math teaching guide relc jan 29 2010
TRANSCRIPT
MATHEMATICS I
General Standard: The learner demonstrates understanding of key concepts and principles of number and number sense as applied to measuring, estimating, graphing, solving equations and inequalities, communicating mathematically and solving problems in real life.
Quarter I: Real Number System, Measurement and Scientific Notation
Topic: Real Number System Time Frame: 20 days
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Stage 1Content Standard: The learner demonstrates understanding of the key concepts of real number system.
Performance Standard: The learner formulates real life problems involving real numbers and solves these using a variety of strategies.
Essential Understanding(s):Daily tasks involving measurement, conversion, estimation and scientific notation make use of real numbers.
Essential Question(s):How useful are real numbers?
The learner will know:
• the real number system• rational and irrational numbers• the importance of order axioms • fundamental operations with real numbers• the application of real numbers to daily life.
The learner will be able to:• apply real numbers in a variety of ways to other disciplines.• identify/give examples of rational and irrational numbers• illustrate rational and irrational numbers in practical
situations• use the appropriate symbolic notation to illustrate the
order axioms.• cite examples/situations where order axiom is applied.• perform the sequence of operations with real numbers
properly.• solve problems in other disciplines such as science, art,
agriculture, etc.
Stage 2Product or Performance Task:Problems formulated1.are real –life related2.involve real numbers, and3.are solved using a variety of strategies.
Evidence at the level of understandingLearner should be able to demonstrate understanding of the real number system using the six (6) facets of understanding:
Explaining how numbers are expressed in different ways. Criteria:
Evidence at the level of performanceAssessment of problems formulated based on the following suggested criteria: real-life related problems problems involve real numbers. problems are solved using a variety of
strategies
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Thorough Coherent Clear
Interpreting similarities between rational and irrational numbers. Criteria: Thorough Illustrative Creative
Applying a variety of techniques in solving daily life problems. Criteria: Appropriate Practical Accurate Relevant
Developing Perspective on the types of real numbers. Criteria: Perceptive Open-minded Sensitive Responsive
Showing Empathy by describing the difficulties one can experience in daily life whenever tedious calculations are done.
Criteria: Open Sensitive
Tools: Rubrics for assessment of problems formulated and solved
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Responsive
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. ExploreInitially, begin with some interesting and challenging exploratory activities on real 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 daily life context.
a. Varied activities in the form of a game, puzzle or storytelling 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 Citing situations where real numbers are applied
b. Assessment by answering accurately the activity sheets containing HOTS questions on real numbers (See attached sheet as sample) c. Journal writing on real number system and its applications to real life. (See attached sheet as sample)
2. Firm UpThese are the enabling activities/experiences that the learner will have to go through to validate understanding on real numbers during the activities in the exploratory phase. These would answer some misconceptions on real numbers that have been encountered in daily life situations.
a. Conduct an investigation considering the following steps:
Giving a list of different numbers Changing the form of the given set of numbers by doing the basic operations and simplifying the results. Analyzing/Observing the results. Classifying numbers as to their types.
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Classifying numbers into rational or irrational. Preparing oral/written reports about the investigation conducted.
b. Give more exercises which may be in problem form, games, puzzles, etc. c. Perform fundamental operations on real numbers and classify results.
3. DeepenActivities in this stage shall provide opportunity for differentiated instruction for the learner to further reflect, revisit, revise and rethink. Moreover, the learner shall express his/her understanding on real numbers and engage in meaningful self-evaluation.
a. Explaining thoroughly the difference between rational and irrational numbers by giving several examples.b. Investigating patterns leading to rational or irrational numbers (both manually and with the use of calculators).c. Identifying the difference between rational and irrational numbers.d. Generalizing and writing a report of what has been discovered about real numbers.
4. TransferApplications of learner’s understanding are demonstrated through culminating activities that reflect meaningful and relevant problems/situations.
Formulating daily life problems involving real numbers using varied activities (e.g. storytelling, simulation, role-playing, flowcharting, etc.)
Constructing scale models of toys, houses, bridges, etc. indicating the use of real numbers.
Resources:
See Appendix
Quarter I : Real Number System, Measurement and Scientific Notation
Topic: Measurement Time Frame: 25 days
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Stage 1Content Standard: The learner demonstrates understanding of the key concepts of measurements.
Performance Standard: The learner formulates real-life problems involving measurements and solves these using a variety of strategies.
Essential Understanding(s):Physical quantities are measured using different measuring devices. The precision and accuracy of measurement depend on the measuring device used.
Essential Question(s): How are different measuring devices useful? How does one know when a measurement is precise? accurate?
The learner will know:• the concept of measurement• the different measuring devices and their respective
uses.• conversion of units of measure.• rounding off numbers• approximation.• how to solve problems involving measurements using
a variety of strategies.
The learner will be able to:• use different tools/devices and units of measures.• cite situations where measuring tools are appropriately
used.• convert units of measure.• round off numbers.• cite real life situations where rounding off numbers is
applied.• approximate measurement by rounding off to its
nearest desired value.• formulate and solve real life problems applying
conversion of units.Stage 2
Product or Performance Task:Problems formulated
1. are real –life related 2. involve measurement and
3. are solved using a variety of strategies.
Evidence at the level of understandingLearner should be able to demonstrate understanding of measurement using the six (6) facets of understanding:
Explaining how to use the calibration model and find its degree of precision. Criteria: Thorough Clear
Evidence at the level of performance
Assessment of problems formulated based on the following suggested criteria: real-life problems problems involve measurement
problems are solved using a variety of strategies
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Accurate Justified
Interpreting through story telling situations that describe the appropriate use and choice of measuring devices. Criteria: Illustrative Accurate Justified Significant
Applying a variety of techniques in posing and solving daily life problems involving measurement Criteria: Appropriate Practical
Revealing Empathy by role-playing the uses of the primitive measuring devices for the people who invented them and discuss how they got accurate results. Criteria: Perceptive Open
Manifesting Self-knowledge by assessing how one can give his/her best solution to a problem/situation on measurement. Criteria: Reflective Responsive
Tools: Rubrics for assessment of problems formulated and solved (See attachment)
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Stage 3Teaching/Learning Sequence:
1. ExploreInitially, begin with some interesting and challenging exploratory activities on measurements 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. Group activities Identifying and using the different measuring devices through games, puzzles, storytelling, etc. Citing real life situations where these measuring devices are used through role playing, simulations, storytelling,
etc. b. Creative group presentations of authentic situations showing the evolution of the different measuring devices. c. Giving oral or written reactions on the group presentations.
