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Math 208, Section 5.5 solutions: The Distributive Property
5. There are six cars traveling together. Each car has 2 people in front and 3 people in
back. Explain how to use this situation to illustrate the distributive property.
To find out how many people are in all six cars, we could use the following expression, which
counts the number of people in all the cars at once:
6 × (2 + 3) = 6 × 5 = 30.
Or, we could count all the people in the front seats and add all the people in the back seats as
follows:
6 × 2 + 6 × 3 = 12 + 18 = 30.
Since both expressions equal the same number, we have shown the distributive property.
6 Draw arrays to help you show why the equations shown are true, without evaluating any
of the products or sums. Explain briefly.
a. (5 + 1) × 7 = 5 × 7 + 1 × 7
5 rows
1 row
In the green box there are 5 rows of 7 squares each, so 5 × 7 squares.
In the orange box there are 1 row of 7 squares, so
1 × 7 squares.
In the whole array there are ( 5 + 1 ) rows of 7 squares each, so there are (5 + 1) × 7 squares.
Therefore,
(5 + 1) × 7 = (5 × 7) + (1 × 7).
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b. 6 × (5 + 2) = 6 × 5 + 6 × 2
In this picture let’s still use the convention that we multiply
(number of rows) × (number of columns).
6 rows
5 columns
2 columns
The green box is 6 rows of 5 squares each; the orange box is 6 rows of 2 squares each.
So, we get (number of green squares) + (number of orange squares) = (6 × 5) + (6 × 2)
On the other hand, there are 6 rows, each consisting of ( 5 + 2) squares.
So the total number of squares is: 6 × (5 + 2)
So we have once again shown that 6 × (5 + 2) = (6 × 5) + (6 × 2) .Voila.
7a. Josh consistently remembers that 7 × 7 = 49, but keeps forgetting 7 × 8. Explain to Josh
how 7 × 7 and 7 × 8 are related. Draw an array to help you show this relationship.
Josh can remember 7 × 8 by observing that 7 × 8 = 7 × (7 + 1) = (7 × 7) + (7 × 1)
= 49 + 7 = 56. The easy way to say this is that 7 × 8 is one more 7 than 7 × 7.
Here is a picture:
oooooooo
oooooooo
oooooooo
oooooooo
oooooooo
oooooooo
oooooooo
oooooooo
To the left of the dotted line there is a 7 × 7 array, and to the right there are 7 more little o’s.
The whole array has 7 × 8 o’s and so once again we see that 7 × 8 is 49 + 7.
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8. Explain how to use the distributive property to make 31 × 25 easy to calculate mentally.
Write an equation that corresponds to your strategy. Without drawing all the detail, draw
a rough picture of an array that illustrates this calculation strategy.
31 × 25 = (3 + 7 × 4) × 25
= 3 × 25 + 7 × 4 × 25
= 75 + 7 × 100
= 75 + 700
= 775
re-writing 31 = 3 + 28 = 3 + 7 × 4
using the distributive property
multiplying
multiplying
adding
7 columns with 4 in each
1 column with 3
Each box represents 25
11. a) Use the distributive property several times to show why
( 10 + 2 ) × ( 10 + 4 ) = 10 × 10 + 10 × 4 + 2 ×10 + 2 × 4
Using the distributive property, we have:
( 10 + 2 ) × ( 10 + 4 ) = [ ( 10 + 2 ) × 10 ] + [ ( 10 + 2 ) × 4 ]
[here we left (10 + 2) alone and just used the distributed multiplication over the sum (10 + 4) ]
= (10 × 10) + (2 × 10) + (10 × 4) + (2 × 4)
[here we used the distributive property twice: we distributed multiplications over the sum
(10 + 2) ]
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b) Draw an array to show why ( 10 + 2 ) × ( 10 + 4 ) = 10 × 10 + 10 × 4 + 2 ×10 + 2 × 4.
10
×
10
10
×
4
2
×
10
2
×
4
c) Relate the steps in your equations in part (a) to your array in part (b).
Using the distributive property, we see that ( 10 + 2 ) × ( 10 + 4 ) = (10 × 10) + (2 × 10) +
(10 × 4) + (2 × 4). So, we split the 12 rows into 10 + 2 rows and split the 14 columns into 10 +
4 columns, and find the area of each sub-rectangle. The sub-rectangles have areas (10 × 10), (2 ×
10), (10 × 4), and (2 × 4), respectively.
12. Ted thinks that because 10 × 10 = 100 and 2 × 5 = 10, he should be able to calculate 12 × 15
by adding 100 + 10 to get 110. Explain to Ted in two different ways that, even though his
method is not correct, his calculations can be part of a correct way to calculate 12 × 15:
Ted’s expressions are included in the right answer but he needs two more terms to have it
correct. This is a very common mistake that children make—they want to multiply only the first
pair of terms, then the last pair of terms, and then add; they forget the “cross-terms.” Let’s do
part (b) first:
b. 12 × 15 = (10 + 2) × (10 + 5) = ( (10 + 2) × 10 ) + ( (10 + 2) × 5 )
= (10 × 10) + (2 × 10) + (10 × 5) + (2 × 5)
So, Ted, you had two of the correct terms, but you need the other two: 2 × 10
10 × 5.
and
a. A picture may help to see where the extra terms came from: We make the 12 by 15 array but
we cut it in slices as follows:
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The light green part represents the 10 × 10 term; the orange part is the 10 × 5 term; the blue part
is the term 2 × 10 ; and the gray part is the term 2 × 5.
To see FOIL in operation we can draw it again:
F
O
I
L