Grade 6 Math worksheet on Matchstick Patterns, challenging students to identify sequences and derive function rules for matchstick arrangements.

Problem: Matchstick Patterns

The task is to complete the table for each sequence of matchstick shapes and determine the function rule for finding the number of matchsticks needed for the shape in the 50th position. Let's solve each part step by step.

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#### Part 1:
Shapes:
- The pattern shows a series of shapes where each subsequent shape adds a consistent number of matchsticks.

Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 7 | 12 | 17 | 22 | 27 | ? | ? | ? | ? | ? |

Observation:
- Shape 1 uses 7 matchsticks.
- Shape 2 uses 12 matchsticks.
- Shape 3 uses 17 matchsticks.
- Each subsequent shape adds 5 matchsticks.

Function Rule:
The number of matchsticks increases linearly by 5 for each new shape. The general form of a linear function is:
\[
\text{Number of Matchsticks} = \text{slope} \times \text{Shape Number} + \text{constant}
\]
Here, the slope is 5, and we can find the constant by using the first shape:
\[
7 = 5 \times 1 + \text{constant} \implies \text{constant} = 2
\]
Thus, the function rule is:
\[
\text{Number of Matchsticks} = 5 \times \text{Shape Number} + 2
\]

Complete the Table:
- For Shape 6: \( 5 \times 6 + 2 = 32 \)
- For Shape 7: \( 5 \times 7 + 2 = 37 \)
- For Shape 8: \( 5 \times 8 + 2 = 42 \)
- For Shape 9: \( 5 \times 9 + 2 = 47 \)
- For Shape 50: \( 5 \times 50 + 2 = 252 \)

Final Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 7 | 12 | 17 | 22 | 27 | 32 | 37 | 42 | 47 | 252 |

Function Rule:
\[
\text{Number of Matchsticks} = 5 \times \text{Shape Number} + 2
\]

---

#### Part 2:
Shapes:
- The pattern shows a series of shapes where each subsequent shape adds a consistent number of matchsticks.

Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 6 | 9 | 12 | 15 | 18 | ? | ? | ? | ? | ? |

Observation:
- Shape 1 uses 6 matchsticks.
- Shape 2 uses 9 matchsticks.
- Shape 3 uses 12 matchsticks.
- Each subsequent shape adds 3 matchsticks.

Function Rule:
The number of matchsticks increases linearly by 3 for each new shape. The general form of a linear function is:
\[
\text{Number of Matchsticks} = \text{slope} \times \text{Shape Number} + \text{constant}
\]
Here, the slope is 3, and we can find the constant by using the first shape:
\[
6 = 3 \times 1 + \text{constant} \implies \text{constant} = 3
\]
Thus, the function rule is:
\[
\text{Number of Matchsticks} = 3 \times \text{Shape Number} + 3
\]

Complete the Table:
- For Shape 6: \( 3 \times 6 + 3 = 21 \)
- For Shape 7: \( 3 \times 7 + 3 = 24 \)
- For Shape 8: \( 3 \times 8 + 3 = 27 \)
- For Shape 9: \( 3 \times 9 + 3 = 30 \)
- For Shape 50: \( 3 \times 50 + 3 = 153 \)

Final Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 153 |

Function Rule:
\[
\text{Number of Matchsticks} = 3 \times \text{Shape Number} + 3
\]

---

#### Part 3:
Shapes:
- The pattern shows a series of shapes where each subsequent shape adds a consistent number of matchsticks.

Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 5 | 8 | 11 | 14 | 17 | ? | ? | ? | ? | ? |

Observation:
- Shape 1 uses 5 matchsticks.
- Shape 2 uses 8 matchsticks.
- Shape 3 uses 11 matchsticks.
- Each subsequent shape adds 3 matchsticks.

Function Rule:
The number of matchsticks increases linearly by 3 for each new shape. The general form of a linear function is:
\[
\text{Number of Matchsticks} = \text{slope} \times \text{Shape Number} + \text{constant}
\]
Here, the slope is 3, and we can find the constant by using the first shape:
\[
5 = 3 \times 1 + \text{constant} \implies \text{constant} = 2
\]
Thus, the function rule is:
\[
\text{Number of Matchsticks} = 3 \times \text{Shape Number} + 2
\]

Complete the Table:
- For Shape 6: \( 3 \times 6 + 2 = 20 \)
- For Shape 7: \( 3 \times 7 + 2 = 23 \)
- For Shape 8: \( 3 \times 8 + 2 = 26 \)
- For Shape 9: \( 3 \times 9 + 2 = 29 \)
- For Shape 50: \( 3 \times 50 + 2 = 152 \)

Final Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 5 | 8 | 11 | 14 | 17 | 20 | 23 | 26 | 29 | 152 |

Function Rule:
\[
\text{Number of Matchsticks} = 3 \times \text{Shape Number} + 2
\]

---

#### Part 4:
Shapes:
- The pattern shows a series of shapes where each subsequent shape adds a consistent number of matchsticks.

Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 6 | 10 | 14 | 18 | 22 | ? | ? | ? | ? | ? |

Observation:
- Shape 1 uses 6 matchsticks.
- Shape 2 uses 10 matchsticks.
- Shape 3 uses 14 matchsticks.
- Each subsequent shape adds 4 matchsticks.

Function Rule:
The number of matchsticks increases linearly by 4 for each new shape. The general form of a linear function is:
\[
\text{Number of Matchsticks} = \text{slope} \times \text{Shape Number} + \text{constant}
\]
Here, the slope is 4, and we can find the constant by using the first shape:
\[
6 = 4 \times 1 + \text{constant} \implies \text{constant} = 2
\]
Thus, the function rule is:
\[
\text{Number of Matchsticks} = 4 \times \text{Shape Number} + 2
\]

Complete the Table:
- For Shape 6: \( 4 \times 6 + 2 = 26 \)
- For Shape 7: \( 4 \times 7 + 2 = 30 \)
- For Shape 8: \( 4 \times 8 + 2 = 34 \)
- For Shape 9: \( 4 \times 9 + 2 = 38 \)
- For Shape 50: \( 4 \times 50 + 2 = 202 \)

Final Table:
| Shapes Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 50 |
|---------------|---|---|---|---|---|---|---|---|---|----|
| Number of Matchsticks | 6 | 10 | 14 | 18 | 22 | 26 | 30 | 34 | 38 | 202 |

Function Rule:
\[
\text{Number of Matchsticks} = 4 \times \text{Shape Number} + 2
\]

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Final Answers:

1. Function Rule: \( 5 \times \text{Shape Number} + 2 \)
- Shape 50: \( \boxed{252} \)

2. Function Rule: \( 3 \times \text{Shape Number} + 3 \)
- Shape 50: \( \boxed{153} \)

3. Function Rule: \( 3 \times \text{Shape Number} + 2 \)
- Shape 50: \( \boxed{152} \)

4. Function Rule: \( 4 \times \text{Shape Number} + 2 \)
- Shape 50: \( \boxed{202} \)