
operations.
 26. A ∪ A 
 27. A ∩ A 
 28. A ∪ U
 29. A ∩ U
 30. ∅ ∩ A
 31. ∅ ∩ U
 32. ∅' ∪ U
 33. ∅' ∪ ∅
 34. ∅' ∩ U'
 35. U ∩ A'
Shade the region that represents each given set in each Venn diagram.
 36. A
  
B
U
A
 37. B'
  
B
U
A
 38. U – A
  
B
U
A
 39. A ∩ B
  
B
U
A


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Let A ⊂ U. The complement of A is the set of elements of U that are not contained in A.  
It is denoted by A' (read as “A complement”). In symbols,
A' = {x|x ∈ U and x ∉ A}.
Key Concept
To find the complement of a set, look for the elements in the universal set that 
are not in the given set. For instance, to find A', remove all the elements of A from U. 
The elements that remain in U are the elements of A'. 
The shaded region in the following diagram represents A'. In a Venn diagram, the 
universal set is represented by a rectangle. Inside the rectangle are the circles that 
represent the sets involving the universal set.
A
A'
U


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30
In the given situation,
B × G = {(b1, g1), (b1, g2), (b1, g3), (b2, g1), (b2, g2), (b2, g3)}.
Therefore, the five players will play a total of 6 matches.
Note that the order of the elements in an ordered pair is important. The ordered 
pairs (a, b) and (c, d) are the same if and only if a = c and b = d. For example, the 
ordered pair (1, 2) is different from (2, 1). Does this mean that A × B is not the same 
as B × A? Study the next example.
examPLe 5
Let C = {a, b, c } and D = {1, 2, 3}. Define each set.
a. C × D b. D × C
 Solutions:
 a. C × D = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}
 b. D × C = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c)}
Based on the previous example, can you say that set product is commutative? 
Explain.
Practice 1-3
Concepts and Skills
Determine whether each statement is always, sometimes, or never true.
 1. ∅' ⊆ U 
 2. A – B = B
 3. A ∪ B = A 
 4. A' ∩ A = ∅
 5. A ⊆ (A ∪ B) 
 6. A ⊆ (A ∩ B)


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S' =  = {1, 3, 5, 7, 9}
 b. T' =  = {0, 1, 4, 6, 8, 9}
 c. U' =  = ∅
Notice that for any set A combined with its complement A', the resulting union 
is the universal set. The following properties hold true for the complement of a set: 
For any set A and a universal set U:
A ∪ A' = U 
A ∩ A' = ∅
Key Concept
Verify these properties using Venn diagrams.


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33
Enrichment Exercises
Show your answer to each item using a Venn diagram.
 51. If A ⊂ B, find A ∩ B.
 52. If A ⊂ B, find A ∪ B.
 53. If A ⊂ B and B ⊂ C, find A ∩ C.
 54. If A and B are disjoint sets and A ⊂ C, find A ∩ C.
 55. If A and B are not disjoint sets and C ⊂ (A ∩ B), find C ∩ (A ∩ B).
Answer each item.
 56. Is (A ∩ B) ∩ C = A ∩ (B ∩ C)? Give an example to support your answer.
 57. If A and B are disjoint sets, is A – B = A? Explain.
 58. If A and B are disjoint sets, is A ∩ B = ∅? Explain.
 59.  If A and B are finite sets that are not disjoint, is n(A ∪ B) = n(A) + n(B)? Explain.
 60. How does set difference differ from set complement?


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34
Problem Solving Involving Sets
Learning  
Objectives
At the end of the 
lesson, you should be 
able to:
•	 illustrate	problems	
involving	sets	using	
Venn	diagrams
•	 solve	problems	
involving	sets
Probe and Learn
A teacher conducts a survey among 50 students. She asked 
them which of the two sports, basketball or volleyball, they 
would like to play. Each student can choose one or two sports, 
or none of the two sports if they do not like to play either sport. 
The teacher finds out that 8 students like to play both basketball 
and volleyball, 24 prefer volleyball, and 32 like basketball. How 
many students do not like basketball or volleyball?
Many problems involving sets can be solved using Venn 
diagrams. Recall that a Venn diagram can help you visualize 
the logical relationships among sets. It is made by drawing a 
rectangle to represent the universal set. Inside the rectangle 
are two or more circles, where each circle represents a specific


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of
f	in
	a	
“bout.”
	F
encers
	use
	blade
	w
ork,
	f
ootwork,	
and
	
strategy
	
to
	
beat
	
their
	
opponents.
Let the set of boys be given by B = {b1, b2}. Let the set of girls be represented by  
G = {g1, g2, g3}. To find the possible matches, pair each boy with each girl. This can 
be illustrated in a diagram as follows:
 g1
b1 g2
 g3
       and       
 g1
b2 g2
 g3
Based on the diagram, the possible pairs are (b1, g1), (b1, g2), (b1, g3), (b2, g1), (b2, g2), 
and (b2, g3).
Such pairing of the elements between two sets is called set product. 
The set product or Cartesian product of two sets A and B is the set of all possible 
ordered pairs (a, b), where a ∈ A and b ∈ B. In symbols, the set product A × B (read as 
“A cross B”) is given by:
A × B = {(a, b)|a ∈ A and b ∈ B}.
Key Concept


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element (2) away from S. Hence, S – T = {0, 2, 4, 6, 8} = {0, 4, 6, 8}.
What elements of S will be removed from T to find T – S ?
examPLe 4
Let J = {a, b, c, d, e} and V = {a, e, i, o, u}.
Define each set.
a. J – V
b. V – J


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35
wherein each region represents a set of elements that share common properties. It 
is important that you know how to illustrate problems using Venn diagrams. Study 
the following illustrations:
A Venn diagram involving two sets A and B is shown.
Α
Α only A ∩ B B only
B
U
(A ∪ B)'
The region where the two circles overlap represents  A ∩ B, as labeled in the 
diagram. It contains the elements that belong to both A and B. The region labeled  
“A only” represents those elements that belong to A but not to B. Similarly, the region 
labeled “B only” represents those elements that belong to B but not to A. On the 
other hand, the entire region covered by both A and B, which is the colored region, 
represents A ∪ B. The uncolored region inside the rectangle represents (A ∪ B)'.
Now study the logical relations among three sets A, B, and C, as shown here.
Α only
C only
B only
U
(A ∩ B ∩ C)
Α B
C
(A ∩ C) − B
(A ∪ B ∪ C)'
(A ∩ B) − C
(B ∩ C) − A


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Practice 1-3
Concepts and Skills
Determine whether each statement is always, sometimes, or never true.
 1. ∅' ⊆ U 
 2. A – B = B
 3. A ∪ B = A 
 4. A' ∩ A = ∅
 5. A ⊆ (A ∪ B) 
 6. A ⊆ (A ∩ B)
 7. (A ∩ B) ⊆ A 
 8. n(A) + n(A') = n(U)
 9. n(A – B) = n(A) – n(B) 
 10. n(A) + n(B) = n(A ∪ B)
 11. If A ⊆ U, then A' ⊆ U. 
 12. If A ≠ ∅, then ∅ ∪ A = ∅.
 13. If A ⊆ U, then A' = U – A. 
 14. If A = B, then A ∪ B = A ∩ B.
 15. If A ~ B, then A and B are disjoint.


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37
 
V
8
24 32
50
B
U
24 − 8
   = 16
32 − 8
  = 24
  Adding the number of students in the shaded regions gives the number 
of students who like either volleyball or basketball. Hence, there are  
16 + 8 + 24 = 48 students who like either volleyball or basketball. This number 
corresponds to n(V ∪ B). So n(V ∪ B)' represents the number of students who 
do not like volleyball or basketball. Since the universal set has a total of 50 
students, then:
 n(V ∪ B)' = 50 – 48 = 2.
  Therefore, 2 of the students surveyed do not like any of the two sports. The 
complete Venn diagram is shown. 
V
8
24 32
50
B
U
24 − 8
    = 16
32 − 8
  = 24
50 – 48 = 2
16 + 8 + 24 = 48
n(V ∩ B) =  number of students 
who like 
both volleyball 
and basketball
n(V ∪ B) = 
 number of students who like 
either
 basketball or volleyball
n(V) = 
 number of 
students who 
like 
volleyball
n(V – B) = 
 number of 
students who like 
volleyball 
only
n(U) =  number of students 
surveyed
n
(B) = 
 number of 
students who


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36
examPLe 1
Out of the 50 students surveyed by the teacher, how many students do not like 
basketball or volleyball?
 Solution:
  If U is the universal set, you can represent the sets in the problem as follows:
  U = {x|x is a student who was surveyed}
  V = {v|v is a student who likes volleyball}
  B = {b|b is a student who likes basketball}
  To construct the Venn diagram for this problem, draw two circles (representing  
sets V and B) inside a rectangle (representing set U). Note that some students 
are elements of both V and B. This implies that V and B intersect. Thus,  
V and B are not disjoint. So you will have the following diagram:
 
V B
U
  In the following diagram, n(U) = 50, n(V) = 24, n(B) = 32, and n(V ∩ B) = 8
 
V24 32
50
B
U
8
  The next step is to determine the number of students who like volleyball only 
and those who like basketball only. Note that the number of students who like 
volleyball only refers to n(V − B). The number of students who like basketball


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39
  The universal set U is the same set of students surveyed in example 1. 
The following Venn diagram illustrates the relationships among sets B, L, 
and T. You can see that the three sets are represented by overlapping circles. 
Can you explain why? 
 
U
L B
T
 Based on the given information, you can describe each region as follows:
students who like 
both badminton 
and lawn tennistotal number of 
students surveyed
students who like lawn 
tennis
students who like both lawn 
tennis and table tennis
students who do not like any 
of the three racket sports
students who like all three 
racket sports
students who like 
badminton 
students who like 
both badminton 
and table tennis
students who like 
table tennis
U
n
B L
T
30 22
50
11
15 10
25
6
  The diagram indicates the number of elements in each labeled set. Consider 
first the cardinality of the intersection of any two sets. In each of the boxed 
numbers in the diagram, the 6 students who like the three racket sports are


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41
Practice 1-4
Concepts and Skills
For items 1 to 8, refer to the following situation:
Jay and Ray are both music enthusiasts. The following Venn diagram shows their 
interests, where J is the set of interests of Jay and R is the set of interests of Ray:
J R
playing the  
guitar
playing the  
pianosinging
Based on the diagram, write True if the statement is true; otherwise, write False.
 1. Jay likes singing.
 2. Ray does not like singing.
 3. Both Ray and Jay like singing.
 4. Jay likes singing while Ray likes playing the piano.
 5. Ray and Jay have exactly the same interests.
 6. Ray and Jay like singing, playing the guitar, or playing the piano.
 7. Both Jay and Ray like singing, while Ray likes playing the guitar.
 8.  Ray likes both singing and playing the guitar but he does not like playing the 
piano.
For items 9 to 25, refer to the following text and diagrams:
A guidance counselor conducts a survey among her 
students about their membership in three clubs—


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40
 
U
B L
T
30 22
50
11
15 10
25
6
5
9 4
15 − 6 = 9 10 − 6 = 4
11 − 6 = 5
  Next solve for the numbers in the unfilled regions in each circle. It is given 
that the total number of elements in B is 30. So far, there are already  
5 + 6 + 9 = 20 elements in B that have been accounted for. This implies that 
there are 10 elements in the unified region of B only. Do the same process for 
sets L and T.
  Lastly consider the universal set that contains 50 students. To get the total 
number of students already indicated in the colored regions, you will have  
10 + 5 + 6 + 9 + 7 + 4 + 6 = 47 students. Therefore, there are 3 students  
(that is, 50 – 47 = 3) who do not belong to sets B, L, and T.
 The following is the complete Venn diagram for the problem:
 
U
B L
T
30 22
50
25
6
5
6
10 7
9 4
22 − (5 + 6 + 4) = 730 − (5 + 6 + 9) = 10
25 − (9 + 6 + 4) = 6
50 − (10 + 5 + 6 + 9 + 7 + 4 + 6) = 3
3
Hence, 3 students do not like to play any of the three racket sports.


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the logical relationships among sets. It is made by drawing a 
rectangle to represent the universal set. Inside the rectangle 
are two or more circles, where each circle represents a specific 
set. The circles divide the universal set into several regions, 
LESSON
1-4


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Now study the logical relations among three sets A, B, and C, as shown here.
Α only
C only
B only
U
(A ∩ B ∩ C)
Α B
C
(A ∩ C) − B
(A ∪ B ∪ C)'
(A ∩ B) − C
(B ∩ C) − A
As you can see in the diagram, each region is specified. The entire colored region 
shows A ∪ B ∪ C. 
A Venn diagram could clearly represent the logical conditions specified in a 
certain problem. That is why it is a useful tool for solving problems involving sets. 
You can now solve the problem posed at the start of the lesson. Study the solution 
in the next example.


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38
Consider the 8 students who prefer both sports. These 8 students are included 
in both the number of students who like basketball and the number of students 
who like volleyball; thus, they are counted twice. Hence, you have to subtract 8 from 
the sum (56) to get a difference of 48. Consequently, there are 50 – 2 = 48 students 
who like either sport. Note that you obtained the same answer (2) as in the previous 
solution. In symbols, you may solve for the number of students who like either sport 
as follows:
24 + 32 – 8 = 48
↓ ↓ ↓ ↓
n(V) + n(B) – n(V ∩ B) = n(V ∪ B)
This equation illustrates the counting formula for sets, which can be stated as 
follows:
If A and B are finite sets, then: 
n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
An equivalent equation to this is:
n(A ∩ B) = n(A) + n(B) – n(A ∪ B).
Key Concept
examPLe 2
A teacher asks the same set of students in example 1 which racket sport 
(badminton, lawn tennis, table tennis) they would like to play. Each student


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and those who like basketball only. Note that the number of students who like 
volleyball only refers to n(V − B). The number of students who like basketball 
only refers to n(B − V).
 Out of 24 elements in V, 8 elements belong to B. Thus,
 n(V – B) = 24 – 8 = 16.
  This means that 16 students like volleyball only. Similarly, subtract 8 from 32 
to find n(B – V). So the number of students who prefer basketball only is:
 n(B – V) = 32 – 8 = 24.


