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• 1 Ignatian Pedagogical Paradigm Lesson Plan (First Entry) Subject: Mathematics 1: Beginning Algebra (I–St. Garnet) Prepared by: Rommel M. Gonzales Date of coverage: 3 Periods June 19, 2008 2 periods June 23, 2008 2 periods June 20, 2008 1 period June 24, 2008 1 period Content Coverage: Chapter 1: The Real Number System 1.1. The Real Number System 1.2. Properties of Real Numbers Learning Objectives Experiences/Strategies/Reflection/Action/Evaluation/Assignment June 19 Objectives: At the end of the two periods, the learners should be able to: PRELECTION: Presentation of the Problem for the Day: “Ann as P150.00. She went to a bookstore to buy 5 pieces of Cartolina at P8.00 each and 3 pieces pentel pen at P32.00 each. Her classmate shared half of the expenses for the Cartolina and P50.00 for the pentel pen. How much was left of Ann’s money?” ACTIVITIES/STRATEGIES (EXPERIENCE) 1. Small Group Activities 1. solve a problem in many ways • The class will be grouped into 11 groups of 4 members each. possible through collaborative work; • Instructions will be given to them. • Guide questions for discussion will also be given out: a. What are the given information in the problem? How much is the money of Ann? How much did her classmate contribute? b. What are the expenses? c. In how many ways can you solve the problem? • The groups will be given 30 minutes to accomplish the tasks. 2. Presentation of Solutions and Processing 2. distinguish from among the (Classroom Discussion) solutions and answers the most • Each group will be asked to post their output on the board. Outputs will be categorized according to similarity reasonable solution/s and correct of final answers, regardless of the different solutions. answer.
• 4 2. Reference: Acelajado, Maxima J., “Algebra: A Problem-Solving Approach (Second Edition).” Belgosa Media Systems, Inc. Makati City, 2002. 3. Student’s Textbook June 23-24 Objectives: At the end of the two periods, the learners should be able to: PRELECTION 1. familiarize with the different 1. Idea organizer outlining the different axioms and fundamental properties will be presented to the class. Students axioms and fundamental properties will be informed that these will be the nature of the numbers that we will be working with, the first of which will be the on real numbers, equality, and set of real numbers. order; ACTIVITIES/STRATEGIES (EXPERIENCE) 1. Work by Dyad 2. define each property of real • Students will work by dyads. Each of them will be given handout containing the properties of real numbers. As numbers; they study each property, they will be asked to make their own examples (at least two) that will further illustrate the property. Also, they will be asked to identify some similarities and differences among properties (or distinct properties), in terms of their application in the four fundamental operations. 2. Classroom Discussion/Processing • Board work: Pairs will be asked to define each property and provide two examples for each. • In the processing of information obtained from previous activities, the following will be used to help students put the different properties in categories. In doing so, students will be helped in gaining familiarization and understanding of the properties (POINT FOR REFLECTION: Familiarization or association can bridge understanding. Remember how a child develops concepts of a particular thing). Write YES or NO. •
• 5 Questions O P E R A T I O N S Addition Subtraction Multiplication Division 1. In which operation/s does closure exist? Show why it exists. 2. In which operation/s does commutativity exist? Show why it exists. 3. In which operation/s does associativity exist? Show why it exists. 4. In which operation/s does identity exist? What is the 3. categorize properties according corresponding identity element? Show why it exists. to their existence in the four fundamental operations; 5. In which operation/s does inverse exist? What is the corresponding inverse element? Show why it exists. 6. Under which operation/s does distributivity exist? Show why it exists. 4. apply the properties to new Following questions may be asked: mathematical statements or a. What does each property mean? Can you give examples? expressions. b. Why do some properties exist under certain operations? c. How does each property operate under certain operations? • Some Reflection Questions: a. Where do we see each of the properties in our day to day experiences? b. What is the identity/image that you would like to project now that you are in the High School? What
