Printable worksheet for estimating square roots, designed for math education.

Let's solve each of these square root estimation problems step by step. The goal is to estimate the square root of a number by finding two perfect squares between which the number lies, and then identifying the integers whose squares are those perfect squares.

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🔍 Strategy:

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

So we need to find two consecutive integers $ a $ and $ b $ such that $ a^2 < x < b^2 $.

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Now let’s go through each one:

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1. $ \sqrt{200} $
$ 14^2 = 196 $, $ 15^2 = 225 $ → $ 196 < 200 < 225 $
✔ So $ \sqrt{200} $ is 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):

| Problem | Answer |
|--------|--------|
| $ \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:

You can estimate square roots by identifying the nearest perfect squares below and above the given number, then taking their square roots (which are whole numbers). This gives you the range in which the actual square root lies.

Let me know if you'd like this turned into a printable answer key!