2. Firm UpThese are the enabling activities/experiences that the learner will have to go through to validate understanding on measurements during the activities in the exploratory phase. These would answer some misconceptions on measurements that have been encountered in real life situations.
a. The learner shall conduct varied activities. Using the given measuring instruments, find the measures of classroom table, blackboard, window frames, etc. 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, puzzles, games, simulation, storytelling etc. c. Performing experiments/activities that will verify formulas for finding areas of plane geometric figures and volumes of solid figures. d. Solving teacher-made problems about measurements.
3. Deepen Activities in this stage shall provide opportunity for differentiated instruction for the learner to further reflect, revisit, revise and rethink. Moreover, the learner shall express his/her understanding on measurements and engage in meaningful self-
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evaluation.
Explaining thoroughly the process/procedure undertaken in every activity on measurement, including the computation part.
Identifying objects whose area/volume can be found using formulas. Exploring the different possibilities of finding the measures of objects with irregular shapes (e.g. football, star,
etc.). Investigating the relationship between the number of square units/cubic units in a given figure and the
area/volume of the given figure using concrete models. Journal writings on the activities undertaken.
4. Transfer
Applications of learner’s understanding are demonstrated through culminating activities that reflect meaningful and relevant problems/situations.
Formulating and solving a daily life situations/problems involving measurements. Writing a report on what he/she has learned about measurement. Improvising instruments used in measuring objects of different forms. Creating miniature models (e.g. dream house, school, thermometer, weather vane, etc.).
Resources:
See Appendix
Quarter 1 : Real Number System, Measurement and Scientific Notation
Topic: Scientific Notation Time Frame: 5 days
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Stage 1Content Standard: The learner demonstrates understanding of the key concepts of scientific notation.
Performance Standard: The learner formulates real-life problems involving scientific notation and solves these using a variety of strategies.
Essential Understanding(s):Big and small quantities can be expressed conveniently in scientific notation.
Essential Question(s): Why are measures of certain quantities expressed in scientific notation? How?
The learner will know: numbers that are expressed in scientific notation. real life measures where scientific notation is applied. the application of scientific notation to different
disciplines.
The learner will be able to: express numbers in scientific notation and vice- versa. solve real life problems involving scientific notation. cite real life situations where scientific notation is
applied. formulate and solve real life problems involving
scientific notation.Stage 2
Product or Performance Task:
Problems formulated 1. are real –life related 2. involve scientific notation and
3. are solved using a variety of strategies.
Evidence at the level of understandingLearner should be able to demonstrate understanding of scientific notation using the six (6) facets of understanding:
Explaining how big and small quantities are expressed in scientific notation. Criteria: Thorough Accurate Justified
Interpreting meaning of scientific notation by considering the size of an atom, distances of planets, etc. Criteria:
Evidence at the level of performance
Assessment of problems formulated based on the following suggested criteria:
real-life problems
problems involve real numbers using scientific notation
problems are solved using a variety of strategies
Tools: Rubrics for assessment of problems formulated and solved
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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
Developing Perspective on other ways to express big and small numbers. Criteria: Appropriate Practical
Stage 3
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Teaching/Learning Sequence:
1. Explore
Initially, begin with some interesting and challenging exploratory activities on scientific notations 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. Group activities such as games, puzzles, storytelling, simulation, role playing, etc. can be used in identifying very big and very small numbers (e.g. 1 000 000 000, 23 000 000 000, 0.000000012, 0.0000001, etc.); recognizing and finding patterns from a given set of numbers; expressing big and small numbers in scientific notations; and citing real life situations when/where scientific notations can be used.
b. Creative group presentations of authentic situations where scientific notations are used. c. Oral and written reactions/comments to the presentations.
2. Firm UpThese are the enabling activities/experiences that the learner will have to go through to validate understanding on scientific notations during the activities in the exploratory phase. These would answer some misconceptions on scientific notations that have been encountered in real life situations. a. The learner shall answer
activity sheets on expressing numbers in scientific notation; and exercises involving fundamental operations using scientific notation.
b. Giving more exercises which may be in problem form, games, puzzles, storytelling, role playing, simulation, etc. c. Solving problems involving scientific notation.
3. DeepenActivities in this stage shall provide opportunity for differentiated instruction for the learner to further reflect, revisit, revise and rethink. Moreover, the learner shall express his/her understanding on scientific notations and engage in meaningful self-evaluation.
Explaining thoroughly the process/procedure undertaken in every activity, including the computation part. Investigating the procedure used in expressing numbers in scientific notation. Journal writing on the usefulness of scientific notations.
4. TransferApplications of learner’s understanding are demonstrated through culminating activities that reflect meaningful and relevant
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problems/situations. Formulating and solving situations/problems that will make use of scientific notation through games, puzzles,
simulations, storytelling, role playing, etc. Writing a report explaining the advantages/disadvantages of using scientific notation. Developing non-linear powerpoint presentation expressing distances, in scientific notation, of planets from the
sun, size of atoms, molecules, etc. using hyperlink.Resources:
See Appendix
Quarter II : Algebraic Expressions, First-Degree Equations and Inequalities in One Variable
Topic: Algebraic Expressions Time Frame: 25 days
Stage 1Content Standard: The learner demonstrates understanding of the key concepts of algebraic expressions.
Performance Standard: The learner models situations using oral, written, graphical and algebraic methods to solve problems involving algebraic expressions.
Essential Understanding(s):Algebraic expressions represent patterns and relationships that guide us in understanding how certain problems can be solved.
Essential Question(s):Why are algebraic expressions useful?
