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 a series of problems asking you to compare rational and irrational numbers using the symbols `<`, `>`, or `=`.

Since I can't view images directly, I’ll help you solve this type of problem step-by-step based on the typical structure of such worksheets. If you can provide the specific values from the worksheet (e.g., the actual numbers in the problems), I can give exact answers. But for now, I’ll walk through how to approach these types of problems and solve a few examples.

---

🔍 Understanding Rational vs. Irrational Numbers

- Rational numbers: Can be expressed as fractions (e.g., 1/2, 0.75, -3). They include integers, terminating decimals, and repeating decimals.
- Irrational numbers: Cannot be written as fractions. Examples: √2, π, √3. Their decimal representations are non-repeating and non-terminating.

---

🎯 Goal:

Compare each pair of numbers using `<`, `>`, or `=`.

---

✔ Strategy:

1. Estimate irrational numbers (like square roots) by finding perfect squares near them.
2. Convert all numbers to decimals (if needed) to compare easily.
3. Compare the values.

Let’s go through some common examples that appear on such worksheets:

---

🔢 Example Problems & Solutions

#### 1. √3 ___ 2
- √3 ≈ 1.732
- 1.732 < 2
✔ Answer: √3 < 2

---

#### 2. 5 ___ √36
- √36 = 6
- 5 < 6
✔ Answer: 5 < √36

---

#### 3. √16 ___ 3
- √16 = 4
- 4 > 3
✔ Answer: √16 > 3

---

#### 4. √20 ___ √36
- √20 ≈ 4.47
- √36 = 6
- 4.47 < 6
✔ Answer: √20 < √36

---

#### 5. 17 ___ √25
- √25 = 5
- 17 > 5
✔ Answer: 17 > √25

---

#### 6. √8 ___ √1
- √8 ≈ 2.828
- √1 = 1
- 2.828 > 1
✔ Answer: √8 > √1

---

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

---

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

---

#### 9. √44 ___ 14.87
- √44 ≈ 6.63
- 6.63 < 14.87
✔ Answer: √44 < 14.87

---

#### 10. √277 ___ 14½
- 14½ = 14.5
- √277: Let's estimate
- 16² = 256
- 17² = 289
- So √277 is between 16 and 17
- Try 16.6² = 275.56
- 16.7² = 278.89 → too big
- So √277 ≈ 16.64
- 16.64 > 14.5
✔ Answer: √277 > 14½

---

#### 11. 13¼ ___ √173
- 13¼ = 13.25
- √173:
- 13² = 169
- 14² = 196
- √173 ≈ 13.53 (since 13.5² = 182.25 — wait, too high)
- 13.4² = 179.56 → too high
- 13.3² = 176.89
- 13.2² = 174.24
- 13.1² = 169.61
- 13.15² = ?
- 13.1² = 169.61
- 13.15² = (13 + 0.15)² = 169 + 2×13×0.15 + 0.0225 = 169 + 3.9 + 0.0225 = 172.9225
- Close!
- 13.16² = 13.15² + 2×13.15×0.01 + 0.0001 ≈ 172.9225 + 0.263 + 0.0001 ≈ 173.1856 → too high
- So √173 ≈ 13.15
- 13.15 > 13.25? No → 13.15 < 13.25
✔ Answer: 13¼ > √173

Wait — let's double-check:

- 13.15² ≈ 172.92 → less than 173
- 13.16² ≈ 173.18 → greater than 173
→ So √173 ≈ 13.153
- 13.153 < 13.25
✔ So 13¼ > √173

---

#### 12. √81 ___ 9⅔
- √81 = 9
- 9⅔ = 9.666...
- 9 < 9.666
✔ Answer: √81 < 9⅔

---

#### 13. 7¼ ___ √53
- 7¼ = 7.25
- √53:
- 7² = 49
- 8² = 64
- √53 ≈ 7.28 (since 7.28² = 53.00)
- 7.25 < 7.28
✔ Answer: 7¼ < √53

---

#### 14. √69 ___ 8½
- 8½ = 8.5
- √69:
- 8² = 64
- 9² = 81
- √69 ≈ 8.306 (since 8.3² = 68.89, 8.31² = 69.0561)
- 8.306 < 8.5
✔ Answer: √69 < 8½

---

#### 15. 10 ___ √100
- √100 = 10
- So 10 = √100
✔ Answer: 10 = √100

---

✔ Summary of Common Techniques:

| Number | How to Estimate |
|--------|------------------|
| √n | Find nearest perfect squares |
| Fractions | Convert to decimals |
| Mixed numbers | Convert to improper fractions or decimals |

---

🧠 Tips:

- Use a calculator if allowed.
- If not, use approximation: e.g., √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, etc.
- Always convert both sides to decimals or same form before comparing.

---

If you can list the exact problems from your worksheet (e.g., “√7 ___ 2.6”), I can give you precise answers with explanations.

Would you like me to solve the rest of the problems from your worksheet? Just copy-paste the list!