**Middle School Number Sense Lesson 22: Using Difference of Squares to Multiply Two Numbers Surrounding a Third Number**

Once again, the title of my post seems to suggest a much more difficult concept than it is. To make things a little clearer, imagine you're asked to do

**19 x 21**. The number that 19 & 21 are surrounding is

**20**. Here we will illustrate DiffSquares1. This concept showed up

**9 times**this year on middle school tests, always between questions 22 & 37 (inclusive), with the median at

**question 23**.

**Number Dojo Level: 167**

This process works best (and is the easiest) when the number directly in the middle of (and surrounded by) the two factors is a multiple of 10. In other words, it is best when

**the average of the two factors is a multiple of 10**. I will show the steps of why this works first, but don't get bogged down in the details. The actual steps performed on a number sense test will be much quicker.

**Why it Works:**

19 x 21 can be rewritten as (20 - 1) x (20 + 1). Using the

**FOIL**method of multiplication:

**F**refers to**First**(20 x 20), which is**400**.**O**refers to**Outside**(20 x 1), which is**20**.**I**refers to**Inside**(-1 x 20), which is**-20**.**L**refers to**Last**(-1 x 1), which is**-1**.

**400 + 20 - 20 - 1**. The O & I always cancel out, so you are left with

**400 - 1**, which is

**399**.

So why is this process called the

**difference of squares**? Because what you were really left with during the FOIL process was:

**20**

^{2}- 1^{2}.Taking the numbers out of the mix, let any two factors be represented by:

FOILing gives you (a x a) + (a x b) + (-b x a) + (-b x b). This can be rewritten as:

^{2}+ ab - ab - b

^{2}.

The ab & -ab cancel out, so only

**a**remains.

^{2}- b^{2}

**EVERY TIME.****How to Solve:**

- Determine the average of the two factors, and square this number.
- Determine the difference between either factor and this average, and square this difference.
- Subtract the result from step 2 from the result from step 1.

**Example 1: 38 x 42**

**40**. 40

^{2}=

**1600**.

2. The difference between 38 and 40 is

**2**. 2

^{2}=

**4**.

3. 1600 - 4 =

**1596**, which is your answer.

**Example 2: 67 x 73**

**70**. 70

^{2}=

**4900**.

2. The difference between 67 and 70 is

**3**. 3

^{2}=

**9**.

3. 4900 - 9 =

**4891**, which is your answer.

**Example 3: 42 x 58**

**50**. 50

^{2}=

**2500**.

2. The difference between 42 and 50 is

**8**. 8

^{2}=

**64**.

3. 2500 - 64 =

**2436**, which is your answer.

**Example 4: 39 x 21**

**30**. 30

^{2}=

**900**.

2. The difference between 39 and 30 is

**9**. 9

^{2}=

**81**.

3. 900 - 81 =

**819**, which is your answer.

**Example 5: 15 x 19**

*(To illustrate that the average number doesn't have to be a multiple of 10 for this to be a useful skill).*

1. The average of 15 and 19 is

**17**. 17

^{2}=

**289**.

2. The difference between 15 and 17 is

**2**. 2

^{2}=

**4**.

3. 289 - 4 =

**285**, which is your answer.

**Here's a free worksheet to help you practice DiffSquares1:**

diffsquares1.pdf |

**Up Next for Middle School: MultMixWhole**