The learner will know: translation of verbal phrases to mathematical
expressions and vice-versa laws on integer exponents operations of algebraic expressions rules on finding special products types of special products special products of two binomials relationships between special products and factors complete factorization of polynomials applications of special products and factors in solving
real life problems
The learner will be able to: translate verbal phrases to mathematical expressions and
vice-versa simplify algebraic expressions using the laws on integer
exponents perform fundamental operations on algebraic expressions explore the product of two binomials and search for
patterns identify special products find special products of two binomials discover the relationships between special products and
factors find the complete factorization of polynomials apply factoring polynomials in solving real life problems
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Stage 2Product or Performance Task:Situations modeling the use of oral, written, graphical and algebraic methods to solve problems involving algebraic expressions
Evidence at the level of understandingThe learner should be able to demonstrate understanding of algebraic expressions using the six (6) facets of understanding:
Explaining how the language of mathematics is used to show /describe real-life situations. Criteria: Clear Coherent Justified
Interpreting representations of mathematical situations Criteria: Illustrative Meaningful
Applying algebraic expressions in daily life situations Criteria: Appropriate Practical Relevant
Developing Perspective on the various ways of writing algebraic expressions and solving a problem Criteria: Critical Insightful Credible
Evidence at the level of performancePerformance assessment of situations involving algebraic expressions based on the following suggested criterion:
Use oral, written, graphical and algebraic methods in modeling situations
Tools:Rubrics of situations modeling the use of oral, written, graphical and algebraic methods
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Showing Empathy to persons who encounter difficulties in the lesson. Criteria: Open Sensitive Responsive
Manifesting Self-knowledge by discussing the best and most effective strategies that one has found for solving problems Criteria: Insightful Clear Coherent
Stage 3Teaching/Learning Sequence
1. ExploreInitially, begin with some interesting and challenging exploratory activities on algebraic expressions 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. Playing “Guess my rule” game and writing mathematical expression for the rule.b. Asking students to surf the internet and look for similar games which they can share to the class.c. Providing students with teacher-made worksheets on translating mathematical expressions to English phrases and
vice-versa. d. Asking students to give their own English phrases and translate them to mathematical expressions and vice-versa.e. Completing teacher-made activity sheets on: evaluating algebraic expressions and addition and subtraction of
algebraic expressions.f. Investigating relationships among integer exponents.g. Manipulating algebra tiles to illustrate the product of two algebraic expressionsh. Finding the product of two algebraic expressions
2. Firm Up
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These are the enabling activities/experiences that the learner will have to go through to validate understanding on algebraic expressions during the activities in the exploratory phase. These would answer some misconceptions on algebraic expressions that have been encountered in real life situations.
Activities such as games, puzzles, manipulative, storytelling, simulation, role playing, etc. can be used in simplifying algebraic expressions,
performing operations on algebraic expressions, finding special products, and
factoring (the reverse process of finding the product).
3. DeepenActivities 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 on algebraic expressions and engage in meaningful self-evaluation.
Summarizing the steps in performing the fundamental operations on algebraic expressions. Citing situations in the environment where the concepts of algebraic expressions and operations are applied. Writing journals on how knowledge of algebraic expressions help in finding solutions to challenging computations.
4. TransferApplication of learner’s understanding on algebraic expressions is demonstrated through culminating activities that reflect relevant and authentic problems/situations.
Applying special products and factors in real life problems (e.g. business, engineering, etc.). 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 AppendixQuarter II : Algebraic Expressions, First-
Degree Equations and Inequalities in One Variable
Topic: First-Degree Equations and Inequalities in One Variable
Time Frame: 25 days
Stage 1
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Content Standard: The learner demonstrates understanding of the key concepts of first-degree equations and inequalities in one variable.
Performance Standard: The learner models situations using oral, written, graphical and algebraic methods to solve problems involving first-degree equations and inequalities in one variable.
Essential Understanding(s): Real-life problems where certain quantities are unknown
can be solved using first-degree equations and inequalities in one variable
Essential Question(s): How can we use first-degree equations and inequalities in
one variable to solve real life problems where certain quantities are unknown?
The learner will know: mathematical expressions, first-degree equations and
inequalities in one variable first-degree equations and inequalities in one variable properties of first-degree equations and inequalities in
one variable applications of first-degree equations and inequalities in
one variable
The learner will be able to: differentiate mathematical expressions from equations and
inequalities identify and describe first-degree equations and
inequalities in one variable give examples of first-degree equations and inequalities in
one variable describe situations where first-degree equations and
inequalities in one variable enumerate and explain the different properties of first-
degree equations and inequalities in one variable give illustrative examples of each property of first-degree
equations and inequalities in one variable 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 first-
degree equations and inequalities in one variable extend, pose, and solve related problems in real life
Stage 2Product or Performance Task:Situations modeling the use of oral, written, graphical and algebraic methods
Evidence at the level of understandingThe learners should be able to demonstrate understanding of first-degree
Evidence at the level of performancePerformance assessment of situations involving first-degree equations and
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to solve problems involving first-degree equations and inequalities in one variable
equations and inequalities in one variable using the six (6) facets of understanding:
Explaining the properties of first-degree equations and inequalities in one variable. Criteria: Clear Coherent Justified
Interpreting mathematical conjectures and arguments involving first-degree equations and inequalities in one variable Criteria: Illustrative Meaningful
Applying first-degree equations and inequalities in one variable in daily life situations Criteria: Appropriate Practical Relevant
Developing Perspective on the various ways of writing first-degree equations and inequalities in one variable in solving a problem Criteria: Critical Insightful Credible
Showing Empathy by describing difficulties one can experience in daily life
inequalities in one variable based on the following suggested criterion.
Use oral, written, graphical and algebraic methods in modeling situations
Tools:Rubrics of situations modeling the use of oral, written, graphical and algebraic methods
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whenever tedious calculations are done without using the concepts of first-degree equations and inequalities in one variable Criteria: Open Sensitive Responsive
Manifesting Self-knowledge by discussing the best and most effective strategies that one has found for solving problems involving first-degree equations and inequalities in one variable Criteria: Insightful Clear Coherent
Stage 3Teaching/Learning Sequence:
1. ExploreInitially, begin with some interesting and challenging exploratory activities on first-degree equations and inequalities in one variable 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. Group activities such as games, puzzles, storytelling, simulation, role playing, etc. Identifying and describing first-degree equations and inequalities in one variable. Citing real-life situations that may be represented using first-degree equations and inequalities in one
variable.
b. Online/offline presentations of authentic situations involving first-degree equations and inequalities in one variable (e.g. ICT tools such as CONSTEL CDs, open-source learning materials, E-TV learning episodes, etc.).
c. Oral and written reactions to online/offline presentations.
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2. Firm UpThese are the enabling activities/experiences that the learner will have to go through to validate understanding on first-degree equations and inequalities in one variable during the activities in the exploratory phase. These would answer some misconceptions on first-degree equations and inequalities in one variable that have been encountered in real life situations.
Group activities such as games, puzzles, storytelling, simulation, role playing etc. Giving exercises on representing situations using first-degree equations and inequalities in one variable. Enumerating, explaining and giving illustrative examples of the properties of first-degree equations and inequalities in
one variable. Solving exercises involving 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.