• 6 was the image you projected when you were younger? Is it worth keeping or changing? Why? 3. Synthesis: In the end students must be able to come up with a generalized approach to looking at the different properties of real numbers. This will be done through a concept map. Properties of Real Numbers Closure Commutative Associative Identity Inverse Distributive Property Property Property Property Property Property Note: Students will continue this map by supplying the operations where the property exists, its definition and one example. EVALUATION: Quiz will follow after the discussion, where students will be asked to identify the property that each statement describes. MATERIALS USED: 1. Manila paper (visual aids containing matrices and concept map) 2. Handouts (taken from a reference material) 3. Reference: Acelajado, Maxima J., “Algebra: A Problem-Solving Approach (Second Edition).” Belgosa Media Systems, Inc. Makati City, 2002. 4. Student’s textbook ASSIGNMENT: Each pair will be asked to make 50 square tiles of two contrasting colors. The dimension of each tile will be 1x1. This will be used for the next topic on Integers. Ignatian Pedagogical Paradigm Lesson Plan (Second Entry) Subject: Mathematics 1: Beginning Algebra (I–St. Garnet) Prepared by: Rommel M. Gonzales Date of coverage: June 25, 2008 2 periods June 26, 2008 2 periods Content Coverage: Chapter 1: The Real Number System
• 10 • Each item is worth 2 points. • The group to garner the highest number of points will emerge the winner. 3. decide which sets of rules for integers are difficult to remember 3. Synthesis statements: and what can be done about the • In multiplying two integers, find the product of their absolute values. difficulty? a. If the integers have the same sign, their product is positive. b. If the integers have different signs, their product is negative. • The quotient of two integers with the same sign is positive real number. • The quotient of two integers with different signs is a negative real number. ACTION: Please refer to the REFLECTION QUESTION part above. Questions: 1. Why do you consider the set/s difficult to remember? Why do you consider the others easier to remember? 2. What would you do so as to remove the difficulty? EVALUATION (Work by Dyad): I. Computational Exercises: Answer exercises found in the textbook, pages 108-110 (even-numbered items only). II. Quiz the following day MATERIALS TO BE USED: 1. Exercise worksheet 2. Textbook 3. Sets of number sentences NOTE: The students were given a problem set on the application of rules for addition and subtraction of integers a week ago which they need to accomplish in one week. Please see attached sheets for the problem set. This is to help students improve their skills on problem solving. As much as possible, a problem will be used as prelection, the so- called problem solving strategy. Ignatian Pedagogical Paradigm Lesson Plan Subject: Mathematics 1: Beginning Algebra (I–St. Garnet) Prepared by: Rommel M. Gonzales
• 11 Date of coverage: 2 Periods July 14, 2008 Monday Content Coverage: Chapter 2: Algebraic Expressions 2.1. Historical Development of Algebra 2.2. Constants, Variables, Exponents, and Algebraic Expressions Learning Objectives Experiences/Strategies/Reflection/Action/Evaluation/Assignment July 14, 2008 Objectives At the end of the two periods, the students should be able to: PRELECTION: 1. discuss the historical Questions: 1. Who was Francois Viete? What was his contribution to the world? development of Algebra and how it 2. Who was Al-Khowarizmi? What was his contribution to the world? has evolved; Tasks (by 4s): Get to know them more. Do the following: 1. Find the picture of these two important people. 2. Find out their family background. Make it brief. 3. Find out their life story. What motivated them to do the things they had done? 4. Reflection: If you are to present their lives to the youth today, what characteristics of each will you highlight which can serve as inspiration for the youth? Note: The teacher will explain to the class on how to come up with the output. ACTIVITIES/STRATEGIES (EXPERIENCE) 1. Classroom Activity (Dyad): Number-Pattern Experiment • Students will be asked to make an experiment that involves just a sheet of paper. By repeatedly folding the paper in half, an interesting number-pattern will emerge. The basic question will be: What is the relationship between the number of folds and the thickness? • The dyads will be guided on how to do the experiment. In the end, they must discover that: The thickness doubles with each successive folds. • Processing will take place as they do the experiment. Questions, as follows, may be asked: a. How thick is the sheet if the paper is not yet folded? How do you define thickness using the paper? What does thickness mean in this experiment? b. How thick is the paper when you fold the paper once? Twice? Thrice? Four times? etc. c. Is it possible to determine the thickness if the paper will be folded 10 times? d. Suppose you will not fold the paper anymore, can you still determine the thickness given the data at hand?
