Chris Mikles
CPM Educational Program
A California Non-profit Corporation
1233 Noonan Drive
Sacramento, CA 95822
(888) 808-4276
fax: (208) 777-8605 email: mikles@cpm.org
www.cpm.org
An Exemplary Mathematics Program
--U.S. Dept. of Education
Making
Mathematics
More Visual
Using
Algebra Tiles
© 2002 CPM Educational Program
page 1
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 (?) ? Explain how you would do
this. Note that "#" is a standard symbol for the word "number".
10
5
6
2
2
7
?
4
-1
3
5
#
#
-4
?
-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)
b)
3
4
c)
-2
12
d)
4
-3
7
[ xy = 12;
x+y = 7 ]
f)
4
[ xy = 6;
x+y = -5 ]
g)
4
[ x = 3;
y=4]
h)
-9
-5
[ y = 1;
x+y = 5 ]
© 2002 CPM Educational Program
[ xy = 45;
x+y = -14 ]
6
9
[ y = -2;
xy = -18 ]
5
[ x = 3;
y=2]
8
1
2
-6
[ x = 8;
x+y = 8.5 ]
i)
7
e)
[ x = -2;
y = -4 ]
j)
1
2
4
[ y = 3.5;
xy = 1.75 ]
page 2
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 side
length of 1. What is the area of:
a)
the big square? [ x2 ]
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.
b) the rectangle? [ x ]
x
From now on we will name each tile by
its area.
SQ-66.
c) the small square? [ 1 ]
x
1
x
1
1
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 like
terms with the students in preparation for
the next few problems. Place this on the
overhead 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
© 2002 CPM Educational Program
page 3
SQ-67.
Summarize the idea of Combining Like Terms in your tool kit. Then
represent the following situations with an algebraic expression.
Example:
Combining tiles that have the same area to
write a simpler expression is called
COMBINING LIKE TERMS.
x
x2
x
1
1
x2 + 2x
+2
We write 2x to show 2(x) or 2 · x.
Represent each of the following situations with an algebraic expression.
38 small squares
20 rectangles
5 large squares
[ 3x + 5 ]
[ 2x2 + 3x + 4 ]
+
[ 5x2 + 20x + 38 ]
=
[ (x2 + 2x + 4) + (x2 + 3x) = 2x2 + 5x + 4 ]
SQ-68.
SQ-71.
You put your rectangle and two small squares with another pile of three rectangles and five
small squares. What is in this new pile? [ 4x + 7 ]
Example: To show that 2x does not
usually equal x2 , you need two rectangles
and one big square.
x x
1
x2
x
1
x
x2
2x
[ Solutions shown below ]
a)
Show that 3x + x ≠ 3x2 .
x
x
1
b)
Show that 2x - x ≠ 2.
© 2002 CPM Educational Program
+x
x
1
1
x
x
x
1
x
x
x
≠
page 4
Unit 2: Area and Subproblems
Today the student problems extend the area work to variable multiplication and the Distributive
Property. Encourage students to use Algebra Tiles to explore 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 4(2x) = 8x to make the day smoother.
MULTIPLYING WITH ALGEBRA TILES
Example:
The dimensions of this rectangle are
x
x by 2x
x
x
x2
x2
Since two large squares cover the area, the area is 2x2 .
We can write the area as a multiplication problem using its dimensions:
x(2x) = 2x2
x
KF-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
Give a brief demonstration of Grouping With Algebra Tiles. Start your demonstration by placing 3
‘x’s and 12 ‘1’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 text. Help the students
interpret the drawings in these problems. Emphasize that the variable x 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 examples below, we read and write the number of
rows 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 is
3x + 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 and
3 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.
© 2002 CPM Educational Program
page 5
KF-56.
Match each geometric figure below with an algebraic expression that describes it. Note: “3 . x”
means “3 times x” and is often written 3x. This represents 3 rows of x.
a)
d)
e)
b)
c)
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 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 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 that
states this relationship. This relationship is known as
The 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 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 expressions. Use Algebra
Tiles 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 ]
© 2002 CPM Educational Program
page 6
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 the
dividing lines for the rectangular tiles and unit squares. For example, we can represent the
rectangle whose dimensions are x + 1 by x + 2 with the generic rectangle shown below:
x
+
x
2
+
2
x
+ 2
x
x
x
x2
2x
+
1
+
1
+
1
x
2
area as a product
area as a sum
2
(x + 1)(x + 2) = x + 2x + 1x + 2 = x 2 + 3x + 2
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:
area as a product = area as a sum.
a)
x
+ 3
b)
x
c)
+ 7
2x
+ 1
x
x
x
+
+
3
Hint: This one has
only two parts.
