8th Grade Common Core Math Worksheets - Free Printable

Let's solve each problem step by step and provide explanations.

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

Solution:
We compare the two numbers:
11 is less than 71.
So, the correct symbol is <.

✔ Answer: $ 11 < 71 $

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2. Compare: 105 ○ 406

Solution:
105 is less than 406.
So, we use <.

✔ Answer: $ 105 < 406 $

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3. Which of the following is a true statement?

[A] 618 > 189
[B] 981 < 967
[C] 139 > 603
[D] 808 < 486

Let’s evaluate each:

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

✔ 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 them:
246, 248, 253, 256

Fourth number is 256

✔ Answer: [B] 256

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

Compare:
- 159 < 519 < 951

So, 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

Order: 169, 619, 961

✔ Answer: 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

Negative numbers: the one closer to zero is greater.

- -7 is farther from zero than -1 → 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.

Between -4 and -3, any number like -3.5, -3.2, -3.8, etc.

✔ Example Answer: $-3.5$ (or any number between -4 and -3)

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11. (a) Arrange these numbers in order from greatest to least: 0, 3, -3, 1/3

List: 3, 1/3, 0, -3

Explanation:
- 3 is greatest
- 1/3 ≈ 0.333 → next
- 0
- -3 is smallest

✔ Answer: $ 3, \frac{1}{3}, 0, -3 $

(b) List the numbers in part (a) that are not integers.

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

Non-integers: $ \frac{1}{3} $

✔ Answer: $ \frac{1}{3} $

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12. If $ \frac{23}{8}, \frac{11}{2}, \frac{9}{6}, \text{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 2.14 $
- $ \frac{23}{8} = 2.875 $
- $ \frac{11}{2} = 5.5 $

So, first (smallest) is $ \frac{9}{6} $

✔ Answer: [B] $ \frac{9}{6} $

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

Compare negative fractions.

First, find common denominator or decimals:

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

On number line: -0.5 is to the right of -0.666..., so it's greater.

So: $ -\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?

[A] 32.18 = 32.180000 → True (adding zeros after decimal doesn’t change value)
[B] 24.30009 < 24.31 → True (24.30009 < 24.31000)
[C] 32.7400 > 32.74 → Is this true?

Wait: 32.7400 = 32.74 → they are equal

So, 32.7400 > 32.74 is false

[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 approximate both.

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

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

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

Try 18.8² = ?
- $ 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 + 0.1^2 = 353.44 + 3.76 + 0.01 = 357.21 $

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

Try 18.85²:
- $ 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 18.83²

Better: Try 18.82²:
- $ 18.82^2 = (18.8 + 0.02)^2 = 353.44 + 2×18.8×0.02 + 0.0004 = 353.44 + 0.752 + 0.0004 = 354.1924 $
- Now 18.84² = 354.1924 + 2×18.82×0.02 ≈ 354.1924 + 0.7528 = 354.9452
- 18.85² ≈ 355.3225 as before

So $ \sqrt{355} \approx 18.84 $ (since 355 - 354.9452 = 0.0548; very close)

So $ \sqrt{355} \approx 18.84 $

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

$ \pi \approx 3.1416 $, so:

$ 6\pi \approx 6 × 3.1416 = 18.8496 $

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

Therefore, $ \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 2, 8, 6 exactly once.

Even number → last digit must be even. All digits are even, so any permutation could work, but must be less than 600.

So, hundreds digit must be less than 6.

Available digits: 2, 6, 8

Digits less than 6: only 2

So, hundreds digit must be 2

Then remaining digits: 6 and 8 → can be arranged as 68 or 86

So possible numbers:
- 268
- 286

Both are even and less than 600.

✔ Answer: Possible answers: 268 or 286

(Any one 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. [B] $ \frac{9}{6} $
13. $ -\frac{1}{2} > -\frac{2}{3} $
14. [C] 32.7400 > 32.74
15. $ \sqrt{355} $ is smaller (≈18.84 vs 6π ≈ 18.85)
16. 268 or 286

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