# CPM Educational Program

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1 CPM Educational Program A California, Non-Profit Corporation Chris Mikles, National Director (888) CPM Courses and Their Core Threads Each course is built around a few core ideas as recommended by the TIMSS Report, which in turn are developed and deepened over the four year program. Specific skills and concepts are tied to these threads and presented in contet--both conceptually and in realistic problems--so that students will see the connections among the ideas and make sense of mathematics. Math (Algebra ) (detailed below) Math 2 (Geometry) Problem Solving Problem Solving Graphing Graphing Writing Equations Ratios Solving Equations Geometric Properties Ratios Algebra Symbol Manipulation Spatial Visualization Conjecture & Eplanation (Proof) Math 3 (Algebra 2) Math 4 (Mathematical Analysis) Problem Solving Problem Solving Representation & Modeling Concepts of Calculus Functions & Graphing Analysis of Models Intersections and Systems Trigonometry Algorithms Advanced Functions Reasoning and Communication Algebraic Fluency & Accuracy CPM Snapshot: Developing the Factoring Thread in Algebra (Math ) Students begin preparing for trinomial factoring with the third problem in the tet (GS-3). The "diamond problems" introduce "sum and product" as a patterning problem in a puzzle format. These problems recur in homework throughout units 0-7. In Unit the tiles are used to introduce combining like terms, followed by the Distributive property in Unit 2. In Unit 2 and subsequent units, the tiles are used to create area models for multiplying algebraic terms. Notice how units 4, 6 and 7 continue to practice the skills students will need to factor trinomials. Toward the end of the first semester tiles and generic rectangles are used to introduce multiplying binomials. After a unit and a half of practicing this multiplication, the course turns to undoing the multiplication, that is, trinomial factoring with a = (factoring a comes in Unit 0, after students have had at least a month to consolidate basic trinomial factoring). Once again tiles, area models, and patterning are used to help concretize and visualize factoring, and the process is tied back to the diamond problems. The basic factoring thread culminates with an investigation in the theme problem, AP-79, in Unit 8. Factoring is revisited in each of the subsequent CPM courses. Answers appear in bold print, as they do in the teacher version of the tet. The references to student "tool kits" is CPM's way of teaching and requiring students to have study notes for each unit. The tool kit icon means "put this in your tool kit---now!" 998 CPM Educational Program Factoring Thread Preview page

2 Unit 0 Getting Started GS-3. (Pre-problem in puzzle form for trinomial factoring) DIAMOND PROBLEMS With your study team, see if you can discover a pattern in the three diamonds below. In the fourth diamond, if you know the numbers (#), can you find the unknowns (?)? Eplain how you would do this. Note that "#" is a standard symbol for the word "number" ? # # - 5? Patterns are an important problem solving skill we use in algebra. The patterns in Diamond Problems will be used later in the course to solve algebraic problems. Copy the Diamond Problems below and use the pattern you discovered to complete each of them. a) 3 4 b) c) 2 7 d) 4 2 e) 8-6 [ y = 2; y = 7 ] [ y = 6; y = -5 ] [ = 3; y = 4 ] [ = 8; y = 8.5 ] [ = -2; y = -4 ] Unit Organizing Data (Introducing manipulatives for symbol manipulation.) SQ-67. Summarize the idea of Combining Like Terms in your tool kit. Then represent the following situations with an algebraic epression. Combining tiles that have the same area to write a simpler epression is called COMBINING LIKE TERMS. Eample: We write 2 to show 2() or 2. Represent each of the following situations with an algebraic epression. a) b) c) 38 small squares 20 rectangles 5 large squares [ ] [ 3 5 ] [ ] 998 CPM Educational Program Factoring Thread Preview page 2