• 12 e. What pattern do you see? 2. Classroom Discussion 2. differentiate constant from • From the pattern, the teacher will start to develop the basic definitions and concepts of the following: variable; a. Constant b. Variable 3. define exponents and algebraic c. Exponents expression; d. Algebraic Expression • To facilitate discussion, the following questions may be asked: a. From the generalized pattern, what value is fixed? What makes it fixed? Why do you consider it fixed? b. What could be other examples of fixed values? c. From the pattern, what is changing? What real life examples that you can think of can be considered changing? d. What could be other term for changing? e. What is the difference between constant and changing? f. From the pattern, how else can we describe n, aside from being a variable? g. Generally and collectively, how do we describe the pattern? 3. Group Activity • Students will be grouped and they will be given tasks involving identification of any of the following: constant, variable, exponent, algebraic expression. 4. Synthesis statements: • Using the terms constant, variable, exponent, and algebraic expression, show their relationships using a simple graphic organizer or a concept map. Include definitions and examples. REFLECTION 4. recognize relevance of the terms, 1. Get the lyric of the song by Jose Mari Chan, entitled “Constant Change.” like constant and variable, in real 2. Get the chance to listen to the song. Try to understand the meaning of the song as you enjoy listening to its life situations. melody and music. 3. Answer the following questions: a. Do you like the song? Why? b. Are there lines which you can relate very much and you find meaningful to you? c. Why do you think the song was entitled “Constant Change”? d. Do you agree that change is something constant?
• 13 ACTION Draw an image that best describes your idea of “Constant Change.” Then write down your responses to the questions in the reflection part. EVALUATION Output of the group activity MATERIALS TO BE USED: 1. Drawing materials 2. Textbooks and references 3. Research materials 4. Lyrics of the song “Constant Change” 5. Cardboard Ignatian Pedagogical Paradigm Lesson Plan Subject: Mathematics 1: Beginning Algebra (I–St. Garnet) Prepared by: Rommel M. Gonzales
• 14 Date of coverage: 3 Periods July 15, 2008 Tuesday 1 period July 16, 2008 Wednesday 2 periods Content Coverage: Chapter 2: Algebraic Expressions 2.2. Laws of Exponents (Note: Extended to the following week because of review of topics discussed by student teacher.) Learning Objectives Experiences/Strategies/Reflection/Action/Evaluation/Assignment July 15, 2008 Objectives At the end of the period, the students should be able to: PRELECTION: The Place Value System • The Place Value System will be presented to the class. • The following questions may be asked: a. How do you describe each place value in the system? b. What does each mean? c. How else is each described? • The teacher will present to them the term “exponents” as a way of describing each place value in the system. ACTIVITIES/STRATEGIES (EXPERIENCE) 1.define exponent, base, and power; 1. Dyad Work: Problem Solving • Present to the class a problem: Pedro Hardinero is constructing a den for his chicken. The measure of the length is the same as the measure of the width. a. If the measure of the width is 5m, what is the measure of the length? b. What kind of figure represents the actual den? c. What will be the area to be covered by the den? • Let the pairs work on the problem. • Call on some pairs to present their solutions on the board. (Different solutions are highly appreciated.) • Processing questions: a. Are the solutions similar or different? Are the answers similar or different? b. How did you get your solution? What operation was involved? c. How did you get the area of the den? 2. Classroom Discussion • Focus discussion on the area of the figure: A = s 2 • The following questions may be asked that will lead to the definition of exponents.
