To find the number of matchsticks in the $n$-th pattern, we look at how the number changes from one step to the next. This is called an arithmetic sequence.
Step-by-step Deduction:
1.
Section A (Squares):
* Pattern 1: 4 matchsticks.
* Pattern 2: 7 matchsticks ($4 + 3$).
* Pattern 3: 10 matchsticks ($7 + 3$).
* The difference is always $+3$.
* Formula: Start with 1 vertical stick on the left, then add groups of 3 for each square.
* Expression: $3n + 1$.
2.
Section B (Triangles on squares):
* Pattern 1: 5 matchsticks.
* Pattern 2: 9 matchsticks ($5 + 4$).
* Pattern 3: 13 matchsticks ($9 + 4$).
* The difference is always $+4$.
* Formula: $4 \times n = 4n$. Since $4(1) = 4$ but we have 5, we add 1.
* Expression: $4n + 1$.
3.
Section C (Hexagons joined vertically/diamonds):
* Pattern 1: 6 matchsticks.
* Pattern 2: 11 matchsticks ($6 + 5$).
* Pattern 3: 16 matchsticks ($11 + 5$).
* The difference is always $+5$.
* Formula: $5 \times n = 5n$. Since $5(1) = 5$ but we have 6, we add 1.
* Expression: $5n + 1$.
4.
Section D (L-shapes):
* Pattern 1: 5 matchsticks.
* Pattern 2: 8 matchsticks ($5 + 3$).
* Pattern 3: 11 matchsticks ($8 + 3$).
* The difference is always $+3$.
* Formula: $3 \times n = 3n$. Since $3(1) = 3$ but we have 5, we add 2.
* Expression: $3n + 2$.
5.
Section E (Hexagons joined horizontally):
* Pattern 1: 6 matchsticks.
* Pattern 2: 11 matchsticks ($6 + 5$).
* Pattern 3: 16 matchsticks ($11 + 5$).
* The difference is always $+5$.
* Formula: $5 \times n = 5n$. Since $5(1) = 5$ but we have 6, we add 1.
* Expression: $5n + 1$.
6.
Section F (Trapezoids/Chevrons):
* Pattern 1: 5 matchsticks.
* Pattern 2: 9 matchsticks ($5 + 4$).
* Pattern 3: 13 matchsticks ($9 + 4$).
* The difference is always $+4$.
* Formula: $4 \times n = 4n$. Since $4(1) = 4$ but we have 5, we add 1.
* Expression: $4n + 1$.
7.
Extension (Square Grids):
* Pattern 1 ($1\times1$): 4 matchsticks.
* Pattern 2 ($2\times2$): 12 matchsticks.
* Pattern 3 ($3\times3$): 24 matchsticks.
* Let's check the logic: In an $n \times n$ grid, there are $(n+1)$ vertical lines of length $n$, and $(n+1)$ horizontal lines of length $n$.
* Total matchsticks = (Vertical sticks) + (Horizontal sticks)
* Vertical sticks = $n(n+1)$
* Horizontal sticks = $n(n+1)$
* Total = $2n(n+1)$ or $2n^2 + 2n$.
* Check for $n=1$: $2(1)(2) = 4$. Correct.
* Check for $n=2$: $2(2)(3) = 12$. Correct.
* Check for $n=3$: $2(3)(4) = 24$. Correct.
* Expression: $2n(n+1)$.
Final Answer:
Section A: $3n + 1$
Section B: $4n + 1$
Section C: $5n + 1$
Section D: $3n + 2$
Section E: $5n + 1$
Section F: $4n + 1$
Extension: $2n(n + 1)$
Parent Tip: Review the logic above to help your child master the concept of sequences worksheet year 7.
