**Middle School Number Sense Lesson 86: Adding Squares**

**--Special Case**

Today's concept took me a few tries to master, but it was well worth the effort. It has become one of my favorite number sense skills because of the amount of time saved in calculating. This concept has appeared

**8 times**on middle school tests this year, with a median placement at

**question # 55**.

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The easiest way to recognize this type of problem is that it doesn't fit the other patterns of adding squares we have already discussed (

**AddSquareTrip**and

**AddSquareDoub**). For the skill to work, the problem will be presented as the sum of the squares of two 2-digit numbers:

**ab**

^{2}+ cd^{2}- The sum of the outer digits (a + d) = 10, and
- The difference of the inner digits (b - c) = 1

**How to Solve:**

- Verify that the two conditions above are met.
- Square each digit in the first 2-digit number and add the squares together.
- Multiply this result by 101.

**Example 1:**

**52**

^{2}+ 15^{2}= ___1. Notice that the two conditions are met: 5 + 5 = 10 and 2 - 1 = 1.

2. Square and then add the digits in

**52**. 5

^{2}+ 2

^{2}= 25 + 4 =

**29**.

3. 29 x 101 =

**2929**; this is your answer.

**Example 2:**

**85**

^{2}+ 42^{2}= ___1. Notice that the two conditions are met: 8 + 2 = 10 and 5 - 4 = 1.

2. Square and then add the digits in

**85**. 8

^{2}+ 5

^{2}= 64 + 25 =

**89**.

3. 89 x 101 =

**8989**; this is your answer.

**Example 3:**

**39**

^{2}+ 87^{2}= ___1. Notice that the two conditions are met: 3 + 7 = 10 and 9 - 8 = 1.

2. Square and then add the digits in

**39**. 3

^{2}+ 9

^{2}= 9 + 81 =

**90**.

3. 90 x 101 =

**9090**; this is your answer.

**Example 4:**

**78**

^{2}+ 73^{2}= ___1. Notice that the two conditions are met: 7 + 3 = 10 and 8 - 7 = 1.

2. Square and then add the digits in

**78**. 7

^{2}+ 8

^{2}= 49 + 64 =

**113**.

3. 113 x 101 =

**11413**; this is your answer.

**Example 5:**

**56**

^{2}+ 46^{2}= ___1.

*Notice that the two conditions are NOT met, but if the order is reversed (*4 + 6 = 10 and 6 - 5 = 1.

**46**), they are. These problems will occasionally be written in this order to throw the student off.^{2}+ 56^{2}2. Square and then add the digits in

**46**. 4

^{2}+ 6

^{2}= 16 + 36 =

**52**.

3. 52 x 101 =

**5252**; this is your answer.

**Up Next for Middle School: AddSeqGeom**