• 15 a. The formula A = s gives us the area of a square figure. How did you use the formula? 2 b. For some of you, you used A = s x s. Does it give the same answer? c. What can we say about s 2 and s x s? • Let’s us try to explore more: a. The volume of a cubic figure is V = s 3 . If the measure of the side is 10 cm, what is its volume? Express solution in two ways possible. b. Find the product: 1. 3 4 2. (−2) 5 3. 10 5 • Then, students will be asked to define the term “exponents” in their own words. • Formal definition will also be presented to them. Other related terms will be defined, such as base, power, exponentiation, exponential expression, and the like. • The teacher will also ask them to observe some patterns in the following: a. (−2) 3 and (−2) 4 b. (−3) 3 and (−3) 4 • Generalization will follow. July 16, 2008 Objectives (original) July 17, 21-23, 2008 Objectives At the end of the two periods, the PRELECTION: students should be able to:  Recall the fundamental definition of exponent: If b is a real number and n is a natural number, then b n = b ⋅ b ⋅ b ⋅ ... ⋅ b , where b appears as a factor n times. b n is read “the nth power of b ” or “ b to the nth power”. Thus, the nth power of b is defined as the product of n factors of b . ACTIVITIES/STRATEGIES (EXPERIENCE) 1. formulate the different laws of exponents, namely: 1. Classroom Discussion (Induction)
• 16 a. Product Law  Present to them the different sets of mathematical expressions. b. Power of the Product Law SET A: Find the area of the rectangle below. c. Power of the Power Law d. Quotient Law 8 in e. Power of Quotient Law 4 in Questions to ask: a. Find the area of the rectangle using the area formula for rectangles. b. How else can we express in the formula the factors 8 in and 4 in, aside from A = (8in)(4in) ? c. What do you observe with the factors written in exponential form? d. Try to expand both factors. What product do you get? e. Can we syncopate the factors and yet get the same product? f. What makes is possible? PRODUCT LAW: x m ⋅ x n = x m + n (Additional examples will be presented to them.) SET B: Find the area of the square figure below. 4 in Questions to ask: a. Find the area of the square using the formula A = s 2 . What is the product? b. Express 4 in exponential form. How would you insert the expression in the formula? c. How would you then get the same product as in using the formula? d. What have you noticed with the expression? e. If similar expressions appear, what will you do? POWER OF THE POWER LAW: ( x m ) n = x mn (Additional examples will be given to them.) SET C: Find the product of 12 2 . Questions to ask: a. Expand 12 2 . Find the product.
• 17 b. Suppose we express 12 as two smaller factors raised to 2. Can we still get the same product? Show in more than one possible way. c. For instance, 12 2 = (6 ⋅ 2) 2 . How would you manipulate the right side of the equation to find the product the easier way without resorting to multiplying 6 and 2? d. For expressions such as (6 ⋅ 2) 2 , what would be a corresponding exponential expression for this? e. In case you encounter similar expressions such as this, what will you do? POWER OF THE PRODUCT LAW: ( x ⋅ y ) n = x n y n (Additional examples will be given to them.)  23  SET D: Suppose we manipulate the expression in SET A and get  2 2  , instead of 2 3 ⋅ 2 2 .    Questions to ask: a. How similar are the two expressions? b. Expand both the numerator and denominator, then simplify. What do you get? 23 2 ⋅ 2 ⋅ 2 c. In the equation = = 2 , what is the exponent of 2? What manipulations can we make in the original 22 2⋅2 expression to get the same result? Why is that possible and when is that possible? d. If you encounter similar expressions such as this, what will you do? x m x m−n QUOTIENT LAW: n = = x m − n , m > n . The assumption of which is x n ≠ 0 . x 1 Extension of the Quotient Law: 23 22  Suppose we reverse the expression from to 3 , what would be the result? Do we get the same result? 22 2 What relationship can you establish? xm 1  State the observation and come up with the rule. n = n−m , m < n . x x  Suppose the exponent in the numerator is lesser than the exponent in the denominator as in the extension example, and you did not follow the rule, what result do you get?  Since the sign of the exponent is negative, is there a way to make it positive since any expression with negative exponents is not simplified? 1 1 xn x −n = ⇔ = = xn . xn x −n 1
