Math worksheet for students to practice identifying and continuing patterns using sequences of shapes made from sticks or tiles.

Let’s solve each question one by one, explaining the pattern and reasoning clearly.

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1) Squares in a row

Pattern: Each term adds one more square to the right.

- Term 1: 1 square → 4 sticks
- Term 2: 2 squares → 7 sticks (shared side saves 1 stick)
- Term 3: 3 squares → 10 sticks
- Term 4: 4 squares → 13 sticks (as shown)

Pattern for sticks:
Each new square adds 3 sticks (since it shares one vertical side with the previous square).

So:
- Term n: 3n + 1 sticks

→ For 6th term: 3×6 + 1 = 19 sticks

✔ Answer: 19 sticks

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2) Houses (square + triangle roof)

Each “house” is made of 1 square (4 sticks) + 1 triangle roof (2 sticks on top, sharing the top side of the square). But when adding a new house, you share a vertical side.

Let’s count:

- Term 1: 1 house → 6 sticks (square: 4, roof: 2 — but roof shares base with square, so total = 4 + 2 = 6)
- Term 2: 2 houses → 6 + 5 = 11 sticks (second house shares 1 vertical side → adds 5 new sticks: 3 for square sides, 2 for roof)
- Term 3: 11 + 5 = 16 sticks
- Term 4: 16 + 5 = 21 sticks

Pattern: Starts at 6, then adds 5 each time.

Formula: Sticks = 5n + 1

Check: n=1 → 5(1)+1=6 ✔️, n=2 → 11 ✔️

→ For 5th term: 5×5 + 1 = 26 sticks

✔ Answer: 26 sticks

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3) Staircase-like shape

Let’s count sticks per term:

Term 1: 3 squares arranged like L-shape → let's count carefully.

Actually, better to count total sticks visually:

- Term 1: 3 squares → 12 sticks? Wait — let’s look again.

Actually, looking at the image:

Term 1: 3 squares → looks like 2 stacked vertically + 1 to the right → total sticks = 10? Let’s find a pattern.

Better approach: Count total sticks per term:

Term 1: 10 sticks
Term 2: 16 sticks
Term 3: 22 sticks
Term 4: 28 sticks

Difference between terms: +6 each time.

So: Sticks = 6n + 4

Check: n=1 → 6+4=10 ✔️, n=2 → 12+4=16 ✔️

We are asked: Which term uses 35 sticks?

Set 6n + 4 = 35
→ 6n = 31
→ n = 31/6 ≈ 5.166 → not an integer

Wait — maybe I miscounted.

Let me recount Term 1:

It’s 3 squares: 2 vertical, 1 horizontal attached to bottom right.

Each square has 4 sides, but shared sides reduce total.

Total unique sticks:

- Vertical lines: 3 columns → left: 2 sticks, middle: 2 sticks, right: 1 stick → total vertical: 5
- Horizontal lines: 3 rows → top: 2 sticks, middle: 2 sticks, bottom: 1 stick → total horizontal: 5

Total sticks: 10 ✔️

Term 2: 6 squares → similar structure extended.

Actually, from image:

Term 1: 3 squares → 10 sticks
Term 2: 6 squares → 16 sticks
Term 3: 9 squares → 22 sticks
Term 4: 12 squares → 28 sticks

So, every term adds 3 squares and 6 sticks.

So formula: Sticks = 6n + 4

Set 6n + 4 = 35 → 6n = 31 → n = 31/6 → not whole number.

But 35 is not divisible by 6 remainder 4? 35 - 4 = 31, not divisible by 6.

So no term uses exactly 35 sticks.

Wait — perhaps I miscounted.

Alternative: Maybe the pattern is Sticks = 6n + 4, as above.

Then possible stick counts: 10, 16, 22, 28, 34, 40...

So 34 is term 5 (6×5 + 4 = 34), 40 is term 6.

35 is not in the sequence.

