**Middle School Number Sense Lesson 59: Estimations with Squaring Numbers**

Early last month (for a high school lesson), we covered estimation with exponents (greater than 2) in

**EstHighPower**. I recommend you review that topic if you haven't recently. Today we're going to back it off a notch for middle school, and we'll work on estimations where 2 is the highest exponent. This concept appeared

**5 times**last year, with a median placement at

**question # 40**.

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These problems have several variations, which I will try to cover in the examples. In many cases, we will be asked to incorporate another skill into our calculation, such as

**AddSequence**,

**Square11-20**, or

**Square21-30**.

*****: When multiplying or squaring numbers

__NOTE__**ending in 0**, it's a good idea to write the answer's (correct number of) zeros first. I do this for two reasons:

- I'm more likely to get the answer's number of digits correct, and
- I can dump those 0's out of my brain and work with smaller numbers to multiply or square.

**Example 1:**

**122**

^{2}= ___1. Just round 122 down to 120 and then square it.

2. We know the answer will have 2 zeros; write them down and then square 12.

3. 12

^{2}=

**144**; write this in front. Your answer is

**14400**.

4. The exact answer is

**14884**, and the acceptable (+/- 5%) range is

**from 14140 to 15628**. We are well within range.

**Example 2:**

**18**

^{2}x 11^{2}= ___1. This is the same as

**(18 x 11)**.

^{2}2. 18 x 11 =

**198**. Round this to 200 and then square it. (Or write down the 4 zeros and then square 2).

3. Your answer is

**40000**.

4. The exact answer is

**39204**. The acceptable (+/- 5%) range is

**from 37244 to 41164**.

**Example 3:**

**28**

^{2}+ 29^{2}+ 30^{2}+ 31^{2}+ 32^{2}= ___1. In this case, we can round each number to 30 and then square. The rounding errors on the low end will be offset by the rounding errors on the high end.

2. Let's

**square 30**and then multiply by 5. 30

^{2}=

**900**.

3. 900 x 5 =

**4500**.

4. The exact answer is

**4510**; the acceptable range is

**from 4285 to 4735**.

**Example 4:**

**14**

^{2}+ 48^{2}= ___1. We already know 14

^{2}=

**196**, and we can round this to 200.

2. We need to be careful rounding 48 to 50 and then squaring it. Instead, let's use an estimate of

**46 x 50**(think of

**DiffSquares1**) to approximate 48

^{2}. 46 x 50 =

**2300**.

3. 200 + 2300 =

**2500**. This is actually the

**exact answer**.

**Example 5:**

**(1 + 2 + 3 + 4 + ... + 20)**

^{2}= ___1. We can determine that the sum of the first 20 consecutive numbers is

**20 x 21/2**, which is 10 x 21 =

**210**.

2. 210

^{2}=

**44100**, which is the exact answer.

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

estsquare.pdf |

**Up Next for Middle School: EstMult3Close**