1. The document outlines a mathematics curriculum covering the real number system over 20 days. It focuses on helping students understand key concepts like rational and irrational numbers through hands-on activities and solving real-life problems. 2. The curriculum is divided into three stages: exploring real numbers through number lines and games, having students firmly grasp operations and order axioms through investigations, and applying the concepts to daily life problems. 3. Student understanding and performance will be assessed through problems they formulate involving real numbers and multiple solution strategies.

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

3. 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 b y describing the
difficulties one can experience in daily life
whenever tedious calculations are done.
Criteria :
Open
Sensitive
Responsive
3

4. 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. gi ve 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.
Gi ve 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
4

5. 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.
5

6. 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 acti vities 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
in vestigating 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.
Gi ve 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.
6

7. 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.
Gi ve 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, Topic : Measurement Time Frame: 25 days
Measurement and Scientific Notation
Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of the key The learner formulates real-life problems involving measurements
concepts of measurements. and solves these using a variety of strategies.
Essential Understanding(s): Essential Question(s):
Physical quantities are measured using different measuring How are different measuring devices useful?
devices. The precision and accuracy of measurement How does one know when a measurement is precise?
depend on the measuring device used. accurate?
The learner will know: The learner will be able to:
• the concept of measurement • use different tools/devices and units of measures.
• the different measuring devices and their respective • cite situations where measuring tools are appropriately used.
uses. • convert units of measure.
• conversion of units of measure. • round off numbers.
• rounding off numbers • cite real life situations where rounding off numbers is applied.
• approximation. • approximate measurement by rounding off to its nearest
7

8. • how to solve problems involving measurements desired value.
using a variety of strategies. • formulate and solve real life problems applying conversion of
units.
Stage 2
Product or Performance Task: Evidence at the level of understanding Evidence at the level of performance
Problems formulated Learner should be able to demonstrate
1. are real –life related understanding of measurement using the Assessment of problems formulated
2. invol ve measurement and six (6) facets of understanding: based on the following suggested criteria:
3. are solved using a variety of • real-life problems
strategies. Explaining how to use the calibration • problems involve measurement
model and find its degree of precision.
Criteria : • problems are solved using a variety of
Thorough strategies
Clear
Accurate Tools: Rubrics for assessment of
Justified problems formulated and solved
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
Revealing Empathy by role-playing the
8

9. 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
b y 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
consideration.
9

10. 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. Sol ving 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
e ven 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.
10

11. 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 acti vity, 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. )
• In vestigate 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.
Activity 21.
Let the students write reaction paper or journal about the different activities being undertaken.
11

12. 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.
Gi ve each group enough time to construct scale models of either houses, toys, bridges, etc. depending on its problem
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.
12

13. 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, Topic: Scientific Notation Time Frame: 5 days
Measurement and Scientific Notation
Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of the key concepts The learner formulates real-life problems involving scientific
of scientific notation. notation and solves these using a variety of strategies.
Essential Understanding(s): Essential Question(s):
Big and small quantities can be expressed conveniently in Wh y are measures of certain quantities expressed in scientific
scientific notation. notation? How?
The learner will know: The learner will be able to:
• numbers that are expressed in scientific notation. • express numbers in scientific notation and vice- versa.
• real life measures where scientific notation is applied. • solve real life problems involving scientific notation.
• the application of scientific notation to different • cite real life situations where scientific notation is applied.
disciplines. • formulate and solve real life problems involving scientific
notation.
Stage 2
Product or Performance Task: Evidence at the level of understanding Evidence at the level of performance
Learner should be able to demonstrate
Problems formulated understanding of scientific notation using Assessment of problems formulated
1. are real –life related the six (6) facets of understanding: based on the following suggested criteria:
2. invol ve scientific notation and real-life problems
3. are solved using a variety of Explaining how big and small quantities
strategies. are expressed in scientific notation. problems involve real numbers using
Criteria: scientific notation
13

14. Thorough
Accurate problems are solved using a variety of
Justified strategies
Interpreting meaning of scientific Tools : Rubrics for assessment of
notation by considering the size of an problems formulated and solved
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
and small quantities.
Criteria:
Sensitive
Perceptive
14

15. 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 = ?
103 = 1 000
102 = 100
101 = 10
100 = 1
15

16. 10-1 = 1
10
10-2 = 1
100
10-3 = 1
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.
Study the table below.
16

17. Set A Set B
Column I Column II Column I Column II
Decimal Form Scientific Scientific
Notation Decimal Form Notation
8 8 x 100 14.325 1.4325 x 101
81 8.1 x 101 143.25 1.4325 x 102
814 8.14 x 102 1 432.5 1.4325 x 103
8 143 8.143 x 103
Set C
Column I Column II
Decimal Form Scientific Notation
0.3768 3.768 x 10-1
0.03768 3.768 x 10-2
0.003768 3.768 x 10-3
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?
17