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42
 9.   How many students are members of all the three clubs? 
 10. How many students are not members of any of the three clubs? 
 11. How many students are members of the Science Club? 
 12. How many students are members of the Math Club? 
 13. How many students are members of the Theater Club? 
 14.  How many students are members of both the Math Club and the Science Club? 
 15.  How many students are members of both the Math Club and the Theater Club? 
 16.  How many students are members of both the Science Club and the Theater Club?
 17.  How many students are members of the Math Club but not of the Theater Club 
and the Science Club? 
 18.  How many students are members of the Science Club but not of the Theater 
Club and the Math Club?
 19.  How many students are members of the Theater Club but not of the Math and 
the Science Club? 
 20.  How many students are members of both the Math Club and the Theater Club 
but not of the Science Club?


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30 22
50
25
6
5
6
10 7
9 4
22 − (5 + 6 + 4) = 730 − (5 + 6 + 9) = 10
25 − (9 + 6 + 4) = 6
50 − (10 + 5 + 6 + 9 + 7 + 4 + 6) = 3
3
Hence, 3 students do not like to play any of the three racket sports.
What other conclusions can you make from the diagram in example 2?


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piano.
For items 9 to 25, refer to the following text and diagrams:
A guidance counselor conducts a survey among her 
students about their membership in three clubs—
Science Club, Math Club, and Theater Club. The 
results of the survey are shown in the Venn diagram 
on the right, where: 
  S = {s|s is a Science Club member};
  M = {m|m is a Math Club member}; and
  T = {t|t is a Theater Club member}.
U
S M
T
3
8
23
8
27 12
12 7


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Key Concept
examPLe 2
A teacher asks the same set of students in example 1 which racket sport 
(badminton, lawn tennis, table tennis) they would like to play. Each student 
chooses at least one of the three racket sports, or none if they do not like to play 
any of the given sports. The results are as follows:
 •  6 students like the three racket sports.  
 •  30 students like badminton.  
 •  22 students like lawn tennis.  
 •  25 students like table tennis.  
 •  10 students like both table tennis and lawn tennis.
 •  15 students like both table tennis and badminton.
 •  11 students like both lawn tennis and badminton.
How many students do not like any of the three racket sports? 
 Solution:
  Three different sets of students can be formed. So you will have the following 
sets:
  B = {b|b is a student who likes badminton}
  L = {w|w is a student who likes lawn tennis}
  T = {t|t is a student who likes table tennis}


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43
 27. How many fishes are both hardy and resistant to diseases?
 28. How many fishes are not hardy? 
 29. How many fishes are not resistant to diseases? 
 30. How many fishes are hardy but are not resistant to diseases?
For items 31 to 35, refer to the following situation:
In a class of 120 students, 45 students are members of dance clubs and 78 
students are members of sports teams. On the other hand, 32 students are not 
members of any of the mentioned groups.
 31. Draw a Venn diagram to illustrate the given information.
  
 32. How many students are members of both dance clubs and sports teams? 
 33. How many students are not members of sports teams?
 34.  How many students are members of dance clubs but are not members of sports 
teams? 
 35.  How many students are either members of dance clubs or members of sports 
teams? 
For items 36 to 40, refer to the following situation:
The science club adviser asks some student members what science courses they


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first the cardinality of the intersection of any two sets. In each of the boxed 
numbers in the diagram, the 6 students who like the three racket sports are 
already counted. Thus, subtract 6 from the boxed number in each of the three 
intersections.


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n(V) = 
 number of 
students who 
like 
volleyball
n(V – B) = 
 number of 
students who like 
volleyball 
only
n(U) =  number of students 
surveyed
n
(B) = 
 number of 
students who 
like 
basketball
n(B – V) = 
 number of students 
who like basketball 
only
n
(V ∪ B)' =  n umber of 
students who like 
neither basketball 
nor volleyball
As shown in the previous example, a Venn diagram organizes data and shows 
the relationships between sets. Other information can also be extracted from the 
diagram, like the number of students who like only one sport.
If you will add the 24 students who prefer volleyball and the 32 students who like 
basketball, you will get 56 students. Note that this sum is greater than the number 
of students surveyed, which does not satisfy the given information in the problem. 
What is wrong with this solution?


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the Science Club? 
 20.  How many students are members of both the Math Club and the Theater Club 
but not of the Science Club? 
 21 .   H ow many students are members of both the Science Club and the Theater 
Club but not of the Math Club?
 22.  How many students are members of both the Science Club and the Math Club 
but not of the Theater Club? 
 23. How many students did the guidance counselor survey? 
 24. How many students are members of exactly one club? 
 2 5.  How many students are members of exactly two clubs?
Applications
Answer each item completely.
For items 26 to 30, refer to the following situation:
In an aquarium, 20 different freshwater fishes are either hardy or resistant to 
diseases. Twelve are hardy and 10 are resistant to diseases.
 26. Draw a Venn diagram to illustrate the given information.


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45
For items 48 and 49, refer to the following situation:
Jose asked his classmates which telecommunication company they subscribed to. 
He found out that 14 of his classmates subscribed to company A and 18 subscribed 
to company B. He also found out that a total of 25 students subscribed to either 
company A or company B. 
 48. Can you tell how many students Jose asked? Explain. 
 49. What other information can you infer from the given data?
 50.  Let A, B, and C be disjoint finite sets. Is n(A ∪ B  ∪ C) = n(A) + n(B) + n(C)? Give an 
example to support your answer.


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46
Write True if the statement is true; otherwise, write False.
 1.  If S = {0, 1}, then {0} ∈ S.
 2. An empty set has no cardinality.
 3.  Any subset of a finite set is finite.
 4.  Any set is a proper subset of itself.
 5.  The set of points on a line is an infinite set.
 6.  The union of two sets is a set that contains their common elements.
 7.  The complement of a set is a subset of the universal set.
 8.  Two sets are equal if they have the same number of elements.
 9.  Two sets A and B are disjoint if no element of A is an element of B.
 10. The elements of a set product are the products of the elements of the sets.
Choose the set relation in column B that corresponds to each pair of sets in  
column A. 
 Column A
 11. A = {g, o, d} and B = {l, o, v, e}
 12. A = {g, o, d} and B = {f, a, i, t, h}
 13. A = {m, a, t, h} and B = {l, o, v, e}
 14.  A = {m, a, t, h} and 
B = {b|b is a distinct letter in “mahatma”}
 15.  A = {m, a, t, h} and


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48
Answer each question.
 21. Which is not a set?
  A.  {m, a, t, h, e, i, c, s}
  B. {x|x is a very large counting number}
  C. {x|x is a month with 32 days}
  D.  {x|x is a bacterium in the human body}
 22.  What method could be more appropriate to describe the set whose elements 
are Monday, April, 1, fish, apple, and love?
  A.  rule method
  B.  roster method
  C.  either A or B
  D.  neither A nor B
 23. Which set is infinite?
  A. all the points on a line
  B.  all the people on Earth
  C.  all the stars in the Milky Way Galaxy
  D.  all the trees in the Black Forest
 24.  Let G = {g, o, d} and H = {x|x is a distinct letter in the word “good”}. Which 
relation is not true?
  A. G ⊆ H
  B. G ~ H
  C. G = H
  D. G' = H
 25. Let W = {0, 1, 2, 3}. Which statement is true?
  A. { } ∈ W
  B.  0 ⊂ W
  C. {0} ⊂ W
  D.  {0} ∈ W
 26. How many subsets does an empty set have?
  A. 0
  B.  1
  C.  2
  D.  3
 27. Let D = {a, b, c, d}. How many subsets of D contain exactly two elements?


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49
For items 31 to 35, refer to the following situation:
The first term of a sequence is 2. Its second term is 4. The succeeding terms 
can be obtained by getting the average of all the preceding terms. Recall that 
the average is equal to the sum of the terms of a set divided by the number of 
terms. For example, the third term is the average of 2 (the first term) and  
4 (the second term). It is given by:
sum of the terms
number of terms  = 
2 + 4
2  = 
6
2  = 3
 31.  Let S be the set of all distinct terms in the sequence. Describe the set using the 
roster method.
 32. Find the cardinality of S.
 33. Classify S as a finite set or an infinite set.
 34. Find the number of subsets of S.
 35. Let T be the set of the first ten counting numbers. Is S ⊂ T ? Explain.
For items 36 to 40, refer to the following diagram:
X
2
5
3
7
Y
U
1
9
Let U = {x|x is a counting number less than 10}.
Perform the indicated operations.
 36. X ∩ Y 
 37. X – Y
 38. X'
 39. Y' 
 40. (X ∪ Y )'


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teams? 
For items 36 to 40, refer to the following situation:
The science club adviser asks some student members what science courses they 
like. Twenty-eight members say they like physics, 35 say they like chemistry, and a 34 
say they like biology. Also, 10 members say they like physics and chemistry, 12 say 
they like biology and chemistry, 9 say they like physics and biology, and 4 say they like 
all three. On the other hand, 10 say they do not like any of the three science courses.
 36. Draw a Venn diagram to illustrate the given information.


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47
Choose the diagram in column B that illustrates each set in column A. 
  Column A            Column B
 16. M' A.
 
N
O
U
M
 17. M – N B.
 
N
O
U
M
 18. (N ∪ O)' C.
 
O
U
M
N
 19. (O ∩ M)' D.
 
O
U
M
N
 20. M ∪ N ∪ O E.
 
O
U
M
N


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44
 37. How many students did the science club adviser ask? 
 38. How many students like both physics and chemistry but not biology? 
 39. How many students like any of the three courses? 
 40. How many students do not like physics? 
For items 41 to 45, refer to the following situation:
A class adviser is arranging the schedules of foreign language classes for  
70 students. Thirty students prefer to study French, 30 prefer Spanish, and 30 
prefer Nihongo. Twelve students prefer to study both French and Nihongo, and of 
these students, 5 prefer to study Spanish as well. Sixteen students prefer to study 
Nihongo only, and 20 prefer Spanish only.
 41. Draw a Venn diagram to illustrate the given information.
  
 42. How many students do not prefer to study Spanish, French, or Nihongo?
 43. How many students prefer to study Spanish and Nihongo but not French? 
 44. How many students prefer to study French and Spanish but not Nihongo?
 45. How many students prefer to study French only?


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50
 41. Draw a Venn diagram to illustrate the given information.
  
 42. H ow many students will sing, act, or dance?
 43. How many students will only act?
 4 4.  How many students will only sing?
 4 5.  How many students will only dance?
 46. How many students w ill act and dance but will not sing?
 47. H ow many students will sing and act but will not dance?
 48. H ow many students will sing and dance but will not act?
 49. How many students will sing or act but will not dance?
 50. How many students will make the props for the play?
Answer each item completely.
 51.  Let C = {x|x is a counting number divisible by 2 or by 3}. Let U be the set of 
counting numbers. Does C' contain a multiple of 9? Explain.
 52.  Let A = {1, 3, 5, 7, 9, 11} and B = {1, 2, 3, 4, 5, 6}. Which elements of A should 
be removed so that A ∩ B = ∅?
 53.  In a school of 1,400 students, 2
7  are grade 7 students. Two-fifths of the grade


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51
The Real Number System
CHAPTER
2
Look closely at the head of the sunflower. You would notice two spirals winding in 
two directions. The number of spirals in each direction is not the same—some sunflowers 
have 21 and 34. Others have 34 and 55, 55 and 89, or 89 and 144. These numbers of 
spirals belong to a special sequence of numbers, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..., 
called the Fibonacci sequence. In this sequence, the next number is found by adding the 
two numbers preceding it. Fibonacci numbers can also be observed in plants, petals, tree 
branches, and fruits. On a pineapple, for instance, there are 8 scales in a diagonal in one 
direction and 13 in another. If you count the number of petals in a flower, you will most 
likely get a Fibonacci number.  
Mathematics is the language of nature. In this chapter, you will learn about the different 
types of numbers and their properties, as you discover the beauty of math and nature.
Lesson 2-1. Real Numbers


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B.  0 ⊂ W
  C. {0} ⊂ W
  D.  {0} ∈ W
 26. How many subsets does an empty set have?
  A. 0
  B.  1
  C.  2
  D.  3
 27. Let D = {a, b, c, d}. How many subsets of D contain exactly two elements?
  A.  2
  B.  4
  C. 6
  D.  8
 28.  Which set operation is performed when combining all the elements of two or 
more sets?
  A.  intersection of sets
  B. set difference
  C. set product
  D.  union of sets
 29.  Let U = {x|x is a rainbow color} and S = {blue, green, yellow, red}. Which set is 
the complement of S?
  A. {orange, pink, red}
  B.  {orange, pink, violet}
  C. {orange, indigo, violet}
  D. {red, blue, yellow, green}
 30.  Suppose that S = {s, m, i, l, e} and J = {j, o, v, i, a, l}. What is the cardinality of  
S × J?
  A.  8
  B.  10
  C.  25
  D.  30


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X
2
5
3
7
Y
U
1
9
Let U = {x|x is a counting number less than 10}.
Perform the indicated operations.
 36. X ∩ Y 
 37. X – Y
 38. X'
 39. Y' 
 40. (X ∪ Y )'
For items 41 to 50, refer to the following situation:
A teacher organizes a musical play. A total of 50 students will participate 
in the play. Suppose that 24 of the students will sing, 25 will act, and 23 
will dance. Also, 16 of the students will sing and dance, 12 will act and dance,  
14 will sing and act, and 6 will sing, act, and dance. Those students who will not  
perform will create the props for the play.


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52
Probe and Learn
A unique property of Fibonacci numbers is seen when dividing 
any two successive numbers. Look at the quotient of each pair of 
successive numbers ( A and B) in the third column  of the 
following table:
A B B
A
2 3 1.5
3 5 1.666666666...
5 8 1.6
8 13 1.625
M M M
144 233 1.618055556...
233 377 1.618025751...
M M M
Note that 
B
A gets closer and closer to a special number called 
the golden ratio, which has a value of 1.6180339887.... The 
golden ratio is denoted by the Greek letter phi φ. Do you know 
that the golden ratio can also be observed on the characteristics 
of many things around you? Different parts of the human body 
also show the golden ratio. For instance, the ratio of the length 
of the forearm to the length of the hand is equal to the value of φ.
Real Numbers
Learning 
Objectives
At the end of the 
lesson, you should be 
able to:
•	 apply	the	
properties	of	 real 	
numbers
•	 identify	the	subsets	
of	 real 	 numbers
•	 classify	real	
numbers
LESSON
2-1


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44. How many students prefer to study French and Spanish but not Nihongo?
 45. How many students prefer to study French only? 
Enrichment Exercises
Answer each item completely.
For items 46 and 47, refer to the following situation:
Suppose that 25 basketball players, 30 volleyball players, and 28 soccer players 
compose a varsity club. 
 46.  If no member plays at least two sports, how many members are there in the 
varsity club?
 47.  Suppose that 5 basketball players are also volleyball players. No soccer player 
is also a volleyball or a basketball player. How many members are there in the 
club?