3. DeepenActivities 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 of first-degree equations and inequalities in one variable 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 first-degree equations and inequalities in one variable.
4. Transfer
Applications of learner’s understanding on first-degree equations and inequalities in one variable 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 Making a flowchart on intelligent digital model applying first-degree equations and inequalities in one variable (e.g.
robotics, software, business model, etc.)
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Resources:
See Appendix
Quarter III : Rational Algebraic Expressions, Linear Equations and Inequalities in Two Variables
Topic: Rational Algebraic Expressions
Time Frame: 25 days
Stage 1Content Standard: The learner demonstrates understanding of the key concepts of rational algebraic expressions.
Performance Standard: The learner presents solutions to problems involving rational algebraic expressions using numerical, physical, and verbal mathematical models or representations.
Essential Understanding(s):Simplifying rational algebraic expressions involve factorization and operations similar to operations on numerical fractions.
Essential Question(s):How can rational expressions be simplified?
The learner will know: fractions in simplest form; operations on fractions; rational algebraic expressions in simplest form; operations on rational algebraic expressions; and applications of rational algebraic expressions.
The learner will be able to: explore problems and describe results using numerical,
physical, and verbal mathematical models or representations;
use his/her reading, listening and visualizing skills to interpret mathematical ideas;
simplify rational algebraic expressions by using various methods/techniques;
perform operations on rational algebraic expressions and justify steps by stating the mathematical properties used;
analyze rational algebraic expressions, formulate relationships and extend them to other cases; and
apply the concept of rational algebraic expressions in solving real life situations.
Stage 2Product or Performance Task:Solutions to problems involving rational algebraic expressions are presented using numerical, physical, and verbal mathematical models or representations.
Evidence at the level of understandingThe learner should be able to demonstrate understanding by covering the six (6) facets of understanding:
Explaining by justifying how one’s answer is
Evidence at the level of performanceAssessment of presentation of solutions to problems involving rational algebraic expressions based on the suggested
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changed to simplest form. Criteria:
ClearCoherentJustified
Interpreting how best procedures for simplifying rational expressions are determined. Criteria:
Illustrative Creative Accurate
Applying the appropriate operations in simplifying rational expressions. Criteria:
AppropriateAccurate
Developing Perspective on how to choose the best solution in simplifying rational expressions
Criteria:CredibleInsightful
Showing Empathy on people’s difficulties in performing operations involving rational expressions. Criteria:
PerceptiveResponsiveSensitive
Manifesting Self-Knowledge in recognizing the best solution to a given situation involving rational expressions.
criterion:
the use of numerical, physical, and verbal mathematical models or representations.
Tools:Rubrics for assessment of solutions to problems
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Criteria:ReflectiveInsightful
Stage 3Teaching/Learning Sequence:
1. ExploreInitially, 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.
Group activities such as games, puzzles, storytelling, role-playing, simulation, etc. Simplifying rational numbers. Identifying rational numbers. Citing real life situations involving rational numbers Differentiating rational algebraic expressions from rational numbers
2. Firm UpThese 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.
Activities such as games, puzzles, storytelling, simulation, role-playing, etc.
Performing operations on rational algebraic expressions. Simplifying rational algebraic expressions . Using rational algebraic expressions to represent situations.
3. Deepen Activities in this stage shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety of
experiences. Moreover, the learner shall express his/her understanding and engage in multidirectional self-assessment.
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.).
Writing the series of steps in simplifying rational algebraic expressions (e.g. application of properties of real numbers,
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the different factoring procedures). Investigating relationship of quantities. e.g. the distance (d) from the fulcrum where 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. TransferApplications 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.
Resources:
See AppendixQuarter III : Rational Algebraic Expressions, Linear
Equations and Inequalities in Two VariablesTopic: Linear Equations and Inequalities in Two Variables
Time Frame: 25 days
Stage 1Content Standard: The learner demonstrates understanding of the key concepts of linear equations and inequalities in two variables.
Performance Standard: The learner presents solutions to problems involving rational algebraic expressions.
Essential Understanding(s):Linear equations show constant rate of change.
Graphs of linear equations show trends which help predict outcomes and make decisions
Essential Question(s):How are linear equations used to communicate relationships between quantities?
How does one know an outcome is favorable? How can mathematics help one find out?
The learner will know: coordinate plane and the terminologies associated with
it. graph of linear equations in two variables equation of a linear equation in two variables application of linear equations in two variables graph of a linear inequality in two variables
The learner will be able to: give the exact location of a point, person or object using maps,
navigation devices, etc. investigate graphs of linear equations in two variables with
deductive arguments and evidences. solve linear equations in two variables graphically . apply mathematical thinking to solve problems in disciplines
such as art, music, science and business.
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formulate and solve real life problems using various representations.
Stage 2Product or Performance Task:Solutions to problems involving linear equations and inequalities in two variables are presented using numerical, physical, and verbal mathematical models or representations.
Evidence at the level of understandingThe learner should be able to demonstrate understanding by covering the six (6) facets of understanding:
Explaining how a statement is translated into mathematical symbols
CriteriaClearCoherent
Interpreting possible relationships of rates of change in a given set of data.
Criteria:RevealingIllustrative
Applying mathematical thinking and modeling to solve problems in other disciplines such as art, music, science and business.
Criteria:PracticalAppropriateAccurate
Developing Perspective on the most likely outcomes that may result from trends shown in graphs
Criteria:Credibleinsightful
Showing Empathy on people experiencing difficulties
Evidence at the level of performance
Assessment of presentation of solution to problems involving linear equations and inequalities in two variables based on the suggested criterion:
Use of numerical, physical, and verbal mathematical models or representations.
Tools:Rubrics for assessment of solutions to problems
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in making decisions without the help of graphs and linear equations
Criteria:PerceptiveResponsiveSensitive
Manifesting Self-Knowledge by sharing insights one may have about how math can help make reasonable judgments and predictions. Criteria:
ReflectiveInsightful
Stage 3Teaching/Learning Sequence:
1. ExploreInitially, 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. Ask follow-up questions that would enhance the critical thinking skill of the learner.
2. Firm UpThese 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.
Group activities such as games, puzzles, storytelling, simulation, role-playing, etc. Finding/locating the coordinates of a point in the coordinate plane;
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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.
3. Deepen Activities in this stage shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety of
experiences. Moreover, the learner shall express his/her understanding on linear equations and inequalities in two variables and engage in multidirectional self-assessment.
