8th Grade Common Core Math Worksheets - Free Printable

Let's solve each problem step by step and explain the reasoning behind the solutions.

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1. Use > or < to write a true sentence: 11 ___ 71

Answer:
$ 11 < 71 $
Explanation: 11 is less than 71, so we use the "<" symbol.

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2. Compare: 105 ○ 406

Answer:
$ 105 < 406 $
Explanation: 105 is less than 406.

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3. Which of the following is a true statement?

- [A] 618 > 189 → True (618 is greater than 189)
- [B] 981 < 967 → False (981 is greater than 967)
- [C] 139 > 603 → False (139 is less than 603)
- [D] 808 < 486 → False (808 is greater than 486)

✔ Correct answer: [A] 618 > 189

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4. If you put the numbers 246, 248, 256, and 253 in order from smallest to largest, which number would be fourth?

Order:
246, 248, 253, 256

So the fourth number is 256.

✔ Answer: [B] 256

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5. Order the numbers 519, 951, and 159 from least to greatest.

Order:
159, 519, 951

✔ Answer: [B] 159, 519, 951

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6. Order the numbers 169, 961, and 619 from least to greatest.

Compare:
- 169 < 619 < 961

✔ Answer: 169, 619, 961

(No multiple choice listed, but correct order is 169, 619, 961.)

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7. Compare: 13,363 ○ 46,943

Clearly, 13,363 < 46,943

✔ Answer: $ 13,363 < 46,943 $

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8. Compare: -7 ○ -1

On the number line, -7 is to the left of -1, so it’s smaller.

So: $ -7 < -1 $

✔ Answer: $ -7 < -1 $

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9. Compare: -5 ○ -6

-5 is greater than -6 because it's closer to zero.

So: $ -5 > -6 $

✔ Answer: [B] >

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10. Name a point between -3 and -4 on a number line.

Any number between -4 and -3 works. For example:

- $ -3.5 $, or $ -3.2 $, or $ -3.9 $, etc.

✔ Example answer: -3.5

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11. (a) Arrange these numbers in order from greatest to least: 0, 3, -3, $ \frac{1}{3} $

List:
- 3 (largest)
- $ \frac{1}{3} $ ≈ 0.333
- 0
- -3 (smallest)

So order: $ 3 > \frac{1}{3} > 0 > -3 $

✔ Answer: 3, $ \frac{1}{3} $, 0, -3

(b) List the numbers that are not integers.

Integers: whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)

- 0 → integer
- 3 → integer
- -3 → integer
- $ \frac{1}{3} $ → not an integer

✔ Answer: $ \frac{1}{3} $

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12. If $ \frac{23}{8} $, $ \frac{11}{2} $, $ \frac{9}{6} $, and $ \frac{15}{7} $ are placed in order from least to greatest, which would be first?

Let’s convert each to decimal:

- $ \frac{23}{8} = 2.875 $
- $ \frac{11}{2} = 5.5 $
- $ \frac{9}{6} = 1.5 $
- $ \frac{15}{7} \approx 2.1429 $

Now order from least to greatest:
- $ \frac{9}{6} = 1.5 $ ← smallest
- $ \frac{15}{7} \approx 2.1429 $
- $ \frac{23}{8} = 2.875 $
- $ \frac{11}{2} = 5.5 $

So the first (least) is $ \frac{9}{6} $

✔ Answer: [A] $ \frac{9}{6} $

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13. Insert =, <, or > to make a true statement: $ -\frac{1}{2} $ [?] $ -\frac{2}{3} $

Compare negative fractions:

- $ \frac{1}{2} = 0.5 $
- $ \frac{2}{3} \approx 0.666... $

So:
- $ -\frac{1}{2} = -0.5 $
- $ -\frac{2}{3} \approx -0.666... $

Since -0.5 is greater than -0.666..., we have:

$ -\frac{1}{2} > -\frac{2}{3} $

✔ Answer: $ -\frac{1}{2} > -\frac{2}{3} $

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14. Which of the following is *not* a true statement?

