make math visual w alg tiles

8/9/2019 Make Math Visual w Alg Tiles

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2002 CPM Educational Program page 1

www.cpm.org

Chris MiklesCPM Educational Program

A California Non-profit Corporation1233 Noonan Drive

Sacramento, CA 95822(888) 808-4276

fax: (208) 777-8605 email: [email protected]

An Exemplary Mathematics Program--U.S. Dept. of Education

MakingakingMathematicsathematicsMore Visualore Visual

UsingsingAlgebra Tileslgebra Tiles


2/14


Unit 0: Working in Teams

We purposely placed Diamond Problems in Unit Zero, before arithmetic with integers isinvestigated in Unit One. Use these problems as an informal assessment of current studentskills. Do not stop and teach the class how to add integers. This will be thoroughly introducedearly 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 fourthdiamond, if you know the numbers (#), can you find the unknowns (?) ? Explain how you would dothis. Note that "#" is a standard symbol for the word "number".

5 2

10

7

2 3

6

5

4

- 5

- 4- 1 #

?

?

#

Patterns are an important problem solving skill we use in algebra. The patterns in Diamond Problemswill 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

[ xy = 12;x+y = 7 ]

b)

- 2 - 3

[ xy = 6;x+y = -5 ]

c)12

7

[ x = 3;y = 4 ]

d)4

1

2

[ x = 8;x+y = 8.5 ]

e)8

- 6

[ x = -2;y = -4 ]

f)

4

4

[ y = 1;x+y = 5 ]

g)

- 9 - 5

[ xy = 45;x+y = -14 ]

h)

9

7

[ y = -2;xy = -18 ]

i)6

5

[ x = 3;y = 2 ]

j)

4

1

2

[ y = 3.5;xy = 1.75 ]


3/14


Unit 1: 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 x and the small square has a sidelength of 1. What is the area of:

a) the big square? [ x2 ] b) the rectangle? [ x ] c) the small square? [ 1 ]

d) Trace one of each of the tiles in yourtool kit. Mark the dimensions along thesides, then write the area of each tile inthe center of the tile and circle it.

From now on we will name each tile byits area. x

x

11

x

1

SQ-66. Check your results with your team members.

a) Find the areas of each tile in problem SQ-65 if x = 4. Find the areas of each tile if x = 6.[ 16, 4, 1; 36, 6, 1 ]

b) Why do you think x is called a variable? [ area varies ]

You will need to model combining liketerms with the students in preparation forthe next few problems. Place this on theoverhead and ask What area do these tiles

represent? [ x2 + 2x + 4 ]

After they name it, place these on the other

side, and ask What is this area?

[ x2 + 3x + 1 ]

Then ask, If we put everything on the screen together, what is the area?

[ 2x2 + 5x + 5 ] Keep modeling examples until the students are comfortable.

Other examples to model: (4x + 2) + (x + 1) = 5x + 3

(2x2 + 2x + 3) + (x2 + 4x + 5) = 2x2 + 6x + 8

(3x2 + x + 4) + (2x2 + 3x + 2) = 5x2 + 4x + 6


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SQ-67. Summarize the idea of Combining Like Terms in your tool kit. Thenrepresent the following situations with an algebraic expression.

Combining tiles that have the same area towrite a simpler expression is calledCOMBINING LIKE TERMS.

Example:

2x + 2x + 2

x2

1

x

x

We write 2x to show 2(x) or 2 x.

1

Represent each of the following situations with an algebraic expression.

38 small squares20 rectangles5 large squares

[ 2x2 + 3x + 4 ] [ 3x + 5 ] [ 5x2 + 20x + 38 ]

+ =

[ (x2 + 2x + 4) + (x2 + 3x) = 2x2 + 5x + 4 ]

SQ-68. You put your rectangle and two small squares with another pile of three rectangles and fivesmall squares. What is in this new pile? [ 4x + 7 ]

SQ-71. Example: To show that 2x does notusually equal x2, you need two rectanglesand one big square.

1

x x

1

2x

x

x

x 2

x 2

[ Solutions shown below ]

a) Show that 3x + x 3x2.x+

1 1 1 1

x xx x x x

x x x

b) Show that 2x - x 2.


