Math worksheet for estimating radicals on a number line.

Let's solve each problem step by step. The goal is to estimate the square root of a number and determine which letter on the number line best represents that value.

We’ll use perfect squares as reference points to estimate the radicals.

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1) √133

- Perfect squares near 133:
- √121 = 11
- √144 = 12
- So, √133 is between 11 and 12.
- 133 is closer to 144 than 121 → so √133 ≈ 11.5 or slightly more.
- On the number line: A=6, B=7, C=9, D=12 → Wait! This number line goes from 6 to 12 with marks at 6,7,8,9,10,11,12.
- But the points are labeled:
- A at 7
- B at 8
- C at 9
- D at 12

Wait — this seems off. Let’s recheck:

> Number line: 6 → A (at 7), B (at 8), C (at 9), D (at 12)

But √133 ≈ 11.5 → should be near 11.5, but the only point above 9 is D at 12.

So the closest point is D, since 11.5 is closer to 12 than to 9.

✔ Answer: D

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2) √135

- √121 = 11, √144 = 12 → √135 is between 11 and 12.
- 135 is closer to 144 → √135 ≈ 11.6
- Number line: 9 to 15, with:
- A at 10
- B at 12
- C at 13
- D at 14
- 11.6 is between 10 and 12 → closer to 12 → so B at 12 is the best choice.

✔ Answer: B

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3) √132

- √121 = 11, √144 = 12 → √132 ≈ ?
- 132 is closer to 121? No: 132 - 121 = 11, 144 - 132 = 12 → actually closer to 121?
Wait: 132 - 121 = 11, 144 - 132 = 12 → yes, closer to 121 → so √132 ≈ 11.5?

But let’s compute better:

Try 11.5² = 132.25 → very close!

So √132 ≈ 11.49 → just under 11.5

Number line: 10 to 16
- A at 11
- B at 12
- C at 13
- D at 15

So √132 ≈ 11.5 → halfway between 11 and 12 → A is at 11, B is at 12 → 11.5 is midway → but since it’s slightly less than 11.5, maybe closer to 11?

Wait: 11.5 is exactly halfway → but 11.49 is almost 11.5 → still closer to 12?

No: 11.5 is midpoint → 11.49 is very close to 11.5, so equally close to 11 and 12, but since the options are discrete points, and 11.49 < 11.5, it's closer to 11.

But let’s check:

- Distance from 11.49 to 11 = 0.49
- Distance to 12 = 0.51 → so closer to 11

So A at 11 is better.

But wait — is A at 11? Yes.

So √132 ≈ 11.49 → closer to 11 than 12 → choose A

✔ Answer: A

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4) √12

- √9 = 3, √16 = 4 → √12 is between 3 and 4
- Try 3.4² = 11.56, 3.5² = 12.25 → so √12 ≈ 3.46
- Number line: -1 to 5
- A at 1
- B at 3
- C at 4
- D at 5

So √12 ≈ 3.46 → between 3 and 4 → closer to 3 or 4?

- Distance to 3: 0.46
- Distance to 4: 0.54 → so closer to 3 → B at 3

✔ Answer: B

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5) √92

- √81 = 9, √100 = 10 → √92 between 9 and 10
- 9.5² = 90.25, 9.6² = 92.16 → so √92 ≈ 9.59
- Number line: 4 to 10
- A at 4
- B at 5
- C at 8
- D at 9

Wait — D is at 9? But √92 ≈ 9.59 → between 9 and 10 → but no point at 10? Only up to D at 9.

Wait — look again:

> Number line: 4 → A(4), B(5), then 6,7,8,9,10 → C at 8, D at 9

Wait: C at 8, D at 9 → but √92 ≈ 9.59 → which is greater than 9, but no point beyond 9.

But D is at 9 → and next is 10, but not labeled.

But 9.59 is between 9 and 10, and the only point at 9 is D.

So even though it's greater than 9, the closest marked point is D at 9, unless there’s a point at 10?

Wait — the number line shows:
```
4 5 6 7 8 9 10
A B C D
```

Wait — no: labels:
- A at 4
- B at 5
- C at 8
- D at 9

But 9.59 is after 9, so D is at 9 → but 9.59 is closer to 10 than to 9?

- Distance to 9: 0.59
- Distance to 10: 0.41 → so closer to 10

But there is no point at 10 — only D at 9.

So we must pick among existing points.

- D is at 9 → distance 0.59
- C is at 8 → distance 1.59 → much farther

So D is the closest point.

Even though 9.59 is closer to 10, 10 is not a labeled point, so D is the best available.

✔ Answer: D

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6) √67

- √64 = 8, √81 = 9 → √67 between 8 and 9
- 8.2² = 67.24 → so √67 ≈ 8.19
- Number line: 6 to 12
- A at 6
- B at 8
- C at 9
- D at 11

So √67 ≈ 8.19 → between 8 and 9 → closer to 8?

- Distance to 8: 0.19
- Distance to 9: 0.81 → so much closer to 8

→ B at 8

✔ Answer: B

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7) √45

- √36 = 6, √49 = 7 → √45 between 6 and 7
- 6.7² = 44.89, 6.71² = ~44.9 → so √45 ≈ 6.71
- Number line: 4 to 10
- A at 7
- B at 8
- C at 9
- D at 10

Wait — A is at 7? But √45 ≈ 6.71 → so between 6 and 7 → but A is at 7, and no point at 6.

So points:
- A at 7
- B at 8
- C at 9
- D at 10

So 6.71 is less than 7, so closest point is A at 7

Distance: |6.71 - 7| = 0.29

Next is B at 8 → 1.29 → much farther

So A is closest.

✔ Answer: A

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8) √17

- √16 = 4, √25 = 5 → √17 between 4 and 5
- 4.1² = 16.81, 4.2² = 17.64 → so √17 ≈ 4.12
- Number line: -1 to 5
- A at -1
- B at 2
- C at 3
- D at 4

Wait — D is at 4? But √17 ≈ 4.12 → so just above 4

Points:
- D at 4
- C at 3
- B at 2
- A at -1

So √17 ≈ 4.12 → closest to D at 4 (distance 0.12)

Is there a point at 5? No — only up to D at 4.

So D is the closest.

✔ Answer: D

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9) √83

- √81 = 9, √100 = 10 → √83 between 9 and 10
- 9.1² = 82.81, 9.2² = 84.64 → so √83 ≈ 9.11
- Number line: 8 to 14
- A at 9
- B at 12
- C at 13
- D at 14

So √83 ≈ 9.11 → very close to 9 → A at 9

Distance: 0.11 → very small

Other points are farther.

✔ Answer: A

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10) √103

- √100 = 10, √121 = 11 → √103 between 10 and 11
- 10.1² = 102.01, 10.2² = 104.04 → so √103 ≈ 10.15
- Number line: 7 to 13
- A at 8
- B at 9
- C at 10
- D at 12

So √103 ≈ 10.15 → between 10 and 11 → closest to 10?

- Distance to 10: 0.15
- Distance to 11: 0.85 → so closer to 10

C is at 10 → so C

✔ Answer: C

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✔ Final Answers:

1. D
2. B
3. A
4. B
5. D
6. B
7. A
8. D
9. A
10. C

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✔ Answer Key:

| # | Answer |
|---|--------|
| 1 | D |
| 2 | B |
| 3 | A |
| 4 | B |
| 5 | D |
| 6 | B |
| 7 | A |
| 8 | D |
| 9 | A |
| 10| C |

Let me know if you'd like a visual explanation or estimation method!