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12. A = {g, o, d} and B = {f, a, i, t, h}
 13. A = {m, a, t, h} and B = {l, o, v, e}
 14.  A = {m, a, t, h} and 
B = {b|b is a distinct letter in “mahatma”}
 15.  A = {m, a, t, h} and 
B = {x|x is a distinct letter in “mathematics”}
Column B
 A. A ~ B
 B. A = B
 C. A ⊂ B
 D. A ∩ B = {o}
 E. A ∩ B = ∅
Chapter Assessment


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be removed so that A ∩ B = ∅?
 53.  In a school of 1,400 students, 2
7  are grade 7 students. Two-fifths of the grade 
7 students are girls. Also, 75 of the girls are into swimming and 87 are into 
gymnastics. Suppose that all the girls are enrolled in at least one of the two 
sports. How many girls are into both swimming and gymnastics?
 54.  A nurse surveyed 26 children. She learned that 14 of the children have dogs, 
9 have cats, and 6 have rabbits. Also, 4 have dogs and cats, 3 have dogs and 
rabbits, and 1 has cats and rabbits. Suppose that none of them have all three 
kinds of pets. How many children do not have any of the three kinds of pets?
 55.  In a certain municipality, a market analyst asked 150 mothers about the 
detergent brand they use. The analyst found out that 102 of the mothers use 
brand A, 70 use brand B, and 40 use brand C. He also found out that 25 use 
brand A and brand B, 27 use brand A and brand C, and 30 use brand C and


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53
1 1.618
The golden ratio is an example of an irrational number. The set of irrational 
numbers is a subset of the set of real numbers. 
Set of Real Numbers
The set of real numbers (R) has many subsets. Refer to the following list of subsets 
of R:
• Numbers that are used in counting objects comprise the set of natural numbers 
(N). So, N = {1, 2, 3, ...}.
• Natural numbers and 0 (zero) comprise the set of whole numbers (W ).  
So, W = {0, 1, 2, 3, ...}.
• Natural numbers, their negatives, and 0 comprise the set of integers (Z ).  
So, Z = {..., –3, –2, –1, 0, 1, 2, 3, ...}.
• Numbers that can be expressed as a ratio of two integers belong to the set 
of rational numbers (Q). Rational numbers can be terminating decimals (like 
0.5 and 0.75), repeating decimals (like 0.333... and 0.2727...), or fractions 
like 
1
2 and 
3
4 .
• Nonterminating and nonrepeating decimals (like 1.6180339887... and 
3.1416...) belong to the set of irrational numbers (Q').
The Venn diagram on the


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54
The Real Number Line 
An important characteristic of the set of real numbers is that they can be  
represented by the points on a number line. In fact, the graph of the set of real numbers 
is the entire number line. Every point on the number line corresponds to a unique 
real number. Look at the following number line:
–4 –2–3 –1 0
–0.5 p–2.3
1 2 3 4
The numbers on the right of 0 are positive real numbers. The numbers on the 
left of 0 are negative real numbers. Some real numbers shown on the number line 
are the positive integers 1, 2, 3, and 4, the negative integers –1, –2, –3, and –4, the 
rational numbers –2.3, –0.5, and 5
2
, and the irrational numbers  and .
Properties of Real Numbers 
The real number system consists of the set of real numbers and the numbers that 
result from binary operations such as the four fundamental operations. Several 
assumptions govern the operations on real numbers. These assumptions are called 
axioms.


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55
A set of numbers is said to be closed under an operation if, upon performing the 
operation on any two elements of the set, the result is also an element of that set. 
The closure property for real numbers guarantees that the sum and the product of 
any two real numbers are also real numbers. Thus, the set of real numbers is said 
to be closed under addition and multiplication.
examPLe 1
Each finger of a human hand is divided 
into four parts by the knuckles. The 
relative lengths of these four parts 
are shown on the right. It could be a 
coincidence, but these four numbers, 
2, 3, 5, and 8 belong to the Fibonacci 
sequence. Suppose that F = {2, 3, 5, 8}. Is 
F closed under addition? 
 Solution:
  To check if set F is closed under addition, add each pair of numbers in the set. 
Then check if each sum also belongs to F. Note that the sum of 2 and 8 is 10, 
which is not an element of F. Thus, F is not closed under addition. 
Commutative Property
Let a and b be real numbers.


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56
examPLe 2
The Philippines is recognized as the world’s center of marine biodiversity. Among 
its marine reservoirs are beautiful coral reefs. These coral reefs serve as shelter to 
more than 900 fish species and 400 coral species in the country. Using the given 
numbers, show that addition and multiplication are commutative. 
 Solutions:
 a. 400 + 900  ? 900 + 400  The two numbers can be added in any 
order without affecting the sum.  1,300  3 1,300
 b. 400 • 900  ? 900 • 400  The two numbers can be multiplied in any 
order without affecting the product.  360,000  3 360,000
Associative Property
Let a, b, and c be real numbers. 
 Axiom 5.
 Associative Property of 
Addition
a + (b + c) = (a + b) + c
 Axiom 6.
 Associative Property of Multiplication
a
 • (b • c) = (a • b) • c
Key Concept
The next example illustrates the Associative Property.
examPLe 3
Show that each equation is correct.
a. 4 + (7 + 1) = (4 + 7) + 1 b. 2 • (3 • 4) = (2 • 3) • 4
 Solutions:


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brand A, 70 use brand B, and 40 use brand C. He also found out that 25 use 
brand A and brand B, 27 use brand A and brand C, and 30 use brand C and 
brand B. Suppose that all of them use at least one of the three brands. Are 
there mothers who use all three brands? Show your answer using a Venn 
diagram.


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58
Existence of an Inverse Element
 Axiom 9.
 Additive Inverse Property
 
    For any real number a
, there exists a real number –a, called the additive 
inverse of a, such that 
 
   a
 + (–a) = 0  and  (–a) + a = 0. 
 Axiom 10.
 Multiplicative Inverse Property
 
    For 
any nonzero real number a, there exists a multiplicative inverse 1
a ,  
such that 
 
   a
 • 1
a = 1  and  1
a • a = 1.
Key Concept
examPLe 5
Find the number that will complete each equation.
a. 7 + ____ = 0 b. 4 • ____ = 1
 Solutions:
 a. 7 + (–7) = 0   The additive inverse of a real number is obtained by changing the  
sign of the number. Hence, –7 is the additive inverse of 7. Equivalently, 
7 is the additive inverse of –7.
 b.  4 • 
1
4 = 1  The multiplicative inverse of a nonzero real number is its reciprocal. 
Therefore, the multiplicative inverse of 4 is 1
4 . Conversely, the 
multiplicative inverse of 1
4 is 4.
examPLe 6
The top four eggplant-producing provinces in the country are Pangasinan,


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result from binary operations such as the four fundamental operations. Several 
assumptions govern the operations on real numbers. These assumptions are called 
axioms. 
Axioms are properties that are assumed to be true without proof.
Key Concept
Study the following axioms for real numbers:
Closure Property
Let a and b be real numbers. 
 Axiom 1.
 Closure Property of 
Addition
 
  The sum
 a + b is a real number.
 Axiom 2.
 Closure Property of Multiplication
 
  The product 
a • b is a real number.
Key Concept


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Then check if each sum also belongs to F. Note that the sum of 2 and 8 is 10, 
which is not an element of F. Thus, F is not closed under addition. 
Commutative Property
Let a and b be real numbers. 
 Axiom 3.
 Commutative Property of 
Addition
a + b = b + a
 Axiom 4.
 Commutative Property of Multiplication
a
 • b = b • a
Key Concept
Note that by the Commutative Property, rearranging the addends does not affect 
the sum. Similarly, rearranging the factors does not affect the product.


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like 
1
2 and 
3
4 .
• Nonterminating and nonrepeating decimals (like 1.6180339887... and 
3.1416...) belong to the set of irrational numbers (Q').
The Venn diagram on the 
right can help you visualize the 
relationships among the set of real 
numbers and its subsets.
Using your knowledge of sets, 
what generalizations can you 
make from the given diagram?
(Q')
Irrational 
Numbers
Rational Numbers (Q)
Real Numbers (R)
Whole 
Numbers (W)
Natural 
Numbers 
(N)
Integers (Z)


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Key Concept
The next example illustrates the Associative Property.
examPLe 3
Show that each equation is correct.
a. 4 + (7 + 1) = (4 + 7) + 1 b. 2 • (3 • 4) = (2 • 3) • 4
 Solutions:
 a.  4 + (7 + 1)  ? (4 + 7) + 1  When adding numbers, regrouping can be done 
without affecting the sum.   4 + 8  ? 11 + 1
   12  3 12
 b.  2 • (3 • 4)  ? (2 • 3) • 4  When multiplying numbers, regrouping can be 
done without affecting the product.  2 • 12  ? 6 • 4
  24  3 24


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types of numbers and their properties, as you discover the beauty of math and nature.
Lesson 2-1. Real Numbers
Lesson 2-2. Integers
Lesson 2-3. Rational Numbers
Lesson 2-4. Irrational Numbers


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Therefore, the multiplicative inverse of 4 is 1
4 . Conversely, the 
multiplicative inverse of 1
4 is 4.
examPLe 6
The top four eggplant-producing provinces in the country are Pangasinan, 
Quezon, Cebu, and Misamis Oriental. Suppose that you planted 4 rows of eggplant 
seedlings, where each row contains 5 seedlings. Then you added 2 more rows with 
the same number of eggplant seedlings per row. How many eggplant seedlings 
did you plant in all?


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59
 Solution:
  Let  represent an eggplant seedling that you planted initially. You already had  
4 rows, where each row contains 5 seedlings, as shown here.
    
    
    
    
  Now let  represent each additional eggplant seedling you planted. Since you 
added 2 rows, you will have the following number of eggplant seedlings:
 
    
    
    
    
    
    
 There are two ways to solve the problem numerically.
 Method A:
  Step 1.  Identify the total number of rows. If you added 2 more rows, then  
there will be 4 + 2 = 6 rows of seedlings.
  Step 2.   Find the total number of seedlings by multiplying the number 
of rows by the number of seedlings per row. Hence, there are a 
total of 5 • 6 = 30 seedlings.
  The solution based on this method may be summarized as follows:
    5 • (4 + 2)  =  5 • 6
     =  30
 Method B:
  Step 1.  Find the original number of seedlings that you planted. There


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  The solution based on this method is summarized as follows:
    5 • 4 + 5 • 2  =  20 + 10
     =  30
  Note that you obtained the same answer as in the previous method. Hence, 
you planted a total of 30 seedlings.
In the previous example, notice that 5 • (4 + 2) = 5 • 4 + 5 • 2. This equation 
illustrates the Distributive Property of Multiplication Over Addition.
Distributive Property of Multiplication Over Addition
Let a, b, and c be real numbers. 
 Axiom 1
1. 
 a
 • (b + c) = a • b + a • c 
Key Concept
This property states that when multiplying a number by a sum of two or more 
numbers, you can multiply first the number by each of the addends. Then add the 
resulting products. You can use this property when multiplying numbers, particularly 
in cases when rewriting one of the factors as a sum could make the computation 
easier. Study the next example. 
examPLe 7
Perform the indicated operations. Apply the Distributive Property to make your 
computations easier.


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Determine the set(s) where each number belongs to. Put a check mark on the correct 
column. You may use a calculator when necessary.
Number N W Z Q Q'
11. 0
12. 2p
13. 51
17
14. 2
15. 16
16. 3.14
17. –13
1
7
18. 0.345
19. –3.8333...
20. 0.12321213...
Write True if the statement is true; otherwise, write False.
 21. N ⊂ W 
 22. Z ⊂ Q
 23. Q' ⊂ Q
 24. Q' = R
 25. R – Q' = Q 
 26. Q ∩ N = Z
 27. Z ∩ W = N
 28. Q ∩ Q' = ∅
 29. N ∩ W = ∅ 
 30. W – N = {0}
Write the property of real numbers that each equation shows.
 31. 
1
2  2 = 1
 32. 
1
2 + 0 = 
1
2
 33. 2  x  y = 2  y  x
 34.  (–1) + 1 = 0


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•	 Subtraction
 If a
, b, and c are real numbers, the difference a – b is the number c such that  
 c
 + b = a.
•	 Division
 
 
If a, b, and c are real numbers, the quotient a
b , where b ≠ 0, is the number c such that  
c • b = a.
Key Concept
examPLe 9
Prove each equation using the definition of subtraction or division. 
a. 100 – 47 = 53 b. 264 ÷ 11 = 24
 Solutions:
 a. 100 – 47 = 53, since 47 + 53 = 100.
 b.  264 ÷ 11 = 24, since 24 • 11 = 264.
Which of the presented axioms hold true for subtraction? For division? Verify 
your answers by giving examples using different pairs of numbers.
Practice 2-1
Concepts and Skills
Write True if the statement is true; otherwise, write False.
 1. 0 is a natural number.
 2.  All numbers have a multiplicative inverse.
 3. No natural number is an irrational number.
 4. The additive identity element of the set of integers is 0.
 5.  A rational number is a quotient of any two integers.
 6. The product of a number and its additive inverse is 1.


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 35. z  (x  y) = (x  y)  z
 36. x + y = y + x
 37. 2  (x  y) = (2  x)  y
 38. x  y + 1 = 1 + x  y
 39. (–1) + 1 = 1 + (–1)
 40. (–4)  2 = 2  (–4)
 41. (a + b)(c + d) = (c + d)(a + b)
 42. (a + b)(c + d) = a(c + d) + b(c + d)
For items 43 to 45, complete the two-column proof.
For any real numbers x, a, b, and c, prove that: 
x  (a + b + c) = x  a + x  b + x  c.
Statement Reason
 x  (a + b + c) = x  [(a + b) + c] 43.
  =  x  (a + b) + x  c 44.
  =  x  a + x  b + x  c 45.
Applications
Answer each item completely.
 46.  Jaymie took an examination that is divided into three parts. She got 13 correct 
answers in the first part, 25 in the second part, and 27 in the third part. Apply 
the properties of real numbers to find her total score in the exam. Solve for the 
answer mentally.
 47.  An elevator can carry a maximum of 9 persons with an average weight of 50 kg 
each. Suppose that a box of items weighs 9 kg. How many boxes of items can 
the elevator carry? Explain your answer.


