**Middle School Number Sense Lesson 36: Difference of Squares**

Early last month we looked at using the "Difference of Squares" concept to multiply two numbers surrounding a third central number (

**DiffSquares1**). You may wish to review that lesson now if it's not fresh in your memory. Today we will go the other direction. This concept appeared a respectable

**18 times**this year (on average, once per test)--at a median (and average) placement of

**question # 48**.

**Number Dojo Level: 173**

This concept centers around the notion that:

**A**

^{2}- B^{2}**(A + B)(A - B)**.

Once this concept is understood, these questions become easier to solve, because usually either

**(A + B)**or

**(A - B)**will be easy to multiply by.

**29**

^{2}- 21^{2}= ___You could memorize (or calculate) each of these squares and then subtract them. OR, since this is number sense, you could save yourself some time.

To solve this problem, change it to:

**(29 + 21)(29 - 21)**

**(50)(8)**,

which is

**400**.

**Example 1:**

**32**

^{2}- 18^{2}= ___1. Change this to

**(32 + 18)(32 - 18)**.

2. Simplifying, this becomes

**50 x 14**. The answer is

**700**.

**Example 2:**

**19**

^{2}- 11^{2}= ___1. Change this to

**(19 + 11)(19 - 11)**.

2. Simplifying, this becomes

**30 x 8**. The answer is

**240**.

**Example 3:**

**43**

^{2}- 7^{2}= ___1. Change this to

**(43 + 7)(43 - 7)**.

2. Simplifying, this becomes

**50 x 36**. The answer is

**1800**.

**Example 4:**

**74**

^{2}- 26^{2}= ___1. Change this to

**(74 + 26)(74 - 26)**.

2. Simplifying, this becomes

**100 x 48**. The answer is

**4800**.

**Example 5:**

**28**

^{2}- 27^{2}+ 26^{2}- 25^{2}= ___1. Change this to

**(28 + 27)(28 - 27) + (26 + 25)(26 - 25)**.

2. Simplifying, this becomes

**(55 x 1) + (51 x 1)**, or

**55 + 51**. The answer is

**106**.

3. Note: In this specific case, the answer is the same as

**28 + 27 + 26 + 25**.

**Example 6:**

**If 63**

^{2}- 37^{2}= 100k, then k = ___1. Change this equation to

**(63 + 37)(63 - 37) = 100k**.

2. Simplifying the left half, this becomes

**100 x 26**.

3. Plug

**100 x 26**back into the left half of the original equation, and you get

**100 x 26 = 100k**. Therefore, k has to be

**26**.

**Example 7:**

**Find k if 18**

^{2}- 15^{2}= 11k. k = ___1. Change this equation to

**(18 + 15)(18 - 15) = 11k**.

2. Simplifying the left half, this becomes

**33 x 3**, which is

**99**.

3. Plug 99 back into the left half of the original equation, and you get

**99 = 11k**. Divide both sides of the equation by 11, and k has to be

**9**.

**Here's a free worksheet to help you practice DiffSquares2:**

diffsquares2.pdf |

**Up Next for Middle School: SolveX**