Check each:

- [A] $ 32.18 = 32.180000 $ → True (adding zeros after decimal doesn’t change value)
- [B] $ 24.30009 < 24.31 $ → True (24.30009 vs 24.31000 → 24.30009 < 24.31)
- [C] $ 32.7400 > 32.74 $ → False!
$ 32.7400 = 32.74 $ — adding zeros doesn’t change value. So this is not true
- [D] $ 29.337 > 29.327 $ → True (337 > 327 in thousandths place)

✘ So [C] is not a true statement.

✔ Answer: [C] 32.7400 > 32.74

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15. $ \sqrt{355} $ and $ 6\pi $. Which number is smaller? Show how you arrived at your answer.

We need to estimate both values.

#### Step 1: Estimate $ \sqrt{355} $

We know:
- $ 18^2 = 324 $
- $ 19^2 = 361 $

So $ \sqrt{355} $ is between 18 and 19.

Try $ 18.8^2 = ? $

- $ 18.8^2 = (18 + 0.8)^2 = 18^2 + 2(18)(0.8) + 0.8^2 = 324 + 28.8 + 0.64 = 353.44 $
- $ 18.9^2 = 18.8^2 + 2(18.8)(0.1) + 0.01 \approx 353.44 + 3.76 + 0.01 = 357.21 $

So $ \sqrt{355} $ is between 18.8 and 18.9.

Try $ 18.85^2 $:
- $ 18.85^2 = (18.8 + 0.05)^2 = 18.8^2 + 2(18.8)(0.05) + 0.05^2 $
- $ = 353.44 + 1.88 + 0.0025 = 355.3225 $

Too high (355.3225 > 355), so try 18.83:
- $ 18.83^2 = ? $
- $ 18.8^2 = 353.44 $
- Add $ 2(18.8)(0.03) = 1.128 $
- Add $ 0.03^2 = 0.0009 $
- Total: $ 353.44 + 1.128 + 0.0009 = 354.5689 $

Still less than 355.

Try 18.85 → already gave ~355.32 → too high.

So $ \sqrt{355} \approx 18.84 $ (interpolate)

But for estimation: $ \sqrt{355} \approx 18.84 $

#### Step 2: Estimate $ 6\pi $

$ \pi \approx 3.1416 $

So $ 6\pi \approx 6 \times 3.1416 = 18.8496 $

#### Compare:
- $ \sqrt{355} \approx 18.84 $
- $ 6\pi \approx 18.8496 $

So $ \sqrt{355} < 6\pi $

✔ Answer: $ \sqrt{355} $ is smaller.

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16. Use the three digits 2, 8, and 6 once each to make an even number less than 600.

We must:
- Use digits 2, 8, 6 exactly once
- Make an even number → last digit must be even (2, 6, or 8)
- Number must be less than 600 → hundreds digit must be less than 6

Possible hundreds digits: only 2 (since 6 and 8 are ≥6)

So hundreds digit = 2

Then remaining digits: 6 and 8

We can form:
- 268 → even? Yes (ends in 8)
- 286 → even? Yes (ends in 6)

Both are even and less than 600.

✔ Answer: 268 or 286 (either is acceptable)

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## ✔ Final Answers Summary:

1. $ 11 < 71 $
2. $ 105 < 406 $
3. [A] 618 > 189
4. [B] 256
5. [B] 159, 519, 951
6. 169, 619, 961
7. $ 13,363 < 46,943 $
8. $ -7 < -1 $
9. [B] >
10. Example: $ -3.5 $
11. (a) $ 3, \frac{1}{3}, 0, -3 $; (b) $ \frac{1}{3} $
12. [A] $ \frac{9}{6} $
13. $ -\frac{1}{2} > -\frac{2}{3} $
14. [C] 32.7400 > 32.74
15. $ \sqrt{355} $ is smaller. (≈18.84 < 18.85 ≈ $ 6\pi $)
16. 268 or 286

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