5/14


Unit 2: Area and Subproblems

Today the student problems extend the area work to variable multiplication and the DistributiveProperty. Encourage students to use Algebra Tiles to explore these ideas. If students need to make thema master for paper models of Algebra Tiles is included as a Resource Page in Unit One. It may behelpful to model 4(2x) = 8x to make the day smoother.

MULTIPLYING WITH ALGEBRA TILES

Example: The dimensions of this rectangle are

x by 2x

Since two large squares cover the area, the area is 2x2.

x

x x

x2

x2

We can write the area as a multiplication problem using its dimensions:

x(2x) = 2x2

F-53.Use the figure at right to answer these questions.

a) What are the dimensions of the rectangle?

[ 2x by 3x ]

b) What is the area of this rectangle? [ 6x2 ]

c) Write the area as a multiplication problem.

[ (2x)(3x) = 6x2]

x

x x x

x

Give a brief demonstration of Grouping With Algebra Tiles. Start your demonstration by placing 3xs and 12 1s on the overhead. Rearrange them into a rectangle and then split the rectangle twoways to show the different possible grouping as shown in the student text. Help the studentsinterpret the drawings in these problems. Emphasize that the variable x is a symbol for ANY striplength one might choose. Develop the concepts of multiplication as grouping and addition ascombining. 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. Whenwe group rectangles and small squares together, as in the examples below, we read and write the number ofrows first, and the contents of the row second. 3x means three rows of x.

For example, all three figures below contain three rectangles and twelve small squares. The total area is3x + 12, as shown in Figure A. In Figure B, the rectangles are grouped, forming 3 rows of x, written as 3(x).Three rows of four small squares are also a group, written as 3(4).

In Figure C, notice that each row contains a rectangle and four small squares, (x + 4).Since three of these rows are represented, we write this as 3(x + 4).

Figure A

3x + 12

Total Area

Figure B

3(x) + 3(4)

3 rows of x and3 rows of 4

Figure C

3(x + 4)

3 rows of (x + 4)

Write down your observations of the different ways to group 3x + 12.


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KF-56. Match each geometric figure below with an algebraic expression that describes it. Note: 3 . xmeans 3 times x and is often written 3x. This represents 3 rows of x.

a)

b)

c)

d)

e)

f)

1. 2(x) + 2(2) [ b ]

2. 3(x + 2) [ a ]

3. 5(x + 1) [ e ]

4. 3(x) + 3(1) [ c ]

5. 2(x + 5) [ d ]

6. 3(x) + 3(4) [ f ]

KF-57 asks students to discover that when the same tiles are grouped differently, their areas arestill equivalent. The Distributive Property is introduced, and will be revisited in depth in Days 7and 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 expressions below.

[ solutions shown below ]

a) 4(x + 3) b) 4(x) + 4(3)

c) Compare the diagrams. How do their areas compare? Write an algebraic equation thatstates this relationship. This relationship is known asThe Distributive Property.[ 4(x + 3) = 4(x) + 4(3) ]

Students may prefer to use Algebra Tiles to rewrite the following expressions. The numbers

were purposely chosen to allow for tile use. There is plenty of time for students to abstract theDistributive Property. Allow teams to investigate this relationship at their own pace. Note thatwe use the name immediately but introduce it formally after students have had time to workwith it.

KF-58. Use the Distributive Property (from KF-57) to rewrite the following expressions. Use AlgebraTiles if necessary.

a) 6(x + 2) [ 6x + 12 ] c) 2(3x + 1) [ 6x + 2 ]

b) 3(x + 4) [ 3x + 12 ] d) 5(x - 3) [ 5x - 15 ]


7/14


Unit 6: Graphing and Systems of Linear Equations

WR-71. 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 thedividing lines for the rectangular tiles and unit squares. For example, we can represent therectangle whose dimensions are x + 1 by x + 2 with the generic rectangle shown below:

x 2

x

1

+

+ x 2

x

1

x2 2x

x 2

+

+

x 2

x

1+

+

area as a product area as a sum

(x + 1)(x + 2) = x2 + 2x + 1x + 2 = x2 + 3x + 2

Complete each of the following generic rectangles without drawing in all the dividing lines for therectangular tiles and unit squares. Then find and record the area of the large rectangle as the sum ofits parts. Write an equation for each completed generic rectangle in the form:

area as a product = area as a sum.

