This document outlines a mathematics curriculum for Quarter 1. It covers the topics of the real number system, measurement, and scientific notation over multiple stages of learning. For the real number system topic, students will learn about rational and irrational numbers through hands-on activities. They will formulate and solve real-life problems involving real numbers. For the measurement topic, students will learn to use different measuring devices and units of measure through experiments and activities. They will pose and solve daily life problems involving measurement. The curriculum aims to help students demonstrate understanding and performance of key concepts through varied activities and assessments.

1. MATHEMATICS I
General Standard: The learner demonstrates understanding of key concepts and
principles of number and number sense as applied to measurement, estimation,
graphing, solving equations and inequalities, communicating mathematically and
solving problems in real life.

2. 2

3. Quarter I: Real Number System,
Measurement and Scientific Notation
Topic: Real Number System Time Frame: 20 days
Stage 1
Content Standard:
The learner demonstrates understanding of key concepts of
real number system.
Performance Standard:
The learner formulates real life problems involving real numbers
and solves these using a variety of strategies.
Essential Understanding(s):
Daily tasks involving measurement, conversion, estimation and
scientific notation making use of real numbers.
Essential Question(s):
How useful are real numbers?
The learner will know:
• the real number system
• rational and irrational numbers
• the importance of order axioms
• fundamental operations with real numbers
• the application of real numbers to daily life.
The learner will be able to:
• apply real numbers in a variety of ways to other disciplines.
• identify/give examples of rational and irrational numbers
• illustrate rational and irrational numbers in practical
situations
• use the appropriate symbolic notation to illustrate the
order axioms.
• cite examples/situations where order axiom is applied.
• perform the sequence of operations with real numbers.
• solve problems in other disciplines such as science, art,
agriculture, etc.
Stage 2
Product or Performance Task:
Problems formulated
1.are real –life related
2.involve real numbers, and
3.are solved using a variety of strategies.
Evidence at the level of understanding
Learner should be able to demonstrate
understanding of the real number system
using the six (6) facets of understanding:
Explaining how numbers are expressed
in different ways.
Criteria:
Thorough
Coherent
Evidence at the level of performance
Assessment of problems formulated
based on the following suggested criteria:
· real-life related problems
· problems involve real numbers.
· problems are solved using a variety of
strategies
Tools: Rubrics for assessment of
problems formulated and solved
3

4. Clear
Interpreting the differences and
similarities between rational and irrational
numbers.
Criteria:
Thorough
Illustrative
Creative
Applying a variety of techniques in
solving daily life problems.
Criteria:
Appropriate
Practical
Accurate
Relevant
Developing Perspective on the types of
real numbers.
Criteria:
Perceptive
Open-minded
Sensitive
Responsive
Showing Empathy by describing the
difficulties one can experience in daily life
whenever tedious calculations are done.
Criteria:
Open
Sensitive
Responsive
4

5. Manifesting Self-knowledge by
assessing how one can give his/her best
solution to a problem/situation.
Criteria:
Reflective
Responsive
Relevant
Stage 3
Teaching/Learning Sequence
1. Explore
At this stage, the teacher should be able to:
a. give the learner hands-on activities on how to identify /name a real number:
· Locating numbers on the number line
· Giving the coordinate of a point on the number line
· Naming a real number between two given numbers.
b. self-evaluate the learner by giving him activity sheets containing questions ( including HOTS) on real numbers .
c. let the learner share what he has learned about real number system through journal writing.
d. allow the learner to apply the concept to real life by solving worded problems involving real numbers.
Activity 1.
Let some selected students line up to form/picture a number line. Make one student, probably the middle one, represents 0.
Use this to determine the coordinate of a point represented by a student on that number line. Let the students explain
their answer. You may introduce this activity as a game. Help the students by giving activity cards with guided instructions.
Activity 2.
Let a student choose a partner. Then, give each pair an activity sheet containing the number line being drawn either on a
graphing paper or activity card. Guided instructions must be given. Let the students play by taking turns in naming a
number between two given numbers. The teacher may initially give the two numbers and must be ready to check if the
number to be given is between the other two. The game may start with whole numbers, then integers, and later on with
rational or irrational numbers. In the end, they must identify the kind of number being inserted.
Activity 3.
Give each student enough time, like 5-10 minutes, to think of a situation and formulate a real life problem involving the
basic operations on real numbers. Then, if they are ready, they will take turn in presenting the problem. Any student can
5

6. give the answer to the problem. The teacher will ask the one who gave the problem if the presented solution is correct.
Activity 4. Guessing Game:
Directions:
1. Think of a four-digit number.
2. Add the digits and subtract the sum from the original number.
3. Encircle one digit.
4. Tell me the digits that are not circled.
5. Then, I’ll tell you what you encircled.
Note: The answer is taken by subtracting the sum of digits that are not circled from a multiple of nine that is greater than but closer
to the sum of the digits.
Example: Let the four-digit number be: 1 472
The sum of the digits is 14. ( from 1+4+7+2)
Subtracting 14 from 1 472, we get 1 458.
Suppose the encircled digit is 8.
The sum of the remaining digits will be 10. from 1+4+5 )
Note that: The multiple of nine that is greater than but closer to 10 is 18.
Subtracting 10 from 18, we get 8.
Hence, the encircled digit is 8.
Activity 5.
Let the students answer an activity sheet where the questions are simple problems they experienced in daily life. They must
explain the solution to the problem.
2. Firm Up
At this stage, the teacher should be able to:
a. ask the students to conduct an investigation considering the following steps:
· Give a list of different numbers
· Change the form of the given set of numbers by doing the basic operations and simplifying the results.
· Analyze/Observe the results
· Classify the numbers as to their types.
· Classify the numbers into rational or irrational
b. perform fundamental operations on real numbers and classify results
c. cite examples/situations where order axiom is applied.
d. solve daily life problems involving different operations on real numbers.
6

7. Activity 6.
Apply cooperative learning. First, group the students into 4. Let each group be working on an activity sheet where one
Is different from the other. Ask each group to investigate a given set of numbers. Guide questions must be given. Expect
them to have analyzed and classified each of the numbers after changing its form. Let them explain the results.
Activity 7.
Ask the students to answer several activities on the operations applied to the set of real numbers including the order
axiom. Let them write the complete solution to each number.
Activity 8.
Let the students answer activities on solving daily life problems involving different operations on real numbers.
3. Deepen
At this stage, the teacher should be able to:
give activities that will provide the learner the opportunity to reflect on, revisit, or rethink the lesson.
a. Explain thoroughly the difference between rational and irrational numbers by giving several examples.
b. Investigate on the relationship (similarity or difference) between rational and irrational numbers.
c. Investigate patterns on rational or irrational numbers.( both manually and with the use of calculators)
d. Generalize and write a report of what has been discovered about real numbers
e. Formulate/Solve problems they experienced in daily life.
Activity 9.
Instruct the students to be ready for an oral/written test which is in the form of a team competition. The test will include
investigating patterns on rational or irrational numbers( they are allowed to use a calculator), similarity or difference
between rational and irrational numbers, problem solving involving the different operations on real numbers.
Activity 10.
Give each student enough time, like 10-15 minutes, to answer activities that will provide them the opportunity to
reflect on or rethink of the lesson on real numbers. It may be in the form of journal writing, or application of the concept
to problems they experienced in daily life.
4. Transfer
At this stage, the teacher should be able to: demonstrate his/her understanding of the topic by :
give activities that will demonstrate students’ understanding of the topic :
· formulate/create problem situations using real numbers
· construct scale models of houses, toys, bridges, etc. indicating the use of real numbers. These will serve as
students’ project for exhibit during math expo.
7

