Worksheet for placing rational and irrational numbers on number lines.

Let's solve the problem step by step. The task is to place rational and irrational numbers on a number line. We’ll go through each set of numbers, approximate or evaluate them where needed, and then place them correctly on the number line.

---

🔹 First Number Line:

Numbers to place:
- 6.5
- 1/4
- π (pi)
- √4
- √7
- 10/2

#### Step 1: Evaluate or simplify each number
- 6.5 → already a decimal → 6.5
- 1/4 = 0.25
- π ≈ 3.1416
- √4 = 2 (since 2 × 2 = 4)
- √7 ≈ 2.6458 (since 2.6² = 6.76, 2.7² = 7.29 → so √7 is between 2.6 and 2.7)
- 10/2 = 5

#### Step 2: Place them on the number line (from 0 to 9)

```
Number Line: 0 1 2 3 4 5 6 7 8 9
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
0.25 2 3.14 5 6.5 2.65
```

Now let’s assign positions:

- 1/4 (0.25) → just after 0, closer to 1
- √4 (2) → exactly at 2
- π (~3.14) → slightly past 3
- √7 (~2.65) → between 2 and 3, closer to 3
- 10/2 (5) → exactly at 5
- 6.5 → halfway between 6 and 7

✔ So on the first number line:
- 1/4 → near 0.25
- √4 → at 2
- √7 → ~2.65
- π → ~3.14
- 10/2 → at 5
- 6.5 → at 6.5

---

🔹 Second Number Line:

Numbers to place:
- 8.50
- 1/2
- -4/-4
- √9
- √36
- 20/4

#### Step 1: Simplify each
- 8.50 → 8.5
- 1/2 = 0.5
- -4/-4 = 1 (negative divided by negative = positive)
- √9 = 3
- √36 = 6
- 20/4 = 5

#### Step 2: Place on number line (0 to 9)

```
Number Line: 0 1 2 3 4 5 6 7 8 9
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
0.5 1 3 5 6 8.5
```

Assign positions:
- 1/2 (0.5) → halfway between 0 and 1
- -4/-4 (1) → at 1
- √9 (3) → at 3
- 20/4 (5) → at 5
- √36 (6) → at 6
- 8.50 (8.5) → halfway between 8 and 9

✔ On second number line:
- 1/2 → 0.5
- -4/-4 → 1
- √9 → 3
- 20/4 → 5
- √36 → 6
- 8.50 → 8.5

---

🔹 Third Number Line:

Numbers to place:
- 7.25
- 1/3
- π
- √5
- √89
- 64/8

#### Step 1: Evaluate/simplify
- 7.25 → already a decimal
- 1/3 ≈ 0.333...
- π ≈ 3.1416
- √5 ≈ 2.236 (since 2.2³ = 4.84, 2.3² = 5.29 → so √5 ≈ 2.24)
- √89 → Let's estimate: 9² = 81, 10² = 100 → √89 is between 9 and 10. Try 9.4² = 88.36, 9.5² = 90.25 → so √89 ≈ 9.43
- 64/8 = 8

#### Step 2: Place on number line (0 to 9)

```
Number Line: 0 1 2 3 4 5 6 7 8 9
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
0.33 2.24 3.14 7.25 8 9.43
```

Assign positions:
- 1/3 (~0.33) → just after 0
- √5 (~2.24) → between 2 and 3, closer to 2
- π (~3.14) → slightly past 3
- 7.25 → between 7 and 8, closer to 7
- 64/8 (8) → at 8
- √89 (~9.43) → beyond 9, but since the number line goes only to 9, it would be just past 9, possibly off the scale

⚠️ Note: √89 ≈ 9.43 > 9 → so it will be outside the range of the number line shown (0–9), so we can mark it just beyond 9.

✔ On third number line:
- 1/3 → ~0.33
- √5 → ~2.24
- π → ~3.14
- 7.25 → between 7 and 8
- 64/8 → at 8
- √89 → just past 9

---

✔ Final Answer Summary:

#### First Number Line:
| Number | Approximate Value | Position |
|--------|-------------------|---------|
| 1/4 | 0.25 | Between 0 and 1 |
| √4 | 2 | At 2 |
| √7 | ~2.65 | Between 2 and 3 |
| π | ~3.14 | Slightly after 3 |
| 10/2 | 5 | At 5 |
| 6.5 | 6.5 | Midway between 6 and 7 |

#### Second Number Line:
| Number | Value | Position |
|-----------|-------|----------|
| 1/2 | 0.5 | Midway between 0 and 1 |
| -4/-4 | 1 | At 1 |
| √9 | 3 | At 3 |
| 20/4 | 5 | At 5 |
| √36 | 6 | At 6 |
| 8.50 | 8.5 | Midway between 8 and 9 |

#### Third Number Line:
| Number | Value | Position |
|----------|-------------|----------|
| 1/3 | ~0.33 | After 0 |
| √5 | ~2.24 | Between 2 and 3 |
| π | ~3.14 | After 3 |
| 7.25 | 7.25 | Between 7 and 8 |
| 64/8 | 8 | At 8 |
| √89 | ~9.43 | Just past 9 (off-scale) |

---

📌 Explanation:

- Rational numbers: Can be expressed as fractions (e.g., 1/4, 10/2, 64/8). They either terminate or repeat.
- Irrational numbers: Cannot be written as simple fractions (e.g., π, √7, √5, √89). Their decimals go on forever without repeating.
- All numbers were approximated to fit on the number line, with care taken for accuracy.

You can now draw these values on the provided number lines using the approximations above.