Group activities using pictures, multimedia presentations, or daily life experiences and observations where concepts of linear equations and inequalities in two variables are applied through the following activities:
Investigating the behavior of graphs in relation to their slopes. Writing 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.) Checking the results using graphics calculators or computers. Analyzing situations represented by linear equations and inequalities in two variables. Journal writing on the behavior of graphs of linear equations and inequalities in two variables.
4. TransferApplications of learner’s understanding on linear equations and inequalities in two variables are demonstrated through culminating activities that reflect meaningful and relevant problems/situations.
a. Design a game map to locate a person in a certain town, a ship in distress, a treasure buried in a mountain slope. b. Construct a miniature model of the learner’s ideal community.
Resources:
See Appendix
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Quarter IV: Systems of Linear Equations and Inequalities in Two Variables
Topic: Systems of Linear Equations and Inequalities in Two Variables
Time Frame: 25 days
Stage 1Content Standard: The learner demonstrates understanding of the key concepts of systems of linear equations and inequalities in two variables.
Performance Standard: The learner creates situations/ problems in real-life involving systems of linear equations and inequalities in two variables, and solves these by applying a variety of strategies
Essential Understanding(s):Unknown numbers in certain real-life problems may be derived from solving systems of linear equations and inequalities in two variables.
Essential Question(s):How is knowledge of systems of linear equations and inequalities in two variables used to solve real life problems?
The learner will know: graphical solution of systems of linear equations and
inequalities in two variables. algebraic solutions of systems of linear equations and in
two variables. applications of systems of linear equations and
inequalities in two variables in problem solving graph a system of linear inequalities and inequalities in
two variables
The learner will be able to: explain thoroughly how systems of linear equations and
inequalities in two variables can be solved graphically and algebraically.
graph with accuracy the solutions of a system of linear equations and inequalities in two variables.
apply a variety of strategies to solve problems involving systems of linear equations and inequalities in two variables.
graph with accuracy the solution set of a system of linear equations and inequalities in two variables.
Stage 2Product or Performance Task:Situations/ Problems created are drawn from real-life and are solved by applying a variety of strategies.
Evidence at the level of understandingThe learner should be able to demonstrate understanding by covering the six (6) facets of understanding:
Explaining and presenting a mathematical analysis of graphs. Criteria:
Clear CoherentJustified
Evidence at the level of performanceAssessment of situations problems created based on the following suggested criteria: problems are drawn from real-
life; problems involve systems of
linear equations and inequalities in two variables; and
29
Interpreting the significance of the way graphs relate with each other.Criteria:
IllustrativeMeaningful
Applying the appropriate solution that would produce best results. Criteria:
AppropriatePracticalUseful
Developing Perspective on the different possible outcomes illustrated by graphs/equations. Criteria:
CriticalInsightful
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:
SensitiveAuthentic
Manifesting Self-Knowledge on the impact of individual accuracy in solving problems on systems of linear equations Criteria:
InsightfulRelevant
problems are solved using a variety of strategies.
Stage 3
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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.
2. Firm UpThese 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.
Graphing 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.
Algebraic solutions of systems of linear equations in two variables. Applying systems of linear equations in two variables in problem solving. Graphing a system of linear inequalities in two variables. Interpreting graphs of systems of linear equations and inequalities through storytelling, simulation, flowchart, etc.
3. Deepen Activities in this stage shall provide opportunity for the learner to reflect, revisit, revise and rethink about a variety of
experiences. Moreover, the learner shall express his/her understanding on systems of linear equations and inequalities in two variables and engage in multidirectional self-assessment.
Investigating the effect of the slopes on the graph of a linear equation in two variables and giving its significance Exploring graphs of inequalities using graphics calculator. Investigating and analyzing critical points on the graphs of systems of linear inequalities.
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Writing reports on the result of the investigation.
4. TransferApplications 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.
Resources:
See Appendix
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Appendix
Resources : ICT Tools
http://www.deped.gov.ph/iSchool Web Board/Math Web Board http://www.deped.gov.ph/iSchool Web Board/skoool.ph http://www.deped.gov.ph/e-turo http://www.deped.gov.ph/BSE/iDEP http://www.pjoedu.wordpress/Philippine Studies/FREE TEXTBOOKS http://www.teacherplanet.com http://www.pil.ph/innovative teachers leadership award http://www.alcob.com/ICT Model School Network http://www.APEC Cyber Academy.com http://www.globalclassroom.net http://www.think.com http://www.rubistar.com Overhead projector/transparencies Computer/PC tablet/Interactive Whiteboard Graphing calculator CONSTEL CD/DVD materials CAI materials in CD/DVD format Intel Teach to the Future/ASEAN SchoolNet/FIT-ED/SMART School/Innovative Teachers Leadership
Award/sKwela/iSchool/Learn.ph/eskwela ng bayan modules in PowerPoint format Authoring-enabled math storytelling enrichment modules ( e.g. Fibo the Frog Mathemajess’yan, etc.) Video clips/tapes Interactive digital games/puzzles ( e.g. eDamath, 3D damath gaming, Cartesian coordinate system, etc.) E-TV episodes (Knowledge Channel, National Geographic Channel, Discovery Channel, BBC, etc.) Online/offline open-source teaching/learning materials Student/teacher-made PowerPoint/spreadsheet/word instructional materials with Hyperlinks Student/teacher-made webcam/digicam math storytelling materials Mathematical simulations Interactive flowchart/concept map/tree diagram/picture problem situation/secret message decoder Dgital mathematical cartoon strips/humor/jokes/poster with hyperlinks eBooks/eTeachers manual with no hyperlinks
33
Digital books/teaching guides with hyperlinks
Manipulatives Games and puzzles (e.g. damath, chess-math, sungkamath, sudoku, cross-number puzzle, etc.) Geoboard/rubber bands/strings Algebra tiles Cuisenaire rod Dice Domino/triangular domino Playing cards Mathemagic biscuit crackers (e.g. opposite sense in integers, etc.)
Print Materials Worksheets Workbooks Textbooks/supplementary reference materials/teachers manual Consumer education/global warming and climate change integration lesson exemplars Remedial/enrichment modules (e.g. EASE modules, distance learning/self-learning package, etc.) Enrichment math storytelling modules with indigenous cartoon characters (e.g. Max the Matrix & Co., etc.) Handouts/Activity Sheets/Cards Mathematical post card / mathematical cartoon strips/humors/quotes/jokes/jingle/slogan/themes/tarpaulin
Supplies and Materials Papers (graphing, bond, pad, manila, colored) School Supplies (Ruler/straightedge, Colored cartolina , Illustration board , Pairs of scissors, Masking tapes
Pentel pen)
34
Attachment to Quarter I
Measurements
(See Explore)
Opening Activity 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.