8. Activity 11.
Group the students into four or five depending on the number of students per class. Each group will then select its
leader. The group will decide on the problem to be presented making use of the set of real numbers. They will
visualize and present the solution to the said problem.
Activity 12.
Give each group enough time to construct scale models of either houses, toys, bridges, etc. depending on its problem
indicating the use of real numbers. This will serve as students’ project for exhibit during math expo.
Resources/Materials:
See Appendix
Quarter I : Real Number System,
Measurement and Scientific Notation
Topic: Measurement Time Frame: 25 days
Stage 1
Content Standard:
The learner demonstrates understanding of the key
concepts of measurements.
Performance Standard:
The learner formulates real-life problems involving measurements
and solves these using a variety of strategies.
Essential Understanding(s):
Physical quantities are measured using different measuring
devices. The precision and accuracy of measurement
depend on the measuring device used.
Essential Question(s):
How are different measuring devices useful?
How does one know when a measurement is precise?
accurate?
The learner will know:
• the concept of measurement
• the different measuring devices and their respective
uses.
• conversion of units of measure.
• rounding off numbers
• approximation.
The learner will be able to:
• use different tools/devices and units of measures.
• cite situations where measuring tools are appropriately
used.
• convert units of measure.
• round off numbers.
• cite real life situations where rounding off numbers is
8

9. • how to solve problems involving measurements
using a variety of strategies.
applied.
• approximate measurement by rounding off to its nearest
desired value.
• formulate and solve real life problems applying conversion
of units.
Stage 2
Product or Performance Task:
Problems formulated
1. are real –life related
2. involve measurement and
3. are solved using a variety of
strategies.
Evidence at the level of understanding
Learner should be able to demonstrate
understanding of measurement using the
six (6) facets of understanding:
Explaining how to use the calibration
model and find its degree of precision.
Criteria:
Thorough
Clear
Accurate
Justified
Interpreting through story telling
situations that describe the appropriate
use and choice of measuring devices.
Criteria:
Illustrative
Accurate
Justified
Significant
Applying a variety of techniques in
posing and solving daily life problems
involving measurement
Criteria:
Appropriate
Practical
Evidence at the level of performance
Assessment of problems formulated
based on the following suggested criteria:
· real-life problems
· problems involve measurement
· problems are solved using a variety of
strategies
Tools: Rubrics for assessment of
problems formulated and solved
9

10. Revealing Empathy by role-playing the
uses of the primitive measuring devices
for the people who invented them and
discuss how they got accurate results.
Criteria:
Perceptive
Open
Manifesting Self-knowledge by
assessing how one can give his/her best
solution to a problem/situation on
measurement.
Criteria:
Reflective
Responsive
Stage 3
Teaching/Learning Sequence:
1. Explore
At this stage, the teacher should be able to start with interesting exploratory activities that will hook and engage the learner
on what is going to happen or where the said pre-activities would lead to.
a. Group activities
· Identify and describe the different measuring devices
· cite real life situations where these measuring devices are used/important.
b. Group presentations of authentic situations showing the evolution of the different measuring devices
c. Reaction paper or journal writing about the group presentation.
Activity 13.
Let the students answer activity sheets on identifying and describing the different measuring devices. This may be done
by presenting a model/actual measuring device ( if available) or just a drawing of these measuring devices. Questions
on who invented the said device may also be included. Questions about how to get accurate results must also be given
10

11. consideration.
Activity 14.
Give each student enough time, like 5-10 minutes to think of situations that describe the appropriate use and choice of
the different measuring devices. Then, let them discuss the situations they listed with their group members. Let each group take
turns in presenting their consolidated story.
Activity 15.
Let each group present authentic situations showing the evolution of the different measuring devices. These
presentations may serve as one of their projects.
Activity 16.
Let the students write reaction paper or journal about the group presentation.
2. Firm Up
At this stage, the teacher should be able to give sample activities or experiences that the learner will have to undergo in
supporting findings in the exploratory activities and for a deeper understanding of the topic.
a. The learner shall conduct an activity
· Using the given measuring instruments, find the measures of classroom table, backboard, window frames, etc.
( Bring the class outdoor and find familiar objects. Perform the same activity.)
· Measuring objects of different shapes.
· Approximating measurements to the nearest unit of measure.
· Estimating and finding actual measurements of objects
· Finding the perimeter and area of plane figures; surface area and volume of solid figures.
· Formulating problems based on the given information.
b. Giving more exercises which may be in problem form.
c. Performing experiments/activities that will verify formulas for finding areas of plane geometric figures and volumes of
solid figures.
d. Solving teacher-made problems about measurements.
Activity 17.
Let the students perform or conduct activities on actual measurements using the different measuring devices.
It may be the measures of classroom table, backboard, window frames, etc. Allow the students to perform the said activity
11

12. even outside the classroom. For linear measurements, let them use different units to compare one unit with that of the
other. Let them do also conversion from one unit to another using the metric converter or a conversion table.
Activity 18.
Extend activity #17 to measuring objects of different shapes or even irregular shapes. Let the students discuss the
similarities and the differences encountered by the groups in getting the measures of the different objects.
Activity 19.
Directions:
a. Group the students into fives.
b. Pose this Activity:
Problem:
How long would it take you to count to one million (1, 2, 3, 4, 5, …, 1 000 000) at the rate of one number per second?
(Assume that you will not stop until the task has been completed)
c. Ask for the answer in more commonly understood units of time, such as days, weeks, months, or years.
d. Allow students to make an estimate/ approximation before they compute.
e. Discuss the results.
3. Deepen
At this stage, the teacher should be able to give activities that will provide the learner the opportunity to reflect on, revisit,
or rethink the lesson.
· Explain thoroughly the process/procedure undertaken in every activity, including the computation part.
· Identify objects whose area/volume can be found using the formulas.
· Explore the possibilities of finding the measures of object of irregular shapes. (football, star, etc. )
· Investigate the relationship between the number of square units/cubic units in a given figure and the
area/volume of the given figure.
· Write a journal on the activities undertaken.
Activity 20.
Let the students answer activity sheets that will identify the formula for the area or volume of a given object. The
questionnaire may be in the form of multiple choice or identification. They may also be asked to explain the step by step
solution on how to apply the formula.
12

13. Activity 21.
Let the students write reaction paper or journal about the different activities being undertaken.
Activity 22.
Ask the students to bring a box full of ping-pong balls to solve the problem below.
Group the students into four members each.
Problem:
A. For a classroom of average size, do you think we could fit one million ping-pong balls?
1. List the assumptions you make in estimating your answer.
2. Find the volume of the box full of ping-pong balls.
3. Use a tape measure and approximate the volume of the classroom.
4. Compare the volume of the classroom with the volume of the box full of ping-pong balls.
5. How many ping-pong balls are there in the box?
B. Do you think one million ping-pong balls could fit into the room? Explain.
4. Transfer
At this stage, the teacher should be able to give the learner activities that will provide him the opportunity to demonstrate
his /her understanding of the topic by:
· formulating and solving a situation/problem.
· writing a report on what he/she has learned about measurement.
· Improvising measuring instruments for finding linear measures of physical objects.
· creating miniature models of your dream house.
Activity 23.
Group the students into four or five depending on the number of students per class. Each group will then select its
leader. The group will decide on the problem to be presented making use measurements. They will visualize and
present the solution to the said problem.
Activity 24.
Give each group enough time to construct scale models of either houses, toys, bridges, etc. depending on its problem
13

