 1.04 The distributive property

Lesson

Let's look at a way to find the sum of or difference between two numbers that uses the greatest common factor (GCF) and the distributive property.

Exploration

For example, let's say we wanted to evaluate $72-48$7248.

First, we can find the greatest common factor (GCF) between the two numbers.

The factors of $48$48 are:

$1,2,3,4,6,8,12,16,24,48$1,2,3,4,6,8,12,16,24,48

The factors of $72$72 are:

$1,2,3,4,6,8,9,12,18,24,36,72$1,2,3,4,6,8,9,12,18,24,36,72

The numbers that appear in both factor lists are:

$1,2,3,4,6,8,12,24$1,2,3,4,6,8,12,24

The largest number in this list is the GCF, $24$24.

Now, we can rewrite the expression as an equivalent multiplication by using the distributive property.

$48=24\times2$48=24×2

$72=24\times3$72=24×3

$72-48=24\times\left(3-2\right)$7248=24×(32)

Finally, we multiply the two integers to find our answer.

$24\times\left(3-2\right)=24\times1$24×(32)=24×1

$24\times1=24$24×1=24

So, $72-48=24$7248=24.

And there you go! Another way to find the sum or difference between to numbers.

Practice questions

Question 1

Consider the difference $96-80$9680 :

1. Find the GCF of $96$96 and $80$80.

2. Complete the gaps such that $96-80$9680 is rewritten as an equivalent multiplication using the distributive property.

 $96-80$96−80 $=$= $16\times\left(\editable{}-5\right)$16×(−5) $=$= $16\times\editable{}$16×

Question 2

Consider $11\left(8-3\right)$11(83).

1. Using the distributive property complete the gap so that $11\left(8-3\right)$11(83) is rewritten as the difference of two integers.

$11\left(8-3\right)=88-\editable{}$11(83)=88

Question 3

Hermione and Yuri both earn $\$1111 per hour in their casual job. In a day where one works for $7$7 hours and the other works for $2$2 hours, complete the number sentence that can be used to evaluate the difference in their wages (with the difference expressed as a positive quantity).

1. Difference in wages = $\editable{}\left(\editable{}-\editable{}\right)$()

Outcomes

MGSE6.NS.4

Find the common multiples of two whole numbers less than or equal to 12 and the common factors of two whole numbers less than or equal to 100.

MGSE6.NS.4a

Find the greatest common factor of 2 whole numbers and use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factors. (GCF)

MGSE6.NS.4b

Apply the least common multiple of two whole numbers less than or equal to 12 to solve real- world problems.