Math worksheet with pattern recognition and number theory exercises.

Let's solve each part of the worksheet step by step and explain the reasoning.

---

1. Fill in the empty blanks. Write a rule to represent the relationship between input and output.

#### Problem 1:
| Input | Output |
|-------|--------|
| 10 | 15 |
| 2 | 7 |
| 4 | |
| 3 | |

Step 1: Look at the given pairs.
- When input = 10, output = 15 → 10 + 5 = 15
- When input = 2, output = 7 → 2 + 5 = 7

So, it looks like:
Output = Input + 5

Apply the rule:
- For input = 4 → Output = 4 + 5 = 9
- For input = 3 → Output = 3 + 5 = 8

✔ Completed Table:
| Input | Output |
|-------|--------|
| 10 | 15 |
| 2 | 7 |
| 4 | 9 |
| 3 | 8 |

Rule: Output = Input + 5

---

#### Problem 2:
| Input | Output |
|-------|--------|
| 8 | 9 |
| 6 | 7 |
| 9 | |
| 10 | |

Step 1: Analyze the pattern.
- Input = 8 → Output = 9 → 8 + 1 = 9
- Input = 6 → Output = 7 → 6 + 1 = 7

So, Output = Input + 1

Apply the rule:
- Input = 9 → Output = 9 + 1 = 10
- Input = 10 → Output = 10 + 1 = 11

✔ Completed Table:
| Input | Output |
|-------|--------|
| 8 | 9 |
| 6 | 7 |
| 9 | 10 |
| 10 | 11 |

Rule: Output = Input + 1

---

Find the pattern.

#### Problem 3:
Sequence: 0, 6, 12, 18, 24, 30, 36, ___

Step 1: Find the difference between consecutive terms.
- 6 - 0 = 6
- 12 - 6 = 6
- 18 - 12 = 6
- 24 - 18 = 6
- 30 - 24 = 6
- 36 - 30 = 6

It increases by 6 each time.

Next term: 36 + 6 = 42

✔ Answer: 42

---

#### Problem 4:
Sequence: 76, 70, 64, 58, 52, 46, 40, ___

Step 1: Find the difference.
- 70 - 76 = -6
- 64 - 70 = -6
- 58 - 64 = -6
- 52 - 58 = -6
- 46 - 52 = -6
- 40 - 46 = -6

Decreasing by 6 each time.

Next term: 40 - 6 = 34

✔ Answer: 34

---

#### Problem 5:
Sequence: 27, 30, 33, 36, 39, 42, 45, ___

Step 1: Difference between terms.
- 30 - 27 = 3
- 33 - 30 = 3
- 36 - 33 = 3
- 39 - 36 = 3
- 42 - 39 = 3
- 45 - 42 = 3

Increases by 3 each time.

Next term: 45 + 3 = 48

✔ Answer: 48

---

Look at the addition table and answer questions 6a and 6b.

The table is:

```
+ | 1 2 3 4
---------
1 | 2 3 4 5
2 | 3 4 5 6
3 | 4 5 6 7
4 | 5 6 7 8
```

We are asked:

#### 6a. If you add an odd number to another odd number, what will the sum always be?

Let’s check examples from the table:
- 1 + 1 = 2 (even)
- 1 + 3 = 4 (even)
- 3 + 3 = 6 (even)
- 3 + 1 = 4 (even)

Odd + Odd = Even

✔ Answer: The sum is always even.

#### 6b. If you add an even number to another even number, what will the sum always be?

From the table:
- 2 + 2 = 4 (even)
- 2 + 4 = 6 (even)
- 4 + 4 = 8 (even)

Even + Even = Even

✔ Answer: The sum is always even.

---

✔ Final Answers Summary:

1. Table 1:
- Rule: Output = Input + 5
- Missing outputs: 9, 8

2. Table 2:
- Rule: Output = Input + 1
- Missing outputs: 10, 11

3. Pattern 3: 42
4. Pattern 4: 34
5. Pattern 5: 48
6a. Odd + Odd = Even
6b. Even + Even = Even

---

Let me know if you'd like this formatted as a printable answer sheet!