14. indicating the use of measurements. Express dimensions of the actual structure to the scale model as ratios.
This will serve as students’ project for exhibit during math expo.
Activity 25.
Using the same grouping, let the students design a game. They will make use of measurements applying the set of real
numbers.
Resources
See Appendix
Quarter 1 : Real Number System,
Measurement and Scientific Notation
Topic: Scientific Notation Time Frame: 5 days
Stage 1
Content Standard:
The learner demonstrates understanding of the key concepts
of scientific notation.
Performance Standard:
The learner formulates real-life problems involving scientific
notation and solves these using a variety of strategies.
Essential Understanding(s):
Big and small quantities can be expressed conveniently in
scientific notation.
Essential Question(s):
Why are measures of certain quantities expressed in scientific
notation? How?
The learner will know:
· numbers that are expressed in scientific notation.
· real life measures where scientific notation is applied.
· the application of scientific notation to different
disciplines.
The learner will be able to:
· express numbers in scientific notation and vice- versa.
· solve real life problems involving scientific notation.
· cite real life situations where scientific notation is
applied.
· formulate and solve real life problems involving
scientific notation.
Stage 2
Product or Performance Task:
Problems formulated
1. are real –life related
Evidence at the level of understanding
Learner should be able to demonstrate
understanding of scientific notation using
the six (6) facets of understanding:
Evidence at the level of performance
Assessment of problems formulated
based on the following suggested criteria:
14

15. 2. involve scientific notation and
3. are solved using a variety of
strategies.
Explaining how big and small quantities
are expressed in scientific notation.
Criteria:
Thorough
Accurate
Justified
Interpreting meaning of scientific
notation by considering the size of an
atom, distances of planets, etc.
Criteria:
Illustrative
Meaningful
Justified
Applying a variety of techniques in
posing and solving daily life problems
involving very large or very small numbers
expressed in scientific notation.
Criteria:
Appropriate
Practical
Accurate
Manifesting Self-knowledge by showing
the usefulness of scientific notation in
solving a problem.
Criteria:
Reflective
Responsive
Showing Empathy to persons who
encounter difficulties in expressing big
real-life problems
problems involve real numbers using
scientific notation
problems are solved using a variety of
strategies
Tools: Rubrics for assessment of
problems formulated and solved
15

16. and small quantities.
Criteria:
Sensitive
Perceptive
Developing Perspective on other ways
to express big and small numbers.
Criteria:
Appropriate
Practical
Stage 3
Teaching/Learning Sequence:
1. Explore
At this stage, the teacher should give interesting exploratory activities that will hook and engage the learner on what is
going to happen or where the said pre-activities would lead to:
a. group activities on
· identifying/recognizing a pattern from a given set of numbers.
· citing real life situations where scientific notation can be used.
b. group presentations of authentic situations where scientific notation are used.
c. giving reactions /comments to the given presentation.
Activity 26.
Directions: Ask students to work individually on this activity. Read the numbers in the table from top to bottom, then
answer the following questions:
1. What pattern do you observe between succeeding numbers?
2 Guess the next term of the sequence.
3. Write down a rule for finding the value of numbers with negative exponents.
10n
106 = ?
105 = ?
104 = ?
16

17. 103 = 1 000
102 = 100
101 = 10
100 = 1
1
10-1 = 10
1
10-2 = 100
1
10-3 = 1000
10-4 = ?
10-5 = ?
10-6 = ?
5. What happens when the pattern continues?
6. Find the relationship between the succeeding numbers.
2. Firm Up
At this stage the teacher shall present sample activities or experiences that the learner will have to undergo for
a deeper understanding of the topic.
a. The learner shall complete
· activity sheets on expressing numbers in scientific notation
· exercises involving fundamental operations using scientific notation.
b. The learner shall solve more exercises involving scientific notations which may be in problem form.
c. Solve problems involving scientific notation.
Activity 27.
Exploration Activity:
Have students answer the worksheet in pairs.
17

18. Study the table below.
Set A Set B
Set C
18
Column I
Decimal Form
8
81
814
8 143
Column II
Scientific
Notation
8 x 100
8.1 x 101
8.14 x 102
8.143 x 103
Column I
Decimal Form
14.325
143.25
1 432.5
Column II
Scientific
Notation
1.4325 x 101
1.4325 x 102
1.4325 x 103

19. Questions:
1. Observe the numbers in Column I and Column II in Set A.
How do the numbers in each pair compare?
How are the numbers in Column I expressed? Column II?
2. For each set, look at the second number.
How does the second number compare with the number in decimal form?
What can you say about the second number in each pair?
3. Observe the position of the decimal point in each number expressed in scientific notation.
Where do you find the decimal point?
Note: If the decimal point appears after the first nonzero digit, such decimal number is in
STANDARD POSITION
4. Discuss your findings with your partner.
19
Column I
Decimal Form
0.3768
0.03768
0.003768
Column II
Scientific Notation
3.768 x 10-1
3.768 x 10-2
3.768 x 10-3

20. 5. Repeat steps 1 to 4 for Sets B and C.
6. When do you say that a number is expressed in scientific notation?
7. Complete this statement: A number is expressed in scientific notation if it is expressed as
the product of a number in standard position and________________.
3. Deepen
At this stage, the teacher must give the learner the opportunity to reflect on, revisit, or rethink the lesson through the following:
· Explain thoroughly the process/procedure undertaken in every activity, including the computation part.
· Investigate on the procedure use in scientific notation for very big numbers and for very small numbers.
· Journal writing on the usefulness of scientific notation.
Activity 28.
A. Sample Problem
A jeepney park charges the following rates: P15.00 for the first hour, P10.00 for the next hour and P5.00 for each
additional hour. How much does the jeepney park charge for six hours?
Solution with the corresponding rubric points:
Let n pesos be the jeepney park charge for 6 hours.
Php15.00 is the charge for the first hour ( 1 point )
Php10.00 is the additional charge for the 2nd hour
Php 5.00 is the charge for each additional hour after 2 hours.
Thus, n = 15 + 10 + 5( 4 ) ( 1 point )
= 15 + 10 + 20
= Php45.00 ( 1 point )
Total points : 3
Scoring Guide (Rubric) for Problem Solving
20

21. B. Problem Solving Activity:
Solve the following .Show all solutions.
Express the answers in scientific notation.
1. A watch ticks four times each second. How many ticks will it make each day?
2.The sun is approximately 1.5 x 1011 m from Earth.
How far from the Earth is the nearest star if it is approximately 300 000 times as far as the sun?
3. A person’s heart beats approximately 72 times per minute.
How many times does a heart beat in an average lifetime of 75 years? (Assume all years have 365 days.).
21
Points Criteria
3 Understood the problem, performed the correct
operation/s, and got the correct answer.
2 Understood the problem, performed the correct
operation/s, and got an incorrect answer
1 Attempted to solve the problem, performed an incorrect
operation/s and got an incorrect answer.
Got the correct answer, but no solutions/wrong solution.
0 No attempt

22. 4. Biologists use the micrometer or the micron to measure short lengths.
One micrometer is equal to 0.001 millimeter. If a cell is 47 micrometers long, what is its length in millimeter?
4. Transfer
Let the learner demonstrate his/her understanding of the topic by:
· formulating and solving a situation/problem that will make use of scientific notation.
· writing a report about the advantages/disadvantages of using scientific notation.
· creating a miniature model , like the solar system indicating the distances(express in scientific notation) of each planet.
Activity 29.
Prepare contest questions. It may be a team competition of 4 members.
Classify the questions as 15 – sec; 30 – sec; and 1- minute.
Each question must be given orally with the equivalent time allotment. Give the points as 2 for the 15 sec, 3 for 30 sec,
and 5 for the 1 minute, respectively. The first 3 highest scorers must be3 declared as winners.
Activity 30.
Let the students create a model solar system. Express the distances in scientific notation. Let them do this project by
group.
Activity 31.
Another project that they could make is to design a game. Give them guide questions in making the game.
Resources (Web sites, Software, etc.)
See Appendix
22

