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# MATH I TEACHER'S GUIDE (SEC)

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### MATH I TEACHER'S GUIDE (SEC)

1. 1. MATHEMATICS IGeneral 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. Quarter I: Real Number System, Topic : Real Number System Time Frame: 20 daysMeasurement and Scientific Notation Stage 1Content Standard: Performance Standard:The learner demonstrates understanding of key concepts of The learner formulates real life problems involving real numbersreal 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level of performanceProblems formulated Learner should be able to demonstrate Assessment of problems formulated1. 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 problems3. 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. 3. ClearInterpreting the differences andsimilarities between rational and irrationalnumbers. Criteria : Thorough Illustrative CreativeApplying a variety of techniques insolving daily life problems. Criteria : Appropriate Practical Accurate RelevantDeveloping Perspective on the types ofreal numbers. Criteria : Perceptive Open-minded Sensitive ResponsiveShowing Empathy b y describing thedifficulties one can experience in daily lifewhenever tedious calculations are done. Criteria : Open Sensitive Responsive 3
4. 4. Manifesting Self-knowledge by assessing how one can give his/her best solution to a problem/situation. Criteria : Reflective Responsive Relevant Stage 3Teaching/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. 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. 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. 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 AppendixQuarter I : Real Number System, Topic : Measurement Time Frame: 25 daysMeasurement and Scientific Notation Stage 1Content Standard: Performance Standard:The learner demonstrates understanding of the key The learner formulates real-life problems involving measurementsconcepts 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. 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level of performanceProblems 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. 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 3Teaching/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. 10. Activity 14. Give each student enough time, like 5-10 minutes to think of situations that describe the appropriate use and choice ofthe different measuring devices. Then, let them discuss the situations they listed with their group members. Let each group taketurns in presenting their consolidated story. Activity 15. Let each group present authentic situations showing the evolution of the different measuring devices. Thesepresentations 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. 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. 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. 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 AppendixQuarter 1 : Real Number System, Topic: Scientific Notation Time Frame: 5 daysMeasurement and Scientific Notation Stage 1Content Standard: Performance Standard:The learner demonstrates understanding of the key concepts The learner formulates real-life problems involving scientificof 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 scientificscientific 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level of performance Learner should be able to demonstrateProblems 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. 14. Thorough Accurate problems are solved using a variety of Justified strategiesInterpreting meaning of scientific Tools : Rubrics for assessment ofnotation by considering the size of an problems formulated and solvedatom, distances of planets, etc. Criteria : Illustrative Meaningful JustifiedApplying a variety of techniques inposing and solving daily life problemsinvolving very large or very small numbersexpressed in scientific notation. Criteria : Appropriate Practical AccurateManifesting Self-knowledge by showingthe usefulness of scientific notation insolving a problem. Criteria: Reflective ResponsiveShowing Empathy to persons whoencounter difficulties in expressing bigand small quantities. Criteria: Sensitive Perceptive 14
15. 15. Developing Perspective on other ways to express big and small numbers. Criteria: Appropriate Practical Stage 3Teaching/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. 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. 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 10-3Questions: 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. 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. 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 attemptB. 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. 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. 21. Quarter II : Algebraic Expressions, First-Degree Topic : Algebraic Expressions Time Frame: 25 days Equations and Inequalities in One Variable Stage 1Content Standard: Performance Standard:The learner demonstrates understanding of the key concepts The learner models situations using oral, written, graphical andof 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level ofSituations modeling the use of oral, The learner should be able to demonstrate performancewritten, graphical and algebraic methods understanding of algebraic expressions using the Performance assessment ofto solve problems involving algebraic six (6) facets of understanding: situations involving algebraic 21
22. 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. 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 3Teaching/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. 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. 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 1Content Standard: Performance Standard:The learner demonstrates understanding of the key The learner models situations using oral, written, graphical andconcepts of first-degree equations and inequalities in one algebraic methods to solve problems involving first-degree equationsvariable. 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 problemscan 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level ofSituations modeling the use of oral, The learners should be able to demonstrate performancewritten, graphical and algebraic methods understanding of first-degree equations and Performance assessment ofto solve problems involving first-degree inequalities using the six (6) facets of situations involving first-degreeequations and inequalities in one variable understanding: equations and inequalities in one 25