Examples:
1. 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.
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2. 7 214
The sum of the digits is 14 (from 7 + 2 + 1 + 4).Subtracting 14 from 7 214, we have 7 200. Suppose the encircled digit is zero. The sum of the remaining digits is 9 (from 7 + 2 + 0).Nine subtracted from 9 is zero.
Therefore, the encircled digit is 0.
(See Firm Up)
Directions:
1. Group the students into fives.2. 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)
3. Ask for the answer in more commonly understood units of time, such as days, weeks, months, or years.
4. Allow students to make an estimate/ approximation before they compute.
5. Discuss the results.
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(See Deepen)
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:
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?6. Do you think one million ping-pong balls could fit into the room? Explain.
(See Transfer)
1. Design a game.
2. Create a model house. Express dimensions of the actual structure to the scale model as ratios.
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Attachment to Quarter I
Scientific Notation
(See Explore)
Opening ActivityNote: Ask students to work individually on this activity.
Directions:1. Read the numbers in the table from top to bottom. 2. What pattern do you observe between succeeding numbers?3. Guess the next term of the sequence. 4. Write down a rule for finding the value of numbers with negative exponents.
10n 106 = ?105 = ?104 = ?103 = 1 000102 = 100101 = 10100 = 1
10-1 = 110
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.
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(See Firm Up)
Exploration Activity:
Have students answer the worksheet in pairs.Study the table below.
Set A Set B
Set C
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Column IDecimal Form
881
8148 143
Column IIScientific Notation
8 x 100
8.1 x 101
8.14 x 102
8.143 x 103
Column I
Decimal Form
14.325143.251 432.5
Column IIScientific Notation
1.4325 x 101
1.4325 x 102
1.4325 x 103
Column I
Decimal Form
0.37680.03768
0.003768
Column II
Scientific Notation
3.768 x 10-1
3.768 x 10-2
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?
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________________ .
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(See Deepen)
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 hourPhp 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 (1point)
Total points : 3
Scoring Guide (Rubric) for Problem Solving
41
Points Criteria3 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 answer1 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.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?
(See Transfer)
1. Design a game.
2. Prepare contest questions. It may be a team competition of 4 members. Classify the questions as 15 – sec; 30 – sec; and 1- minute
3. Create a model solar system. Express the distances in scientific notation.
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Attachment to Quarter IIAlgebraic Expressions
(See Explore)
Manipulating algebra tiles
Preparation of algebra tiles must be done the day before taking up this activity. The learner recalls briefly the process of finding the area of a square and of a rectangle.
Example: 6 6 Area = (6)2 = 36 units 2
3 2 2 Area = (2)(3) = 6 units2
The learner explores by manipulating algebra tiles. (A video tape or CD on the manipulation of Algebra tiles may be helpful).
(See Firm Up)
Employ the “pair square” structure of cooperative learning. By using algebra tiles, have the learners find the products of two algebraic expressions by giving three (3) exercises for each case of special product/formula.
Example: Finding the products of the following.
1. (x + 4) (x + 4) 6. (x – y) (x – y) 11. (2x + 2) (x – 4) 2. (2x + 3) (2x + 3) 7. (x + y) (x – y) 12. (x – 3) (3x + 1) 3. (x = y) (x = y) 8. (2x + y) (2x - y) 4. (x – 2) (x – 2) 9. (x + 5) (x - 5) 5. (3x – 5) (3x – 5) 10. (x + 3) (x + 5)
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(See Deepen)
The learners investigate the relationship between the products and their factors and discover the following patterns of special products and their factors.
1. (a + b) (2a + 3b) Product of algebraic expressions with like terms 2. (a + b)2 Product of the squares of the sum of two terms 3. (a – b)2 Product of the squares of the difference of two terms 4. (a + b) (a – b) Product of the sum and difference of two terms
The learners write journals on generating rules on finding special products.
(See Transfer) Using models, present a problem solving plan
Problem/Situation: You and your friends are at a restaurant. You ordered several Php 28.00 plates of pancit, puto and banana cue. Your total expenses is Php 221.48 which includes 13% VAT. Use modeling to find the number of plates you ordered.
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=
Attachment to Quarter II
First-Degree Equations and Inequalities in One Variable
(See Explore)
Showing one after the other the following figures (or physical model of each figure) to the learners.
The following questions may be posed to the learners:
1. What does the first figure tell?2. What other figure would mean the same as the first figure? Explain.3. In the second figure, what would balance the two cylinders? Why?4. How many cylinders would balance the cubes in Figure 3? Explain.
Figure 1
Figure 2
Figure 3
Figure 4
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5. Do you think the cubes in Figure 3 could also be balanced by the same number of cubes? If No, explain. If Yes, how many of these cubes are needed?
6. In general, what do the three figures indicate?7. In Figure 4, what must be placed inside the box to show that the two sides of the scale balance are equal? Explain your
answer. (Assume that the box has no weight.)8. If Figure 4 illustrates an equation, how would you represent the big box?9. What equation would represent the given situation?10.How will you solve the equation?
Playing Bingo game on special products and factors
This is a bingo game on special products and their factors. To play this game, each learner must have constructed a 3 x 3 matrix on a piece of paper (prepared the day before). The learner selects nine (9) different numbers from the list of answers written on the blackboard by the teacher and place these nine (9) numbers in the bingo matrix (in ink to prevent cheating) in any order.
After each student has completed his bingo card, the teacher writes one algebraic expression on the blackboard, flash card, acetate to be shown on the overhead projector or read aloud. The teacher then gives the students time to factor the given algebraic expression, locate the answer on the board and encircle the corresponding number on the bingo card if it is there. The teacher presents another algebraic expression to be factored and this goes on until all prepared algebraic expressions are factored. To win the game, a learner must get three (3) answers in a row twice (vertically, horizontally, or diagonally) before declaring “Bingo!”.
(See Firm Up)
Group Activity 1:
Let each group of learners study the following equations and answer the questions that follow.