x + 3

x

+

5

a)x + 7

x

+3

b)2x + 1

x

Hint: This one hasonly two parts.

c)

[ (x+3)(x+5) = x2 + 8x + 15 ][ (x+7)(x+3) = x2 + 10x + 21 ][ x(2x+1) = 2x2 + x ]

WR-72. Carefully read this information about binomials. Then add a description ofbinomials and the example of multiplying binomials to your tool kit.

These are examples ofBINOMIALS:

x + 2 7 - 5x 2x - 7 (3x2 - 17)

These are NOT binomials: 2x 3x2 -5xy - 2x + 9

We can use generic rectangles to find various products. We call this processMULTIPLYING BINOMIALS. For example, multiply (2x + 5)(x + 3):

2x 2

2x + 5

x

+

3

5x

6x 15

2x2

+ 11x + 15(2x + 5)(x + 3) =

area as a product area as a sum

2x + 5

x

+

3


8/14


Unit 8: Factoring Quadratics

AP-3. Write an algebraic equation for the area of each of the following rectangles as shown in theexample below.

1 1

1

x 2 + 5x + 6(x + 3)(x + 2) =

Example: x + 3

x

+

2xx

x x

1 1

x

1

x2

product sum

a) c) e)

[ (x + 3)(x + 4) =

x2

+ 7x + 12 ]

[ (x + 1)(x + 1) =

x2

+ 2x + 1 ]

[ (x + 1)(2x + 3) =

2x2

+ 5x + 3 ]

AP-11. Find the dimensions of each of the following generic rectangles. The parts are not necessarilydrawn to scale. Use Guess and Check to write the area of each as both a sum and a product asin the example.

Example:

x2 3x

2x 6

x + 3

x2 3x

2x 6

x+2

x + 5x + 6 (x+2)(x+3)=2a)

x2

5x

3x 15

c)x

26x

3x 18

e)

x2

5x

2x 10

(x + 5)(x + 3) (x + 6)(x + 3) (x + 5)(x + 2)

b)

x2

3x

4x

12

d)

2x2

10x

f)

x2 4xy

4xy 16y 2

(x + 4)(x + 3) 2x(x + 5) (x + 4y)(x + 4y)


9/14


AP-10. Summarize the following information in your tool kit. Then answerthe questions that follow.

FACTORING QUADRATICS

Yesterday, you solved problems in the form of (length)(width) = area. Today wewill be working backwards from the area and find the dimensions. This is calledFACTORING QUADRATICS.

Using this fact, you can show that x2 + 5x + 6 = (x + 3)(x + 2) 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-2, AP-3, and theexample above. Make a drawing to represent each equation. Label each part to show why thefollowing equations are true. Write the area equation below each of your drawings.

a) x2 + 7x + 6 = (x + 6)(x + 1) c) x2 + 3x + 2 = (x + 2)(x + 1)

b) x2 + 4x + 4 = (x + 2)(x + 2) d) 2x2 + 5x + 3 = (2x + 3)(x + 1)

An effective visual way to move to the generic rectangle is to assemble one of the problems on theoverhead 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 willnow begin to use them to represent the composite rectangles to factor.

x 2 4x

2x 8


10/14


AP-18. USING ALGEBRA TILES TO FACTOR

What if we knew the area of a rectangle and we wanted to find the dimensions? We would haveto work backwards. Start with the area represented by x2 + 6x + 8. Normally, we would not besure whether the expression represents the area of a rectangle. One way to find out is to useAlgebra Tiles to try to form a rectangle.

You may find it easier to record the rectangle without drawing all the tiles. You may draw ageneric rectangle instead. Write the dimensions along the edges and the area in each of the

smaller parts as shown below.

Example:

x2 4x

2x 8

x + 4

x2 4x

2x 8

x+2

We can see that the rectangle with area x2 + 6x + 8 has dimensions (x + 2) and (x + 4).