5
[ (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 of
binomials and the example of multiplying binomials to your tool kit.
These are examples of BINOMIALS:
x+2
7 - 5x
These are NOT binomials:
(3x2 - 17)
2x - 7
3x2
2x
-5xy - 2x + 9
We can use generic rectangles to find various products. We call this process
MULTIPLYING BINOMIALS. For example, multiply (2x + 5)(x + 3):
2x
+ 5
x
+
3
© 2002 CPM Educational Program
2x
x
+
3
+
5
2x 2
5x
6x
15
(2x + 5)(x + 3) = 2x2 + 11x + 15
area as a product
area as a sum
page 7
Unit 8: Factoring Quadratics
AP-3.
Write an algebraic equation for the area of each of the following rectangles as shown in the
example below.
Example:
x
+ 3
x
x2
xxx
+
2
x
x
1 1 1
1 1 1
x 2 + 5x + 6
sum
(x + 3)(x + 2) =
product
a)
c)
e)
[ (x + 3)(x + 4) =
x2 + 7x + 12 ]
AP-11.
[ (x + 1)(x + 1) =
x2 + 2x + 1 ]
[ (x + 1)(2x + 3) =
2x2 + 5x + 3 ]
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 example.
Example:
x2
3x
2x
6
x
+
2
2
=
x + 5x + 6
a)
c)
x2
5x
3x
15
d)
x2
4x
3x
12
(x + 4)(x + 3)
© 2002 CPM Educational Program
+ 3
x2
3x
2x
6
(x+2)(x+3)
e)
x2
6x
3x
18
(x + 6)(x + 3)
(x + 5)(x + 3)
b)
x
2
2x
x2
5x
2x
10
(x + 5)(x + 2)
f)
10x
x2
4xy
4xy
2x(x + 5)
16y 2
(x + 4y)(x + 4y)
page 8
AP-10.
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 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 the
example 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)
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 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.
x2
2x
© 2002 CPM Educational Program
4x
8
page 9
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 have
to work backwards. Start with the area represented by x2 + 6x + 8. Normally, we would not be
sure whether the expression 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.
Example:
x2
4x
2x
8
x
+
2
x
+ 4
x2
4x
2x
8
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 complete
picture or a generic rectangle and write the dimensions algebraically as in the example above. Be
sure you have written both the product and the sum.
a)
b)
c)
AP-19.
x2 + 6x +
8
x2 + 5x +
4
x2 + 7x +
6
[ (x + 4)(x + 2) ]
d)
x2 + 7x + 12
[ (x + 3)(x + 4) ]
[ (x + 1)(x + 4) ]
e)
2x2 + 8x
[ (x + 1)(x + 6) ]
f)
2x2 + 5x + 3
[ x(2x + 8) or
2x(x + 4) ]
[ (2x + 3)(x + 1) ]
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 x2 + 8x + 12. Draw a
generic rectangle and fill in the parts we know as
shown at right.
x
x
+
+
x2
12
We know the sum of the areas of the two
unlabeled parts must be 8x, but we do not know
how to split the 8x between the two parts. The
8x could be split into sums of 7x + 1x, or
6x + 2x, or 3x + 5x, or 4x + 4x. However, we also know that the numbers that go in
the two ovals must have a product of 12.
product
a)
Use the information above to write and solve a Diamond Problem
to 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) ]
© 2002 CPM Educational Program
8x
sum
page 10
AP-31. We have seen cases in which only two types of tiles are given. Read the example
below and add an example 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
Example:
2x 2
x
10x
+ 5
2x 2
2x
10x
2x 2 + 10x = 2x(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 is
common 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.
AP-70.
a)
x2 + 7x [ x(x + 7) ]
c)
3x + 6
b) 3x2 + 6x [ 3x(x + 2) or 3(x2 + 2x) or x(3x + 6) ]
[ 3(x + 2) ]
Some expressions an be factored more than once. Add this example to your
tool kit. Then factor the polynomials following the tool kit box.
