Eighth Grade Comparing Rational and Irrational Numbers Activity - Free Printable

It looks like you've uploaded a worksheet titled "Comparing Rational and Irrational Numbers", which contains two pages of problems. The task is to compare rational and irrational numbers using the symbols:
- `<` (less than)
- `>` (greater than)
- `=` (equal to)

Since I can't view images directly, I’ll help you solve this based on the text you’ve provided or by assuming standard problems of this type.

Let me walk through how to solve such problems and provide solutions for typical examples found in these worksheets.

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🔍 Understanding the Concepts:

- Rational numbers: Can be expressed as fractions (e.g., ½, 0.75, 3.2), including integers and terminating/repeating decimals.
- Irrational numbers: Cannot be written as fractions; their decimal form is non-repeating and non-terminating (e.g., √2 ≈ 1.414..., π).

We often approximate irrational numbers (like square roots) to compare them with rational numbers.

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✔ Step-by-Step Strategy:

1. Estimate irrational numbers (especially square roots).
2. Convert all numbers to decimal form if needed.
3. Compare the decimals.
4. Use `<`, `>`, or `=` accordingly.

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Let’s go through some sample problems from your worksheet (based on common ones):

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📘 Example Problems & Solutions

#### 1. √2 ___ 2
- √2 ≈ 1.414
- 1.414 < 2
✔ Answer: <

#### 2. 5 ___ √56
- √56 ≈ 7.483
- 5 < 7.483
✔ Answer: <

#### 3. √10 ___ 3
- √10 ≈ 3.162
- 3.162 > 3
✔ Answer: >

#### 4. √50 ___ √36
- √50 ≈ 7.07, √36 = 6
- 7.07 > 6
✔ Answer: >

#### 5. 17 ___ √17
- √17 ≈ 4.123
- 17 > 4.123
✔ Answer: >

#### 6. √6 ___ √5
- √6 ≈ 2.45, √5 ≈ 2.236
- 2.45 > 2.236
✔ Answer: >

#### 7. -0.8 ___ √1
- √1 = 1
- -0.8 < 1
✔ Answer: <

#### 8. 19.07 ___ √361
- √361 = 19
- 19.07 > 19
✔ Answer: >

#### 9. √100 ___ 14.87
- √100 = 10
- 10 < 14.87
✔ Answer: <

#### 10. √197 ___ 14½
- 14½ = 14.5
- √197 ≈ 14.035
- 14.035 < 14.5
✔ Answer: <

#### 11. √277 ___ 16⅓
- 16⅓ ≈ 16.333
- √277 ≈ 16.643
- 16.643 > 16.333
✔ Answer: >

#### 12. 13¼ ___ √176
- 13¼ = 13.25
- √176 ≈ 13.266
- 13.25 < 13.266
✔ Answer: <

#### 13. √405 ___ 9⅔
- 9⅔ ≈ 9.666
- √405 ≈ 20.12
- 20.12 > 9.666
✔ Answer: >

#### 14. 7⅔ ___ √51
- 7⅔ ≈ 7.666
- √51 ≈ 7.141
- 7.666 > 7.141
✔ Answer: >

#### 15. 7⅖ ___ √52
- 7⅖ = 7.4
- √52 ≈ 7.211
- 7.4 > 7.211
✔ Answer: >

#### 16. √89 ___ √81
- √89 ≈ 9.43, √81 = 9
- 9.43 > 9
✔ Answer: >

#### 17. √90 ___ 9.5
- √90 ≈ 9.487
- 9.487 < 9.5
✔ Answer: <

#### 18. √70 ___ 8⅝
- 8⅝ = 8.625
- √70 ≈ 8.367
- 8.367 < 8.625
✔ Answer: <

#### 19. √100 ___ 10
- √100 = 10
- 10 = 10
✔ Answer: =

#### 20. √225 ___ 15
- √225 = 15
- 15 = 15
✔ Answer: =

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✔ Final Tips:

- Always approximate square roots using a calculator or estimation (e.g., √100 = 10, √121 = 11).
- For mixed numbers, convert to decimals (e.g., 14½ = 14.5).
- Negative numbers are always less than positive ones.

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📝 Summary Table (Example Answers):

| Problem | Answer |
|--------|--------|
| 1. √2 ___ 2 | < |
| 2. 5 ___ √56 | < |
| 3. √10 ___ 3 | > |
| 4. √50 ___ √36 | > |
| 5. 17 ___ √17 | > |
| 6. √6 ___ √5 | > |
| 7. -0.8 ___ √1 | < |
| 8. 19.07 ___ √361 | > |
| 9. √100 ___ 14.87 | < |
| 10. √197 ___ 14½ | < |
| 11. √277 ___ 16⅓ | > |
| 12. 13¼ ___ √176 | < |
| 13. √405 ___ 9⅔ | > |
| 14. 7⅔ ___ √51 | > |
| 15. 7⅖ ___ √52 | > |
| 16. √89 ___ √81 | > |
| 17. √90 ___ 9.5 | < |
| 18. √70 ___ 8⅝ | < |
| 19. √100 ___ 10 | = |
| 20. √225 ___ 15 | = |

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If you’d like, you can type out the exact problems from your worksheet, and I’ll give you the exact answers with explanations!