Use Algebra Tiles to build rectangles with each of the following areas. Draw the completepicture or a generic rectangle and write the dimensions algebraically as in the example above. Besure you have written both the product and the sum.

a x2 + 6x + [ (x + 4)(x + 2) ] d x2 + 7x + 12 [ (x + 3)(x + 4) ]

2 [ (x + 1)(x + 4) ] 2 [ x(2x + 8) or

c x2 + 7x + [ (x + 1)(x + 6) ] f 2x2 + 5x + 3 [ (2x + 3)(x + 1) ]

AP-19. USING DIAMOND PROBLEMS TO FACTOR

Using Guess and Check is not the only way tofind the dimensions of a rectangle when weknow its area. Patterns will help us find anothermethod. Start with x2 + 8x + 12. Draw ageneric rectangle and fill in the parts we know asshown at right.

We know the sum of the areas of the twounlabeled parts must be 8x, but we do not knowhow to split the 8x between the two parts. The

8x could be split into sums of 7x + 1x, or

x +

x

+

x2

12

8x

6x + 2x, or 3x + 5x, or 4x + 4x. However, we also know that the numbers that go inthe two ovals must have a product of 12.

a) Use the information above to write and solve a Diamond Problemto help us decide how the 8x should be split.[ product of 12, sum of 8; 2, 6 ]

b) Complete the generic rectangle and label the dimensions.[ (x + 2)(x + 6) ]

product

sum


11/14


AP-31. We have seen cases in which only two types of tiles are given. Read the examplebelow and add an example of the Greatest Common Factor to your tool kit. Thenuse a generic rectangle to find the factors of each of the polynomials below. In otherwords, find the dimensions of each rectangle with the given area.

GREATEST COMMON FACTOR

2x2

10x

Example:

2x2 + 10x = 2x(x + 5)

2x2

10x2x

x 5+

For 2x2 + 10x, 2x is called the GREATEST COMMON FACTOR.Although the diagram could have dimensions 2(x2 + 5x), x(2x + 10), or 2x(x+ 5), we usually choose 2x(x + 5) because the 2x is the largest factor that iscommon to both 2x2 and 10x. Unless directed otherwise, when told to factor,you should always find the greatest common factor, then examine the

parentheses to see if any further factoring is possible.

a) x2 + 7x [ x(x + 7) ] b) 3x2 + 6x [ 3x(x + 2) or 3(x2 + 2x) or x(3x + 6) ]

c) 3x + 6 [ 3(x + 2) ]

AP-70. Some expressions an be factored more than once. Add this example to yourtool kit. Then factor the polynomials following the tool kit box.

FACTORING COMPLETELY

Example:Factor 3x3 - 6x2 - 45x as completely as possible.

3x 3x3 -6x2 -45x 3x 3x 3 -6x2 -45x

x2 -2x -15

We can factor 3x3 - 6x2 - 45x as (3x)(x2 - 2x - 15).

However, x2 - 2x - 15 factors to (x + 3) (x - 5).

Thus, the complete factoring of 3x3 - 6x2 - 45x is 3x(x + 3)(x - 5).

Notice that the greatest common factor, 3x, is removed first.

Discuss this example with your study team and record how to determine if apolynomial is completely factored.

Factor each of the following polynomials as completely as possible. Consider these kinds of problemsas another example of subproblems. Always look for the greatest common factor first and write it as aproduct with the remaining polynomial. Then continue factoring the polynomial, if possible.

a) 5x2 + 15x- 20 [ 5(x2 + 3x - 4) = 5(x + 4)(x - 1) ]

b) x2y - 3xy - 10y [ y(x2 - 3x - 10) = y(x - 5)(x + 2) ]

c) 2x2 - 50 [ 2(x2 - 25) = 2(x - 5)(x + 5) ]


12/14


AP-79. THE AMUSEMENT PARK PROBLEM

The city planning commission is reviewingthe master plan of the proposed AmusementPark coming to our city. Your job is to helpthe Amusement Park planners design the landspace.

Based on their projected daily attendance,

the planning commission requires 15 rows ofparking. The rectangular rows will be of thesame length as the Amusement Park.Depending on funding, the Park size

may change so planners are assuming the park will be square and have a length of x. Theparking will be adjacent to two sides of the park as shown below.