23. Quarter II : Algebraic Expressions, First-Degree
Equations and Inequalities in One Variable
Topic: Algebraic Expressions Time Frame: 25 days
Stage 1
Content Standard:
The learner demonstrates understanding of the key concepts
of algebraic expressions.
Performance Standard:
The learner models situations using oral, written, graphical and
algebraic methods to solve problems involving algebraic
expressions.
Essential Understanding(s):
Algebraic expressions represent patterns and relationships that
guide us in understanding how certain problems can be solved.
Essential Question(s):
Why are algebraic expressions useful?
The learner will know:
· translation of verbal phrases to mathematical
expressions and vice-versa
· laws on integer exponents
· operations of algebraic expressions
· rules on finding special products
· types of special products
· special products of two binomials
· relationships between special products and factors
· complete factorization of polynomials
· applications of special products and factors in solving
real life problems
The learner will be able to:
· translate verbal phrases to mathematical expressions and
vice-versa
· simplify algebraic expressions using the laws on integer
exponents
· perform fundamental operations on algebraic expressions
· explore the product of two binomials and search for
patterns
· identify special products
· find special products of two binomials
· discover the relationships between special products and
factors
· find the complete factorization of polynomials
· apply factoring polynomials in solving real life problems
Stage 2
Product or Performance Task: Evidence at the level of understanding Evidence at the level of
23

24. Situations modeling the use of oral,
written, graphical and algebraic methods
to solve problems involving algebraic
expressions
The learner should be able to demonstrate
understanding of algebraic expressions using the
six (6) facets of understanding:
Explaining how the language of mathematics is
used to show /describe real-life situations.
Criteria:
Clear
Coherent
Justified
Interpreting representations of mathematical
situations
Criteria:
Illustrative
Meaningful
Applying algebraic expressions in daily life
situations
Criteria:
Appropriate
Practical
Relevant
Developing Perspective on the various ways of
writing algebraic expressions and solving a
problem
Criteria:
Critical
Insightful
Credible
Showing Empathy to persons who encounter
difficulties in the lesson.
performance
Performance assessment of
situations involving algebraic
expressions based on the
following suggested criterion:
· Use oral, written, graphical
and algebraic methods in
modeling situations
Tools:
Rubrics of situations modeling
the use of oral, written, graphical
and algebraic methods
24

25. Criteria:
Open
Sensitive
Responsive
Manifesting Self-knowledge by discussing the
best and most effective strategies that one has
found for solving problems
Criteria:
Insightful
Clear
Coherent
Stage 3
Teaching/Learning Sequence
1. Explore
Initially, let the teacher begin with some interesting and challenging exploratory activities that will make the learner aware of
what is going to happen or where the said pre-activities would lead to through meaningful and relevant real life context.
· Playing “Guess my rule” game and writing mathematical expression for the rule
· Ask students to surf the internet and look for similar games which they can share to the class.
· Provide students with worksheets on translating mathematical expressions to English phrases and vice-versa
· Ask students to give their own English phrases and translate them to mathematical expressions and vice-versa
· Completing teacher-made activity sheets on evaluating algebraic expressions
· Completing teacher-made activity sheets on addition and subtraction of algebraic expressions
· Investigating relationships among integer exponents
· Finding the product of algebraic expressions
.
2. Firm Up
25
These are the enabling activities/ experiences that the learner will have to go through for the learner to understand
· Simplifying algebraic expressions
· Performing operations on algebraic expressions
· Finding special products
· Factoring, the reverse process of finding the product

26. 3. Deepen
Activities in this stage shall provide opportunity for differentiated instruction for the learner to reflect,
revisit, revise and rethink. Further, the learner shall express his/her understanding and engage in
meaningful self-evaluation.
4. Transfer
Resources
· Summarizing the steps in performing the fundamental operations on algebraic expressions
· Writing journals on how knowledge of algebraic expressions help in finding solutions to challenging
computations
· Citing situations in the environment where the concepts of algebraic expressions and operations
are applied
See Appendix
26
Learner’s understanding is demonstrated through culminating activities that reflect relevant and authentic
problems/situations.
· Applying special products and factors in real life problems
· Creating/posing and solving problems using a variety of strategies
· Presenting a problem solving plan using models
· Making a flowchart on intelligent digital model applying algebraic expressions (e.g. robotics,
software, etc.)

27. Quarter II : Algebraic Expressions, First-Degree
Equations and Inequalities in One Variable
Topic: First-Degree Equations and
Inequalities in One Variable
Time Frame: 25 days
Stage 1
Content Standard:
The learner demonstrates understanding of the key
concepts of first-degree equations and inequalities in one
variable.
Performance Standard:
The learner models situations using oral, written, graphical and
algebraic methods to solve problems involving first-degree equations
and inequalities in one variable.
Essential Understanding(s):
Real-life problems where certain quantities are unknown
can be solved using equations and inequalities in one
variable.
Essential Question(s):
How can we use equations and inequalities to solve real life problems
where certain quantities are unknown?
The learner will know:
· mathematical expressions, equations and
inequalities
· linear equations and inequalities
· properties of equations and inequalities
· applications of first-degree equations and
inequalities
The learner will be able to:
· differentiate mathematical expressions from equations and
inequalities
· identify and describe linear equations and inequalities in one
variable
· give examples of linear equations and inequalities in one
variable
· describe situations where equations and inequalities are used
· enumerate and explain the different properties of equations
and inequalities
· give illustrative examples of each property of equations and
inequalities
· apply the properties of equations and inequalities in solving
first-degree equations and inequalities in one variable
· verify and explain the solutions to problems involving
equations and inequalities
· extend, pose, and solve related problems in real life
Stage 2
27

28. Product or Performance Task:
Situations modeling the use of oral,
written, graphical and algebraic methods
to solve problems involving first-degree
equations and inequalities in one variable
Evidence at the level of understanding
The learners should be able to demonstrate
understanding of first-degree equations and
inequalities using the six (6) facets of
understanding:
Explaining the properties of first-degree
equations and inequalities in one variable.
Criteria:
Clear
Coherent
Justified
Interpreting mathematical conjectures and
arguments involving first-degree equations and
inequalities in one variable
Criteria:
Illustrative
Meaningful
Applying first-degree equations and inequalities
in one variable in daily life situations
Criteria:
Appropriate
Practical
Relevant
Developing Perspective on the various ways of
writing first-degree equations and inequalities in
one variable in solving a problem
Criteria:
Critical
Insightful
Credible
Showing Empathy by describing difficulties one
can experience in daily life whenever tedious
Evidence at the level of
performance
Performance assessment of
situations involving first-degree
equations and inequalities in one
variable based on the following
suggested criterion.
· Use oral, written, graphical
and algebraic methods in
modeling situations
Tools:
Rubrics of situations modeling the
use of oral, written, graphical and
algebraic methods
28

29. calculations are done without using the
concepts of first-degree equations and
inequalities in one variable
Criteria:
Open
Sensitive
Responsive
Manifesting Self-knowledge by discussing the
best and most effective strategies that one has
found for solving problems involving first-degree
equations and inequalities in one variable
Criteria:
Insightful
Clear
Coherent
Stage 3
Teaching/Learning Sequence:
1. Explore
2. Firm Up
29
Start with interesting exploratory activities that will hook and engage the learner on what is going to happen or where
the said pre-activities would lead to:
· Group activities/games and puzzles on:
· identifying and describing linear equations and inequalities
· citing real life situations involving linear equations and inequalities
· Online/offline presentations of authentic situations involving linear equations and inequalities (e.g. ICT
tools CONSTEL CDs, open-source learning materials, E-TV learning episodes, etc.
Giving reactions to online/offline presentations
These are the enabling activities or experiences that the learner will have to undergo in supporting findings in the
exploratory activities in order to equip them for meaningful understanding.
· Giving exercises on representing situations using linear equations and inequalities
· Group activity on enumerating, explaining and giving illustrative examples of the properties of equations
and inequalities
· Solving exercises on first-degree equations and inequalities in one variable where the properties are
applied
· Verifying solutions using scientific calculator/computer