1. 13 + 5 = 13 + 52. If 3 + 7 = 2 + 8, then 2 + 8 = 3 + 7.3. If 6 + 9 = 3 + 12 and 3 + 12 = 2 + 13, then 6 + 9 = 2 + 134. If 5 + 9 = 3 + 11, then 5 + 9 + 7 = 3 + 11 + 7.5. If 18 – 4 = 6 + 8, then (18 – 4) – 5 = (6 + 8) – 5.6. If 9 + 15 = 25 – 1, then 7(9 + 15) = 7(25 – 1)
46
7. If 4 x 12 = 16 x 3, then (4 x 12) ÷ 2 = (16 x 3) ÷ 2.
The following questions may be posed to the learners:
1. What does each mathematical statement mean?
The different properties of equality are as follows:
PROPERTIES OF EQUALITY
a.Reflexive PropertyFor any real number a, a = a is always true. This means that any number is equal to itself.
b.Symmetric PropertyFor any real numbers a and b, if a = b, then b = a. Interchanging the left and right members of the equation does not change their value.
c.Transitive PropertyFor any real numbers a, b, and c, if a = b and b = c, then a = c. Two quantities that equal to the same quantity are equal.
d.Addition Property of EqualityFor any real numbers a, b, and c, if a = b, then a + c = b + c and c + a = c + b. If the same number is added to both sides of an equation, the sums are equal.
e.Multiplication Property of EqualityFor any real numbers a, b, and c, if a = b, then ac = bc and ca = ba. If the same number is multiplied to both sides of an equation, the products are equal.
2. Describe how the property of equality is illustrated in each mathematical statement.3. Give two illustrative examples of each property of equality.
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(See Deepen)
A. Group Activity:
Let each group of learners find the value of x in each equation using any method familiar to them.
1. x + 5 = 9; x = ____2. x – 4 = 8; x = ____3. x + 12 = 24; x = ____4. x – 13 = -7; x = ____5. x + 15 = -8; x = ____6. 3 x = 12; x = ____
7.
x4 = 24; x = ____
8. 5 x = -45; x = ____
9.
x−3 = 8; x = ____
10.
2x5 =
415 ; x = ____
B. The following questions may be posed:
1. How did you find the value of x in each equation?2. What does the value of x mean in each equation?3. How will you check if the value of x satisfies the equation? Verify the solutions you got.4. Suppose you apply the different properties of equality in solving the equations, do you think you will get the same value of x?
Why/Why not? 5. How did you apply the different properties of equality in solving each equation?
48
C. Class discussion on the results of the activity follows.
1. What property of equality is illustrated by each of the following?
a. x + 6 = x + 6b. If 2x – 5 = 4x + 7, then 4x + 7 = 2x – 5 c. If 3x + 5 = 12 and 4x – 9 = 3x + 5, then 4x – 9 = 12d. If 5x – 4 = 9, then 5x – 4 – 7 = 9 – 7
e. If
x+53
=4 x, then
3( x+53 )=3(4 x )
f. If 7x + 12 = 17, then 7x + 12 – 10 = 17 – 10
g. If 12x = 48, then
12 x12
=4812
h. If 7 + 3x = 4x – 5, then 7 + 3x – 2x = 4x – 5 – 2x
i. If
2x+36
−4=2 x−37
+5 , then
2x+36
−4+4=2 x−37
+5+4
j. If 3(5x – 4) = 18, then
3(5 x−4 )3
=183
2. What must be the numbers in the boxes below to make the equation true?
One possible set of numbers are the solutions to the equations below. Solve the equations by applying the different properties of equality and write your answers in the boxes provided.
a. x + 15 = 22b. 5x = -15c. 3x + 8 = 32d. x – 11 = -9
x + ÷ - + x ÷ + - = -13a b c d e f g h i j
49
e.
x2=2
f. 7 – 3x = -2
g.
5 x3
=−10
h. 6x – 11 = -23
i.
x+32
=6
j. 4(x – 17) = -28
3. Solve each of the following equations and write the properties of equality used.
a. x + 5 = 27b. m – 8 = 45c. 5p = 65
d .m7
=−4
e. 2n – 7 = 39f. 3a – 7 = 14g. 4(x – 5) = 20
h .m+2
5=16
i .a−3
2−9=8
j. 9 – 3(x + 5) = 27
Other suggested activities:
1. PowerPoint with Hyperlinks presentation illustrating the different properties of equality and how they are applied in solving first degree equations.
50
2. Practice exercises in solving first-degree equations applying the different properties of equality. Use worksheets.
3. Journal writing on situations or experiences involving equations and inequalities that need to be valued by every learner.
Sample Activity: Journal writing
Write a journal of the history of your savings account for the month of December, 2009. Include the tabular record, the formula for each week savings, and the formula for the total deposit for December.
Solution: Note. The rubric distribution of points is indicated.
A. My initial deposit for the month of December was Php200.00. On the second week of the month, I deposited 50 more than twice my initial deposit. Thus, I deposited Php450.00 on the second week. On the 3rd week, I deposited half of
1 point the amount I deposited on the second week, which is Php225.00. My last deposit for the month of December is equal to the sum of my initial deposit and my first deposit, which is Php650.00.
Below is the table showing my deposits for the month of December and then formulas.
B. 1. Let first week be x, C.2. Second week: 2x + 50
2 points 3. Third week: x/2 1 point4. Fourth week: x + (2x + 50)
D. Hence, my deposit is:A = x + (2x +50) + x/2 + [x + (2x+50)]. 1 point
So, the total score is 5 points.
(See Transfer) Creating/posing and solving problems
1. Santino is running for class president. By 11:00 AM on election day he has 35 votes and his opponent has 45 votes. Thirty-five more students will be voting. Let x represent the number of students (of the 35) who vote for Santino.
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Dates Amount Cumulative SumFirst Week Php200.00Second Week Php450.00 Php650.00Third Week Php225.00 Php875.00Last Week Php650.00 Php1 525.00
a. Write an inequality that shows the value of x that will allow Santino to win the election.b. What is the smallest value of x that is a solution of the inequality?
2. Kudzu is a type of Japanese vine that grows at a rate of 26.4 cm per day during summer. On August 1, the length of 1 vine was 1,320 cm. What was the length on July 1? Use verbal and algebraic models to solve the problem.
52
Attachment to Quarter III
Rational Algebraic Expressions
(See Explore)
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?
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a.
2436 d.
−4554
b.
3556 e.
12−18
c.
−3248
6. When do you say that a fraction is in its simplest form?
(See Firm Up)
Group Activity 1:
Mr. Gabriel has a farmland which he subdivided equally among his 6 children and 22 grandchildren.