18. 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.
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.
• In vestigate 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
18

19. Scoring Guide (Rubric) 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
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.).
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?
19

20. 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
20

21. Quarter II : Algebraic Expressions, First-Degree Topic : Algebraic Expressions Time Frame: 25 days
Equations and Inequalities in One Variable
Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of the key concepts The learner models situations using oral, written, graphical and
of algebraic expressions. algebraic methods to solve problems involving algebraic
expressions.
Essential Understanding(s): Essential Question(s):
Algebraic expressions represent patterns and relationships that Why are algebraic expressions useful?
guide us in understanding how certain problems can be solved.
The learner will know: The learner will be able to:
• translation of verbal phrases to mathematical • translate verbal phrases to mathematical expressions and
expressions and vice-versa vice-versa
• laws on integer exponents • simplify algebraic expressions using the laws on integer
• operations of algebraic expressions exponents
• rules on finding special products • perform fundamental operations on algebraic expressions
• types of special products • explore the product of two binomials and search for
• special products of two binomials patterns
• relationships between special products and factors • identify special products
• complete factorization of polynomials • find special products of two binomials
• applications of special products and factors in solving • discover the relationships between special products and
real life problems 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
Situations modeling the use of oral, The learner should be able to demonstrate performance
written, graphical and algebraic methods understanding of algebraic expressions using the Performance assessment of
to solve problems involving algebraic six (6) facets of understanding: situations involving algebraic
21

22. expressions expressions based on the
Explaining how the language of mathematics is following suggested criterion:
used to show /describe real-life situations. • Use oral, written, graphical
Criteria : and algebraic methods in
Clear modeling situations
Coherent
Justified Tools :
Rubrics of situations modeling
Interpreting representations of mathematical the use of oral, written, graphical
situations and algebraic methods
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.
Criteria :
Open
Sensitive
22

23. 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
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
23

24. 3. Deepen
Acti vities 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.
• 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
4. Transfer
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.)
Resources
See Appendix
24

25. Quarter II : Algebraic Expressions, First-Degree Topic : First-Degree Equations and Time Frame: 25 days
Equations and Inequalities in One Variable Inequalities in One Variable
Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of the key The learner models situations using oral, written, graphical and
concepts of first-degree equations and inequalities in one algebraic methods to solve problems involving first-degree equations
variable. and inequalities in one variable.
Essential Understanding(s): Essential Question(s):
Real-life problems where certain quantities are unknown How can we use equations and inequalities to solve real life problems
can be solved using equations and inequalities in one where certain quantities are unknown?
variable.
The learner will know: The learner will be able to:
• mathematical expressions, equations and • differentiate mathematical expressions from equations and
inequalities inequalities
• linear equations and inequalities • identify and describe linear equations and inequalities in one
• properties of equations and inequalities variable
• applications of first-degree equations and • give examples of linear equations and inequalities in one variable
inequalities • 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
Product or Performance Task: Evidence at the level of understanding Evidence at the level of
Situations modeling the use of oral, The learners should be able to demonstrate performance
written, graphical and algebraic methods understanding of first-degree equations and Performance assessment of
to solve problems involving first-degree inequalities using the six (6) facets of situations involving first-degree
equations and inequalities in one variable understanding: equations and inequalities in one
25

26. Explaining the properties of first-degree variable based on the following
equations and inequalities in one variable. suggested criterion.
Criteria :
Clear • Use oral, written, graphical and
Coherent algebraic methods in modeling
Justified situations
Interpreting mathematical conjectures and Tools :
arguments involving first-degree equations and Rubrics of situations modeling the
inequalities in one variable use of oral, written, graphical and
Criteria : algebraic methods
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 b y describing difficulties one
can experience in daily life whenever tedious
calculations are done without using the
concepts of first-degree equations and
inequalities in one variable
Criteria :
Open
26

27. 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
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
2. Firm Up
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
• Solving problems involving first-degree equations and inequalities in one variable
27

28. 3. Deepen
Acti vities 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
inequalities in one variable
• 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 valued by
every learner
4. Transfer
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.)
Resources
See Appendix
28

