Matchstick Patterns worksheet for Grade 6 Algebra, helping students identify sequences and function rules.

Let's solve each of the four matchstick pattern problems step by step. We'll complete the table for each sequence and determine the function rule (a linear formula) to find the number of matchsticks needed for the 20th shape.

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🔷 Problem 1: Triangles in a Row

Pattern: Each new triangle shares one side with the previous one.

| Shape Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 20 |
|--------------|---|---|---|---|---|---|---|---|---|----|
| Matchsticks | 3 | 5 | 7 | 9 | 11 | ? | ? | ? | ? | ? |

#### Step 1: Find the pattern
- Shape 1: 3 sticks
- Shape 2: 5 sticks → +2 from previous
- Shape 3: 7 → +2
- So, it increases by 2 each time.

This is an arithmetic sequence:
- First term = 3
- Common difference = 2

So, the formula is:

> Number of Matchsticks = 2 × Shape Number + 1

Check:
- For 1: 2×1 + 1 = 3 ✔
- For 2: 2×2 + 1 = 5 ✔
- For 5: 2×5 + 1 = 11 ✔

Now fill in the rest:
- Shape 6: 2×6 + 1 = 13
- Shape 7: 2×7 + 1 = 15
- Shape 8: 2×8 + 1 = 17
- Shape 9: 2×9 + 1 = 19
- Shape 20: 2×20 + 1 = 41

✔ Function Rule:
Number of Matchsticks = 2 × Shape Number + 1

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🔷 Problem 2: Houses (Triangle on Top of Square)

Each "house" has a square base and a triangular roof.

| Shape Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 20 |
|--------------|---|---|---|---|---|---|---|---|---|----|
| Matchsticks | 5 | 9 | 13 | 17 | ? | ? | ? | ? | ? | ? |

#### Step 1: Look at the pattern
- Shape 1: 5
- Shape 2: 9 → +4
- Shape 3: 13 → +4
- Shape 4: 17 → +4

So, common difference = 4

Arithmetic sequence:
- First term = 5
- Difference = 4

Formula:
> Number of Matchsticks = 4 × Shape Number + 1

Check:
- 1: 4×1 + 1 = 5 ✔
- 2: 4×2 + 1 = 9 ✔
- 4: 4×4 + 1 = 17 ✔

Now:
- Shape 5: 4×5 + 1 = 21
- Shape 6: 4×6 + 1 = 25
- Shape 7: 4×7 + 1 = 29
- Shape 8: 4×8 + 1 = 33
- Shape 9: 4×9 + 1 = 37
- Shape 20: 4×20 + 1 = 81

✔ Function Rule:
Number of Matchsticks = 4 × Shape Number + 1

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🔷 Problem 3: Squares in a Row

Each new square shares a side with the previous one.

| Shape Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 20 |
|--------------|---|---|---|---|---|---|---|---|---|----|
| Matchsticks | 4 | 7 | 10 | 13 | ? | ? | ? | ? | ? | ? |

#### Step 1: Pattern
- Shape 1: 4
- Shape 2: 7 → +3
- Shape 3: 10 → +3
- Shape 4: 13 → +3

Common difference = 3

So:
> Number of Matchsticks = 3 × Shape Number + 1

Check:
- 1: 3×1 + 1 = 4 ✔
- 2: 3×2 + 1 = 7 ✔
- 4: 3×4 + 1 = 13 ✔

Now:
- Shape 5: 3×5 + 1 = 16
- Shape 6: 3×6 + 1 = 19
- Shape 7: 3×7 + 1 = 22
- Shape 8: 3×8 + 1 = 25
- Shape 9: 3×9 + 1 = 28
- Shape 20: 3×20 + 1 = 61

✔ Function Rule:
Number of Matchsticks = 3 × Shape Number + 1

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🔷 Problem 4: Houses (Same as Problem 2 but different structure?)

Wait — let’s look closely.

In Problem 4, each shape looks like a house, but the first house uses 6 sticks, then second uses 11, third 16, fourth 21.

| Shape Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 20 |
|--------------|---|---|---|---|---|---|---|---|---|----|
| Matchsticks | 6 | 11 | 16 | 21 | ? | ? | ? | ? | ? | ? |

#### Step 1: Pattern
- Shape 1: 6
- Shape 2: 11 → +5
- Shape 3: 16 → +5
- Shape 4: 21 → +5

So, common difference = 5

Arithmetic sequence:
- First term = 6
- Difference = 5

Formula:
> Number of Matchsticks = 5 × Shape Number + 1

Check:
- 1: 5×1 + 1 = 6 ✔
- 2: 5×2 + 1 = 11 ✔
- 3: 5×3 + 1 = 16 ✔
- 4: 5×4 + 1 = 21 ✔

Now:
- Shape 5: 5×5 + 1 = 26
- Shape 6: 5×6 + 1 = 31
- Shape 7: 5×7 + 1 = 36
- Shape 8: 5×8 + 1 = 41
- Shape 9: 5×9 + 1 = 46
- Shape 20: 5×20 + 1 = 101

✔ Function Rule:
Number of Matchsticks = 5 × Shape Number + 1

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✔ Final Answers:

#### Problem 1
- Table:
- 6: 13, 7: 15, 8: 17, 9: 19, 20: 41
- Function Rule: 2 × Shape Number + 1

#### Problem 2
- Table:
- 5: 21, 6: 25, 7: 29, 8: 33, 9: 37, 20: 81
- Function Rule: 4 × Shape Number + 1

#### Problem 3
- Table:
- 5: 16, 6: 19, 7: 22, 8: 25, 9: 28, 20: 61
- Function Rule: 3 × Shape Number + 1

#### Problem 4
- Table:
- 5: 26, 6: 31, 7: 36, 8: 41, 9: 46, 20: 101
- Function Rule: 5 × Shape Number + 1

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📌 Summary of Function Rules:

| Problem | Function Rule |
|--------|----------------|
| 1 | 2 × n + 1 |
| 2 | 4 × n + 1 |
| 3 | 3 × n + 1 |
| 4 | 5 × n + 1 |

Where n = Shape Number

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