• 18 (Additional examples will be given to them.) SET E: Use the given expression in Set C. This time express two numbers as a quotient of 12, then raise it to 2. Questions to ask: 2 24  24  a. Take one instance, then raise to the power of 2. We have   . Then find the product by expressing the 2  2  factors as quotients. 2  24  b. What would be the exponential expression equivalent to   ?  2  c. In case you encounter similar expressions such as this, what will you do? 2  22  d. Suppose the expression is  3 3  , will the law apply?    n p  x xn  xn  x np POWER OF THE QUOTIENT LAW:   = n . Similarly,  y  m y  = mp .    y   y (Additional examples will be given to them.) 2. apply the laws of exponents to different exercise items and 2. Dyad Work: Solving Exercises situations;  A set of exercises will be presented to them for each dyad to work. Sample items are as follows: A. Evaluate each exponential expression. 0  3a −5 b 2  1. (−9) 2 6. (x )11 5 11. (−5 x y )(−6 x y ) 4 7 11 3 −2 14. (4 x ) 17.   12a 3 b − 4     3 8 x 20 24 x 3 y 5  30a 14 b 8  2. − 9 0 −6 4 7. ( x ) 12. 15. 18.   10a 17 b −2   2x 4 32 x 7 y −9   −2 25a 13b 4  5x 3  3. 3 ⋅ 3 2 3 4 2 8. (6 x ) 13. 16.   y   − 5a 2 b 3   4. x 11 ⋅ x 27 9. (−3 x 2 y 5 ) 2 x 30 5. 10. (3x 4 )(2 x 7 ) x10
• 19  Problem Solving: Approximately 2 × 10 4 people run in New York City Marathon each year. Each runner runs a distance of 26 miles. Write the total distance covered by all the runners (assuming that each person completes the marathon) in scientific notation. 3. Individual Work: Writing in Mathematics (Journal Writing to replace Reflection-Action)  Essay. Answer each of the following. 1. Describe what it means to raise a number to a power. In your description, include a discussion of the difference between − 5 2 and (−5) 2 . 58 2. Explain the power of the power rule for exponents. Use 2 in your explanation. 5 −5 3. Why is (−3 x )(2 x ) not simplified? What must be done to simplify the expression? 2 4. Synthesis:  In their index cards, the students will be asked to write down important notes about exponents. Remember! −n 1 xm 1. x = 5. = x m−n xn x n 2. x 0 = 1 6. ( xy ) n = x n y n n a an 3. x ⋅ x = x m n m+ n 7.   = n b b 4. ( x m ) n = x mn EVALUATION  Quiz (separate sheet)  Exercise Output
• 20 MATERIALS TO BE USED: 1. Exercise worksheets 2. Textbook/References 3. Acetates and other visual aids 4. Power Point Presentations where appropriate taken from Internet sites Ignatian Pedagogical Paradigm Lesson Plan Subject: Mathematics 1: Beginning Algebra (I–St. Garnet) Prepared by: Rommel M. Gonzales
• 21 Date of coverage: 3 Periods August 07, 2008 2 periods August 08, 2008 1 period Content Coverage: Chapter 2: Algebraic Expressions 2.3 Evaluating Algebraic Expressions Learning Objectives Experiences/Strategies/Reflection/Action/Evaluation/Assignment August 07, 2008 Objectives: At the end of 2 periods, the learners should be able to: PRELECTION: Presentation of the Problem for the Day and the questions. The teacher will clarify important points of the problem. Mrs. Alvarez wants to keep track of her students’ scores on the latest examinations. She is preparing a program (in this case a formula) that will give each student’s score on the examination after she enters just the number of items that a student missed. To do this, an algebraic expression for the test score is needed. Questions: 1. There are 100 points on the examination. Each item is worth 5 points. What quantity varies from student to student and could be used to find each student’s score? 2. Define the variable you will use. Write an algebraic expression for the examination score. 3. Using your expression in number 2, complete the table.