26. 26. Explaining the properties of first-degree variable based on the followingequations and inequalities in one variable. suggested criterion. Criteria : Clear • Use oral, written, graphical and Coherent algebraic methods in modeling Justified situationsInterpreting mathematical conjectures and Tools :arguments involving first-degree equations and Rubrics of situations modeling theinequalities in one variable use of oral, written, graphical and Criteria : algebraic methods Illustrative MeaningfulApplying first-degree equations and inequalitiesin one variable in daily life situations Criteria : Appropriate Practical RelevantDeveloping Perspective on the various ways ofwriting first-degree equations and inequalities inone variable in solving a problem Criteria : Critical Insightful CredibleShowing Empathy b y describing difficulties onecan experience in daily life whenever tediouscalculations are done without using theconcepts of first-degree equations andinequalities in one variable Criteria : Open 26
27. 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 3Teaching/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. 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. 29. Quarter III : Rational Algebraic Expressions, Linear Topic : Rational Algebraic Time Frame: 25 days Equations and Inequalities in Two Variables Expressions Stage 1Content Standard: Performance Standard:The learner demonstrates understanding of key concepts of The learner presents solutions to problems involving rationalrational 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level ofSolutions to problems involving rational The learner should be able to demonstrate performancealgebraic expressions are presented understanding by covering the six (6) facets of Assessment of presentation ofusing numerical, physical, and verbal understanding: solutions to problems involvingmathematical 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. 30. Coherent physical, and verbal Justified mathematical models or representations.Interpreting how best procedures for simplifyingrational expressions are determined. Tools : Criteria : Rubrics for assessment of Illustrative solutions to problems Creative AccurateApplying the appropriate operations in simplifyingrational expressions. Criteria : Appropriate AccurateDeveloping Perspective on how to choose the bestsolution in simplifying rational expressions Criteria : Credible InsightfulShowing Empathy on people’s difficulties inperforming operations involving rationalexpressions. Criteria : Perceptive Responsive SensitiveManifesting Self-Knowledge in recognizing thebest solution to a given situation involving rationalexpressions. Criteria: Reflective Insightful 30
31. 31. Stage 3Teaching/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. 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. 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. 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 ofrational 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 GamesResources: See AppendixQuarter 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. 35. Stage 1Content Standard: Performance Standard:The learner demonstrates understanding of key concepts of The learner presents solutions to problems involving linearlinear equations and inequalities in two variables. equations and equalities in two variables using numerical, physical, and verbal mathematical models or representationsEssential 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 predictoutcomes 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level ofSolutions to problems involving linear The learner should be able to demonstrate performanceequations and inequalities in two variables understanding by covering the six (6) facets ofare presented using numerical, physical, understanding: Assessment of presentationand verbal mathematical models or of solution to problemsrepresentations. 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. 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 ofproblems in other disciplines such as art, music, solutions to problemsscience and business. Criteria: Practical Appropriate AccurateDeveloping Perspective on the most likely outcomesthat may result from trends shown in graphs Criteria: Credible insightfulShowing Empathy on people experiencing difficultiesin making decisions without the help of graphs andlinear equations Criteria: Perceptive Responsive SensitiveManifesting Self-Knowledge b y sharing insights onemay have about how math can help make reasonablejudgments and predictions. Criteria : Reflective Insightful 36
37. 37. Stage 3Teaching/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. 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. 39. Using the definition of slope, trace the path using the slopes listed below. A correct solution will trace the route to thetreasure. 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 7Activity 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 analysisActivity 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. 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. 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. 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. Heres 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. 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. 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 1Content Standard: Performance Standard:The learner demonstrates understanding of key concepts of The learner creates situations/ problems in real-life involvingsystems 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 strategiesEssential Understanding(s): Essential Question(s):Unknown numbers in certain real-life problems may be derived How is knowledge of systems of linear equations and inequalitiesfrom 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 2Product or Performance Task: Evidence at the level of understanding Evidence at the level ofSituations/ Problems created are drawn The learner should be able to demonstrate performancefrom real-life and are solved by applying understanding by covering the six (6) facets of Assessment of situationsa 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. 45. Justified • problems are solved using aInterpreting the significance of the way graphs variety of strategies.relate with each other.Criteria: Illustrative MeaningfulApplying the appropriate solution that wouldproduce best results. Criteria : Appropriate Practical UsefulDeveloping Perspective on the different possibleoutcomes illustrated by graphs/equations. Criteria : Critical InsightfulShowing Empathy on problems that may resultwhen systems of linear equations are not properlysolved and the unknown number is not correctlydetermined Criteria : Sensitive AuthenticManifesting Self-Knowledge on the impact ofindividual accuracy in solving problems on systemsof linear equations Criteria : Insightful Relevant 45
46. 46. Stage 3Teaching/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. 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. 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