FACTORING COMPLETELY
Example: Factor 3x3 - 6x2 - 45x as completely as possible.
x2
-2x
3x
3x 3
-6x 2
-45x
3x
3x 3
-6x 2
-15
-45x
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 a
polynomial is completely factored.
Factor each of the following polynomials as completely as possible. Consider these kinds of problems
as another example 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)
5x2 + 15x - 20
[ 5(x2 + 3x - 4) = 5(x + 4)(x - 1) ]
b) x2 y - 3xy - 10y
[ y(x2 - 3x - 10) = y(x - 5)(x + 2) ]
c)
2x2 - 50
[ 2(x2 - 25) = 2(x - 5)(x + 5) ]
© 2002 CPM Educational Program
page 11
AP-79. THE AMUSEMENT PARK PROBLEM
The city planning commission is reviewing
the master plan of the proposed Amusement
Park coming to our city. Your job is to help
the Amusement Park planners design the land
space.
Based on their projected daily attendance,
the planning commission requires 15 rows of
parking. The rectangular rows will be of the
same 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. The
parking will be adjacent to two sides of the park as shown below.
a)
Your task is to list all the possible configurations of
land use with the 15 rows of parking. Find the areas
of the picnic space for each configuration. Use the
techniques you have learned in this unit. There is
more than one way to approach this problem, so
show all your work.
[ Area = 14, 26, 36, 44, 50, 54, 56 ]
x + ?
x Amusement
Park
+
?
parking
Our city requires all development plans to include “green
space” or planted area for sitting and picnicking. See the plan
below.
parking
picnic area
b)
Record the configuration with the minimum and maximum picnic area. Write an equation
for each that includes the dimensions and the total area for the project. Verify your
solutions 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 by
adding 11 more rows. Assume that the new plan will add 11 additional rows of parking in
such a way that the maximum original green space from part (b) will triple. Show all your
work. Record your final solution as an equation describing the area of the total = product
of 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, find
x. Use the dimensions from part (c) to write an equation and solve for the side of the
Park. [ 2208 = (x + 12)(x + 14); using Guess & Check x = 34 ]
© 2002 CPM Educational Program
page 12
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.
x + 4
8
2
4
6
x2
4x
2x
8
x x2
+
2 2x
4x
(x + 4)(x + 2)
8
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:
2
2x + 7x + 3
multiply
6
6
?
?
2x 6x
1
6
1x
7
7
x+3
2x 2x2 6x
+
1 1x 3
2
3
(2x + 1)(x + 3)
Try this problem: 5x2 - 13x + 6.
?
30
?
5x 2
?
-13
?
5x
?
6
?
(5x - 3)( ? )
-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
b)
3x2 + x - 2
[ (x + 1)(3x - 2) ]
e)
5x2 + 13x + 6 [ (5x + 3)(x + 2) ]
c)
2x2 - 3x - 5
[ (2x - 5)(x + 1) ]
© 2002 CPM Educational Program
[ (x + 5)(x - 9) ]
page 13
Unit 12: 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 expression. This
technique is called COMPLETING THE SQUARE. For example, start with the
polynomial: x2 + 8x + 12:
x2
x
x
x
x
x
x
x x
1
1
1
1
1
1
1
1
1
1
x
First, put these tiles together in the
usual arrangement and you can see a
“square that needs completing.”
x
a) How many small squares are
needed to complete this square?
[ Four ]
1
1
+
4
2
x
+
4
x
b) Draw a neutral field beside the
tiles. Does this neutral field affect
the value of our tiles? [ No ]
The equation now reads:
x
+ 4
2
x
Neutral Field
+
x2 + 8x + 12 + 0
4
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? [ -4 ]
x
x
+ 4
Adjusted
Neutral Field
2
x
+
(x2 + 8x + 12 + 4) + ( − 4)
complete square neutral field
4
d) Combining like terms,
x2 + 8x + 16 + - 4
x
e) Factoring the trinomial square,
(x + 4)2 - 4
x
So, x2 + 8x + 12 = (x + 4)2 - 4
+ 4
2
x
Net change to
Neutral Field
+
4
`
© 2002 CPM Educational Program
page 14