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You can apply the Associative and the Commutative Properties when doing 
mental computations. When adding three or more numbers, you may first combine 
the addends that add up to 10 or to any multiple of 10 to make computations easier. 
Look at the following illustration:
7   +  4  +  3   =  14
10
When multiplying three or more numbers, you may first multiply factors that give 
a product of 10 or any multiple of 10. Refer to the following illustration:
5   •  7  •   2   =  70
10
Other properties of real numbers include the existence of the identity element 
and the inverse element. Study the following axioms:
Existence of an Identity Element
Let a be any real number. 
 Axiom 7.
 Additive Identity Property
 
    There exists a real number 0, called the 
additive identity element, such that 
 
   a
 + 0 = a  and  0 + a = a.
 Axiom 8. 
 Multiplicative Identity Property
 
    There exists a real number 1, 
called the multiplicative identity element , 
such that 
 
   a


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The solution based on this method may be summarized as follows:
    5 • (4 + 2)  =  5 • 6
     =  30
 Method B:
  Step 1.  Find the original number of seedlings that you planted. There 
are 4 rows, where each row contains 5 eggplant seedlings. Thus, 
there are 5 • 4 = 20 seedlings.
  Step 2.  Find the number of seedlings you added. You added 2 rows, 
with 5 eggplant seedlings in every row. This means that you 
added 5 • 2 = 10 seedlings.
  Step 3.  Find the total number of seedlings. Add the results in steps 1 
and 2. So you will have 20 + 10 = 30 seedlings.


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For items 49 to 51, consider the following situation:
 Janice buys 12 cupcakes that cost P8.50 each. She also buys 8 muffins that cost 
P11 each. Instead of doing the conventional multiplication process, she computes 
for the total price of the cupcakes and muffins as shown here. Write the properties 
of real numbers that she used.
8.5(10 + 2) + 11(10 – 2) = 85 + 17 + 110 − 22 49.   
= 85 + 110 + 22 – 5 – 22 Commutative Property of 
Addition
= 195 – 5 + 22 + (−22) Commutative Property of 
Addition
= 190 + 0 50.   
= 190 51.   
Enrichment Exercises
Answer each item completely.
 52. Is any integer a rational number? Justify your answer.
 53.  Is the set of odd whole numbers closed under addition? Why or why not?
 54. Consider the following equation: 
  (3 • 4) + (5 • 6) = 3 • 4 + 5 • 6. 
  Can you say that 3 • (4 + 5) = 3 • 4 + 5? Justify your answer.
 55.  Can you find values for a, b, and c that will make the given equation true. 
Explain your answer?
  ab – ac = a – bc


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4. The additive identity element of the set of integers is 0.
 5.  A rational number is a quotient of any two integers.
 6. The product of a number and its additive inverse is 1.
 7.  The set of counting numbers is not closed under division.
 8. Subtraction is associative, but it is not commutative.
 9. Subtracting the additive identity element from any number results in the given 
number.
 10. The union of the sets of integers and rational numbers is the set of real numbers.


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easier. Study the next example. 
examPLe 7
Perform the indicated operations. Apply the Distributive Property to make your 
computations easier.
a. 120 • 64 + 120 • 36
b. 80 • 104
 Solutions:
 a. 120 • 64 + 120 • 36  = 120 • (64 + 36)
   = 120 • 100
   = 12,000
   Notice that 64 and 36 are added first to get 100, which can be easily 
multiplied by 120. 
 b.  To find the product of 80 • 104, you may think of 104 as 100 + 4 since it 
is easier to multiply a number by multiples of 10.
   80 • 104  = 80 • (100 + 4)
    = 80 • 100 + 80 • 4
    = 8,000 + 320
    = 8,320


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Learning 
Objectives
At the end of the 
lesson, you should be 
able to:
•	 define	integers
•	 locate	integers	on	
a	number	line
•	 compare	and	order	
integers
•	 find	the	absolute	
value	of	 an 	 integer	
or	expression	
involving	integers
•	 perform	operations	 
on	integers
Probe and Learn
The Marianas Trench in the Pacific Ocean is considered as 
the deepest part of the ocean. It has a depth of about 10,994 m.  
The Philippines’ deepest point, which is also the world’s second 
deepest spot, is the Mindanao Trench. It is about 10,540 m 
below sea level. Like ocean depths, heights of mountains are also 
measured from sea level. Mount Apo, the highest mountain in 
the Philippines, has a height of 2,954 m. Mount Everest, the 
highest mountain in the world, has a height of 8,848 m.
Ocean depths are measured using sound navigation and 
ranging (sonar). This is done by sending sound waves from a ship 
towards the ocean bottom. The time it takes for sound waves to


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each. Suppose that a box of items weighs 9 kg. How many boxes of items can 
the elevator carry? Explain your answer.
 48.  Bob teaches 4 hours a day, 4 days a week. His hourly rate is P150. How many 
weeks should he work to earn at least  7,000?


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examPLe 1
Express each numerical value as an integer.
a.  The 2018 General Wholesale Price Index (GWPI) in the Philippines dropped by 
about 2% during a certain month. 
b.  The 2013 functional literacy rate of the country grew by about 8 percentage 
points from 2008.
c. Fuelwood consumption in rural areas declined by 6%.
 Answers:
 a.  –2  The word dropped suggests a negative integer.
 b. 8 The word grew indicates a positive integer.
 c. –6 Why?
examPLe 2
The following table shows the differences in the grades of a student from one 
grading period to the next grading period:
 Subject Difference
English +2
Filipino 0
Science –1
Math –3
Araling Panlipunan +4
Plot each integer on a number line. Label the points properly.
 Solutions:
  Use the following letters to label the points: E for English, F for Filipino, S for 
Science, M for Math, and A for Araling Panlipunan. So you will have:
 –4–5 –2–3 –1 0 1 2 3 4 5
M S F E A


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a
 + 0 = a  and  0 + a = a.
 Axiom 8. 
 Multiplicative Identity Property
 
    There exists a real number 1, 
called the multiplicative identity element , 
such that 
 
   a
 • 1 = a  and  1 • a = a.
Key Concept
examPLe 4
Find the number that will complete each equation.
a. 4 + ____ = 4 b. 19 • ____ = 19
 Solutions:
 a. 4 + 0 = 4  Adding 0 to a number gives a sum equal to that number.
 b. 19 • 1 = 19   Multiplying a number by 1 gives a product equal to that number.


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You have seen that applying the properties of real numbers could help make your 
computations easier. You can also use these basic properties to derive or prove other 
properties of real numbers.
Properties that can be proven using axioms are called theorems.
Key Concept
There are no absolute rules in proving theorems. To prove a theorem, you may 
write a sequence of statements, each of which follows directly from the preceding 
statement. Each statement must be supported by a reason, which can be an axiom, 
a definition, or a previously proven theorem. A clear understanding of the properties 
of real numbers would be a big help when proving theorems. Study the next example.
examPLe 8
Prove that c • (a + b) = c • b + c • a, for any real numbers a, b, and c.
 Solution:
  To prove the given property, manipulate the expression on one side of 
the equation to yield the expression on the other side. In this case, you


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of a mountain when it is brought at the top of the mountain. Such measures of 
heights and depths are good examples of the existence of positive and negative values. 
Look at the following number line. Describe the labeled numbers on the left of 0. 
How do these numbers compare with the set of natural numbers?
–5 –4 –3 –2 –1 1 2 3 4 5 0
Set of Integers
The set of integers, which is denoted as Z, is composed of the natural numbers, their 
additive inverses, and 0.
Key Concept
The set of integers may be described as follows: 
Z = {…, –3, –2, –1, 0, 1, 2, 3, ...}.
Note that the set of integers is the union of three disjoint sets, namely, the set of 
positive integers Z+ = {1, 2, 3, 4, 5, ...}, the set containing 0, and the set of negative 
integers Z− = {…, –5, –4, –3, –2, –1}. It may be represented as follows:
Z = Z– ∪ {0} ∪ Z+.
Note that the set of positive integers is equal to the set of natural numbers.


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Comparing and Ordering Integers
Look at the following number line: 
–5 –4 –3 –2 –1 1 2 3 4 5 0
A number line can help you in comparing and ordering integers. Note that the 
values of integers are increasing from left to right on the number line. This means 
that the integer on the right of a given integer is greater, and any other integer on its 
left is lesser in value. Also, remember that any negative integer is less than 0, while 
any positive integer is greater than 0. Can you say that any positive integer is greater 
than any negative integer?
examPLe 3
Refer to the given table in the previous example. In which subject does the student 
have the greatest increase in grades? How about the greatest decrease?
 Solution:
  You can use the number line you made in the previous example to arrange the 
integers from least to greatest. Consider that the integers are written in order 
from left to right on the number line.
M S F E A
–3 –1 0 2 4


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Absolute Value
The distance of a number from 0 on a number line is called the absolute value of the 
number. The absolute value of an integer n is written as |n|.
Key Concept
Note that since the absolute value refers to a distance, it is always nonnegative. 
|n|  =  distance of n from 0
0 n
Let n be any integer.
• |n|  =   n if n is a positive integer.
• |n|   =  0 if n equals 0.
• |n|  =   –n if n is a negative integer.
examPLe 5
Find |4|, |–4|, and |0|.
 Solutions:
  First, locate each integer on the number line. Then count how many units 
away each integer is from 0.
 –4–5 –2–3 –1 0 1 2 3 4 5
4 units 4 units
 Since 4 is 4 units away from 0, then |4|   =  4.
 Since –4 is 4 units away from 0, then |–4|   =  4.
 Notice that |0| = 0. Why?
Operations on Integers
Like with whole numbers, performing operations on integers follows certain rules. 
Your knowledge of the concept of absolute value will be of great help in understanding


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Ocean depths are measured using sound navigation and 
ranging (sonar). This is done by sending sound waves from a ship 
towards the ocean bottom. The time it takes for sound waves to 
travel to the seafloor and back to the ship is accurately measured 
and recorded. On the other hand, a surveying altimeter is used to 
measure the height of mountains. It directly records the altitude 
IntegersLESSON
2-2


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 Solution:
  Represent each integer using counters. Then remove any zero pair, if there are 
any. The number that represents the remaining counters is the sum.
 –34 + (–7)
 Hence, 4 + (–7)  =  –3.
In examples 6 and 7, you found out that (–4) + 7 = 3 and 4 + (–7) = –3. Notice that 
the sum in each of the given equations is the difference of the absolute values of the 
addends, and the sign of the sum is the sign of the addend with the greater absolute 
value. This shows the next rule.
When adding integers with unlike signs, subtract their absolute values. Then attach the 
sign of the integer with the greater absolute value to the sum.
Key Concept
The rule means that the sum of a positive integer and a negative integer may be 
either negative or positive, depending on the sign of the addend that has the greater 
absolute value. Can their sum be equal to 0?
examPLe 8
Find the sum of (−9) + 4.
 Solution:
  |–9| – |4|  = 5  Find the difference of the absolute values of the integers.


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 Solutions:
 a. (–4) + 7
  
u Start at –4.
v
 Move 7 units to the right.
–
4–5 –2–3 –1 0 1 2 3 4 5
  Thus, (–4) + 7 = 3.
 b. (−4) + (−7)
  
v Move 7 units to the left.
u	Start at –4.
–
12–13 –10–11 –9 –8 –7 –6 –5 –4 –3
  Thus, (–4) + (–7) = –11.
In item b of the previous example, you found out that (–4) + (–7) = –11. This 
means that the sum of two negative integers is also negative. Use the following rule 
when adding integers with like signs:
When adding integers with like signs, add their absolute values. Then attach the common 
sign to the sum.
Key Concept
Another way to illustrate addition of integers is by using counters. Counters are 
useful in showing the rule in adding integers with different signs. 
Let  represent +1. Let  represent −1. Note that a  and a  make a zero pair. 
Since a zero pair has a value of 0, you can remove zero pairs without affecting the 
resulting answer. Study next example.
examPLe 7
Find the sum of 4 + (–7) using counters.


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Solution:
  To prove the given property, manipulate the expression on one side of 
the equation to yield the expression on the other side. In this case, you 
may start with the left side of the equation. Then apply the properties 
of real numbers to end up with the expression on the right side.  
You may use a two-column proof. The first column gives the statements, while 
the second column gives the corresponding reasons. So you will have the 
following:
Statement Reason
c • (a + b)  =  c • (b + a) Commutative Property of Addition
=  c • b + c • a Distributive Property
  Since you already obtained the expression on the right side of the equation, 
you have shown that the property is true. 
Have you noticed that the previously presented properties of real numbers involve 
only addition and multiplication? Can you explain why?
In earlier grade levels, you have learned that subtraction is the inverse of


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integers Z− = {…, –5, –4, –3, –2, –1}. It may be represented as follows:
Z = Z– ∪ {0} ∪ Z+.
Note that the set of positive integers is equal to the set of natural numbers. 
Mountain heights and ocean depths are measured with respect to sea level, which 
may be referred to as the reference point. Thus, the sea level may be represented  
by 0. The phrases above sea level and below sea level  indicate values that have 
opposite signs. Distances below sea level may be represented by negative numbers. 
Distances above sea level may be represented by positive numbers. Hence, the given 
ocean depths and mountain heights may be written using signed numbers as follows:
Place Distance from  
Sea Level (in meters)
Above Sea Level
Mount Everest  8,848
Mount Apo  2,954
Below Sea Level
Mindanao Trench  –10,540
Marianas Trench  –10,994
Positive numbers may be written without a plus sign (+). Thus, +4 is the same 
as 4. Negative numbers are written with a minus sign (–) before them. For example,


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integers from least to greatest. Consider that the integers are written in order 
from left to right on the number line.
M S F E A
–3 –1 0 2 4
  Therefore, the greatest increase is 4, which is in Araling Panlipunan. The 
greatest decrease is –3, which is in Math.
To compare two integers, the following symbols may be used:
Symbol Meaning
= is equal to
> is greater than
< is less than
examPLe 4
Compare each pair of integers using >, <, or =.
a. 1 _____ 5 b. –1 _____ –2 c. –4 _____ 0
 Solutions:
 a. 1 < 5  1 is located on the left side of 5 on the number line.
 b.  –1 > –2  –1 is located on the right side of –2 on the number line.
 c. –4 < 0  Why?


