# Mathematics More Visual Using Algebra Tiles

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1 Chris Mikles CPM Educational Program A California Non-profit Corporation 33 Noonan Drive Sacramento, CA 958 (888) fa: (08) An Eemplary Mathematics Program --U.S. Dept. of Education Making Mathematics More Visual Using Algebra Tiles 00 CPM Educational Program page

2 Unit 0: Working in Teams We purposely placed Diamond Problems in Unit Zero, before arithmetic with integers is investigated in Unit One. Use these problems as an informal assessment of current student skills. Do not stop and teach the class how to add integers. This will be thoroughly introduced early in Unit One. GS-3. 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 b) c) 7 d) e) 8-6 [ y = ; y = 7 ] [ y = 6; y = -5 ] [ = 3; y = ] [ = 8; y = 8.5 ] [ = -; y = - ] f) g) h) 9 7 i) 6 5 j) [ y = ; y = 5 ] [ y = 5; y = - ] [ y = -; y = -8 ] [ = 3; y = ] [ y = 3.5; y =.75 ] 00 CPM Educational Program page

3 Unit : Organizing Data Model SQ-65 with the class. SQ-65. You have made or have been provided with sets of tiles of three sizes. We will call these "algebra tiles". Suppose the big square has a side length of and the small square has a side length of. What is the area of: a) the big square? [ ] b) the rectangle? [ ] c) the small square? [ ] d) Trace one of each of the tiles in your tool kit. Mark the dimensions along the sides, then write the area of each tile in the center of the tile and circle it. From now on we will name each tile by its area. SQ-66. Check your results with your team members. a) Find the areas of each tile in problem SQ-65 if =. Find the areas of each tile if = 6. [ 6,, ; 36, 6, ] b) Why do you think is called a variable? [ area varies ] You will need to model combining like terms with the students in preparation for the net few problems. Place this on the overhead and ask What area do these tiles represent? [ ] After they name it, place these on the other side, and ask What is this area? [ 3 ] Then ask, If we put everything on the screen together, what is the area? [ 5 5 ] Keep modeling eamples until the students are comfortable. Other eamples to model: ( ) ( ) = 5 3 ( 3) ( 5) = 6 8 (3 ) ( 3 ) = CPM Educational Program page 3

4 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 to show () or. Represent each of the following situations with an algebraic epression. 38 small squares 0 rectangles 5 large squares [ 3 ] [ 3 5 ] [ ] = [ ( ) ( 3) = 5 ] SQ-68. You put your rectangle and two small squares with another pile of three rectangles and five small squares. What is in this new pile? [ 7 ] SQ-7. Eample: To show that does not usually equal, you need two rectangles and one big square. a) Show that 3 3. b) Show that -. [ Solutions shown below ] 00 CPM Educational Program page

5 Unit : Area and Subproblems Today the student problems etend the area work to variable multiplication and the Distributive Property. Encourage students to use Algebra Tiles to eplore these ideas. If students need to make them, a master for paper models of Algebra Tiles is included as a Resource Page in Unit One. It may be helpful to model () = 8 to make the day smoother. Eample: MULTIPLYING WITH ALGEBRA TILES The dimensions of this rectangle are by Since two large squares cover the area, the area is. We can write the area as a multiplication problem using its dimensions: () = KF-53.Use the figure at right to answer these questions. a) What are the dimensions of the rectangle? [ by 3 ] b) What is the area of this rectangle? [ 6 ] c) Write the area as a multiplication problem. [ ()(3) = 6 ] Give a brief demonstration of Grouping With Algebra Tiles. Start your demonstration by placing 3 s and s on the overhead. Rearrange them into a rectangle and then split the rectangle two ways to show the different possible grouping as shown in the student tet. Help the students interpret the drawings in these problems. Emphasize that the variable is a symbol for ANY strip length one might choose. Develop the concepts of multiplication as grouping and addition as combining. Be sure that students summarize their observations at the end of KF-55. KF-55. GROUPING WITH ALGEBRA TILES In order to develop good algebraic skills, we must first establish how to work with our Algebra Tiles. When we group rectangles and small squares together, as in the eamples below, we read and write the number of rows first, and the contents of the row second. 3 means three rows of. For eample, all three figures below contain three rectangles and twelve small squares. The total area is 3, as shown in Figure A. In Figure B, the rectangles are grouped, forming 3 rows of, written as 3(). Three rows of four small squares are also a group, written as 3(). In Figure C, notice that each row contains a rectangle and four small squares, ( ). Since three of these rows are represented, we write this as 3( ). Figure A 3 Total Area Figure B 3() 3() 3 rows of and 3 rows of Write down your observations of the different ways to group 3. Figure C 3( ) 3 rows of ( ) 00 CPM Educational Program page 5