30. 3. Deepen
Activities in this stage shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety of
experiences. Moreover, the learner shall express his/her understanding and engage in multidirectional self-assessment.
· Making and evaluating mathematical conjectures and arguments involving first-degree equations and
· Investigating solutions to problems related to first-degree equations and inequalities in one variable
· Writing journals on situations or experiences involving equations and inequalities that need to be
4. Transfer
30
inequalities in one variable
valued by every learner
Applications of learner’s understanding are demonstrated through culminating activities that reflect meaningful and
relevant problems/situations.
· Applying mathematical thinking and modeling to solve problems in other disciplines such as art, music,
science, business, etc.
· Creating/posing and solving problems involving linear equations and inequalities in one variable using a
variety of strategies
· Using models present a problem solving plan on linear equations and inequalities in one variable using
models
Making a flowchart on intelligent digital model applying first-degree equations and inequalities in one variable (e.g.
robotics, software, business model, etc.)

31. Resources
See Appendix
Quarter III : Rational Algebraic Expressions, Linear
Equations and Inequalities in Two Variables
Topic: Rational Algebraic
Expressions
Time Frame: 25 days
Stage 1
Content Standard:
The learner demonstrates understanding of key concepts of
rational algebraic expressions.
Performance Standard:
The learner presents solutions to problems involving rational
algebraic expressions using numerical, physical, and verbal
mathematical models or representations.
Essential Understanding(s):
Simplifying rational algebraic expressions involve factorization
and operations similar to operations on numerical fractions.
Essential Question(s):
How can rational algebraic expressions be simplified?
The learner will know:
· fractions in simplest form;
· operations on fractions;
· rational algebraic expressions in simplest form;
· operations on rational algebraic expressions; and
· applications of rational algebraic expressions.
The learner will be able to:
· explore problems and describe results using numerical,
physical, and verbal mathematical models or
representations;
· use his/her reading, listening and visualizing skills to
interpret mathematical ideas;
· simplify rational algebraic expressions by using various
methods/techniques;
· perform operations on rational algebraic expressions and
justify steps by stating the mathematical properties used;
· analyze rational algebraic expressions, formulate
relationships and extend them to other cases; and
· apply the concept of rational algebraic expressions in
solving real life situations.
Stage 2
Product or Performance Task:
Solutions to problems involving rational
Evidence at the level of understanding
The learner should be able to demonstrate
Evidence at the level of
performance
31

32. algebraic expressions are presented
using numerical, physical, and verbal
mathematical models or representations.
understanding by covering the six (6) facets of
understanding:
Explaining by justifying how one’s answer is
changed to simplest form.
Criteria:
Clear
Coherent
Justified
Interpreting how best procedures for simplifying
rational expressions are determined.
Criteria:
Illustrative
Creative
Accurate
Applying the appropriate operations in simplifying
rational expressions.
Criteria:
Appropriate
Accurate
Developing Perspective on how to choose the best
solution in simplifying rational expressions
Criteria:
Credible
Insightful
Showing Empathy on people’s difficulties in
performing operations involving rational
expressions.
Criteria:
Perceptive
Responsive
Sensitive
Assessment of presentation of
solutions to problems involving
rational algebraic expressions
based on the suggested
criterion:
· the use of numerical,
physical, and verbal
mathematical models or
representations.
Tools:
Rubrics for assessment of
solutions to problems
32

33. Manifesting Self-Knowledge in recognizing the
best solution to a given situation involving rational
expressions.
Criteria:
Reflective
Insightful
Stage 3
33

34. Teaching/Learning Sequence:
1. Explore
Activity 1:
Show the following figures on the board and ask students to observe them. (Physical models could also be used.)
Ask the following questions:
1. What can you say about the 3 figures?
2. What does each shaded part represent?
3. If the 3 figures have the same sizes, how are the three shaded parts related?
4. How would you show that the three shaded parts are equal or the fractions representing them are equivalent?
5. How would you simplify the following fractions?
34
Initially, begin with some interesting and challenging exploratory activities on rational numbers that will make the
learner aware of what is going to happen or where the said pre-activities would lead to through meaningful and
relevant real life context.
Let the students:
a. explore problems and describe results using numerical, physical and verbal mathematical models or
representations.
b. use his/her reading, listening and visualizing skills to interpret mathematical ideas.
c. simplify rational numbers by using various methods/techniques.

35. 24 b. 56
a. 36
-32
35 c. 48
-45 e. 18
d. 54
12
-
6. When do you say that a fraction is in its simplest form?
2. Firm Up
Activity 2:
1. Present a problem in real life.
Mr. Gabriel has a farmland which he subdivided equally among his 6 children and 22 grandchildren.
a. How would you represent the area of Mr. Gabriel’s farmland?
b. If one-third of Mr. Gabriel’s farmland is given to his children, what expression represents the part of the land they
would receive? How about the part of the land each child would receive?
c. If the remaining part will be shared by the grandchildren, what expression represents the part of the land they would
receive? How about the part of the land each grandchild would receive?
d. How would you describe the expressions you got in (c) and (d)?
e. How would you compare these expressions with the fractions which you already studied before?
f. How would you differentiate rational numbers from rational expressions?
g. Which of the following are rational algebraic expressions? Explain your answer.
35
These are the enabling activities/experiences that the learner will have to go through to validate understanding on
rational algebraic expressions during the activities in the exploratory phase. These would answer some
misconceptions on rational algebraic expressions that have been encountered in real life situations.
At this phase, the students should be able to:
a. apply the concept of rational algebraic expressions by presenting problems in real life.
b. describe solutions using numerical, physical and verbal mathematical models or representations.
c. simplify rational algebraic expressions by using various methods/techniques.
d. perform operations on rational algebraic expressions.
e. explain results and make the necessary justification of each steps used by stating the mathematical properties
applied.

36. x f. 5
4
x -
a. 2
x ; x = 5
+
x
5
-
b.
x 4
g. ( 2 )( 3
) 4 4
2
2
-
+ +
x x
2 5 6
- -
x x
- +
x x
3 h.
c. 7
3 4 +12 2 -6
x x x
x
3
d.
5 i. 4 +4
2x
x
e.
8 3 27
x j.
-
x
-
2 3
3 3
x -
3
x
2. Let the students discuss the results of the activity.
3. Deepen
Activity 3:
1. Show the following are rational algebraic expressions.
x b. 2
4
x -
a. 2
2x + 4 c.
x 9
d.
6 9
2
2
-
+ +
x x
x x
2 8
+ +
7 12
2
x x
2
+
Ask the following questions:
36
Activities in this phase shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety of
experiences. Moreover, the learner shall express his/her understanding and engage in multidirectional self-assessment.
The students should be able to:
a. explain/write the series of steps in simplifying rational algebraic expressions (e.g. application of properties of real
numbers, the different factoring procedures).
b. use pictures, multimedia presentations, or daily life experiences and observations where concepts of rational
algebraic expressions are applied (e.g. business, science, industry, etc.)