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Operations on Integers
Like with whole numbers, performing operations on integers follows certain rules. 
Your knowledge of the concept of absolute value will be of great help in understanding 
these rules. Study how addition, subtraction, multiplication, and division on integers 
are performed. 
Do you know?
You	can 	think 	of	the 	a bsolute 	value 	of	a 	
number	as 	the 	number 	itself , 	b ut 	without 	
the	sign.	For	example,	|–47|		=		47.


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Note that the addends in the equation include positive and negative integers. 
Recall that addition of real numbers is commutative and associative. Thus, you can 
group together and add integers of the same sign. Then add the resulting sums. So 
you will have:
 N = (1,426 + 320) + [(–292) + (–585)]
  =  1,746 + (−877)
  =  869
The sum is positive. This means that Amy has a profit of P869 in those four 
weeks.
Subtraction of Integers
Different cooperatives help 
people through their loan programs. 
Suppose that Susan borrows P10,000 
from a credit cooperative to start a 
small business. After a few months, 
she pays P5,000 back. How much 
does she still owe the cooperative, 
assuming no interest will be applied?
An amount of borrowed money 
implies a negative number. In this 
problem, the amount Susan borrows can be represented by –10,000. The act of 
paying back may be expressed by subtracting a negative number. Since Susan pays


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absolute value. Can their sum be equal to 0?
examPLe 8
Find the sum of (−9) + 4.
 Solution:
  |–9| – |4|  = 5  Find the difference of the absolute values of the integers.
  (–9) + 4  = –5   Attach to the sum the sign of the integer with the greater absolute 
value. Since –9 has the greater absolute value, the sign of the sum is 
negative.
You can now solve the problem posed at the start of the section on adding integers. 
The problem asked for Amy’s gain or loss from her sari-sari store in four weeks. To 
solve it, you have to find the value of N in the following equation:
N = 1,426 + (–292) + (–585) + 320.


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examPLe 9
Find each difference.
a. (–5) – (–2) b. (–5) – 2
 Solutions:
 a. (–5) – (–2)
  
u Start at –5.
v Move 2 units to the right.
–5–6 –3–4 –2 –1 0 1
  Thus, (–5) – (–2) = –3.
 b. (–5) – 2
  
u Start at –5.
v Move 2 units to the left.
–7–8 –5–6 –4 –3 –2 –1
  Thus, (–5) – 2 = –7.
In item a of the previous example, notice that the number line also shows  
(–5) + 2. Thus, 
(–5) – (–2) = (–5) + 2.
In item b, the number line also shows (–5) + (–2). Thus,
(–5) – 2 = (–5) + (–2).
These illustrate that you can rewrite a subtraction problem into an addition 
problem. This is done by changing the subtrahend to its additive inverse, and then 
changing the minus sign to a plus sign. Therefore, the rule for subtracting integers 
may be stated as follows: 
When subtracting integers, change the subtrahend to its additive inverse, and change the 
operation to addition. Then follow the rules in adding integers. 
Key Concept


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Study the following illustration:
(–5)  +    2  =  –3
(–5) – (–2)
w Add the integers.
u
 Get the additive inverse.v
 Change to addition.
examPLe 10
Subtract.
a. 7 – (–9)
b. 6 – 18
c. –3 – 12
d. –9 – (–4)
 Solutions:
 a. 7 – (–9) = 7 + 9 = 16 
 b. 6 – 18 = 6 + (–18) = –12
 c. –3 – 12 = –3 + (–12) = –15 
 d. –9 – (–4) = –9 + 4 = –5
In general, if a and b are real numbers, the following equivalence relations exist:
	 •	 a + (–b) = a − b
	 •	 a – (+b) = a − b
	 •	 a − (−b) = a + b
Key Concept
Multiplication of Integers 
Suppose that a bus travels at an average 
rate of 80 kph. It decreases its speed by  
5 kph after every hour. What is its speed 
after 4 hours? 
To find the speed of the bus after  
4 hours, determine by how much the bus 
has decreased its speed after 4 hours. 
A decrease of 5 kph in speed may be 
represented by –5. Now determine how 
many times the bus decreases its speed in 
4 hours. Since its speed decreases every 
hour, the bus decreases its speed 4 times.


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only addition and multiplication? Can you explain why?
In earlier grade levels, you have learned that subtraction is the inverse of 
addition. You have also learned that division is the inverse of multiplication. Study 
how subtraction and division are defined based on their inverses.


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4 • (–5) –20
Hence, the speed of the bus after 4 hours is given by:
80 + (–20) = 60 kph.
You may also use counters to verify the following rules in multiplying integers:
When multiplying integers, get the product of their absolute values. Then apply the 
following rules for the sign of the product:
 •
 The product of two integers with like signs is positive. 
 •
 The product of two integers with unlike signs is negative. 
Key Concept
examPLe 11
Multiply.
a. (–15)(–8) b. 9 • (–12)
 Solutions:
 a. (–15)(–8) = 120   The two factors have like signs. Hence, their product is positive.
 b. 9 • (–12) = –108   The two factors have unlike signs. Hence, their product is negative.
When multiplying three or more integers, get the product of their absolute values. 
Then determine the sign of the product by counting the negative factors. An odd 
number of negative factors gives a negative product. An even number of negative 
factors gives a positive product. Can you explain why?
examPLe 12


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Marianas Trench  –10,994
Positive numbers may be written without a plus sign (+). Thus, +4 is the same 
as 4. Negative numbers are written with a minus sign (–) before them. For example, 
negative 4 is written as –4. Zero is neither positive nor negative, so it is written 
without a + or – sign.


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71
Addition of Integers
You can find sari-sari stores anywhere 
in the Philippines. Amy owns a sari-sari 
store. Every week, she lists down her profits 
and losses. The following is a summary of 
her record for four weeks:
Week 1 a profit of  1,426
Week 2 a loss of  292
Week 3 a loss of  585
Week 4 a profit of  320
Did Amy earn or lose money in those 
four weeks? How much did she earn or 
lose?
Profits imply positive numbers, while losses imply negative numbers. Hence, each 
amount in the table may be represented by an integer. You can solve the problem 
by finding the sum of the integers. Let N represent the sum. So you will have the 
following equation:
N = 1,426 + (–292) + (–585) + 320 
How do you solve for N? First, identify the rules in adding integers. Remember 
that positive integers are counting numbers. This means that the sum of two positive 
integers is also positive.
How about the sum of two negative integers? Or the sum of a positive integer and


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implies a negative number. In this 
problem, the amount Susan borrows can be represented by –10,000. The act of 
paying back may be expressed by subtracting a negative number. Since Susan pays 
P5,000 back, you have to subtract –5,000 from –10,000.
Let N be the amount that Susan still has to pay. Hence,
N = (–10,000) – (–5,000).
To find N, you need to subtract integers. Like in adding integers, you may use 
counters in subtracting integers. For this problem, let  represent P1,000.
–10,000 – (–5,000) = –5,000–10,000
Therefore, Susan still owes the cooperative P5,000.
You may also use a number line to subtract integers. First, locate the minuend 
on the number line. Suppose that n is the subtrahend.
• If n is negative, move |n| units to the right of the minuend.
• If n is positive, move |n| units to the left of the minuend.
Consider the next examples.


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78
Division of Integers
Baguio City is considered as the 
summer capital of the Philippines due 
to its relatively low temperature. Its 
temperatures range from 15°C to 26°C. 
On a particular day in December 2018, the 
temperature reading in Baguio at 3 p.m. 
is 21°C. Suppose that the temperature 
decreases by 2°C every hour. At what  
time will the temperature reading be 
15°C? 
A decrease in temperature implies 
a negative number. If the temperature 
drops from 21°C to 15°C, then the 
total decrease in temperature may be 
represented by 15 – 21 = –6, or −6°C. After how many hours will the temperature be 
15°C? Recall that the temperature decreases by 2°C every hour. A decrease of 2°C 
may be represented by −2.
To solve this problem, you have to divide integers; that is, divide –6 by –2. Since 
division is the inverse of multiplication, the same rules of signs apply. You can verify 
these rules using counters.


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Practice 2-2
Concepts and Skills
Express each numerical value using integers.
 1. Today is warmer than yesterday by 5°C.
 2.  Joy lost 2 kg after months of jogging.
 3. John deposited P20,000 in his savings account.
 4. Mary needs to run 5 more meters to reach the finish line.
 5. The mystical Mount Banahaw is 2,169 m above sea level.
Write True if the statement is true; otherwise, write False.
 6.  The additive inverse of 4 is –4.
 7. The quotient of any two integers is an integer.
 8. The absolute value of any integer is positive.
 9.  The product of three negative integers is negative.
 10. The sum of two integers is greater than each of the addends.
 11. The quotient of a positive number and its additive inverse is –1.
 12.  If a is a negative number and b is a positive number, then a • (–b)  
is positive.
 13.  If the product of two integers is positive, then one of the integers could be 
negative.
Perform the indicated operations.
 14. 36 + 9
 15. 36 + (–9)
 16. (–36) + 9


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number of negative factors gives a negative product. An even number of negative 
factors gives a positive product. Can you explain why?
examPLe 12
Find the product of (–5)(–2)(–3).
 Solution:
 5 • 2 • 3 = 30  Multiply the absolute values of the given integers.
 (–5)(–2)(–3) = –30  The product is negative since there are three negative factors.


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81
 47.  What is the change in the number of enrollees in each grade level from  
SY 2009−2010 to SY 2010−2011? Express each change as an integer.
Grade Level Change in the Number 
of Enrollees
Grade 7
Grade 8
Grade 9
Grade 10
 48.  Plot the data you obtained in the previous item on a number line. Use the grade 
levels to label the points.
 49. In which grade level is the increase in the number of enrollees greatest? 
 50. In which grade level is the decrease in the number of enrollees greatest?
 51.  Which level has a greater change in the number of enrollees from SY 2009−2010 
to SY 2010−2011: grade 8 or grade 9? By how many students? 
 52.  What is the change in the total number of enrollees from SY 2009−2010 to  
SY 2010−2011?
 53.  What is the average change in the number of enrollees from SY 2009−2010 to 
SY 2010−2011?
Answer each item completely.
 54.  What is the distance between the lowest point and the highest point in the


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80
Write the letter of the statement that best describes each pair of quantities in 
the two columns.
  A.  The quantity in column 1 is greater than the quantity in column 2.
  B.  The quantity in column 2 is greater than the quantity in column 1.
  C.  The two quantities are equal. 
  D.  The two quantities cannot be compared.
Column 1 Column 2
 37. |0| 0 • (–5)
 38. –(–1) |–3| – 2
 39. 5 • (–4) (–80) ÷ (–4)
 40. 15 – (–5) –15 + 5
 41. n |n|
Find the next two terms in each sequence.
 42. –16, –11, –6, –1, 4, , 
 43. 729, –243, 81, –27, 9, , 
 44. 1, –1, 2, –2, 4, –4, 8, –8, , 
 45. –2, 8, 2, 5, 6, 2, 10, –1, , 
 46. 0, 0, 1, –1, 2, –4, 3, –9, , 
Applications
For items 47 to 53, refer to the following table that shows the number of enrollees  
in a certain high school for the school year (SY) 2009−2010 and SY 2010−2011.
Grade Level SY 2009–2010 SY 2010–2011
Grade 7 258 342
Grade 8 310 295
Grade 9 365 370
Grade 10 375 360


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represented by –5. Now determine how 
many times the bus decreases its speed in 
4 hours. Since its speed decreases every 
hour, the bus decreases its speed 4 times. 
To find the total decrease in speed after  
4 hours, get the product of 4 • (–5). You may 
solve this using counters. Let  represent 
–1.


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82
Enrichment Exercises
Answer each item completely.
 58.  If |x| < |y|, is it true that x < y ? Justify your answer.
 59.  Consider the equation | x| + | y| = | x + y|. Could x and y  have different 
signs?
 60.  Suppose that a negative number is subtracted from a positive number. Is the 
difference positive or negative? Explain your answer.


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division is the inverse of multiplication, the same rules of signs apply. You can verify 
these rules using counters.
When dividing integers, get the quotient of their absolute values. Then apply the following 
rules for the sign of the quotient:
 •
 The quotient of two integers with like signs is positive.
 •
 The quotient of two integers with unlike signs is negative.
Key Concept
Therefore, the solution to the previous problem is:
(–6) ÷ (–2) = 3.
This means that it will take 3 hours for the temperature to decrease from 21°C to 
15°C. Hence, the temperature reading will be 15°C at 6 p.m.
examPLe 13
Divide.
a. (–15) ÷ (–3) b. 132 ÷ (–12)
 Solutions:
 a. (–15) ÷ (–3) = 5   The two integers have like signs. Hence, their quotient is positive.
 b. 132 ÷ (–12) = –11   The two integers have unlike signs. Hence, their quotient is 
negative.