6 KF-56. Match each geometric figure below with an algebraic epression that describes it. Note: 3. means 3 times and is often written 3. This represents 3 rows of. a) b) c) d) e) f). () () [ b ]. 3( ) [ a ] 3. 5( ) [ e ]. 3() 3() [ c ] 5. ( 5) [ d ] 6. 3() 3() [ f ] KF-57 asks students to discover that when the same tiles are grouped differently, their areas are still equivalent. The Distributive Property is introduced, and will be revisited in depth in Days 7 and 8. The Distributive Property will be added to the tool kit in KF-78. KF-57. Sketch the geometric figure represented by each of the algebraic epressions below. [ solutions shown below ] a) ( 3) b) () (3) c) Compare the diagrams. How do their areas compare? Write an algebraic equation that states this relationship. This relationship is known as The Distributive Property. [ ( 3) = () (3) ] Students may prefer to use Algebra Tiles to rewrite the following epressions. The numbers were purposely chosen to allow for tile use. There is plenty of time for students to abstract the Distributive Property. Allow teams to investigate this relationship at their own pace. Note that we use the name immediately but introduce it formally after students have had time to work with it. KF-58. Use the Distributive Property (from KF-57) to rewrite the following epressions. Use Algebra Tiles if necessary. a) 6( ) [ 6 ] c) (3 ) [ 6 ] b) 3( ) [ 3 ] d) 5( - 3) [ 5-5 ] 00 CPM Educational Program page 6

7 Unit 6: Graphing and Systems of Linear Equations 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 with the generic rectangle shown below: area as a product area as a sum ( )( ) = = 3 Complete each of the following generic rectangles without drawing in all the dividing lines for the rectangular tiles and unit squares. Then find and record the area of the large rectangle as the sum of its parts. Write an equation for each completed generic rectangle in the form: a) 3 area as a product = area as a sum. b) 7 c) 5 3 Hint: This one has only two parts. [ (3)(5) = 8 5 ][ (7)(3) = 0 ][ () = ] WR-7. 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: (3-7) These are NOT binomials: 3-5y - 9 We can use generic rectangles to find various products. We call this process MULTIPLYING BINOMIALS. For eample, multiply ( 5)( 3): ( 5)( 3) = area as a product 5 area as a sum 00 CPM Educational Program 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 ( 3)( ) = product 5 6 sum a) c) e) [ ( 3)( ) = 7 ] [ ( )( ) = ] [ ( )( 3) = 5 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 as in the eample. Eample: = 3 6 ()(3) a) 5 c) 6 e) ( 5)( 3) ( 6)( 3) ( 5)( ) b) d) 0 f) y 3 y 6y ( )( 3) ( 5) ( y)( y) 00 CPM Educational Program page 8

9 AP-0. Summarize the following information in your tool kit. Then answer the questions that follow. FACTORING QUADRATICS Yesterday, you solved problems in the form of (length)(width) = area. Today we will be working backwards from the area and find the dimensions. This is called FACTORING QUADRATICS. Using this fact, you can show that 5 6 = ( 3)( ) because area as a sum area as a product Use your tiles and arrange each of the areas below into a rectangle as shown in AP-, AP-3, and the eample above. Make a drawing to represent each equation. Label each part to show why the following equations are true. Write the area equation below each of your drawings. a) 7 6 = ( 6)( ) c) 3 = ( )( ) b) = ( )( ) d) 5 3 = ( 3)( ) An effective visual way to move to the generic rectangle is to assemble one of the problems on the overhead projector with the tiles. As the students watch, draw the generic rectangle, remove the tiles, fill in the symbols, then factor. Students have used generic rectangles to multiply and will now begin to use them to represent the composite rectangles to factor CPM Educational Program page 9