37. a. Which of the above expressions are expressed in simplest form? Why?
b. Why do you say that the others are not written in their simplest form?
c. How would you simplify these expressions? Give the steps.
d. What mathematics concepts or ideas would you apply to simplify the expressions?
e. How would you apply these mathematics concepts or ideas in simplifying rational algebraic expressions?
f. Express the given rational algebraic expressions in simplest form.
2. Check for understanding:
a. Ask the question: Is x-2 +1 a rational algebraic expression? Why?
b. Cite situations that could be represented by rational algebraic expressions. What expressions represent these situations?
Activity 4:
Perform group activities using pictures, multimedia presentations, or daily life experiences and observations where concepts of
rational algebraic expressions are applied (e.g. business, science, industry, etc.) such as:
· Investigating relationship of quantities. e.g. the distance (d) from the fulcrum of a person of weight (w) on a see-saw.
· Finding the actual size of the rooms of a building from a scale model.
· Designing a scale model for a given classroom size.
4. Transfer
Activity 5:
1. Design a scale model of a structure. Express dimensions of the actual structure to the scale model as ratios.
2. Design Games
Resources:
See Appendix
Quarter III : Rational Algebraic Expressions, Linear
Equations and Inequalities in Two Variables
Topic: Linear Equations and
Inequalities in Two Variables
Time Frame: 25 days
37
Applications of learner’s understanding on first-degree equations and inequalities in one variable are demonstrated
through culminating activities (e.g. Math Exhibits/Expo) that reflect meaningful and relevant problems/situations.
· Designing a scale model of your dream house
· Constructing miniature models, e.g. buildings, playground, amusement parks, ships, etc.

38. Stage 1
Content Standard:
The learner demonstrates understanding of key concepts of
linear equations and inequalities in two variables.
Performance Standard:
The learner presents solutions to problems involving linear
equations and equalities in two variables using numerical,
physical, and verbal mathematical models or representations
Essential Understanding(s):
Linear equations show constant rate of change.
Graphs of linear equations show trends which help predict
outcomes and make decisions
Essential Question(s):
How are linear equations used to communicate relationships
between quantities?
How does one know an outcome is favorable? How can
mathematics help one find out?
The learner will know:
· coordinate plane and the terminologies associated with
it.
· graph of linear equations in two variables
· equation of a linear equation in two variables
· application of linear equations in two variables
· graph of a linear inequality in two variables
The learner will be able to:
· give the exact location of a point, person or object using
maps, navigation devices, etc.
· investigate graphs of linear equations in two variables with
deductive arguments and evidences.
· solve linear equations in two variables graphically .
· apply mathematical thinking to solve problems in disciplines
such as art, music, science and business.
· formulate and solve real life problems using various
representations.
Stage 2
Product or Performance Task:
Solutions to problems involving linear
equations and inequalities in two variables
are presented using numerical, physical,
and verbal mathematical models or
representations.
Evidence at the level of understanding
The learner should be able to demonstrate
understanding by covering the six (6) facets of
understanding:
Explaining how a statement is translated into
mathematical symbols
Criteria
Clear
Coherent
Evidence at the level of
performance
Assessment of presentation of
solution to problems involving
linear equations and
inequalities in two variables
based on the suggested
criterion:
38

39. Interpreting possible relationships of rates of change
in a given set of data.
Criteria:
Revealing
Illustrative
Applying mathematical thinking and modeling to solve
problems in other disciplines such as art, music,
science and business.
Criteria:
Practical
Appropriate
Accurate
Developing Perspective on the most likely outcomes
that may result from trends shown in graphs
Criteria:
Credible
insightful
Showing Empathy on people experiencing difficulties
in making decisions without the help of graphs and
linear equations
Criteria:
Perceptive
Responsive
Sensitive
Manifesting Self-Knowledge by sharing insights one
may have about how math can help make reasonable
judgments and predictions.
Criteria:
Reflective
Insightful
· Use of numerical, physical,
and verbal mathematical
models or representations.
Tools:
Rubrics for assessment of
solutions to problems
39

40. Stage 3
Teaching/Learning Sequence:
1. Explore
Initially, begin with some interesting and challenging exploratory activities on linear equations and inequalities in two
variables that will make the learner aware of what is going to happen or where the said pre-activities would lead to
through meaningful and relevant real life context.
Activity 1:
a. Start with an activity that would assess the learner’s knowledge in naming places or locations.
· Identifying classmates’ location through a seat plan
· Treasure hunting
· Map reading
b. Allow students to translate the variables used into mathematical symbols.
c. Ask follow-up questions that would enhance the critical thinking skill of the learner.
1. Ask the students to arrange their seats to form columns and rows. Tell those seating in the front row to number their
seats as illustrated:
Column
Row 1
Row 2
Row 3
2. Guide them to name the seats of their classmates as (r, c), where r stands for row and c for column.
Example: The seat where Luisa who is seated on the 3rd row, and on the 2nd column is named as (3, 2)
3. After the activity, pose the question:
a. Who is seated at (2, 4)?
b. Are there other instances where locations of places are named?
4. Write experiences encountered if locations of persons, objects, places are not clearly defined.
40
1 2 3 4 5
L

41. 2. Firm Up
Activity 2: Treasure Hunt with Slopes
On a grid paper, mark points that would lead to the treasure.
41
These are the enabling activities/experiences that the learner will have to go through to validate understanding on
linear equations and inequalities during the activities in the exploratory phase. These would answer some
misconceptions on linear equations and inequalities in two variables that have been encountered in real life situations.
At this phase, the students should be able to perform group activities such as games, puzzles, storytelling,
simulation, role-playing, etc in:
· finding/locating the coordinates of a point in the coordinate plane;
· solving for the slopes of two points (e.g. measuring the steepness of stairs/inclined objects);
· finding linear equations using the forms: slope and y-intercept, slope and a point, two points
· graphing linear equations and inequalities; and
· solving real life situations.
Start here ●
●

42. Using the definition of slope, trace the path using the slopes listed below. A correct solution will trace the route to the treasure.
1. 3 5. 2 9. 3
2
1 6. -3 10. 3
2. 4
-1
-2 7. 3
3. 5
1 11. 5
-3
-4 12. 6
4. 6 8. 7
Activity 3: Group Activity
1. Provide students with activity sheets.
2. Send groups of students to measure the length and height of the steps of the stairs of each building in their school.
3. Let other groups measure inclined objects. Mark considered different points in the object and ask students to
measure the vertical and horizontal distances of these points.
4. Represent the measurement/distances as ratios (vertical distance to the horizontal distance). Let them compare the
ratios.
5. Allow them to discuss their findings in class.
6. Introduce the concept of slope.
7. Ask learners’ experiences encountered on situations similar to:
· steps of stairs unevenly spaced.
· going up on an inclined plane with different gradients.
8. Investigate the effects caused by these phenomena.
9. Make the necessary analysis
Activity 4: Solve the problem
Loida who lives in Baguio City usually takes note of temperature readings in degrees Celsius. When she visited her mother in
Chicago, she found that temperature was reported in degrees Fahrenheit. Being used to the Celsius readings, she converts
temperatures using the formula oC = 5 (F – 32).
9
a. Using a thermometer, demonstrate to the class how each scale corresponds to the other.
42

43. b. Make various representations using tables or graphs. Let x be the Celsius scale and y the Fahrenheit scale. (C, F)
c. Solve for the Fahrenheit reading of 40 0C.
d. Analyze the formula in converting 0F to 0C, F = 5
9 C + 32. If in item b we represented x as C and y as F, then transform the
equation using x and y.
9 and 32 in our graph. What do these numbers represent?
e. In the transformation recognize the role of constant numbers 5
f. Given these two formulas, which would you consider using?
3. Deepen
Activity 5:
Use a graphing calculator to discuss:
a. the behavior of the graphs of linear equations in two variables if different slopes are used
b. how a linear equation can be graphed using the forms:
· slope and a point.
· the x – and - y intercepts.
· two points.
· slope and y intercept.
Activity 6:
43
Activities in this stage shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety of
experiences. Moreover, the learner shall express his/her understanding on linear equations and inequalities in two
variables and engage in multidirectional self-assessment.
At this phase, the students should be able to:
· investigate the behavior of graphs in relation to their slopes.
· write the steps in graphing linear equations and inequalities in two variables. (Imagine that you are writing
the steps for someone who has never experienced this concept before.)
· check the results using graphics calculators or computers.
· analyze situations represented by linear equations and inequalities in two variables.
· write journals on the behavior of graphs of linear equations and inequalities in two variables.

