**Middle School Number Sense Lesson 44: Square Roots**

Square root problems are some of my favorite on number sense tests--I guess because they require a little sleuthing around. Please understand that, unless there is an * in front of the problem, you will be required to give the

*exact answer*, which will

*always*be an

*integer*for these problems. This concept appeared

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

**question # 44**.

Before we get into today's lesson, it might be helpful for you to review

**EstSquareRoot**, which we covered last month. This will help you understand how many digits will be in each answer, etc.

**Number Dojo Level: 164**

Sometimes, these questions will be testing our memorization of perfect squares--which is the reverse of

**Square11-20**and

**Square21-30**:

**Example 1: √361 = ___**

- 19 x 19 = 361, so the answer is
**19**.

**Example 2: √529 = ___**

- 23 x 23 = 529, so the answer is
**23**.

Other times, we need to calculate the square root combining skills we learned with

**CubeRoot**and

**EstSquareRoot**.

**How to Solve:**

- Break the number into pairs, starting at the right. (If you have an odd number of digits under the radicand, the first number by itself is its own "pair"). Your answer will have as many digits as the number of resulting pairs.
- Look at the first pair (or digit). Let's call this
**P**. Think of the largest perfect square that is**less than or equal to P**(and let's call this**L**). Take the square root of**L**and write this result down as the first digit of your answer. - Now think of the smallest perfect square that is
**greater than**the first pair (and let's call it**G**). In your mind, place these numbers in order:**L < P < G**. On a number line, think about where that P lands (how far between L & G). If it's about halfway, your 2nd digit will be**4, 5, or 6**. If P is close to L, your 2nd digit will be**0, 1, 2, or 3**. If P is closer to G, your 2nd digit will be**7, 8, or 9**. - Now look at the last digit of your original number. If it is
**0 or 5**, then the last digit of your answer will be the same number (because 0 x 0 =**0**and 5 x 5 = 2**5**). If the last digit of your original number is**1**, then your answer ends in**1 or 9**(because 1 x 1 =**1**and 9 x 9 = 8**1**). If If the last digit of your original number is**4**, then your answer ends in**2 or 8**(because 2 x 2 =**4**and 8 x 8 = 6**4**). If the last digit of your original number is**6**, then your answer ends in**4 or 6**(because 4 x 4 = 1**6**and 6 x 6 = 3**6**). If the last digit of your original number is**9**, then your answer ends in**3 or 7**(because 3 x 3 =**9**and 7 x 7 = 4**9**).

**Example 3: √1764 = ___**

- The number has 2 pairs:
**17**and**64**. So your answer will have 2 digits. - Look at 17. The largest perfect square less than 17 is
**16**, and its square root is**4**. Write this down. - The smallest perfect square greater than 17 is
**25**. 16 < 17 < 25. 17 is pretty close to 16, so your 2nd digit is**0, 1, 2, or 3**. - The last digit of your original number is 4, so your answer ends in
**2 or 8**. It has to be 2 (as determined in step 3). Your answer is**42**.

**Example 4: √5329 = ___**

- The number has 2 pairs:
**53**and**29**. So your answer will have 2 digits. - Look at 53. The largest perfect square less than 53 is
**49**, and its square root is**7**. Write this down. - The smallest perfect square greater than 53 is
**64**. 49 < 53 < 64. 53 is pretty close to 49, so your 2nd digit is**0, 1, 2, or 3**. - The last digit of your original number is 9, so your answer ends in
**3 or 7**. It has to be 3 (as determined in step 3). So your answer is**73**.

**Example 5: √7921 = ___**

- The number has 2 pairs: 79 and 21. Your answer will have 2 digits.
- Look at 79. The largest perfect square less than 79 is
**64**, and its square root is**8**. Write this down. - The smallest perfect square greater than 79 is
**81**. 64 < 79 < 81. 79 is in the high end of that range, so your 2nd digit is**7, 8, or 9**. - The last digit was 1, so your answer ends in
**1 or 9**. It has to be 9 (as determined in step 3). So your answer is**89**.

**Example 6: √4096 = ___**

- The number has 2 pairs: 40 and 96. Your answer will have 2 digits.
- Look at 40. The largest perfect square less than 40 is
**36**, so your answer starts with**6**. Write this down. - The smallest perfect square greater than 40 is
**49**. 36 < 40 < 49. 40 is sort of in the middle of that range, so your answer ends in**4, 5, or 6**. - The last digit was 6, so your answer ends in
**4 or 6**. 40 was less than halfway between 36 & 49, so use a 4. Your answer is**64**.

**Example 7: Find the number of integers between √13 and √91 on the number line. ___**

- This requires a little memorization & a little logic.
- The smallest perfect square greater than 13 is
**16**, and its square root is**4**. - The largest perfect square less than 91 is
**81**, and its square root is**9**. √13 <**√16 < √81**< √91 - Your integers are between
**4**and**9**inclusive (4, 5, 6, 7, 8, and 9), so the answer is**6**.

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

squareroot.pdf |

**Up Next for Middle School: AddSeqEven**