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is positive.
 13.  If the product of two integers is positive, then one of the integers could be 
negative.
Perform the indicated operations.
 14. 36 + 9
 15. 36 + (–9)
 16. (–36) + 9
 17. (–36) + (–9)
 18. 36 – 9
 19. 36 – (–9) 
 20. (–36) – 9 
 21. (–36) – (–9)
 22. 36 • 9 
 23. 36 • (–9) 
 24. (–36) • 9 
 25. (–36) • (–9)
 26. 36 ÷ 9 
 27. 36 ÷ (–9) 
 28. (–36) ÷ 9 
 29. (–36) ÷ (–9)
 30. 5 • [10 – (–2)]
 31. (–9) + 10 + (–11) 
 32. 100 – (–25) – 50 
 33. 20 • (–25) ÷ (–5) 
 34. 15 • 30 + (–10) 
 35. 30 ÷ (–2) – (–15)
 36. 3 • [17 – (–13)] ÷ (–2)


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83
Learning 
Objectives
At the end of the 
lesson, you should be 
able to:
•	 define	rational	
numbers
•	 graph	rational	
numbers	on	a	
number	line	
•	 compare	and	order	
rational	numbers
•	 state	the	density	
property	of	 rational 	
numbers
•	 express	fractions	as	
decimals,	and	vice	
versa
•	 perform	operations	
involving	rational	
numbers
•	 simplify	complex	
fractions
•	 solve	word	problems	
involving	rational	
numbers
Rational Numbers
Probe and Learn
Plants grow new cells in spirals so that the new leaves do 
not block the older leaves from getting sunlight. Each new cell 
is formed about 137.5º away from the previous cell. This angle 
is called the golden angle. In geometry, the golden angle  is the 
smaller of the two angles formed by sectioning the circumference 
of a circle based on the golden ratio.
The number 137.5 is the same as the fraction 
275
2 , which is 
a quotient of two integers. Recall that the set of integers is closed


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SY 2010−2011?
Answer each item completely.
 54.  What is the distance between the lowest point and the highest point in the 
Philippines? Refer to the data given at the start of this lesson.
 55.  In a 20-item test, a student receives 5 points for each question answered 
correctly. Two points are subtracted from the total score for each question 
answered incorrectly. One point is subtracted for each unanswered question. 
If a student has 10 correct answers and 6 incorrect answers, what is his total 
score in the test?
 56.  The Philippine national debt rose to about 4 trillion pesos in 2008. Suppose 
that there are approximately 75 million Filipinos. What is the average debt of 
each Filipino?
 57.  Nora records the temperatures in her place at different times. At midnight, 
the temperature is 15°C. By 4 a.m., it drops by 6°C. It then begins to rise at a 
rate of 2°C every hour until noon, when the rate of increase becomes 3°C every


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Set of Rational Numbers
The set of rational numbers, denoted by Q, is composed of numbers that can be 
expressed as a ratio of two integers. In symbols,
Q = 
  
p
q  p and q are integers, and q ≠ 0 .
Key Concept
From the given definition, you can say that fractions are rational numbers. Consider 
the fraction 
1
2, which is the ratio of the integers 1 and 2. Dividing 1 by 2 gives a decimal 
quotient of 0.5. Note that any fraction can be expressed either as a terminating decimal 
or a nonterminating but repeating decimal. 
Fractions and Decimals
A terminating decimal consists of a finite number of decimal digits. Numbers, 
such as 4, 1.2345, and 0.987654321, that have finite nonzero digits on the right of 
the decimal point are terminating decimals. 
A nonterminating but repeating decimal has an infinitely repeating digit or group 
of digits. An ellipsis is used to denote that the decimal has an infinite number of


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85
 Solutions:
a. Divide 1 by 4.
   
  Thus, 1
4 = 0.25, which  
is a terminating decimal.
b. Divide 1 by 3.
   
  Therefore, 
1
3 = 0.3, which is a 
nonterminating but repeating 
decimal.
examPLe 2
Some scientific studies say that smiling requires fewer muscles than frowning 
does—17 muscles to smile and 43 to frown. Write a fraction representing the 
number of needed muscles to smile as a part of the number of needed muscles to 
frown. Then express the fraction as a decimal.
 Solution:
  The number of needed muscles to smile as a part of the number of needed 
muscles to frown can be represented as . To write this fraction as a decimal, 
divide 17 by 43. The quotient is the nonterminating but repeating decimal 
0.3953488…. Observe that the decimal does not seem to show a pattern in 
its first 7 digits. For a decimal like this, you may round it off to the nearest 
hundredths as 0.4.
Conversely, you can write any terminating or nonterminating but repeating


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examPLe 4
Simplify 
30
45 .
 Solution:
  First, find the prime factorization of 30 and 45. Then cancel out all of their 
common factors. 
 
Since 2 and 3 are relatively prime, 
2
3 is the simplest form of 
30
45 .
Now, how do you write a nonterminating but repeating decimal as a fraction? 
Note that in this lesson, you will only consider decimals that are made up 
entirely of repeating digits. You will not deal with decimals that have a combination 
of nonrepeating digits and repeating digits. In the next example, you will learn a 
method for converting nonterminating but repeating decimals into fractions.
examPLe 5
Express each decimal as a fraction in simplest form.
a. 0.333… b. 0.545454…
 Solutions:
 a. 
f 3 is the repeating digit.
f
 W
rite a 9 for every repeating digit.
f
 Simplify 3
9.
0.333...
 b.  0.545454… f 54 is the repeating group of digits.
f
 W
rite a 9 for every repeating digit.
f
 Simplify 54
99.


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examPLe 3
Express each decimal as a fraction. Write your answer in simplest form.
a. –1.2 b. 0.625
 Solutions:
 a. –1.2 is read as “negative 1 and 2-tenths.”
   
–1.2 = –1
2
10
 = –1
1
5
← The single decimal place in –1.2 corresponds to 1 zero.
  Hence, –1.2 = –1
1
5.
 b. 0.625 is read as “625-thousandths.”
   
← The three decimal places in 0.625 correspond to 3 zeros.0.625 = 
625
1,000
 
= 
5
8
  Therefore, 0.625 = 
5
8.
You can express a rational number as a fraction in infinitely many ways. How do 
you know if a fraction is already in its simplest form?
A fraction is in simplest form if its numerator and denominator are relatively 
prime; that is, their greatest common factor (GCF) is 1. For example, the fraction 
5
7 
is in simplest form since 5 and 7 are relatively prime. 
To simplify fractions, you can apply the Fundamental Law of Fractions.
The Fundamental Law of Fractions
If a, b, and c are real numbers, b ≠ 0 and c ≠ 0, then ac
bc = a
b.
Key Concept


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examPLe 6
Plot each rational number on a number line.
a. b. –2.3
 Solutions:
 a. You know that 
   To plot  on a number line, divide the units of the number line into 
halves.
  
–1 0 1 2 3 4 5
7
2
 b. –2.3
   To plot –2.3 on a number line, divide the units of the number line into 
tenths.
  
–4 –3
–2.3
–2 –1 0
Comparing Rational Numbers
Can you tell how many rational numbers there are between 
1
4  and 
1
3? The answer 
can be explained using the next property of rational numbers.
Density Property of Rational Numbers
Between any two rational numbers, there is always another rational number.
Key Concept
This means that between any two rational numbers, there are infinitely many 
rational numbers. 
Knowing the positions of rational numbers or their decimal representations on a 
number line may help you compare fractions. However, you may not need to plot the 
numbers on a number line to be able to compare them. The following are the rules 
when comparing rational numbers:


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the temperature is 15°C. By 4 a.m., it drops by 6°C. It then begins to rise at a 
rate of 2°C every hour until noon, when the rate of increase becomes 3°C every 
hour. At 4 p.m., the temperature begins to decrease by 2°C every hour. What 
is the temperature at 6 p.m.?


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its first 7 digits. For a decimal like this, you may round it off to the nearest 
hundredths as 0.4.
Conversely, you can write any terminating or nonterminating but repeating 
decimal as a fraction. Study the next examples.


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of a circle based on the golden ratio.
The number 137.5 is the same as the fraction 
275
2 , which is 
a quotient of two integers. Recall that the set of integers is closed 
under addition, subtraction, and multiplication, but not under 
division. For example, the quotient of the integers 275 and 2 is 
not an integer. Such quotients are part of another subset of the 
set of real numbers—the set of rational numbers. 
LESSON
2-3


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number line may help you compare fractions. However, you may not need to plot the 
numbers on a number line to be able to compare them. The following are the rules 
when comparing rational numbers:
•  When comparing two positive rational numbers, compare as you would with 
whole numbers.


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f
 W
rite a 9 for every repeating digit.
f
 Simplify 3
9.
0.333...
 b.  0.545454… f 54 is the repeating group of digits.
f
 W
rite a 9 for every repeating digit.
f
 Simplify 54
99.
Knowing how to express rational numbers as fractions or as decimals can help 
you plot them on a number line. Study the next example.


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To simplify fractions, you can apply the Fundamental Law of Fractions.
The Fundamental Law of Fractions
If a, b, and c are real numbers, b ≠ 0 and c ≠ 0, then ac
bc = a
b.
Key Concept
This property means that if the numerator and the denominator of a fraction  
have a common factor (c), you can cancel out the common factor without changing 
the value of the fraction. Also, when you cancel the GCF of the terms of the fraction, 
the result is the simplest form of the fraction.


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that positive integers are counting numbers. This means that the sum of two positive 
integers is also positive.
How about the sum of two negative integers? Or the sum of a positive integer and 
a negative integer? You could better understand how to add integers if you use a 
number line. Study how it could be done. 
First, locate one of the addends on the number line. Suppose that n is the other 
addend.
If n is positive, move |n| units to the right of the first addend.
If n is negative, move |n| units to the left of the first addend.
Study the next example.
examPLe 6
Find each sum.
a. (–4) + 7 b. (−4) + (−7)
A sari-sari store is a small convenience store where 
you can buy items by piece or by small packages.


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A nonterminating but repeating decimal has an infinitely repeating digit or group 
of digits. An ellipsis is used to denote that the decimal has an infinite number of 
digits. For example, 0.33333... has an infinite number of 3s on the right of the 
decimal point, while the decimal 0.456456456456… has an infinitely repeating 
group of digits, 456. The repeating digit or group of digits may also be written once 
with a bar on top, called a vinculum. Thus, you can rewrite 0.3333... as 0. 3 and 
0.456456... as 0.456.
examPLe 1
Write each fraction as a decimal.
a. 1
4
b. 1
3


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•  Any negative rational number is less than any positive rational  number.
•  When comparing two negative rational numbers, the greater the absolute 
value of a number, the lesser the number.
examPLe 7
Compare each pair of numbers using <, >, or =.
a. 5.05 ____ 5.005 b. –2.5 ____ 1.5 c. –6.4 ____ –10.2
 Solutions:
 a. 5.05 > 5.005   The hundredths digit of 5.05 is greater than the hundredths digit of 5.005
 b. –2.5 < 1.5  Any negative rational number is less than any positive rational number.
 c. –6.4 > –10.2  Since |–6.4| < |–10.2|, then –6.4 > –10.2.
You have learned that to compare fractions, you may need to rename the 
fractions as similar fractions using the Least Common Denominator (LCD) as their 
denominator. Recall that the LCD of the fractions is the least common multiple of the 
denominators. You may then compare the numerators of the similar fractions. The 
fraction with a greater numerator is the greater fraction.
examPLe 8
Compare 
4
9  and 
5
12 .
 Solution:


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Ordering Rational Numbers
You already know how to compare rational numbers. Now you are ready to order 
rational numbers from least to greatest, or vice versa. Study the next example.
examPLe 10
Order the following rational numbers from least to greatest:
1.5, 
10
7 , 1
7
8 , 
4
9, 0.3, –0.3, –
1
3
 Solution:
  Observe that two of the given numbers are negative. Recall that negative 
numbers are less than positive numbers. So, which is less, –0.3 or –
1
3? To 
answer it, write the numbers in the same form, say decimal form. You may 
express the fraction –
1
3  as –0.3. Thus, –
1
3 < –0.3.
  Now consider the five remaining numbers: 1.5, 
10
7 , 1
7
8, 
4
9, and 0.3. You can 
see that 0.3 < 1.5, since both are in decimal form.
  Of the three remaining fractions, 
4
9 is the least since it is a proper fraction, 
which means it is less than 1.
  On the other hand, 
10
7  = 1
3
7, which is less than 1
7
8. Thus, the three fractions 
may be written from least to greatest as 
4
9, 1
3
7, 1


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90
Another way to compare fractions is by performing cross-multiplication. The 
following diagrams illustrate how cross-products are used to compare the fractions 
a
b and 
c
d:
a • d  >  c • b
a
b   
> 
 
c
d
a • d  <  c • b
a
b  
< 
 
c
d
a • d  =  c • b
a
b  
= 
 
c
d
Why can you compare fractions by comparing the resulting cross-products?
examPLe 9
Compare each pair of fractions by getting their cross-products.
a. 5
8  ––––– 6
7 b. –  5
11  ––––– – 4
9 
 Solutions:
 a. Compare the cross-products.
  
5 • 7  =  35  <  6 • 8  =  48
5
8   
< 
 
6
7
  Since 35 < 48, then .
 b.  Notice that both fractions are negative. You can write a negative rational 
number in any of the following forms:
  
–a
b , a
–b , or – a
b .
  Hence, you may rewrite – 5
11 as –5
11 and – 4
9  as –4
9 . 
  (–5) • 9  =  –45  <  (–4) • 11  =  –44
  Since –45 < –44, then .


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Addition and Subtraction of Rational Numbers
Recall that when adding or subtracting decimals, you have to align the decimal 
points and the digits having the same place value.
examPLe 11
Add or subtract.
a. 10.23 + 5.3 + 27.456 b. 11.346 + (–3.09) c. –20.5 – 7.12
 Solutions:
 a.  10.23
 
5.3
+
 27.456
 42.986
b.  11.346
+
 (–3.09) 
 
 8.256
c.  – 20.5
–
 7.12
   –20.50
+
 
(–7.12)
 –27.62
Recall that when adding or subtracting similar fractions, add or subtract the 
numerators. Then copy the denominator. Express the sum or difference in simplest 
form, if necessary.
examPLe 12
Add or subtract.
a. b. 
 Solutions:
 a.  = 
   =  Rename 9
7  as 1 2
7.
 b.     Rename 4 4
15  as 319
15. Why?
   
  Thus, , which may be written in simplest form as .


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To find the sum or the difference of dissimilar fractions, rename the fractions using 
the LCD. Then follow the rules in adding and subtracting similar fractions. 
examPLe 13
Add or subtract.
a. b. c. 
 Solutions:
 a.  =   The LCD of 10 and 15 is 30.
   =  
 b.  =   Rename 10 1
2  as 9 3
2 
   =  
   =  
 c.  =   Rewrite it into an addition problem.
   =  
   =  
Now study the next example that shows how to solve a word problem involving 
addition and subtraction of fractions.
examPLe 14
Among the Philippines’s major agricultural exports are canned pineapples. Jose 
owns a pineapple farm. He wants to fence his rectangular farm, which has a 
length of  ft. and a width of 21 1
3  ft. Suppose that fencing wires are sold in 
rolls of 20 yards. How many rolls of wire should he buy? How many feet of wire 
will be left over?