Our city requires all development plans to include greenspace or planted area for sitting and picnicking. See the planbelow.

a) Your task is to list all the possible configurations ofland use with the 15 rows of parking. Find the areasof the picnic space for each configuration. Use thetechniques you have learned in this unit. There ismore than one way to approach this problem, soshow all your work.[ Area = 14, 26, 36, 44, 50, 54, 56 ]

x

+

x + ?

Amusement

Park

parking

picnic area

?

parking

b) Record the configuration with the minimum and maximum picnic area. Write an equationfor each that includes the dimensions and the total area for the project. Verify yoursolutions before moving to part (c).[ (x + 14)(x + 1) = x2 + 15x + 14; (x + 7)(x + 8) = x2 + 15x + 56 ]

c) The Park is expected to be a success and the planners decide to expand the parking lot byadding 11 more rows. Assume that the new plan will add 11 additional rows of parking insuch a way that the maximum original green space from part (b) will triple. Show all yourwork. Record your final solution as an equation describing the area of the total = productof the new dimensions.[ (x2 + 26x + 168 = (x + 12)(x + 14) ]

d) If the total area for the expanded Park, parking and picnic area is 2208 square units, findx. Use the dimensions from part (c) to write an equation and solve for the side of thePark. [ 2208 = (x + 12)(x + 14); using Guess & Check x = 34 ]


13/14


Unit 10: Exponents and Quadratics

YS-1. Add this information to your tool kit.

EXTENDING FACTORING

In earlier units we used Diamond Problems to help factor sums like x2 + 6x + 8.

2 4

6

8x2 4x

2x 8

x2 4x

2x 8

x + 4

x+2

(x + 4)(x + 2)

We can modify the diamond method slightly to factor problems that are a little

different in that they no longer have a 1 in front of the x2. For example, factor:

2x + 7x + 32

6

7

? ?

6

7

6 12x 6x

1x 3

2 2x 6x

1x 3

22x+1

x + 3

multiply

(2x + 1)(x + 3)

Try this problem: 5x2 - 13x + 6.

? ?

2 ?

?

? ?

30

-13 6

5x5x

-3(5x - 3)( ? )

[ (x - 2) ]

YS-2. Factor each of the following quadratics using the modified diamond procedure.

a) 3x2 + 7x + 2 [ (3x + 1)(x + 2) ] d) x2 - 4x - 45 [ (x + 5)(x - 9) ]

b) 3x2 + x - 2 [ (x + 1)(3x - 2) ] e) 5x2 + 13x + 6 [ (5x + 3)(x + 2) ]

c) 2x2 - 3x - 5 [ (2x - 5)(x + 1) ]


14/14


Unit 12: More about Quadratic Equations

RS-67. Taking notes is always an important study tool. Take careful notes andrecord 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 theoriginal polynomial. However, by using a neutral field, we can take any number oftiles and create a square without changing the value of the original expression. Thistechnique is called COMPLETING THE SQUARE. For example, start with thepolynomial: x2 + 8x + 12:

x2

x x x x x x x x 1 1 1 1 1 1

1 1 1 1 1 1

First, put these tiles together in theusual arrangement and you can see asquare that needs completing.

a) How many small squares areneeded to complete this square?[ Four ]

x

xx

+

+

4

4

2

b) Draw a neutral field beside thetiles. Does this neutral field affectthe value of our tiles? [ No ]The equation now reads:

x2 + 8x + 12 + 0

c) To complete the square, we are

going to need to move tiles fromthe neutral field to the square.When we take the necessary fourpositive tiles that complete thesquare, what is the value of theformerly neutral field? [ -4 ]

(x2+8x +12 + 4) + ( 4)

complete square neutral field

d) Combining like terms,

x2 + 8x + 16 + - 4

e) Factoring the trinomial square,

(x + 4)2 - 4

So, x2 + 8x + 12 = (x + 4)2 - 4

`

x

xx

+

+

4

4

2Neutral Field

x

xx

+

+

4

4

2Neutral Field

x

xx

+

+

4

4

2Neutral Field

Adjusted

Net change to

make math visual w alg tiles

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