Chris Yuen. Algebra 1 Factoring. Early High School 8-10 Time Span: 5 instructional days

Transcription

1 1 Chris Yuen Algebra 1 Factoring Early High School 8-10 Time Span: 5 instructional days Materials: Algebra Tiles and TI-83 Plus Calculator. AMSCO Math A Chapter 18 Factoring. All mathematics material and homework assignments are from that tetbook. Description: This 5-day unit is about factoring. We start out with a review of the multiplication of algebraic epressions, and then we move into division of algebraic epression as a transition to factoring. Algebra Tiles is heavily used and toward the end of the lesson, TI-83 plus calculator will also be used. In addition to technology and manipulatives being used in this unit, there are several writing prompts for students to reflect what they learn and epand their thinking and development of the concepts. Day 1: Multiplication of algebraic epressions Day 2: Division of algebraic epressions (Part I) Day 3: Division of algebraic epressions (Part II) Day 4: Factoring polynomials (Part I) Day 5: Factoring polynomials (Part II) Day Objectives: 1) Student will be able to competently multiply algebraic epressions. 2) Student will be able to use algebraic tiles to divide algebraic epressions. Zero pairs are used is Day 3. 3) Student will be able to factor polynomials, and compare how multiplication and factoring are related. NCTM Standards (9 12): Represent and analyze mathematical situations and structures using algebraic symbols. 1) understand the meaning of equivalent forms of epressions 2) use symbolic algebra to represent and eplain mathematical relationships 3) build new mathematical knowledge through problem solving 4) apply and adapt a variety of appropriate strategies to solve problems 5) monitor and reflect on the process of mathematical problem solving

2 NYS Math A Core Curriculum: Key Idea 3 Operation 1) Use multiplication and division with real numbers and algebraic epressions. 2) Multiplication of polynomials: powers, products of monomials and binomials 3) Division of polynomials by monomials 4) Factoring: common monomials, binomial factors of trinomials 2

3 3 LESSON 1: PRODUCT OF TWO BINOMIALS This lesson is designed for students to review the product of two binomials. Specifically, students in my school building are likely to have been taught to find the product using FOIL instead of using algebra tiles. Therefore, this lesson gives the students an opportunity to develop ability using algebra tiles. Zero pairs are emphasized is this lesson. This lesson is the first of the series, preparing students for the upcoming lessons on factoring. Anticipatory Activity: This lesson begins with a question/challenge. Consider how to find the area of the following etendable square bracket. Before etension, it is a square, and for the moment, let s define the measurement of the side of this square shape bracket cm. The sides of the square bracket can be etended and after etension, the bracket is rectangular with its length being maimally etended by 3 cm and its width by 2 cm. Problem 1: What is, in an algebraic epression, the area of the original square bracket? Problem 2: What is the new area of the maimally etended bracket? How would one represent this area in an algebraic epression? cm cm 3 cm cm EXTEND cm 2 cm Hint: Consider the area of the etended rectangle to be the total of four smaller areas:

4 4 cm cm 3 cm cm EXTEND cm 2 cm When students are struggling, with the area of the etended rectangle, the hint is useful to remind the students. Also, emphasize that rectangles have two pairs of congruent sides. After the students work on the two problems, the discussion of the solution is used to introduce to them on the product of binomials. Developmental Activity: This activity requires students to develop the skills to find the product of two binomials using algebra tiles. Teacher will illustrate the following eamples to the students: Eample 1: Epress (2)(3)++ as a trinomial This eample is directly from the problem of the day in the anticipatory activity. This eample is useful because students could relate from the previous activity to this lesson. Eample 2: Epress (4)(5)-+ as a trinomial This eample shows how algebra tiles could represent positive and negative numbers. Zero pairs emphasized here.

5 5 Eample 3: Epress (5)(4)+- as a trinomial This is another eample showing how algebra tiles could represent positive and negative numbers and how zero pairs play a role here. The commutative property of multiplication is emphasized in this eample. Eample 4: Epress (23)(4)-- as a trinomial This eample serves two purposes: 1) is to acquaint student with the product of two negative numbers, and 2A 2) is to show how algebraic tile represents term. Eample 5: Epress (3)(3)-+ as a binomial. -3 This eample shows that not all product of binomial results in a trinomial. +3 After the eamples are shown to students, students are to try out, using algebra tiles, the following problems. Students are given time to work on those problems. Teacher is to go around to help out struggling students. The concept of zero pair will be emphasized. Selected students are to draw their solutions on the board.

6 6 Sample problems for try out: Epress the following five epressions as polynomials: (2)(5)++ 1) (4)(4)-+ 2) (21)(3)-+ 3) (5)(2)-- 4) (25)(31)+- 5) Closing Activity: This activity is to mirror the anticipatory activity with different etensions, with a built-in writing component for student to reflect on what they have developed in this lesson. Consider how to find the area of the following etendable square bracket. Before etension, it is a square, and for the moment, let s define the measurement of the side of this square shape bracket cm. The sides of the square bracket can be etended and after etension, the bracket is rectangular with its length being maimally etended by 3 cm and its width by 2 cm. Problem 1: What is, in an algebraic epression, the area of the original square bracket? Problem 2: What is the new area of the maimally etended bracket? How would one represent this area in an algebraic epression? Problem 3: How did you come up with these algebraic epressions? Write a paragraph eplaining your algebra tile model. Does zero pair apply in this question? Why or why not? cm cm 5 cm cm EXTEND cm 3 cm Homework assignments will be given.