• 22 Student Number of Items Missed Examination Score a. Alvin 2 b. Gio 4 1. describe the process on evaluating algebraic expressions; c. Homer 5 d. Madel 3 e. Rose 6 f. Any student x 4. How could the expression in number 2 help Mrs. Alvarez to compute grades? As a student, could you find a use for this expression? Explain. ACTIVITIES/STRATEGIES (EXPERIENCE)
• 24 MATERIALS TO BE USED: 5. USAid Instructional Materials 6. Manila Paper 7. Crayons 8. Masking tape Ignatian Pedagogical Paradigm Lesson Plan Subject: Mathematics 1: Beginning Algebra (I–St. Garnet) Prepared by: Rommel M. Gonzales, MAST Date of coverage: August 11 – 13, 2008 5 periods Content Coverage: Chapter 3: Operations on Polynomials 3.1. Addition and Subtraction of Polynomials
• 25 Learning Objectives Experiences/Strategies/Reflection/Action/Evaluation/Assignment August 11, 2008 Objectives: At the end of 2 periods, the learners should be able to: PRELECTION: Presentation of the Problem for the Day and the questions. The teacher will clarify important points of the problem. This will be worked by pairs. Martin has a rectangular fishpond. In the afternoon, he finds time walking along the dike of his fishpond. The length is (14 x + 5) meters and the width is (5 x − 2) meters. What is the distance around the pond? Questions: 1. Draw the rectangular fishpond according to how you imagine it. Indicate the measures of the sides. 2. What does “distance around the pond” mean? ACTIVITIES/STRATEGIES (EXPERIENCE) 1. add/subtract polynomials 1. Classroom Discussion: Presentation of Solutions and Processing of the Problem Posted in the Prelection Part vertically and horizontally; • Some pairs will be asked to show their drawing of the fishpond. • For the processing, the following questions may be asked: 1. How did you decide to draw the fishpond that way? 2. How did you interpret “distance around the pond”? 3. How did you find the distance around the pond? Are there different solutions and answers? 4. How did you resolve difficulties? • The class will be asked of the concept behind their solution. • Then the facilitator will now check the solutions presented on the board. • The following questions will be asked: 1. How do you add/subtract polynomials? 2. What rules can you formulate in adding or subtracting polynomials? • The students will also be presented two ways in adding and subtracting polynomials – horizontal and vertical methods. 2. Pair Work: Exercises A. Add the following polynomials.
• 26 1. (3a 5 − 9a 3 + 4a 2 ) + (−8a 5 + 8a 3 + 2) 6. (4k 3 + k 2 + k ) + (2k 3−4k 2 − 3k ) 2. 2c 2 − 4 + 8 − c 2 7. (3 p 2 + 2 p − 5) + (7 p 2 − 4 p 3 + 3 p) 3. 6 + 3 p − ( 2 p + 1) − (2 p + 9) 8. (2a 2 + 3a − 1) + (4a 2 + 5a + 6) − 6m 3 + 2m 2 + 5m 4. 6m 2 n − 8mn 2 + 3mn 2 − 7 m 2 n 9. 8m 3 + 4m 2 − 6m − 3m 3 + 2m 2 − 7 m 5. ( y 3 + 3 y 2 + 2) + (4 y 3 − 3 y 2 + 2 y + 1) 1. describe the process on B. Subtract the following polynomials. evaluating algebraic expressions; 1. 8a − (3a + 4) − (5a − 3) 4. (q 4 − 2q 2 + 10) − (3q 4 + 5q 2 − 5) 2. (3r + 8) − (2r − 5) 5. ( z 5 + 3 z 2 + 2 z ) − ( 4 z 5 + 2 z 2 − 5 z ) 3. (2a 2 + 3a − 1) − (4a 2 + 5a + 6) 6. (5t 3 − 3t 2 + 2t ) − ( 4t 3 + 2t 2 + 3t ) 3. Synthesis: • When adding polynomials, simply combine like terms. But more than just the combining of like terms, learners should not forget the basic rules of adding integers. • When subtracting two polynomials, add the first polynomial and the negative of the second polynomial. The basic rule applies: Change the sign of the subtrahend, then proceed to addition. REFLECTION: When combining terms, only those that are like can be combined. On similar stance, accomplishing a group task necessitates a common orientation of the goal which the group has to achieve. Otherwise, if there are some members who do not believe in concerted efforts, then the goal simply cannot be reached. 1. How do you assess your working dynamics in the different group activities? 2. Was it helping or not? Why? ACTION: Suggest at least two ways on how you think your group dynamics be improved.
• 27 EVALUATION: Analyze each of the items, then perform the indicated operations. 1. Subtract 4 y 2 − 2 y + 3 from 7 y 2 − 6 y + 5 . 2. Subtract − (−4 x + 2 z 2 + 3m) from [(2 z 2 − 3 x + m) + ( z 2 − 2m)] . 3. (−4m 2 + 3n 2 − 5n) − [(3m 2 − 5n 2 + 2n) + (−3m 2 ) + 4n 2 ] 4. − [2 p − (3 p − 6)] − [(5 p − (8 − 9 p)) + 4 p ] MATERIALS TO BE USED: 1. Reference materials 2. Exercise items 3. Textbook