✔ Answer: No term is made with 35 sticks. The sequence increases by 6 each time, starting at 10: 10, 16, 22, 28, 34, 40... 35 is not in this list.

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4) Hexagons in a row

Each hexagon has 6 sides. When joined, they share one side.

- Term 1: 1 hexagon → 6 sticks
- Term 2: 2 hexagons → 6 + 5 = 11 sticks (share 1 side)
- Term 3: 3 hexagons → 11 + 5 = 16 sticks
- Term 4: 4 hexagons → 21 sticks

Pattern: Sticks = 5n + 1

Check: n=1 → 6 ✔️, n=2 → 11 ✔️

Question: Can a term be made using 51 sticks?

Set 5n + 1 = 51 → 5n = 50 → n = 10 → YES!

✔ Answer: Yes, the 10th term uses 51 sticks.

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5) “Wavy” shapes (like folded paper or zigzag)

Count sticks:

Term 1: 1 “unit” → looks like 6 sticks? Let’s count.

Actually, each “segment” seems to add 4 sticks.

Term 1: 6 sticks
Term 2: 10 sticks
Term 3: 14 sticks
Term 4: 18 sticks

Pattern: +4 each time → Sticks = 4n + 2

Check: n=1 → 6 ✔️, n=2 → 10 ✔️

For 10th term: 4×10 + 2 = 42 sticks

✔ Answer: 42 sticks

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6) Cross with expanding center

White tiles only (ignore gray center for now).

Term 1: 4 white tiles (arms of cross)
Term 2: 8 white tiles
Term 3: 12 white tiles
Term 4: 16 white tiles

Pattern: White tiles = 4n

Question: Can a term be made using 19 white tiles?

Set 4n = 19 → n = 19/4 = 4.75 → not an integer

So no term has exactly 19 white tiles.

✔ Answer: No, because the number of white tiles is always a multiple of 4 (4, 8, 12, 16, 20...), and 19 is not divisible by 4.

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7) Dots in L-shape

Term 1: 3 dots (2 vertical, 1 horizontal)
Term 2: 6 dots (3 vertical, 3 horizontal? Wait — image shows 2 vertical, 4 horizontal? Let’s count:

Actually:

Term 1: 3 dots
Term 2: 6 dots
Term 3: 9 dots
Term 4: 12 dots

Looks like 3n dots

Confirm:

Term 1: 3 × 1 = 3 ✔️
Term 2: 3 × 2 = 6 ✔️
Term 3: 9 ✔️
Term 4: 12 ✔️

So for 12th term: 3 × 12 = 36 dots

✔ Answer: 36 dots

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8) Staircase of squares

This pattern is different because:

- In all previous patterns, the number of shapes (squares, hexagons, etc.) increases linearly with the term number (n).
- In this pattern, the number of squares increases quadratically.

Let’s see:

Term 1: 1 square
Term 2: 3 squares (1+2)
Term 3: 6 squares (1+2+3)
Term 4: 10 squares (1+2+3+4)

This is the triangular number sequence: n(n+1)/2

So while other sequences grow linearly (e.g., 4n, 5n+1, 3n), this one grows quadratically.

Also, visually, it’s building diagonally, adding a new “row” each time, whereas others are extending in a line.

✔ Answer: This pattern is different because the number of squares follows a quadratic rule (n(n+1)/2), while the others follow linear rules. It grows faster and forms a triangular staircase shape instead of a straight line or simple repetition.

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## ✔ Final Answers Summary:

1) 19 sticks
2) 26 sticks
3) No term uses 35 sticks (sequence: 10,16,22,28,34,40...)
4) Yes, 10th term uses 51 sticks
5) 42 sticks
6) No, 19 is not a multiple of 4
7) 36 dots
8) It follows a quadratic growth (triangular numbers), unlike the linear growth of others.

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Let me know if you’d like diagrams or step-by-step drawings for any part!