44. In the given graphs below, identify parallel and intersecting lines.
a. Identify the y-intercepts of each line.
b. Solve for the slope using two points on the line.
c. Critically examine the graphs and compare the slopes of the lines. Which graphs have the same slope and which do
not have? Make deductions.
d. Which pair of graphs intersects and which do not? Give the point of intersection of these pair of graphs.
e. Tell the class how these graphs give meaning in real life and relate advantages and disadvantages if all things intersect
or are parallel from each other.
Activity 7: Analyze and solve the following problems:
1. Lisa baked 10 cakes in 5 hours. She worked for 2 ½ more hours and had baked a total of 15 cakes.
How many cakes had she baked after 10 hours?
a. Know what is asked.
b. Make a table to show the number of cakes she baked in certain number of hours.
No. of hours (x) 5
44
a
b
c

45. No. of cakes she baked (y) 10
c. Write an equation to show the rule.
2. How much exercise is the right amount? Most health experts suggest that you should exercise to the point where your heart
rate reaches a target level based on your age. Here's a rule that is suggested:
To find your target heart rate or pulse, subtract your age from 220.
a. Write an equation for the relationship.
b. Find your target rate using the equation.
c. What is the target heart rate for a 5-year old child?
Procedure:
1. Know what is asked.
a. Your target heart rate
b. The target heart rate of a 5-year old child
2. Make a table to show the target heart rate of your group mates.
Age(x)
Target heart rate(y)
3. Solve the problem.
4. State the rule or the equation involved.
4. Transfer
45
Applications of learner’s understanding on linear equations and inequalities in two variables are demonstrated through
culminating activities that reflect meaningful and relevant problems/situations.
The students should be able to:
a. construct a miniature model of the learner’s ideal community.
b. design a game map to locate a person in a certain town, a ship in distress, a treasure buried in a mountain slope.
c. find the amount of fencing material needed to enclose a vegetable plot, flower garden, etc.

46. Activity 7:
Design a Model Community
a. Group students into four members.
b. Provide each group with an activity sheet, grid paper, pictures of houses, hospital or clinic, church, school, etc.
c. Let them design their ideal community by placing the cutout pictures at the corner of each grid.
d. Ask them to place their house at a strategic place. Let the location of the house be the reference point or (0, 0).
e. Tell them to give the coordinates of the structures they have placed on the grid paper in relation to their house.
f. Develop guide questions that would provide insights about what has been learned in the activity.
Activity 8:
On a grid paper, design a game map where students find location of:
a. a buried treasure.
b. a boat in distress.
c. a particular animal.
d. the tallest tree.
Activity 9:
Design a competition. (Graphing Calculator Competition)
a. Form students into groups
b. Construct questions about the topics discussed. Solutions to the problems should make use of the resources of a
graphing calculator.
c. Other students may act as runners/scorers in the competition.
d. Math teachers may be invited to act as judge.
e. Provide incentive to winning groups.
Resources:
See Appendix
46

47. Quarter IV: Systems of Linear Equations and
Inequalities in Two Variables
Topic: Systems of Linear Equations
and Inequalities in Two Variables
Time Frame: 25 days
Stage 1
Content Standard:
The learner demonstrates understanding of key concepts of
systems of linear equations and inequalities in two variables.
Performance Standard:
The learner creates situations/ problems in real-life involving
systems of linear equations and inequalities in two variables, and
solves these by applying a variety of strategies
Essential Understanding(s):
Unknown numbers in certain real-life problems may be derived
from solving systems of linear equations and inequalities in two
variables.
Essential Question(s):
How is knowledge of systems of linear equations and inequalities
in two variables used to solve real life problems?
The learner will know:
· graphical solution of systems of linear equations and
inequalities in two variables.
· algebraic solutions of systems of linear equations and in
two variables.
· applications of systems of linear equations and
inequalities in two variables in problem solving
· graph a system of linear inequalities and inequalities in
two variables
The learner will be able to:
· explain thoroughly how systems of linear equations and
inequalities in two variables can be solved graphically and
algebraically.
· graph with accuracy the solutions of a system of linear
equations and inequalities in two variables.
· apply a variety of strategies to solve problems involving
systems of linear equations and inequalities in two
variables.
· graph with accuracy the solution set of a system of linear
equations and inequalities in two variables.
Stage 2
Product or Performance Task:
Situations/ Problems created are drawn
from real-life and are solved by applying
a variety of strategies.
Evidence at the level of understanding
The learner should be able to demonstrate
understanding by covering the six (6) facets of
understanding:
Explaining and presenting a mathematical analysis
of graphs.
Criteria:
Clear
Coherent
Evidence at the level of
performance
Assessment of situations
problems created based on the
following suggested criteria:
· problems are drawn from real-life;
· problems involve systems of
linear equations and inequalities
in two variables; and
47

48. Justified
Interpreting the significance of the way graphs
relate with each other.
Criteria:
Illustrative
Meaningful
Applying the appropriate solution that would
produce best results.
Criteria:
Appropriate
Practical
Useful
Developing Perspective on the different possible
outcomes illustrated by graphs/equations.
Criteria:
Critical
Insightful
Showing Empathy on problems that may result
when systems of linear equations are not properly
solved and the unknown number is not correctly
determined
Criteria:
Sensitive
Authentic
Manifesting Self-Knowledge on the impact of
individual accuracy in solving problems on systems
of linear equations
Criteria:
Insightful
Relevant
· problems are solved using a
variety of strategies.
48

49. Stage 3
Teaching/Learning Sequence:
1. Explore
Activity 1:
1. Pose a Problem:
Andrea who lives in a condominium in Makati plans to avail herself of one of the parking packages being offered by the
condominium’s management
Package A: Space Rental: P3000 per year
Monthly Dues: P200 per month
Package B: Space Rental: P4000 per year
Monthly Dues: P100 per month
2. Analyze the situations by graphing the two packages in one coordinate plane.
3. Ask the Questions:
Which parking package should she avail? Why? When is one package cheaper?
4. Introduce the system of linear equations in two variables and discuss related lessons.
49
Initially, begin with some interesting and challenging exploratory activities on linear equations and inequalities in
two variables that will make the learner aware of what is going to happen or where the said pre-activities would
lead to through meaningful and relevant real life context.
a. Start with an activity that would assess the learner’s knowledge on linear equations in two variables.
· Problem – posing activity
· Outdoor activity (e.g. measuring the horizontal and vertical distances of the steps of stairways,
inclined plane, etc.)
· Games, puzzles, storytelling, role-playing, simulation. etc.
· Video/powerpoint presentations that would show representations of linear equations in two variables
b. Ask follow-up questions that would enhance the critical thinking skill of the learner.
c. Pose questions that would link linear equations in two variables and systems of linear equations in two
variables.