3 SQ-7. Eample: To show that 2 does not usually equal 2, you need two rectangles and one big square [ Solutions shown below ] a) Show that b) Show that 2-2. Unit 2 Area and Subproblems (Area and perimeter visual basis for multiplying.) Eample: MULTIPLYING WITH ALGEBRA TILES The dimensions of this rectangle are by Since two large squares cover the area, the area is 2 2. We can write the area as a multiplication problem using its dimensions: (2) = 2 2 KF-53. Use the figure at right to answer these questions. a) What are the dimensions of the rectangle? [ 2 by 3 ] b) What is the area of this rectangle? [ 6 2 ] c) Write the area as a multiplication problem. [ (2)(3) = 6 2 ] KF-74. Show two methods of finding the area of the rectangles below. Be sure to write down and label your subproblems. a) 7 9 b) [ 48 ] 8 [ 280 ] 998 CPM Educational Program Factoring Thread Preview page 3

4 KF-78. In your own words, add the following information to your tool kit. THE DISTRIBUTIVE PROPERTY With two rectangles (2) and eight small squares (8), we can represent the product 2( 4) by grouping the tiles together. Below right, a GENERIC RECTANGLE represents the same product as if its parts were the dimensions (length and width) of a rectangle. Study these representations and find their similarities and differences. Then record both eamples in your tool kit ( 4) = = 2 8 In general, when you use tiles and rectangles to multiply two or more terms by the same number, variable, or mied epression, you can see the multiplying pattern we have been using. The pattern is called the DISTRIBUTIVE PROPERTY and can be written as a rule: a(b c) = ab ac. KF-87. Find the total area of the rectangle at right in as many different ways as you can. Eplain your methods. Are there more than two ways to solve this problem? [ = 224; (0 6)(0 4) = 224 ] Unit 4 Writing & Solving Equations (Practice continues.) CP-52. Use Distributive Property to rewrite each epression. Multiply in parts (a) and (b). Factor in parts (c) and (d). a) 2(3-2) [ 6-4 ] c) 5 5 [ 5( 3) ] b) 2(5 3) [ ] d) 2 4 [ ( 4) ] 998 CPM Educational Program Factoring Thread Preview page 4

5 Unit 6 Graphing and Systems of Linear Equations (Etend model to binomials.) WR-66. Write the area as a sum and as a product. a) c) 5 3 [ (3)(6) = =208 ] b) 4 2 [ 3( 5) = ] d) Find the perimeters of the three rectangles above. [ 2( 4) = 2 8 ] [ 58, 2 2, 8 0 ] WR-67. Copy and solve these Diamond Problems: product # # sum a) 5 0 [ y = 50; y = 5 ] b) c) d) 6y 2 6y 2 2y 2y 3y 7y [ = 3y; sum = 5y ] [ prod = 6y 2 ; sum = 5y ] [ y and 6y ] WR-70. In this semester we have developed several big ideas. One of these ideas has been the Distributive Property and multiplication of variables. In the net two days we will epand our understanding of multiplication to include pairs of sums. Use your Algebra Tiles to construct each of the following rectangles with the specified dimensions. Draw the complete picture of each rectangle on your paper, then write its area as a product and as a sum. Label everything as shown in the eample below. 4 4 Eample: 2 ( )( 4) = product sum a) 3 b) 2 c) 6 5 [ ( 5)( 3) = ] 4 [ ( 2)( 4) = ] [ ( 6) = 2 6 ] 998 CPM Educational Program Factoring Thread Preview page 5

6 WR-7. We can make our work drawing tiled rectangles easier by not filling in the whole picture. That is, we can show a generic rectangle by using an outline instead of drawing in all the dividing lines for the rectangular tiles and unit squares. For eample, we can represent the rectangle whose dimensions are by 2 with the generic rectangle shown below: area as a product area as a sum ( )( 2) = = In the last step we found the area of each of the parts of the generic rectangle (the four interior rectangles) by multiplying each length and width, and then recorded the area of each part as a sum. (The complete problem continues with eercises like those in WR-70.) WR-72. Carefully read this information about binomials. Then add a description of binomials and the eample of multiplying binomials to your tool kit. These are eamples of BINOMIALS: These are NOT binomials: (3 2-7) y Write a description of a binomial that would be clear to a new student. We can use generic rectangles to find various products. We call this process MULTIPLYING BINOMIALS. For eample, multiply (2 5)( 3): (2 5)( 3) = area as a product area as a sum Note that a generic rectangle helps us organize the problem. It does not have to be drawn accurately or to scale. In the eample above, you can see that the region representing 5 is probably not as big as it should be compared to the region for CPM Educational Program Factoring Thread Preview page 6