29. Quarter III : Rational Algebraic Expressions, Linear Topic : Rational Algebraic Time Frame: 25 days
Equations and Inequalities in Two Variables Expressions
Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of key concepts of The learner presents solutions to problems involving rational
rational algebraic expressions. algebraic expressions using numerical, physical, and verbal
mathematical models or representations.
Essential Understanding(s): Essential Question(s):
Simplifying rational algebraic expressions involve factorization How can rational algebraic expressions be simplified?
and operations similar to operations on numerical fractions.
The learner will know: The learner will be able to:
• fractions in simplest form; • explore problems and describe results using numerical,
• operations on fractions; physical, and verbal mathematical models or
• rational algebraic expressions in simplest form; representations;
• operations on rational algebraic expressions; and • use his/her reading, listening and visualizing skills to
• applications of rational algebraic expressions. 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: Evidence at the level of understanding Evidence at the level of
Solutions to problems involving rational The learner should be able to demonstrate performance
algebraic expressions are presented understanding by covering the six (6) facets of Assessment of presentation of
using numerical, physical, and verbal understanding: solutions to problems involving
mathematical models or representations. rational algebraic expressions
Explaining b y justifying how one’s answer is based on the suggested
changed to simplest form. criterion:
Criteria :
Clear • the use of numerical,
29

30. Coherent physical, and verbal
Justified mathematical models or
representations.
Interpreting how best procedures for simplifying
rational expressions are determined. Tools :
Criteria : Rubrics for assessment of
Illustrative solutions to problems
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
Manifesting Self-Knowledge in recognizing the
best solution to a given situation involving rational
expressions.
Criteria:
Reflective
Insightful
30

31. Stage 3
Teaching/Learning Sequence :
1. Explore
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.
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?
31

32. a. 24 b. 35 c. − 32
36 56 48
d. 45 e. 12
−
54 − 18
6. When do you say that a fraction is in its simplest form?
2. Firm Up
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.
e. perform operations on rational algebraic expressions.
f. explain results and make the necessary justification of each steps used by stating the mathematical properties
applied.
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.
32

33. a. 4x f. x + 5 ;x=5
x − 2 x −5
b. x2 − 4 g. x2 −5x + 6
x2 + 4x + 4 (x − 2 )(x − 3 )
c. 3 h. 3 x 4 + 12 x 2 − 6 x
7 3x
d. 5 i. 4
+ 4
2x x
e. 8 x 3 − 27 j. 3x
3
2x −3 x−3
2. Let the students discuss the results of the activity.
3. Deepen
Acti vities 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.)
Activity 3:
1. Show the following are rational algebraic expressions.
a. 4x b. 2x + 4 c. x2 − 9 d. x 2 + 7 x + 12
x −2 2 x2 + 6x+9 2 x 2 + 8x
Ask the following questions:
33

34. 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 e xpressions?
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
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.
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 Topic : Linear Equations and Time Frame: 25 days
Equations and Inequalities in Two Variables Inequalities in Two Variables
34

35. Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of key concepts of The learner presents solutions to problems involving linear
linear equations and inequalities in two variables. equations and equalities in two variables using numerical,
physical, and verbal mathematical models or representations
Essential Understanding(s): Essential Question(s):
Linear equations show constant rate of change. How are linear equations used to communicate relationships
between quantities?
Graphs of linear equations show trends which help predict
outcomes and make decisions How does one know an outcome is favorable? How can
mathematics help one find out?
The learner will know: The learner will be able to:
• coordinate plane and the terminologies associated with • give the exact location of a point, person or object using maps,
it. navigation devices, etc.
• graph of linear equations in two variables • investigate graphs of linear equations in two variables with
• equation of a linear equation in two variables deductive arguments and evidences.
• application of linear equations in two variables • solve linear equations in two variables graphically .
• graph of a linear inequality in two variables • 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: Evidence at the level of understanding Evidence at the level of
Solutions to problems involving linear The learner should be able to demonstrate performance
equations and inequalities in two variables understanding by covering the six (6) facets of
are presented using numerical, physical, understanding: Assessment of presentation
and verbal mathematical models or of solution to problems
representations. Explaining how a statement is translated into involving linear equations and
mathematical symbols inequalities in two variables
Criteria based on the suggested
Clear criterion:
Coherent
35

36. Interpreting possible relationships of rates of change • Use of numerical, physical,
in a given set of data. and verbal mathematical
Criteria: models or representations.
Revealing
Illustrative
Tools :
Applying mathematical thinking and modeling to solve Rubrics for assessment of
problems in other disciplines such as art, music, solutions to problems
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 b y sharing insights one
may have about how math can help make reasonable
judgments and predictions.
Criteria :
Reflective
Insightful
36

37. 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.
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.
Activity 1:
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 1 2 3 4 5
Row 2
Row 3 L
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.
37

38. 2. Firm Up
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.
Activity 2: Treasure Hunt with Slopes
On a grid paper, mark points that would lead to the treasure.
●
Start here ●
38

39. 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
2. 1 6. -3 10. − 1
4 3
3. − 2 7. 1 11. − 3
5 3 5
4. 6 8. − 4 12. 6
7
Activity 3 : Group Acti vity
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
5
temperatures using the formula oC = (F – 32).
9
39