1. How would you represent the area of Mr. Gabriel’s farmland?2. 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?3. 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? 4. How would you describe the expressions you got in (c) and (d)?5. How would you compare these expressions with the fractions which you already studied before?6. How would you differentiate rational numbers from rational expressions?7. Which of the following are rational algebraic expressions? Explain your answer.
a.
4 xx−2 f.
x+5x−5 ; x = 5
b.
x2−4x2+4 x+4 g.
x2−5 x+6( x−2 ) (x−3 )
54
c.
37 h.
3x 4+12 x2−6x3 x
d.
√52x i.
4x+4
e.
8 x3−272x−3 j.
3x√3
x−3
Class Discussion: (Results of the activity)
(See Deepen)
a. The following are rational algebraic expressions.
a.4 xx−2 b.
2x+42 c.
x2−9x2+6 x+9 d.
x2+7 x+122 x2+8 x
1. Which of the above expressions are expressed in simplest form? Why?2. Why do you say that the others are not written in their simplest form?3. How would you simplify these expressions? Give the steps.4. What mathematics concepts or ideas would you apply to simplify the expressions? 5. How would you apply these mathematics concepts or ideas in simplifying rational algebraic expressions? 6. Express the given rational algebraic expressions in their simplest form.
b. Is x−2+1 a rational algebraic expression? Why?
c. Cite situations that could be represented by rational algebraic expressions. What expressions represent these situations?
(See Transfer)
1. Design a scale model of a structure. Express dimensions of the actual structure to the scale model as ratios. 2. Design Games
55
21 3 4 5
L
Attachment to Quarter III
Linear Equations and Inequalities in Two Variables
(See Explore)
1. Ask the students to arrange their seats to form columns and rows. Tell those seating in the front row to number their seats as illustrated:
Column
Row 1 Row 2 Row 3
2. Guide them to name the seats of their classmates as (r, c), where r stands for row and c for column.
Example: The seat where Luisa who is seated on the 3rd row, and on the 2nd column is named as (3, 2)
3. After the activity, pose the question:
a. Who is seated at (2, 4)? b. Are there other instances where locations of places are named?
(See Firm Up)
Discuss:
a. the Cartesian coordinate plane. b. coordinates of a point on a Cartesian coordinate plane
(See Deepen)
Write experiences encountered if locations of persons, objects, places are not clearly defined.
56
(See Transfer)
1. 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.
2. 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.
3. Give the coordinates where pictures of the habitat of different species (pond, farm) are to be posted. Let students post the fishes, ducks, birds, cow, etc in their proper habitat.
57
Attachment to Quarter III
Linear Equations and Inequalities in Two Variables
(See Explore)
Group Activity:
a. Provide students with activity sheets.b. Send groups of students to measure the length and height of the steps of the stairs of each building in their school. c. 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. d. Represent the measurement/distances as ratios (vertical distance to the horizontal distance). Let them compare the ratios.e. Allow them to discuss their findings in class.
(See Firm Up)
1. Introduce the concept of slope.
2. Discuss:
a. the behavior of the graphs of linear equations in two variables if different slopes are used. Use a graphing calculatorb. 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.
58
(See Deepen)
Third QuarterA. 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
Rubric Scoring Guide
B.
1. Ask learners’ experiences encountered on situations similar to: steps of stairs unevenly spaced. going up on an inclined plane with different gradients.
2. Investigate the effects caused by these phenomena. 3. Make the necessary analysis
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Points Criteria3 The graph is correct and Properly labeled2 Graph is correct, but not labeled properly1 Graph is incorrect0 No attempt
Start here●
●
(See Transfer)
1. 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.
2. Treasure Hunt with SlopesOn a grid paper, mark points that would lead to the treasure.
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. 32
2.12 6. -3 10.
−13
3.−2
3 7. 13 11. 4
4. 6 8. -1
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Attachment to Quarter IV
System of Linear Equations and Inequalities
(See Explore)
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 yearMonthly Dues: P200 per month
Package B: Space Rental: P4000 per yearMonthly Dues: P100 per month
2. Ask the Questions:
Which parking package should she avail? Why? When is one package cheaper?
3. Introduce the system of linear equations in two variables and discuss related lessons.
(See Firm Up)
Divide the class into three groups. Each group will be assigned to graph a system of linear equations using any method of their choice and to answer the following guide questions. Two representatives from each group will then be asked to present their work to the class.
Sketch the graph of the given system of equations. Identify the slopes and y-intercepts of the two lines in the system and then answer the questions that follow.
Group 1: {x + y=10¿ ¿¿¿
Group 2: {x − y=−1 ¿¿¿¿
Group 3: {x + y=3¿ ¿¿¿
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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?
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?
(See Deepen)
Technology Integration:
Teach the students how to use the graphics calculator to investigate the graph of a given system of linear equations.
1. Identify the slopes and y-intercepts of the lines in each of the following systems and then identify what kind of system it is.
a.{x + y=4 ¿ ¿¿¿
b. {2 x =− y + 3¿ ¿¿¿
c.{3 x − 5 y=10¿ ¿¿¿
d. { y =2x − 3 ¿¿¿¿
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-3
-2
-1
0
1
2
3
4
5
y
-4 -2 2 4x
-4
-2
0
2
4
y
-4 -2 2 4 6 8 10x
-4
-2
0
2
4
y
-6 -4 -2 2 4x
-4
-2
0
2
4
y
-4 -2 2 4x
2. Analyze the graphs. What kind of system of linear equations is represented by each graph?
a. b.
c. d.
(See Transfer)
Allow students to design competitions, puzzles, games, etc.
y
x
y
x
y y
x x
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Rubrics: 1. Scoring Guide (Rubric) for Problem Solving 2. Rubric Scoring Guide for Graphing
3. Rubrics Scoring Guide for Journal Writing
Points Description
5 Writes a clearly stated main idea, topic and presents supporting details in a logical order. The journal is written with correct use of conventions (grammar, punctuation, capitalization, and spelling)
4Writes clearly stated main idea, topic and presents supporting details in a logical order. The details may not be as complete as it could be. The journal is written with generally correct use of conventions.
3Writes a clearly stated main idea, topic but presents some unrelated details. There are few errors in the use of conventions.
2 Writes a main idea, topic but not clearly stated. Details may not be presented in a logical order, or some of the information may be inaccurate. The journal may include some errors in the use of conventions.
1 No accurate understanding of topic/subject.
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Points Criteria3 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 answer1 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
Points Criteria
3The graph is correct.Properly labeled
2Graph is correct, but not labeled properly
1 Graph is incorrect0 No attempt