7 Sample Algebra Tiles 7

8 8 LESSON 2: DIVISION OF ALGEBRAIC EXPRESSIONS (PART I) This lesson is designed to prepare students for factoring through a transition of division of algebraic epressions. To develop division, this lesson continues to employ the algebra tile model for the reasoning. To not complicate the subject matter, the division that will be developed here will not involve any remainder. Zero pairs will not be introduced until Lesson 3. By the end of the lesson, student will be able to divide simple polynomial, using algebra tiles as a tool. Anticipatory Activity: This lesson begins with a review from the previous lesson. This review serves three purposes: to reacquaint students with the algebra tile model, to acquaint students with three epressions which will be used in the developmental activity, and to provide an opportunity for the teacher to informally evaluate students 3(2)+ on the previous 2(43)- lesson s material. (3)(25)-- Consider the following three epressions:,, and. How are they presented using algebra tiles? Developmental Activity: This activity requires students to develop the division skills using algebra tiles. Teacher will illustrate the following eamples to the students:

9 9 Eample 1: Find the quotient which 36+ is divided by Eample 2: Simplify The problem is etremely useful as the first eample because one can actually divide 36+ into three equal portions. The discussion of this eample is twofold: one is to show the students that the meaning of division could be etended to division of algebraic epressions, and two is to reason out that if 3 represents the width of a rectangle with area of 3 + 6, then + 2 must be the length of the rectangle This problem is deliberately worded differently from the first eample so that students can get used to the variety of instruction. The premise of this problem is similar to the first eample of this lesson: one is to show the students that the meaning of division could be etended to division of algebraic epressions, and two is to reason out that if 2 represents the width of a rectangle with area of 8 6, then 4 3 must be the length of the rectangle. Eample 3: Find the unknown side of a rectangle if one side is 3, and the area of the rectangle is. 2-3 Again, this eample uses a different set of instruction. This eample departs from the original two eamples that division has to do with dividing something into equal portions, and this eample has to do with the area of a rectangle and given a measurement of one side, the measurement of the other side can be found through division. -5

10 Eample 4: Divide by First Trial: How will these tile fit into the rectangle? 1 1 This trial shows students that if the tiles are not strategically placed, then the rectangle would not emerge. This is a good place to ask students for ideas on how to be more strategic. 1 1 Second Trial: This does not look like a rectangle. When students idea lead to some success, the teacher could use those ideas to start a second trial. The key is to let the students to lead a in depth discussion on solving an intricate problem. This could be a good start. How will the rest of the tiles fit into a rectangle?

11 11 Second Trial Continued: The ten "" tiles are in, but there are 24 more "1" tiles. Where are there supposed to be? One may suggest that the rest of the si 2 tiles to be place vertically, right net to the tile. This is the time to let the student discover where the rest of the 24 1 tiles should be placed. Second Trial Final: to the question When the final layout of the rectangle is presented, it is useful to point out to students that there are generally four sections to the rectangle, including two sections of tiles in different orientations. Also, it is time to redirect students attention back to what eactly the question asks. Many students at this point may lose sight of the question. Therefore, it is important to have students reread the question again, in order to find out that 6+ is the answer Students are to try out the following divisions in class: ) Find the quotient: ) Simplify:. 3) Find the unknown side of a rectangle if one side is epressed as + 3, and the area of the rectangle is ) Divide by 6-. The problems for students to try out are similar to the eamples given earlier. The instructions are varied to acquaint the students that all of the different instructions shares similar mathematical meanings. Teacher is to help students out at this portion of the developmental activities. This is also a time for the teacher to evaluate to what etent students have developed the material up to this point.

12 12 Closing Activity: This activity is to bring what is developed in this lesson to a closure, with a built-in writing component for student to reflect on what they have developed in this lesson. Students are to write in their journal in their own words, answering the following questions: 1) What did you learn today? 2) Eplain how you come up with the answer in question number 3? 3) Did the algebra tiles help? 4) In what way did you have to organize tiles for question 3 to form a rectangle? Homework assignments will be given.

13 13 LESSON 3: DIVISION OF ALGEBRAIC EXPRESSIONS (PART II) This lesson is designed to further prepare students for factoring through division of algebraic epressions. In addition to the skills introduced in lesson 2, the division in this lesson will include problems dealing with zero pairs. By the end of this lesson, student will be able to divide using algebra tiles, including using zero pairs as a strategy. Anticipatory Activity: Student will take a quiz (not included here) as to check the concept development so far. Questions are based on division without using zero pairs. Solution will be discussed to ensure all students are ready to move on to division with zero pairs. Developmental Activity: Eample 1: What is the length of the rectangle with an area represented as and the width is represented by + 1?