50. 2. Firm Up
Activity 2:
Divide the class into three groups. Each group will be assigned to graph a system of linear equations using any method of
their choice and to answer the following guide questions. Two representatives from each group will then be asked to present
their work to the class.
Sketch the graph of the given system of equations. Identify the slopes and y-intercepts of the two lines in the system and
then answer the questions that follow.
Group 1:
î í ì
x y
+ =
- =
10
6
x y
Group 2:
î í ì
x y
- = -
- =
3
1
x y
Group 3:
î í ì
x y
+ =
3
3 x + 3 y
=
9
Guide Questions:
1. What is the slope of the first line in your system? second line?
2. What is the y-intercept of the first line in your system? second line?
50
These are the enabling activities/experiences that the learner will have to go through to validate understanding on
systems of linear equations and inequalities in two variables during the activities in the exploratory phase. These
would answer some misconceptions on systems of linear equations and inequalities in two variables that have been
encountered in real life situations.
At this phase, the students should be able to:
· graph solution of systems of linear equations in two variables, e.g. graph showing the gains and losses of a
business firm, income and expenses of a middle income family, etc.
· solve for the algebraic solutions of systems of linear equations in two variables.
· apply systems of linear equations in two variables in problem solving.
· graph a system of linear inequalities in two variables.
· interpret graphs of systems of linear equations and inequalities through storytelling, simulation, flowchart,
etc.

51. 3. What do you notice about the slopes and y-intercepts of the linear equations in your system?
4. Describe the graph that you sketched. What kind of lines is formed?
After each group presentation, discuss the different kinds of systems of linear equations. Let the students identify the
characteristics of each system based on their previous activity.
Use the following questions as guide.
What can you say about the slopes and the y-intercepts of:
1. consistent system of linear equations in two variables?
2. inconsistent system of linear equations in two variables?
3. dependent system of linear equations in two variables?
3. Deepen
Activity 3:
Technology Integration:
Teach the students how to use the graphics calculator to investigate the graph of a given system of linear equations.
51
Activities in this stage shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety
of experiences. Moreover, the learner shall express his/her understanding on systems of linear equations and
inequalities in two variables and engage in multidirectional self-assessment.
The students should be able to:
· Investigate the effect of the slopes on the graph of a linear equation in two variables and giving its
significance
· Explore graphs of inequalities using graphics calculator.
· Investigate and analyzing critical points on the graphs of systems of linear inequalities.
· Write reports on the result of the investigation.

52. -4 2- y---123012345 2 x 4
-6 -4x -2 --02424 y 2 4
1. Identify the slopes and y-intercepts of the lines in each of the following systems and then identify what kind of system it is.
a.
î í ì
x y
+ =
4
3 x + 3 y
=
12
b.
î í ì
= - +
2 3
y = - x
+
1
x y
- =
3 5 10
y x
ì
- = -
ïí
1 3
c. ( 2
) 5
ïî
x y
d.
î í ì
y x
= -
2 1
2 3
y = x
+
2. Analyze the graphs. What kind of system of linear equations is represented by each graph?
a. b.
c. d.
52
y
x
y
x
y
y
x
x

53. 4. Transfer
Activity 4:
Allow students to design competitions, puzzles, games, etc.
Resources:
See Appendix
53
Applications of learner’s understanding on linear equations and inequalities in two variables are demonstrated
through culminating activities that reflect meaningful and relevant problems/situations.
· Applying the best decisions out of a given situation (e.g. choosing between membership packages offered
by two video rentals/cell phone companies, best time to plant crops that would produce more harvest, etc.).
· Designing games and puzzles which would use systems of linear equations.

54. Appendix
Resources :
ICT Tools
· http://www.deped.gov.ph/iSchool Web Board/Math Web Board
· http://www.deped.gov.ph/iSchool Web Board/skoool.ph
· http://www.deped.gov.ph/e-turo
· http://www.deped.gov.ph/BSE/iDEP
· http://www.pjoedu.wordpress/Philippine Studies/FREE TEXTBOOKS
· http://www.teacherplanet.com
· http://www.pil.ph/innovative teachers leadership award
· http://www.alcob.com/ICT Model School Network
· http://www.APEC Cyber Academy.com
· http://www.globalclassroom.net
· http://www.think.com
· http://www.rubistar.com
· Overhead projector/transparencies
· Computer/PC tablet/Interactive Whiteboard
· Graphing calculator
· CONSTEL CD/DVD materials
· CAI materials in CD/DVD format
· Intel Teach to the Future/ASEAN SchoolNet/FIT-ED/SMART School/Innovative Teachers Leadership
Award/sKwela/iSchool/Learn.ph/eskwela ng bayan modules in PowerPoint format
· Authoring-enabled math storytelling enrichment modules ( e.g. Fibo the Frog Mathemajess’yan, etc.)
· Video clips/tapes
· Interactive digital games/puzzles ( e.g. eDamath, 3D damath gaming, Cartesian coordinate system, etc.)
· E-TV episodes (Knowledge Channel, National Geographic Channel, Discovery Channel, BBC, etc.)
· Online/offline open-source teaching/learning materials
· Student/teacher-made PowerPoint/spreadsheet/word instructional materials with Hyperlinks
· Student/teacher-made webcam/digicam math storytelling materials
· Mathematical simulations
54

55. · Interactive flowchart/concept map/tree diagram/picture problem situation/secret message decoder
· Dgital mathematical cartoon strips/humor/jokes/poster with hyperlinks
· eBooks/eTeachers manual with no hyperlinks
· Digital books/teaching guides with hyperlinks
Manipulatives
· Games and puzzles (e.g. damath, chess-math, sungkamath, sudoku, cross-number puzzle, etc.)
· Geoboard/rubber bands/strings
· Algebra tiles
· Cuisenaire rod
· Dice
· Domino/triangular domino
· Playing cards
· Mathemagic biscuit crackers (e.g. opposite sense in integers, etc.)
Print Materials
· Worksheets
· Workbooks
· Textbooks/supplementary reference materials/teachers manual
· Consumer education/global warming and climate change integration lesson exemplars
· Remedial/enrichment modules (e.g. EASE modules, distance learning/self-learning package, etc.)
· Enrichment math storytelling modules with indigenous cartoon characters (e.g. Max the Matrix & Co., etc.)
· Handouts/Activity Sheets/Cards
· Mathematical post card / mathematical cartoon strips/humors/quotes/jokes/jingle/slogan/themes/tarpaulin
Supplies and Materials
· Papers (graphing, bond, pad, manila, colored)
· School Supplies (Ruler/straightedge, Colored cartolina , Illustration board , Pairs of scissors, Masking tapes
Pentel pen)
55

56. Rubrics
1. Scoring Guide for Problem Solving
Points Criteria
3 Understood the problem, performed the correct
operation/s, and got the correct answer.
2 Understood the problem, performed the correct
operation/s, and got an incorrect answer
1 Attempted to solve the problem, performed an
incorrect operation/s and got an incorrect
answer.
Got the correct answer, but no solutions/wrong
solution.
0 No attempt
3. Scoring Guide for Journal Writing
Points Description
5 Writes a clearly stated main idea, topic and presents supporting details in a logical order. The
journal is written with correct use of conventions (grammar, punctuation, capitalization, and spelling)
56

57. 4
Writes clearly stated main idea, topic and presents supporting details in a logical order. The details
may not be as complete as it could be. The journal is written with generally correct use of
conventions.
3 Writes a clearly stated main idea, topic but presents some unrelated details. There are few errors in
the use of conventions.
2
Writes a main idea, topic but not clearly stated. Details may not be presented in a logical order, or
some of the information may be inaccurate. The journal may include some errors in the use of
conventions.
1 No accurate understanding of topic/subject.
57

58. 4. Scoring Guide for Graphing:
Sample Problem: (Note the correct distribution of rubric points.)
Graph the equation y =2x +1.
Solution: m = 2 ; rise of 2 units and run of 1 unit. 1 point
b = 1
Y
y =2x +1
2 points
X
Points Criteria
3 The graph is correct and Properly labeled
2 Graph is correct, but not labeled properly
1 Graph is incorrect
0 No attempt
58