# Using Algebra Tiles from Polynomials to Factoring

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2 -4. DIAMOND PROBLEMS Finding and using a pattern is an important problem-solving skill you will use in algebra. The patterns in Diamond Problems will be used later in the course to solve other types of algebraic problems. Look for a pattern in the first three diamonds below. For the fourth diamond, eplain how you could find the missing numbers (?) if you know the two numbers (#) ? # #? Copy the Diamond Problems below onto your paper. Then use the pattern you discovered to complete each one. a. b. c. d. e Your teacher will distribute a set of algebra tiles for your team to use during this course. As you eplore the tiles, address the following questions with your team. Be prepared to share your responses with the class. How many different shapes are there? What are all of the different shapes? How are the shapes different? How are they the same? How are the shapes related? Which fit together and which do not? 2-4. Build each collection of tiles represented below. Then name the collection using a simpler algebraic epression, if possible. If it is not possible to simplify the epression, eplain why not. a y b y y + 5 c y 2 + 4y + y d. 3y y y 2 + 4y + CPM Educational Program 203, all rights reserved 2

3 3-65. Write the area as a product and as a sum for the composite rectangle shown at right Now eamine the following diagram. How is it similar to the set of tiles in problem 5-22? How is it different? Talk with your teammates and write down all of your observations Use a generic rectangle to multiply the following epressions. Write each solution both as a sum and as a product. a. (2 + 5)( + 6) b. (m! 3)(3m + 5) c. (2 +)(! 5) d. (3! 5y)(2 + y) 8-2. The process of changing a sum to a product is called factoring. Can every epression be factored? That is, does every sum have a product that can be represented with tiles? Investigate this question by building rectangles with algebra tiles for the following epressions. For each one, write the area as a sum and as a product. If you cannot build a rectangle, be prepared to convince the class that no rectangle eists (and thus the epression cannot be factored a b c d. 2y y 2 + 3y 8-3. Work with your team to find the sum and the product for the following generic rectangles. Are there any special strategies you discovered that can help you determine the dimensions of the rectangle? Be sure to share these strategies with your teammates. a. b. c. 2 5!2y!6!9! y CPM Educational Program 203, all rights reserved 3

4 8-4. While working on problem 8-3, Casey noticed a pattern with the diagonals of each generic rectangle. However, just before she shared her pattern with the rest of her team, she was called out of class! The drawing on her paper looked like the diagram at right. Can you figure out what the two diagonals have in common? FACTORING QUADRATIC EXPRESSIONS To develop a method for factoring without algebra tiles, first study how to factor with algebra tiles, and then look for connections within a generic rectangle. a. Using algebra tiles, factor ; that is, use the tiles to build a rectangle, and then write its area as a product. b. To factor with tiles (like you did in part (a)), you need to determine how the tiles need to be arranged to form a rectangle. Using a generic rectangle to factor requires a different process. Miguel wants to use a generic rectangle to factor He knows that 3 2 and 8 go into the rectangle in the locations shown at right. Finish the rectangle by deciding how to place the ten -terms. Then write the area as a product c. Kelly wants to find a shortcut to factor She knows that 2 2 and 6 go into the rectangle in the locations shown at right. She also remembers Casey s pattern for diagonals. Without actually factoring yet, what do you know about the missing two parts of the generic rectangle?? 2 2 6? d. To complete Kelly s generic rectangle, you need two - terms that have a sum of 7 and a product of 2 2. Create and solve a Diamond Problem that represents this situation. product e. Use your results from the Diamond Problem to complete the generic rectangle for , and then write the area as a product of factors. sum CPM Educational Program 203, all rights reserved 4

7 OK, said Jessica, and she arranged her tiles on an equation mat as shown at right. Oh, said Jessica. I need 6 small unit tiles to fill in the corner! + + y But you only have 0, Anita reminded her. Right, I only have ten, Jessica replied. She drew the outline of the whole square and said: Oh, I get it! To complete the square, I need to add si tiles to each side of the equation: y Oh, I see, said Anita. You started with y = , but now you can rewrite it as y + 6 = ( + 4) 2. Thank you so much, Anita! Now I can easily write the function in graphing form, y = ( + 4) 2! 6. How can you use your graphing calculator to verify that y = and y = ( + 4) 2! 6 are equivalent functions? CPM Educational Program 203, all rights reserved 7

8 8-0. Write each function in graphing form, then state the verte and y-intercept of each parabola. a. f () = b. f () = Help Jessica with a new problem. She needs to complete the square to write y = in graphing form. a. Draw tiles to help her figure out how to make this epression into a square. Does she have too few or too many unit squares this time? Write her function in graphing form. b. Find the verte and the -intercepts. What happened? What does that mean? c. Algebraically find the y-intercept. Sketch the graph How could you complete the square to change f () = into graphing form? How would you split the five -tiles into two equal parts? Jessica decided to use force! She cut one tile in half, as shown below. Then she added her two small unit tiles Figure A Figure B a. How many small unit tiles are missing from Jessica s square? b. Write the graphing form of the function, name the verte and y-intercept, and sketch the graph. CPM Educational Program 203, all rights reserved 8

10 Algebra Tiles Resource Page Page 2 of 3 CPM Educational Program 203, all rights reserved 0

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