32
Garden Plots and Multiplication of Binomials
The purpose of this lesson is to explain the multiplication problem (x+2)(2x+3).
The visual model can help you understand and remember how to multiply such expressions.
Consider a terraced garden plot. The farmer wants to make rectangular beds for tall vegetables
with beds for herbs on 2 sides. A rectangular part of the plot reserved for tall vegetables is to be
twice as long as it is wide. There will be a 3 feet wide area added on to a short end of the
rectangle for herbs, and an area 2 feet wide added along a long side, also for herbs. See
diagram.
a. Mark on the diagram where the 2 foot width is.
b. Mark the 3 foot width.
c. What is the length of the whole rectangular plot (in terms of x)? ____________
d. What is the width of the whole plot? ____________
e. Write an expression for the perimeter of the whole plot and simplify it. _________________
f. Write an expression for the area of the whole plot using
the expressions for length and width you just found. ________________________
Now divide the plot up into pieces like this:
g. Write an expression for each of the four parts of the total area.
___________
___________
___________
___________
h. Find the area of the whole rectangular plot by adding up the areas of these 4 parts.
The next step is the important. If you do this, and really think about what you are saying, you will have a
visual model to help you to remember how to multiply expressions like these:
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
33
i. Explain how what you have done illustrates the correct multiplication of (x + 2)(2x + 3).
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
34
Homework after the Garden Plot multiplication investigation
1. a. Draw your own diagram to illustrate what is described here. A parking area at the
university was x meters wide and 4 times that long. Along one of the shorter ends, a 2-meter
width was added for bicycle racks. A 1-meter width was added along one of the longer sides
for a sidewalk. In the 2 meter by 1 meter section where these two additions intersected,
bougainvillea was planted.
b. Label all the lengths in your diagram. Find the area of each section (parking area, bicycle
area, sidewalk, and bougainvillea area).
c. Write an expression for the length of the complete rectangle.
Write an expression for its width.
d. Write two expressions to represent the total area, one using its length and its width, and
the other using the areas of the 4 separate sections found above.
(a+b)(c+d)
2. (x+2)(2x-3)
First write the expression using
the distributive law once:
(a+b)c + (a+b)d
Multiply, by using
the distributive law
again
(x+2)2x + (x+2)(-3)
2x2 + 4x – 3x – 6
Simplify the
result
3. (a-7)(2a+3)
4. (y+7)(z+3)
5. (x+2y)(x-2y)
6. Multiply and simplify (0.5 x + 10)(2 x − 4)
7. Imagine a square field of area 1 square mile. Suppose that a fraction, x, of a mile is
removed from its length and the same fraction is removed from its width (x is between 0 and 1).
Draw a diagram. Then write two expressions for the remaining area.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
35
Teaching Guide for
Garden Plots and Multiplication of Binomials
Introduction: This activity provides a visual model to help students understand the process of
multiplying two binomials followed by homework practice.
Answers and teaching suggestions:
a&b. Be sure students mark the 2 foot width and the 3 foot width on the diagram.
c. The length of the whole rectangular plot is 2x + 3
d. The width of the whole plot is x + 2.
e. An expression for the perimeter of the whole plot might be 2[(2x+3)+(x+2)] = 6x+10
f. An expression for the area of the whole plot using the expressions for length and width is
(2x+3).(x+2).
g. Dividing the plot up into pieces as shown gives four areas of, respectively:
2x . x = 2x2.
3 . x = 3x
6
4x
3x
2x2
2x . 2 = 4x
3 .2 = 6
h. Adding the areas of the 4 parts illustrates (2x+3).(x+2) = 2x2 +7x + 6
i. To respond to the instruction to Explain how what you have done illustrates the correct
multiplication of (x + 2)(2x + 3), a student might write something like the following:
“We found an area for 4 different sections of the rectangle. The area of each section was
found by multiplying 2 distinct terms in the product, one from inside each binomial. When
you add the area of all the sections, you have the whole area, the product of the two
binomials.”
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
36
Homework after the Garden Plot multiplication investigation
1. a and b.
4x
1
2
x
x
The parking area is (x meters).(4x meters) = 4x2 square meters
The bicycle area is 2x square meters
The sidewalk is 4x square meters
The bougainvillea area is 2 square meters
c. The complete rectangle has length 4x+2. Its width is x+1
d. Two expressions representing the total area are (4x+2)(x+1) and the sum of the areas of
the 4 separate sections, 4x2+6x+2. Thus, we see that (4x+2)(x+1)= 4x2+6x+2.
First write the expression using
the distributive law once:
(a+b)c + (a+b)d
Multiply, by using
the distributive law
again
Simplify the
result
2. (x+2)(2x-3)
(x+2)2x + (x+2)(-3)
2x2 + 4x – 3x – 6
2x2 + x – 6
3. (a-7)(2a+3)
(a-7)(2a)+(a-7)3
2a2 -14a+3a-21
2a2 -11a-21
4. (y+7)(z+3)
(y+7)z+ (y+7)3
yz+7z+3y+21
(x+2y)(x)+ (x+2y)(-2y)
x2 +2xy-2xy-4y2
(a+b)(c+d)
5. (x+2y)(x-2y)
x2 -4y2
6. (0.5 x + 10)(2 x − 4) = x 2 + 18 x − 40
7. Note that because the field is known to be square, it must be 1 mile by 1 mile. The remaining
area is (1-x)(1-x) = 1 – 2x + x2. That is, 1 square mile minus 2[x(1-x)] sq.miles minus x2 sq.
miles. OR 1 square mile minus two areas of x sq. miles each but since that subtracts an area of
+ x2 sq. miles out twice, we add it back once.
1
x
x
x
1-x
1-x
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
37
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.