14 14 Eample 2: Simplify Using both eamples 1 and 2, a discussion will be held about zero pairs. Questions such as the following will be used: 1) In eample 1, it seems that there are -4 instead of -3. Is the algebraic tile representation correct? 2) Similarly in eample 2, it seems that there are -3 instead of -2. How is -2 accounted for? Students are to try out the following divisions in class: ) Divide by ) Simplify: When students complete all three problems, they may present their solution on the board. Reinforcement of the concept about will take place during student presentation. Immediately following the presentation will be a closing activity. Closing Activity: Students will be posed with two intriguing problems. The closing is used as a preview for lesson 4. 3) Find the unknown side of a rectangle if one side is epressed as + 6, and the area of the rectangle is. Suppose there are two epressions each representing an area of a rectangle. Find their dimensions of the rectangles: ) ) The problems selected here is to intrigue students to think hard about what is about to come in lesson 4. In this case, they are both factoring problems, and they look very similar. I epect students to be able to work on problem 1, but they may struggle with problem 2. Homework will be to continue working on those two problems. Additional homework will be assigned.

15 15 2ABC++ LESSON 4: FACTORING BINOMIALS AND Trinomial where A = 1 This lesson is to genuinely develop factoring, without the assisted help on division. Problems used in this lesson are binomials and trinomials. The goal for this lesson will be that student leaving the classroom with factoring skills, and be able to check their result. This is the lesson that achieve the unit goal. Anticipatory Activity: A discussion about the previous lessons closing will be conducted. Recalling: Students present their solutions. Are the solutions very similar, just as similar as the questions? Why are the solutions so different? This activity is designed to lead students to think about how to factor without letting the appearances fooling them. Developmental Activities: The solutions will be worked out for the above two problems: first, using algebra tiles, and second, using the table function in the TI-83 Plus calculator: y=++ At the y= menu: Enter and 2(6)(4)y=++. Table starts at = 0 with increasing interval of 1. Student can check with the results that y1 equals y2. Repeat the same process for problem 2. Enter y1 as the trinomial form and y2 as the factored form. Check with the table function. Students are to work on the several problems below in two different ways: first, using algebra tiles and second, one using TI-83 Plus to check the result. Suppose there are two epressions each representing an area of a rectangle. Find their dimensions of the rectangles: ) ) Factor the followings: 23-1) ) ) ) ) 216-6)

16 16 I selected those si problems because two of them are binomials; two pairs are similar to the etents that the only the signs are different. One pair is a perfect square, which serves as a discussion piece in closing activities. Closing Activities: Student will present the problems on the board. Three students per problem, working on 1) the algebraic solutions, 2) the algebra tile solution, and 3) check the solution using the TI-83. This lesson ends with a writing activity. The 2 prompts are as follows: 1) In the above si problems, questions 2 and 3 are considered perfect squares, and questions 1, 4, 5, and 6 are not. Eamine your algebra tiles solutions, and eplain in your own words, why some of them are perfect squares and why some of them are not. 2) Of the three portions the factoring process, working out algebraic solutions, working out algebra tile solutions, and checking solutions using a table. Which method you enjoy the most, and which you enjoy the least? Give some reasons to support your response. Homework will be assigned.

17 17 2ABC++ LESSON 5: FACTORING BINOMIALS AND Trinomial This lesson ends the unit by finally develop student to factor general binomials and trinomials. It is not intended to be a heavy lesson as some of the students may need to catch up with the skills involving algebra tiles and using TI-83 plus. The format is similar to the previous lesson. Students are still working in a group of three, with each factoring 2ABC++ problems. The only difference is that problems here will involve trinomials in a form of where A does not necessary equal to 1. By the end of the lesson, students will be able to factor skillfully. Anticipatory Activity: A reflection to the previous lesson s writing activity. Student will read out loud their writings. This is a time to review the previous lesson and to let students to hear others comments. Homework questions and answer session will be held at this point. Developmental Activities: Setting remains the same as the previous lesson: one student works on the problem, one student works on algebra tiles, and one student work the table feature on TI-83 Plus. When the solution is acceptable to all three students, put the solution up on the board, and students are responsible for presenting the solution. The developmental activity is meant to be not as instructionally heavy as the other lesson because some students may need to catch up with various skills. Closing Activity: In the activity, the students working in groups will be facilitated by the teacher, working on the following factoring problems: ) ) ) 249-4) ) Finally, to finish this unit, students would be eposed to some binomials and trinomials that are not factorable. The writing prompt will be as follows: Consider or, they are not factorable using algebra tiles. If you don t believe it, try arranging the tiles in several different attempts. Why aren t they factorable? Can you name 5 factorable and 5 unfactorable polynomials? Be prepare to bring them in for the net class as we will play factoring marathon. No Homework will be assigned. Student will continue working on the prompt.