7 WR-82. Algebra tiles can be used to build a series of squares. After you build each square, write the area as a product, sketch the tiles, and write the areas as a sum. Organize your work in a table as shown below. a) Complete this chart for the first 6 squares. The first square is by, the second square is 2 by 2, and so on. Area as a Product Sketch Area as a Sum ( ) [ ] [ ] [ ] etc. b) In the rectangles you drew, eplain where you see the squares. [ opposite corners. ] c) Describe all the patterns you found in the table. [ answers vary; look at perfect square constants, patterns with the middle coefficient ] d) Using your patterns, and without drawing a sketch, fill in the chart s net two products and areas. e) Without drawing a sketch, what would the 0th product and area be? [ ( 0) 2 = ] f) Add two more perfect square products and areas to your table without drawing a sketch. Unit 7 Slopes and Rates of Change (Practice and consolidation before factoring.) BR-30. Copy these generic rectangles on your paper. Multiply or factor to rewrite these epressions. a) b) 3-5 [ ] [ ] BR-67. Write a shorter epression for each of the following polynomials by simplifying. a) ( ) - (3 2 6) b) 3( ) 2( ) ( ) [ ] [ 6 6, or 6( ) ] BR-68. Multiply. You may find the sketch of a generic rectangle useful. a) ( 2)( 8) [ ] b) ( - 2)( 8) [ ] 998 CPM Educational Program Factoring Thread Preview page 7

8 Unit 8 Factoring Quadratics AP-3. Write an algebraic equation for the area of each of the following rectangles as shown in the eample below. Eample: ( 3)( 2) = product sum a) c) e) [ ( 3)( 4) = ] [ ( )( ) = 2 2 ] [ ( )(2 3) = ] AP-. Find the dimensions of each of the following generic rectangles. The parts are not necessarily drawn to scale. Use Guess and Check to write the area of each as both a sum and a product. a) 2 5 c) 2 6 e) ( 5)( 3) ( 6)( 3) ( 5)( 2) 998 CPM Educational Program Factoring Thread Preview page 8

9 AP-8. USING ALGEBRA TILES TO FACTOR What if we knew the area of a rectangle and we wanted to find the dimensions? We would have to work backwards. Start with the area represented by Normally, we would not be sure whether the epression represents the area of a rectangle. One way to find out is to use Algebra Tiles to try to form a rectangle. You may find it easier to record the rectangle without drawing all the tiles. You may draw a generic rectangle instead. Write the dimensions along the edges and the area in each of the smaller parts as shown below. Eample: We can see that the rectangle with area has dimensions ( 2) and ( 4). Use Algebra Tiles to build rectangles with each of the following areas. Draw the complete picture or a generic rectangle and write the dimensions algebraically as in the eample above. Be sure you have written both the product and the sum. a) [ ( 4)( 2) ] d) [ ( 3)( 4) ] b) [ ( )( 4) ] e) [ (2 8) or AP-9. USING DIAMOND PROBLEMS TO FACTOR Using Guess and Check is not the only way to find the dimensions of a rectangle when we know its area. Patterns will help us find another method. Start with Draw a generic rectangle and fill in the parts we know as shown at right. 2 2 We know the sum of the areas of the two unlabeled parts must be 8, but we do 8 not know how to split the 8 between the two parts. The 8 could be split into sums of 7, or 6 2, or 3 5, or 4 4. However, we also know that the numbers that go in the two ovals must have a product of 2. a) Use the information above to write and solve a Diamond Problem to help us decide how the 8 should be split. [ product of 2, sum of 8; 2, 6 ] b) Complete the generic rectangle and label the dimensions. [ ( 2)( 6) ] product sum 998 CPM Educational Program Factoring Thread Preview page 9

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