Square Roots math worksheets for math instruction, math ... - Free Printable

You're working on an “Estimating Square Roots” worksheet. The goal is to estimate the square root of each given number by finding the two consecutive perfect squares it lies between — and therefore, the two consecutive integers that its square root lies between.

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🔢 How to Solve:

1. Identify perfect squares near the given number.
2. Find the largest perfect square less than the number.
3. Find the smallest perfect square greater than the number.
4. Take the square roots of those perfect squares — these are your two consecutive integers.
5. Write:
> √(number) is between [smaller integer] and [larger integer]

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✔ Let’s solve each one step-by-step:

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#### 1. √100 → between ___ and ___
- Perfect squares: 81 (9²), 100 (10²), 121 (11²)
- √100 = 10 exactly, so it’s between 10 and 10? But since the worksheet asks for “between”, and 100 is a perfect square, we can say:
> Between 10 and 10 — but typically in estimation, if it's exact, you might still write 10 and 11 or just note it's exact. However, since 100 is 10², and the next integer is 11, technically it’s *between 9 and 11*? No — better to think:

Actually, the instruction says “use your square root estimation skills” — so even for perfect squares, we follow the pattern.

But looking at the example given in the image:
> √60 is between 7 and 8 → because 7²=49 < 60 < 64=8²

So for √100:
100 = 10² → so it’s exactly 10, but since we’re estimating “between”, we should look at the integers around it? But 10 is exact.

However, in most curricula, when estimating square roots, even perfect squares are placed between the same integer twice — or sometimes skipped. But to be consistent with the format, let’s do:

> √100 is between 10 and 10

But that seems odd. Alternatively, perhaps the worksheet expects:

> Since 100 is a perfect square, maybe leave it as 10 and 10, or check if it’s a trick.

Wait — actually, looking at the worksheet again, it has:

> √100 → between ___ and ___

And since 100 = 10², and 9² = 81, 11² = 121 — so 100 is not *between* two different integers — it’s *equal* to one.

But in estimation practice, we often still say:

> √100 is between 10 and 10

Or maybe the worksheet allows us to write the same number twice.

✔ Answer: 10 and 10

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#### 2. √121 → between ___ and ___
Same logic: 121 = 11² → so:

✔ Answer: 11 and 11

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#### 3. √140 → between ___ and ___
Find perfect squares around 140:
- 11² = 121
- 12² = 144
→ 121 < 140 < 144 → so √140 is between 11 and 12

✔ Answer: 11 and 12

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#### 4. √170 → between ___ and ___
- 13² = 169
- 14² = 196
→ 169 < 170 < 196 → so √170 is between 13 and 14

✔ Answer: 13 and 14

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#### 5. √190 → between ___ and ___
- 13² = 169
- 14² = 196
→ 169 < 190 < 196 → so between 13 and 14

✔ Answer: 13 and 14

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#### 6. √200 → between ___ and ___
- 14² = 196
- 15² = 225
→ 196 < 200 < 225 → so between 14 and 15

✔ Answer: 14 and 15

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#### 7. √225 → between ___ and ___
225 = 15² → exact

✔ Answer: 15 and 15

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#### 8. √240 → between ___ and ___
- 15² = 225
- 16² = 256
→ 225 < 240 < 256 → so between 15 and 16

✔ Answer: 15 and 16

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#### 9. √250 → between ___ and ___
- 15² = 225
- 16² = 256
→ 225 < 250 < 256 → between 15 and 16

✔ Answer: 15 and 16

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#### 10. √260 → between ___ and ___
Still between 16²=256 and 17²=289 → 256 < 260 < 289 → between 16 and 17

✔ Answer: 16 and 17

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#### 11. √7 → between ___ and ___
- 2² = 4
- 3² = 9
→ 4 < 7 < 9 → so between 2 and 3

✔ Answer: 2 and 3

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#### 12. √11 → between ___ and ___
- 3² = 9
- 4² = 16
→ 9 < 11 < 16 → between 3 and 4

✔ Answer: 3 and 4

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#### 13. √130 → between ___ and ___
- 11² = 121
- 12² = 144
→ 121 < 130 < 144 → between 11 and 12

✔ Answer: 11 and 12

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#### 14. √150 → between ___ and ___
- 12² = 144
- 13² = 169
→ 144 < 150 < 169 → between 12 and 13

✔ Answer: 12 and 13

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#### 15. √160 → between ___ and ___
- 12² = 144
- 13² = 169
→ 144 < 160 < 169 → between 12 and 13

✔ Answer: 12 and 13

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#### 16. √180 → between ___ and ___
- 13² = 169
- 14² = 196
→ 169 < 180 < 196 → between 13 and 14

✔ Answer: 13 and 14

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📝 Final Answers (Fill in the blanks):

| Expression | Between |
|------------|-------------|
| √100 | 10 and 10 |
| √121 | 11 and 11 |
| √140 | 11 and 12 |
| √170 | 13 and 14 |
| √190 | 13 and 14 |
| √200 | 14 and 15 |
| √225 | 15 and 15 |
| √240 | 15 and 16 |
| √250 | 15 and 16 |
| √260 | 16 and 17 |
| √7 | 2 and 3 |
| √11 | 3 and 4 |
| √130 | 11 and 12 |
| √150 | 12 and 13 |
| √160 | 12 and 13 |
| √180 | 13 and 14 |

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💡 Tip for Estimation:

If you’re unsure, list out squares from 1² to 20² — it helps quickly find which two integers bracket your number.

Example:
1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100,
11²=121, 12²=144, 13²=169, 14²=196, 15²=225, 16²=256, 17²=289, etc.

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Let me know if you want to estimate decimal values too (like √10 ≈ 3.16), but for this worksheet, integers only are needed!

✔ You’ve got this! Just match the number to the two perfect squares around it.