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examPLe 15
Multiply.
a. 24.68 × 0.5 b. (–213.6) × (–2.5)
 Solutions:
 a.  24.68
×
 
0.5
 
12.340
←  2 decimal places
← 1 decimal place
← 3 decimal places
  You may write the product as 12.34. Why?
 b.   –213.6
×
 
  (–2.5)
 
534.00
←  1 decimal place
← 1 decimal place
← 2 decimal places
  The product is positive. Why?
Recall that when multiplying fractions, multiply the numerators to get the 
numerator of the product. Then multiply the denominators to get the denominator 
of the product.
If the numerator and the denominator of the fractions that you need to multiply 
have common factors, you may cancel out the common factors first before multiplying 
the fractions.
When multiplying mixed numbers, write each mixed number as an improper 
fraction before multiplying.
examPLe 16
Multiply. Write your answer in simplest form.
a. b. 
 Solutions:
 a.  Write –6 6
7 as – 48
7 .
    Since the factors have unlike signs, attach a negative sign 
to the product.


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94
 Solution:
  To find the length of the required wire to fence his rectangular farm, find 
the perimeter of the farm. Recall that you can solve for the perimeter P of a 
rectangle by adding the measures of all its sides.
 P  =  l + l + w + w, where l  =  length and w  =  width
 It is given that l = 32
1
4  ft. and w = 21
1
3 ft. So you will have the following:
 
  Thus, Jose needs 107
1
6 ft. of fencing wire. Recall that a roll of fencing wire is 
20 yards long. Since 1 yd. = 3 ft., then 20 yd. = 60 ft. Thus, he needs 2 rolls of 
fencing wire, with a total length of 120 ft.
  To find how much of the wire will be left over after fencing  his farm, subtract 
107
1
6  ft. from 120 ft.
 
 Therefore, 12
5
6 ft. of wire will be left over.
Multiplication of Rational Numbers
Recall that when multiplying decimals, multiply the numbers as you would 
with whole numbers. To determine the position of the decimal point in the product,


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denominators. You may then compare the numerators of the similar fractions. The 
fraction with a greater numerator is the greater fraction.
examPLe 8
Compare 
4
9  and 
5
12 .
 Solution:
  The LCD of 
4
9 and 
5
12  is 36. Find fractions equivalent to the given fractions 
using the LCD as denominator.
 
× 4
× 4   
and
  
× 3
× 3  
 Compare the numerators of the similar fractions. Since 16 > 15, then .


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which means it is less than 1.
  On the other hand, 
10
7  = 1
3
7, which is less than 1
7
8. Thus, the three fractions 
may be written from least to greatest as 
4
9, 1
3
7, 1
7
8. 
  Now consider 0.3 and 1.5. Is 0.3 less than 
4
9? You can express 0.3 as 
3
10 , 
which is less than 
4
9. Since 1.5 = 1
1
2, then 1
3
7 < 1
1
2 < 1
7
8. Thus, the  
following is the correct order of the numbers from least to greatest:
 –
1
3, –0.3, 0.3, 
4
9, 
10
7 , 1.5, 1
7
8


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97
Study the next example that shows how to solve a word problem involving division 
of decimals.
examPLe 18
Lightning is a massive flow of electric current in the atmosphere. A bolt of lightning 
can reach temperatures approaching 30,000°C. It can contain one billion volts of 
electricity. 
In electric devices, the amount of electric current can be computed using the 
formula I = 
V
R , where I is the electric current (in amperes), V is the voltage  
(in volts), and R is the resistance (in ohms). Compute the electric current flowing 
through a light bulb with a resistance of 10.5 ohms and that is plugged on a  
110-volt outlet.
 Solution:
  To solve for the amount of electricity flowing through the light bulb, substitute 
the given values of V and R in the formula for I, then solve.
 It is given that R = 10.5 ohms and V = 110 volts. So you will have: 
 I = 
110
10.5
 To find I, divide 110 by 10.5.
 
  Hence, the amount of electricity flowing through the light bulb is about


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99
examPLe 20
In 2003, Dagupan City in Pangasinan is the record holder of the world’s longest 
barbecue. The barbecue measures about 3,300 ft. It was composed of milkfish 
placed together on grills arranged side by side. Suppose that a milkfish has an 
average width of 3
3
4  in. About how many pieces of milkfish were used for the 
barbecue?
 Solution:
  To find the number of milkfish, divide the total length of the barbecue by the 
width of each milkfish. Since the units are not the same, you may convert 
3,300 ft. to inches. Since 1 ft. = 12 in., then 3,300 ft. = 39,600 in.
 Now divide 39,600 by 3 3
4.
 
 Therefore, there were about 10,560 pieces of milkfish in the longest b arbecue.
examPLe 21
Simplify the following complex fraction: .
 Solution:
  One way to simplify the given expression is to express the numerator as a 
single fraction, and then to treat the fraction bar as a division symbol.
  
Add the fractions in the numerator.
  
The fraction bar means ÷.


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96
 b.  Write –2 5
8   as  – 21
8   and  –3 3
7  as  – 24
7 .
    
     Cancel out the common factors.
examPLe 17
Jam is a call center agent. She earns P85 per hour for the first 8 hours of 
work. She also earns an overtime pay of 1.5 times the hourly rate for each hour in 
excess of 8 hours. Suppose that Jam works for 10 hours each day from Monday 
to Wednesday. She also works for 11 hours each day on Thursday and Friday. 
How much is her salary for the given 5-day period?
 Solution:
  To find Jam’s salary for the given period, compute for her salary for each 
specified day. It is given that for the first 8 hours, she is paid P85 per hour. 
Her overtime pay per hour is P85 • 1.5 = P127.50.
Day Hours  
of Work 
Pay for the First 8 Hours 
(in pesos)
Overtime Pay 
(in pesos)
Monday 10 85 • 8 = 680 127.50 • 2 = 255
Tuesday 10 85 • 8 = 680 127.50 • 2 = 255
Wednesday 10 85 • 8 = 680 127.50 • 2 = 255
Thursday 11 85 • 8 = 680 127.50 • 3 = 382.50
Friday 11 85 • 8 = 680 127.50 • 3 = 382.50


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Multiplication of Rational Numbers
Recall that when multiplying decimals, multiply the numbers as you would 
with whole numbers. To determine the position of the decimal point in the product, 
count the decimal places in the factors. The total number of decimal places in the 
factors is equal to the number of decimal places in the product. Study the given 
example.


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100
  
Multiply the numerator by the reciprocal 
of the divisor, and then simplify.
Practice 2-3
Concepts and Skills
Give one rational number between each pair of rational numbers.
 1.  and
  
2.  and –0.428
Plot each set of rational numbers on a number line. Then order the numbers from 
least to greatest.
 
3.
 
–1.9, 2.5,
 
4
3,
 
17
9  
 
4.  
1
4, 
3
7, 
4
19,
 
7
20 
 5.  0.16, 0.1625, 0.163, 0.16
Write each decimal as a fraction. Express your answer in simplest form.
 6.  0.8
 7. –6.75
 8.  10.35
 9. 5.66…
 10. –0.3636…
Write each fraction as a decimal.
 
11. 
5
12
 12. –
4
9
 13. 
18
7
 14. 3
1
2
 15. –10
3
4


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101
Write each fraction in simplest form.
 
16. 
24
54
 17. 
3
51
 18. 
–72
98
 19. –
56
108
 20. 
26
143
Perform the indicated operations. Express your answer in simplest form. 
 21.  30. 
 22.  31. 
 23.  32. 
 24.  33. 
 25.  34. 
 26.  35. (–5.5)(1.09 + 9.91)
 27.  36. 
 28.  37. 
 29.


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98
Recall that when dividing fractions, multiply the dividend by the reciprocal or 
multiplicative inverse of the divisor.
.
Study the next example about solving a word problem involving division of 
fractions.
examPLe 19
LJ, an interior designer, uses patterns to create quilts. 
One of her designs is shown on the right. What part of 
the square quilt is the white triangle?
 Solution:
  Note that the largest part in her quilt is 
1
2 . The next 
largest part is obtained by dividing the remaining 
part of the triangle into 2; that is:
  
1
2 ÷ 2 =  
1
2 • 
1
2
   =  
1
4
 The next fraction is also obtained in the same way; that is:
 
1
4 ÷ 2 = 
1
4 • 
1
2 = 
1
8
 So, to solve for the required fraction, divide 
1
64 by 2.
 
1
64 ÷ 2 = 
1
64 • 
1
2 = 
1
128
 Hence, the white triangle is 
1
128  of the square quilt.
Simplifying Complex Fractions 
In some cases, a fraction may contain one or more fractions in its numerator 
and/or denominator. Such a fraction is called a complex fraction.


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102
Applications
Answer each item completely.
 38. Mitch scores 45 out of 62 in a Math exam. She scores 36 out of 50 items in an 
English exam. In which subject is her score better?
 39. Engineers set standard measurements for hand tools such as wrenches of 
different sizes. Engineer Joy has four wrenches with sizes 
9
16, 
11
16, 
1
2, and 
5
8 
of an inch. Suppose that she needs the wrench with the second largest size. 
Which of the wrenches should she use?
 40. A school year usually has 200 days. There are 365 days in 1 ordinary year. What 
fraction could represent a school year as part of an ordinary year? Express the 
fraction in simplest form. 
 41. The brain of a newborn baby weighs about 400 g. Suppose that a newborn 
baby weighs 3.5 kg. What fraction of the baby’s weight is his or her brain? 
 42. A hair strand is approximately 
1
450 in. thick. What is the equivalent value of 
this fraction in decimal form?


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103
 47.  Mr. Angeles shared 
1
2  of his estate with his wife. Also, he shared 
1
8 with each 
of his two children, and 
1
16 with each of his three grandchildren. He donated 
the remaining P30,000 worth of property to a charity. What is the value of his 
entire estate?
 48.  A carpenter will cut a 42
3
4-in. long board into 5 pieces of equal lengths.  
If 
1
16 in. of board wears down in every cut, how long will each piece be?
 49. Suppose that 0.408 kg of grapes costs P65.28. How much is 1 kg of grapes?
 50.  The following table shows the monthly average exchange rate of US dollar to 
peso in the first six months of the year 2016:
Month Peso Equivalent 
of 1 US dollar
January 47.73
February 47.45
March 46.64
April 46.20
May 46.67
June 43.32
  What is the average exchange rate for the first six months of that year?
Enrichment Exercises
Answer each item completely.
 51. Can you find unique integers a, b, and c such that ? If yes, give 
a possible set of integers. If no, explain why.


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105
Learning 
Objectives
At the end of the 
lesson, you should be 
able to:
•	 define	irrational	
numbers
•	 locate	an	irrational	
number	on	a	
number	line
•	 find	the	square	
roots	of	 positive 	
rational	numbers
Probe and Learn
Rainbows are among the most beautiful natural phenomena. 
A rainbow is formed when light shines through water droplets, 
usually in the form of rain, and diffracts into different colors. A 
rainbow is not a 360º full circle; it is just a 42º arc around a spot 
opposite the Sun. 
To explain natural phenomena that exhibit circles or spheres, 
the number p (pi) is usually used. This number is an example of 
an irrational number.
Using a scientific calculator, you can 
get the value of p  by pressing the p  button. 
A 12-digit scientific calculator will display 
3.14159265359, while a 15-digit calculator will 
show 3.14159265358979. Notice that when you 
use a calculator that can display more digits, the 
number of displayed digits of p also increases.


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It is given that R = 10.5 ohms and V = 110 volts. So you will have: 
 I = 
110
10.5
 To find I, divide 110 by 10.5.
 
  Hence, the amount of electricity flowing through the light bulb is about  
10.48 amperes.


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Tuesday 10 85 • 8 = 680 127.50 • 2 = 255
Wednesday 10 85 • 8 = 680 127.50 • 2 = 255
Thursday 11 85 • 8 = 680 127.50 • 3 = 382.50
Friday 11 85 • 8 = 680 127.50 • 3 = 382.50
Total 3,400 1,530
  Therefore, Jam’s total salary for the given period is P3,400 + P1,530 = P4,930. 
Could you think of another way of solving the previous problem?
Division of Rational Numbers 
Recall the following steps when dividing a number by a decimal: 
Step 1. Make the divisor a whole number. You may do this by moving the decimal 
point of the divisor to the right until you have a whole number divisor. 
Step 2. Move the decimal point of the dividend the same number of places to the 
right. 
Step 3. Divide as you would with whole numbers. Note that the decimal point of 
the quotient is placed directly above the decimal point of the dividend.


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106
Set of Irrational Numbers
The set of irrational numbers is the set of numbers that cannot be expressed as p
q, 
where p and q are integers, and q ≠ 0.
Key Concept
To better understand the set of irrational numbers, you need to learn how to find 
the square roots of positive rational numbers.
Square Roots of Positive Rational Numbers
Interesting patterns may be discovered by representing numbers with figures. 
Look at the following figure: 
Observe how dots were drawn inside each square. Do you notice a pattern? Draw 
dots in the next two squares based on the pattern. Then complete the following table:
Number of Dots along 
Each Side of the Square
Total Number  
of Dots 
1 1
2 4
3 9
4 16
5
6
7
8
9
10
 
The total number of dots in each square represents a perfect square. The 
number of dots in each row or column in the square is called a square root of the 
perfect square. For example, 16 is a perfect square and 4 is its square root.
Do you know?


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104
 53. Find the sum of the following expression:
  .
 54. Consider the following equation:
  
  Is the equation correct? Explain.
 55. The following figure is called a magic square:
6
1
4 11
1
4
7
1
2
8
3
4
  Complete the magic square. Each row, column, and diagonal must add up to 
the same number.