40. a. Using a thermometer, demonstrate to the class how each scale corresponds to the other.
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.
9
d. Analyze the formula in converting 0F to 0C, F = C + 32. If in item b we represented x as C and y as F, then transform the
5
equation using x and y.
9
e. In the transformation recognize the role of constant numbers and 32 in our graph. What do these numbers represent?
5
f. Given these two formulas, which would you consider using?
3. Deepen
Acti vities 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.
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.
40

41. Activity 6:
In the given graphs below, identify parallel and intersecting lines.
a
b
c
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.
41

42. No. of hours (x) 5
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
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.
42

43. 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
43

44. Quarter IV: Systems of Linear Equations and Topic : Systems of Linear Equations Time Frame: 25 days
Inequalities in Two Variables and Inequalities in Two Variables
Stage 1
Content Standard: Performance Standard:
The learner demonstrates understanding of key concepts of The learner creates situations/ problems in real-life involving
systems of linear equations and inequalities in two variables. systems of linear equations and inequalities in two variables, and
solves these by applying a variety of strategies
Essential Understanding(s): Essential Question(s):
Unknown numbers in certain real-life problems may be derived How is knowledge of systems of linear equations and inequalities
from solving systems of linear equations and inequalities in two in two variables used to solve real life problems?
variables.
The learner will know: The learner will be able to:
• graphical solution of systems of linear equations and • explain thoroughly how systems of linear equations and
inequalities in two variables. inequalities in two variables can be solved graphically and
• algebraic solutions of systems of linear equations and in algebraically.
two variables. • graph with accuracy the solutions of a system of linear
• applications of systems of linear equations and equations and inequalities in two variables.
inequalities in two variables in problem solving • apply a variety of strategies to solve problems involving
• graph a system of linear inequalities and inequalities in systems of linear equations and inequalities in two
two variables variables.
• graph with accuracy the solution set of a system of linear
equations and inequalities in two variables.
Stage 2
Product or Performance Task: Evidence at the level of understanding Evidence at the level of
Situations/ Problems created are drawn The learner should be able to demonstrate performance
from real-life and are solved by applying understanding by covering the six (6) facets of Assessment of situations
a variety of strategies. understanding: problems created based on the
following suggested criteria:
Explaining and presenting a mathematical analysis • problems are drawn from real-
of graphs. life;
Criteria : • problems involve systems of
Clear linear equations and inequalities
Coherent in two variables; and
44

45. Justified
• problems are solved using a
Interpreting the significance of the way graphs variety of strategies.
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
45

46. 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.
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.
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.
46

47. 2. Firm Up
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.
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.
x + y = 10 x − y = − 1 x + y = 3
Group 1:  Group 2:  Group 3: 
x − y = 6 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?
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?
47

48. 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
Acti vities 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.
Activity 3:
Technology Integration:
Teach the students how to use the graphics calculator to investigate the graph of a given system of linear equations.
48

49. 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.
x + y = 4 2 x = − y + 3
a.  b. 
3 x + 3 y = 12  y = −x + 1
3 x − 5 y = 10
  y = 2x − 3
c.  3 d. 
 y − 1 = 5 (x − 2)

 y = 2x + 1
2. Analyze the graphs. What kind of system of linear equations is represented by each graph?
a. y b. 5
y
4
3
4 y
2
y 1
2 x -4 -2 0 2 x 4
-1
x -2
-4 -2 0 2 4 6 8 10
x -3
-2
-4
c. d.
y
4
y 4
y
y 2
2
x 0
-6 -4 -2 2 4
x
x -4 -2 0 2
x
4
-2
-2
-4
-4
49

50. 4. Transfer
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.
Activity 4:
Allow students to design competitions, puzzles, games, etc.
Resources:
See Appendix
50

51. 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 C yber 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
• Interactive flowchart/concept map/tree diagram/picture problem situation/secret message decoder
• Dgital mathematical cartoon strips/humor/jokes/poster with hyperlinks
51

52. • 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/Acti vity 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, Ma sking tapes
Pentel pen)
52

53. 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
Writes a clearly stated main idea, topic and presents supporting details in a logical order. The
5
journal is written with correct use of conventions (grammar, punctuation, capitalization, and spelling)
Writes clearly stated main idea, topic and presents supporting details in a logical order. The details
4 may not be as complete as it could be. The journal is written with generally correct use of
conventions.
Writes a clearly stated main idea, topic but presents some unrelated details. There are few errors in
3
the use of conventions.
Writes a main idea, topic but not clearly stated. Details may not be presented in a logical order, or
2 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.
53

54. 4. Scoring Guide for Graphing:
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
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
54