Estimating Square Roots worksheet with practice problems for students to fill in the blanks.

Let's solve each square root estimation problem step by step. The goal is to estimate the square root of a number by finding two consecutive whole numbers between which the square root lies.

We do this by identifying perfect squares close to the given number and using them as bounds.

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🔢 Step-by-step Method:

For any number $ x $, find two perfect squares such that:
$$
a^2 < x < b^2 \Rightarrow \sqrt{x} \text{ is between } a \text{ and } b
$$

Let’s go through each one:

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1. $ \sqrt{200} $
$ 14^2 = 196 $, $ 15^2 = 225 $ → $ 196 < 200 < 225 $
✔ Between 14 and 15

2. $ \sqrt{37} $
$ 6^2 = 36 $, $ 7^2 = 49 $ → $ 36 < 37 < 49 $
✔ Between 6 and 7

3. $ \sqrt{31} $
$ 5^2 = 25 $, $ 6^2 = 36 $ → $ 25 < 31 < 36 $
✔ Between 5 and 6

4. $ \sqrt{2} $
$ 1^2 = 1 $, $ 2^2 = 4 $ → $ 1 < 2 < 4 $
✔ Between 1 and 2

5. $ \sqrt{134} $
$ 11^2 = 121 $, $ 12^2 = 144 $ → $ 121 < 134 < 144 $
✔ Between 11 and 12

6. $ \sqrt{350} $
$ 18^2 = 324 $, $ 19^2 = 361 $ → $ 324 < 350 < 361 $
✔ Between 18 and 19

7. $ \sqrt{8} $
$ 2^2 = 4 $, $ 3^2 = 9 $ → $ 4 < 8 < 9 $
✔ Between 2 and 3

8. $ \sqrt{500} $
$ 22^2 = 484 $, $ 23^2 = 529 $ → $ 484 < 500 < 529 $
✔ Between 22 and 23

9. $ \sqrt{80} $
$ 8^2 = 64 $, $ 9^2 = 81 $ → $ 64 < 80 < 81 $
✔ Between 8 and 9

10. $ \sqrt{60} $
$ 7^2 = 49 $, $ 8^2 = 64 $ → $ 49 < 60 < 64 $
✔ Between 7 and 8

11. $ \sqrt{111} $
$ 10^2 = 100 $, $ 11^2 = 121 $ → $ 100 < 111 < 121 $
✔ Between 10 and 11

12. $ \sqrt{230} $
$ 15^2 = 225 $, $ 16^2 = 256 $ → $ 225 < 230 < 256 $
✔ Between 15 and 16

13. $ \sqrt{320} $
$ 17^2 = 289 $, $ 18^2 = 324 $ → $ 289 < 320 < 324 $
✔ Between 17 and 18

14. $ \sqrt{190} $
$ 13^2 = 169 $, $ 14^2 = 196 $ → $ 169 < 190 < 196 $
✔ Between 13 and 14

15. $ \sqrt{3} $
$ 1^2 = 1 $, $ 2^2 = 4 $ → $ 1 < 3 < 4 $
✔ Between 1 and 2

16. $ \sqrt{5} $
$ 2^2 = 4 $, $ 3^2 = 9 $ → $ 4 < 5 < 9 $
✔ Between 2 and 3

17. $ \sqrt{150} $
$ 12^2 = 144 $, $ 13^2 = 169 $ → $ 144 < 150 < 169 $
✔ Between 12 and 13

18. $ \sqrt{115} $
$ 10^2 = 100 $, $ 11^2 = 121 $ → $ 100 < 115 < 121 $
✔ Between 10 and 11

19. $ \sqrt{70} $
$ 8^2 = 64 $, $ 9^2 = 81 $ → $ 64 < 70 < 81 $
✔ Between 8 and 9

20. $ \sqrt{90} $
$ 9^2 = 81 $, $ 10^2 = 100 $ → $ 81 < 90 < 100 $
✔ Between 9 and 10

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✔ Final Answers (Fill in the blanks):

| Expression | Between |
|----------------|---------------|
| $ \sqrt{200} $ | 14 and 15 |
| $ \sqrt{37} $ | 6 and 7 |
| $ \sqrt{31} $ | 5 and 6 |
| $ \sqrt{2} $ | 1 and 2 |
| $ \sqrt{134} $ | 11 and 12 |
| $ \sqrt{350} $ | 18 and 19 |
| $ \sqrt{8} $ | 2 and 3 |
| $ \sqrt{500} $ | 22 and 23 |
| $ \sqrt{80} $ | 8 and 9 |
| $ \sqrt{60} $ | 7 and 8 |
| $ \sqrt{111} $ | 10 and 11 |
| $ \sqrt{230} $ | 15 and 16 |
| $ \sqrt{320} $ | 17 and 18 |
| $ \sqrt{190} $ | 13 and 14 |
| $ \sqrt{3} $ | 1 and 2 |
| $ \sqrt{5} $ | 2 and 3 |
| $ \sqrt{150} $ | 12 and 13 |
| $ \sqrt{115} $ | 10 and 11 |
| $ \sqrt{70} $ | 8 and 9 |
| $ \sqrt{90} $ | 9 and 10 |

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📝 Summary:

To estimate square roots:
- Find the perfect squares just below and above the number.
- Take their square roots — those are your bounds.

This method works well for approximations without a calculator!

Let me know if you'd like decimal estimates too!