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baby weighs 3.5 kg. What fraction of the baby’s weight is his or her brain? 
 42. A hair strand is approximately 
1
450 in. thick. What is the equivalent value of 
this fraction in decimal form?
 43. Jose has 4 pieces of wire. Two of the pieces of wire measure 9 in. long each. The 
other 2 pieces measure 1
1
4  ft. each. If he puts the pieces of wire together, how 
long is the wire in inches?
 44.  Dennis has finished painting 
1
3 of his room using 2
1
2 cans of paint. How many 
more cans of paint will he need to finish painting his room?
 45.  The fuel tank of Trina’s car is 
2
3 full. To travel to her destination, she needs 
a 
4
5-full tank. What part of the fuel tank does she need to fill to reach her 
destination?
 46. There are 18 baskets to be filled with different kinds of fruits. Suppose that 
there are 1
1
2 dozen pineapples, 6 dozen apples, 3 dozen oranges, 7
1
2 dozen 
bananas, and 9 dozen mangoes. If each basket should have the same contents,


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Simplifying Complex Fractions 
In some cases, a fraction may contain one or more fractions in its numerator 
and/or denominator. Such a fraction is called a complex fraction.
Complex fractions may involve a series or combination of operations on rational 
numbers. In such cases, the GEMDAS rule may be applied. Recall that GEMDAS  
stands for Grouping Symbols, Exponents, Multiplication, Division, Addition, and 
Subtraction. Study the next examples.
1
2
1
4
1
81
16
1
32
?1
64


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Enrichment Exercises
Answer each item completely.
 51. Can you find unique integers a, b, and c such that ? If yes, give 
a possible set of integers. If no, explain why.
 52.  Look at the following method of renaming fractions:
  Can you find other fractions for which this method works? Also, give an example 
for which this method does not work and explain why.


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3.14159265359, while a 15-digit calculator will 
show 3.14159265358979. Notice that when you 
use a calculator that can display more digits, the 
number of displayed digits of p also increases. 
This indicates that p is nonterminating. Recall 
that nonterminating but repeating decimals 
are classified as rational numbers. Take note, 
however, that nonterminating decimals, such 
as p, that do not show a repeating pattern of 
digits are irrational numbers.
Irrational NumbersLESSON
2-4


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107
If a and b are nonnegative numbers and b2 = a, then b is a square root of a. In symbols, 
it can be written as:
 a = b.
Key Concept
Consider  = 4, which is true since 4 2 = 4 • 4 = 16. The number 16 is called 
the radicand. The number 4 is the square root. However, note that (–4)(–4) is also 
equal to 16. Does this mean that –4 is also a square root of 16? 
The expression  is used to refer specifically to the principal square root of x, 
or the nonnegative square root of x. While it is true that (–4)(–4) = 16, –4 is not the 
principal square root of 16, but the negative square root of 16.
  = 4 →  4 is the principal square root of 16.
 –  = −4 →   –4 is the negative square root of 16.
Perfect squares also include decimals and fractions. An example is 0.04 since 
0.04 = (0.2)2. Another example is 
9
16 since .
examPLe 1
Find each indicated square root.
a. b. 
 Solutions:
 a.  152 = 225
 b.  –  indicates the negative square root. 
examPLe 2


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108
 Replacing A with 400 in the given formula gives . 
  Since 400 = 202 and 400 = (–20)2, then s has two possible values: 20 or –20. 
However, since s represents the measure of each side of the room, it cannot be 
negative. Therefore, each side of the room measures 20 ft.
You have seen that extracting the square root of a perfect square yields a rational 
number. Consider another situation.
A dressmaker designs a cover for square tables that 
measure 2 ft. on each side. She drafts the design shown 
on the right.
Since the square table measures 2 ft. on each side, 
the amount of cloth that she needs for each cover is  
4 sq. ft. Connecting the midpoints of the sides of the 
square forms a small square, which is the shaded part. 
Such small square has an area of 2 sq. ft. What could be 
the measure of each side of the smaller square?
Since the area of the small square is 2 sq. ft., the measure of its side must be a


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110
 Step 2. Divide the radicand by any of the square roots you obtained in step 1. 
Suppose that you divide 8 by 3.
   8 ÷ 3 ≈ 2.67
   (The quotient may be rounded to the nearest hundredths.)
 Step 3.  Find the average of the quotient in step 2 and the square root you 
used as a divisor. 
   So get the average of 2.67 (which is the quotient) and 3 (which is the 
divisor).
   
 Step 4.  Check if the value you obtained in step 3 is a good estimation of the 
required square root. To do this, get the square of the obtained value. 
Compare it with the given radicand by getting the absolute value of 
the difference of the result and the radicand. 
   (2.835)2 ≈ 8.0372
   The difference between this result and the radicand is: 
   .
   You can check if you can still find a closer estimation. Do the next 
step.
 Step 5.  Repeat steps 2 and 3, but this time, use the value you obtained in 
step 3 as the divisor.
   So divide 8 by 2.835.
   8 ÷ 2.835 ≈ 2.822


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number of dots in each row or column in the square is called a square root of the 
perfect square. For example, 16 is a perfect square and 4 is its square root.
Do you know?
Some	number s 	are 	reg arded 	as 	happy 
numbers.	 Consider 	 the 	 number 	 13. 	
Adding	the 	squares 	of	the 	digits 	of	13 	
gives	12	+	 32		= 		10. 	R epeating 	the 	process 	
with	t he 	digits 	of	10 	yields 	12	+	 02		= 		1. 	Since	
the	resulting 	number 	is 	1, 	then 	13 	is 	a 	
happy	number . 	F or 	an y 	ha ppy 	number , 	the 	
resulting	number 	should 	be 	1 	after 	a 	finite 	
number	of	 steps.
How	man y 	ha ppy 	number s 	less 	than 	50 	
can	you	find?


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there are 1
1
2 dozen pineapples, 6 dozen apples, 3 dozen oranges, 7
1
2 dozen 
bananas, and 9 dozen mangoes. If each basket should have the same contents, 
how many pieces of each fruit must be placed in each basket?


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111
Knowing how to approximate square roots of non-perfect squares can help you 
plot irrational numbers on a number line. From the previous example, since  is 
approximately equal to 2.829, you can plot it on the number line as follows:
0 1 2 3
8
examPLe 5
One of Manila’s major attractions is 
the breathtaking sunset view along the 
Baywalk at Manila Bay. The line that 
seems to separate the bay and the sky is 
called the horizon. How far you can see 
to the horizon can be determined using 
the formula , where d is the 
distance you can see (in miles), and h is 
the height (in feet) your eyes are from the 
ground. Suppose that your eyes are about 
5 ft. from the ground. Estimate how far 
you can see to the horizon during a sunset 
if you are standing along the Baywalk. 
 Solution:
 Use the formula , where h = 5.
 
 Thus, you can see to the horizon up to a distance of about 2.7 miles.


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109
examPLe 3
Find a whole number approximation for 70.
 Solution:
  Since 82 = 64 and 9 2 = 81, you can say that 70 is between the two perfect 
squares, 64 and 81. Hence, 70 is between 8 and 9.
 
 64 <  70  <  81
 82 < 70  <  92
 8 <   <  9
  Between 8 and 9, which is a closer whole number approximation of 70? 
In the following 9 × 9 grid, you can see that the 70 shaded squares represent 
a number that is closer to 64 squares than to 81 squares.
Thus, the whole number approximation for 70 is 8. 
For a close approximation of a square root up to a specific number of decimal 
places, you may follow the steps shown in the next example. 
examPLe 4
Approximate 8  up to two decimal places.
 Solution: 
 Step 1.  Determine between which two perfect squares the radicand lies. 
Then get the square roots of such perfect squares.
    The radicand is 8, and 8 is between the perfect squares 4 and 9.
    4 < 8 < 9
    22 < 8 < 32
    2 < 8  < 3
   Therefore, 8  is between 2 and 3.


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112
Practice 2-4
Concepts and Skills
Without using a calculator, determine whether each rational number is a perfect 
square or not.
 1. 16
 2. 20
 3.  121
 4. 225
 5.  1,000
 6. 
 7. 
 8. 0.0004
 9.  0.0404
 10. 4.0004
Find each square root without using a calculator.
 11. 
 12. 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 20. 
Without using a calculator, find the nearest whole number approximation for each 
irrational number. 
 21. 
 22. 
 23. 
 24. 
 25. 
Without using a calculator, find an approximation for each square root up to the  
indicated place value. 
 26.  to the nearest tenths
 27.  to the nearest hundredths
 28.  to the nearest tenths


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0.04 = (0.2)2. Another example is 
9
16 since .
examPLe 1
Find each indicated square root.
a. b. 
 Solutions:
 a.  152 = 225
 b.  –  indicates the negative square root. 
examPLe 2
Four hundred pieces of 1-square-foot tiles are used for the floor tiling of a square 
room. What is the measure of each side of the room?
 Solution:
  It is given that 400 pieces of 1-square-foot tiles are 
used for floor tiling. Thus, the area of the room is  
400 square feet (sq. ft.). To find the measure of each 
side of the square room, use the formula A = s2, where 
A is the area and s is the measure of each side of the 
square. But since you need to find s, you must get the 
square root of A; that is, s = .
400 sq. ft.
s ft.
s ft.


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Such small square has an area of 2 sq. ft. What could be 
the measure of each side of the smaller square?
Since the area of the small square is 2 sq. ft., the measure of its side must be a 
number s which, when squared, gives 2; that is, s2 = 2. What is the value of s? 
You may try to guess and check possible values of s. For example, you know that 
12 = 1 and 22 = 4. Since s2 = 2, and 2 is between 1 (or 1 2) and 4 (or 22), you can say 
that s is between 1 and 2. Note, however, that no integer lies between 1 and 2. This 
implies that s is not an integer.
Recall that a rational number can be written as a ratio of two integers, where 
the denominator is not 0. The equation s2 = 2 is equivalent to s = 2
s . If 2
s  is a rational 
number, then the denominator s has to be an integer. However, you already know that 
no such integer exists. This means that s is neither an integer nor a rational number.
Problems like this involve irrational numbers. The solution to this problem is


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step.
 Step 5.  Repeat steps 2 and 3, but this time, use the value you obtained in 
step 3 as the divisor.
   So divide 8 by 2.835.
   8 ÷ 2.835 ≈ 2.822
   Get the average of 2.835 and 2.822.
   
 Step 6.  Repeat step 4. Then compare the two estimates you obtained. Choose 
the estimate whose square is closer to the radicand. To do this, get 
the absolute value of the difference of each square and the radicand.
   (2.829)2 = 8.0032
   Between 8.0372 and 8.0032, the number closer to 8 is 8.0032 since:
   |8.0032 – 8| = 0.0032   and   |8.0372 – 8| = 0.0372.
   Thus, you can say that 8 is approximately equal to 2.829. When you 
compute  using a calculator, you will get 2.828427125. As you can 
see, you obtained a close approximation for .


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114
Find the next two numbers in each sequence.
 41.  , 
 42.  , 
 43. –1, 1, –9, 9, –25, 25, –49, 49, , 
 44. , , 
 45. –1, 0, 0, 1, 1, 2, 4, 5, 25, 26, , 
Applications
Answer each item completely.
 46.  The Great Pyramid of Khufu is known as 
the largest pyramid ever built in Egypt. 
Its base forms a nearly perfect square 
with an area of 52,900 m2. What is the 
measure of each side of its base?
 47.  Jon is reading a certain page of a book. The square of the page is 961. On 
which pages is Jon’s book open? 
 48.  A farmer grows cabbages in his square-shaped farm. Each cabbage takes  
1 sq. ft. of land area in his farm. He planted 256 cabbages. How many cabbages 
are planted along each side of his farm?
 49.  A square pigpen has an area of 576 sq. ft. How many feet of fencing material is 
required to enclose it?
 50.  A man is at the top of a mountain and his eyes are approximately 500 ft. above 
the ground. How far can he see to the horizon? Use the formula in example 5


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 29.  to the nearest tenths
 30.  to the nearest hundredths
Answer each question. Choose the letter of your answer. 
 31. Which statement is false?
  A. 
  B. 
  C. 
  D. 
 32. What is ?
  A. –14   B.  –13   C.  13   D.  14
 33. Between which two integers does  lie?
  A.  13 and 14
  B.  14 and 15
  C.  16 and 17
  D.  17 and 18
 34. Which number must be a perfect square?
  A. sum of two perfect squares
  B.  product of two odd numbers
  C.  product of two perfect squares
  D.  product of an even number and a perfect square
 35. Which number has a square root that is not a real number?
  A. a counting number
  B.  a negative number
  C.  a positive irrational number
  D.  zero
For items 36 to 40, find the value of each expression.
36. 
  A.  24   B.  34   C.  36   D.  44
 37. 
  A. 2.02   B.  2.2   C.  20.02   D.  20.2
 38. 
  A.  0.08   B.  0.088   C.  0.8   D. 0.88
 39. 
  A.  3   B.  4   C.  5   D.  7
 40. 
  A.  5   B.  15   C.  25   D.  125


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 9. What is the value of ?
  A. 22   B.  28   C.  32   D.  38
 10. Which decimal is greater than 0.25 but less than ?
  A.  0.249   B.   C.   D.  0.29
 11.  A fruit dealer delivers 4 crates of mangoes. He unloads the heaviest crate from 
his van last. Which crate does he unload last?
  A. crate A weighing 27.875 kg
  B.  crate B weighing 28.105 kg 
  C.  crate C weighing 27.984 kg
  D.  crate D weighing 28.052 kg
 12.  Larry withdraws P1,000 from his bank account. Which expression represents 
the amount in his transaction?
  A.  –1,000   B.  1,000   C.  |–1,000|   D.  |1,000|
 13.  A recipe that makes 12 servings requires  cup of melted butter. Suppose that 
you will prepare a recipe that makes only 9 servings. How many cups of melted 
butter do you need?
  A.  cup   B.  cup   C.  cup   D.  cup
 14.  On a particular day, the exchange rate of euro (€) to Philippine peso ( P) is  
€1 = P65.825. The next day, the peso equivalent of €1 is 15 centavos less. How


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Answer each question without using a calculator. Choose your answer.
 1.  What is the simplest form of ?
  A.   B.   C.   D. 
 2.  Which is equivalent to ?
  A.   B.   C.   D. 
 3.  What is the LCD of 
9
28 and 
13
91?
  A.  4   B.  7   C.  28   D.  364
 4.  Which is a rational number?
  A. 
  B.  4.56565653...
  C.  0.123456...
  D.  5.3613613613...
 5.  Which is an irrational number?
  A.  3.123123…
  B.  3.14
  C. 
  D. 
 6.  Which set of fractions is arranged from greatest to least?
  A. 
  B. 
  C. 
  D. 
 7.  Which set is not a subset of the set of rational numbers?
  A. set of terminating decimals
  B.  set of integers
  C.  set of non-terminating decimals
  D.  set of whole numbers
 8.  Which set is closed under division?
  A.  set of counting numbers
  B.  set of integers
  C.  set of non-zero rational numbers
  D.  set of whole numbers
Chapter Assessment