10 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 6 8. 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: 8 8 We can see that the rectangle with area 6 8 has dimensions ( ) and ( ). 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. AP-9. a) 6 8 b) 5 c) 7 6 [ ( )( ) ] d) 7 [ ( 3)( ) ] [ ( )( ) ] e) 8 [ ( 8) or ( ) ] [ ( )( 6) ] f) 5 3 [ ( 3)( ) ] 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 8. Draw a generic rectangle and fill in the parts we know as shown at right. We know the sum of the areas of the two unlabeled parts must be 8, but we do not know how to split the 8 between the two parts. The 8 could be split into sums of 7, or 6, or 3 5, or. However, we also know that the numbers that go in the two ovals must have a product of. a) Use the information above to write and solve a Diamond Problem to help us decide how the 8 should be split. [ product of, sum of 8;, 6 ] b) Complete the generic rectangle and label the dimensions. [ ( )( 6) ] product sum 8 00 CPM Educational Program page 0

11 AP-3. We have seen cases in which only two types of tiles are given. Read the eample below and add an eample of the Greatest Common Factor to your tool kit. Then use a generic rectangle to find the factors of each of the polynomials below. In other words, find the dimensions of each rectangle with the given area. GREATEST COMMON FACTOR Eample: = ( 5) For 0, is called the GREATEST COMMON FACTOR. Although the diagram could have dimensions ( 5), ( 0), or ( 5), we usually choose ( 5) because the is the largest factor that is common to both and 0. Unless directed otherwise, when told to factor, you should always find the greatest common factor, then eamine the parentheses to see if any further factoring is possible. a) 7 [ ( 7) ] b) 3 6 [ 3( ) or 3( ) or (3 6) ] c) 3 6 [ 3( ) ] AP-70. Some epressions an be factored more than once. Add this eample to your tool kit. Then factor the polynomials following the tool kit bo. FACTORING COMPLETELY Eample: Factor as completely as possible We can factor as (3)( - - 5). However, factors to ( 3) ( - 5). Thus, the complete factoring of is 3( 3)( - 5). Notice that the greatest common factor, 3, is removed first. Discuss this eample with your study team and record how to determine if a polynomial is completely factored. Factor each of the following polynomials as completely as possible. Consider these kinds of problems as another eample of subproblems. Always look for the greatest common factor first and write it as a product with the remaining polynomial. Then continue factoring the polynomial, if possible. a) [ 5( 3 - ) = 5( )( - ) ] b) y - 3y - 0y [ y( - 3-0) = y( - 5)( ) ] c) - 50 [ ( - 5) = ( - 5)( 5) ] 00 CPM Educational Program page

13 Unit 0: Eponents and Quadratics YS-. Add this information to your tool kit. EXTENDING FACTORING In earlier units we used Diamond Problems to help factor sums like ( )( ) We can modify the diamond method slightly to factor problems that are a little different in that they no longer have a in front of the. For eample, factor: 7 3 multiply 6?? ( )( 3) Try this problem: ??? -3? 6 5-3?? (5-3)(? ) [ ( - ) ] YS-. Factor each of the following quadratics using the modified diamond procedure. a) 3 7 [ (3 )( ) ] d) [ ( 5)( - 9) ] b) 3 - [ ( )(3 - ) ] e) [ (5 3)( ) ] c) [ ( - 5)( ) ] 00 CPM Educational Program page 3

14 Unit : More about Quadratic Equations RS-67. Taking notes is always an important study tool. Take careful notes and record sketches as your read this problem with your study team. COMPLETING THE SQUARE In problem RS-58, we added tiles to form a square. This changed the value of the original polynomial. However, by using a neutral field, we can take any number of tiles and create a square without changing the value of the original epression. This technique is called COMPLETING THE SQUARE. For eample, start with the polynomial: 8 : First, put these tiles together in the usual arrangement and you can see a square that needs completing. a) How many small squares are needed to complete this square? [ Four ] b) Draw a neutral field beside the tiles. Does this neutral field affect the value of our tiles? [ No ] The equation now reads: 8 0 c) To complete the square, we are going to need to move tiles from the neutral field to the square. When we take the necessary four positive tiles that complete the square, what is the value of the formerly neutral field? [ - ] ( 8 ) ( ) complete square neutral field Neutral Field Adjusted Neutral Field d) Combining like terms, e) Factoring the trinomial square, ( ) - So, 8 = ( ) - Net change to Neutral Field ` 00 CPM Educational Program page

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