Continuing Sequecnes from Patterns Worksheet | Fun and Engaging ... - Free Printable

Here are the solutions to the problems on the worksheet, worked out step by step.

1) Squares Sequence
* Pattern: Each new square shares one side with the previous one.
* Term 1: 4 sticks
* Term 2: 7 sticks (add 3)
* Term 3: 10 sticks (add 3)
* Rule: $3n + 1$ (where $n$ is the term number).
* Question: How many sticks for the 6th term?
* Calculation: $3 \times 6 + 1 = 18 + 1 = 19$.

2) Houses Sequence
* Pattern: Each new house adds a roof and walls but shares a wall.
* Term 1: 6 sticks
* Term 2: 11 sticks (add 5)
* Term 3: 16 sticks (add 5)
* Rule: $5n + 1$.
* Question: How many sticks for the 5th term?
* Calculation: $5 \times 5 + 1 = 25 + 1 = 26$.

3) L-Shape Squares Sequence
* Pattern: We add an "L" shape of squares each time. Let's count the sticks (lines).
* Term 1: 10 sticks
* Term 2: 17 sticks (add 7)
* Term 3: 24 sticks (add 7)
* Rule: $7n + 3$.
* Question: Which term is made with 35 sticks?
* Equation: $7n + 3 = 35$
* Subtract 3 from both sides: $7n = 32$
* Divide by 7: $n = 32 / 7 \approx 4.57$
* Since the term number must be a whole number, no term uses exactly 35 sticks. (Term 4 would use 31 sticks, and Term 5 would use 38 sticks).

4) Hexagons Sequence
* Pattern: Hexagons share one side.
* Term 1: 6 sticks
* Term 2: 11 sticks (add 5)
* Term 3: 16 sticks (add 5)
* Rule: $5n + 1$.
* Question: Can a term be made using 51 sticks?
* Equation: $5n + 1 = 51$
* Subtract 1: $5n = 50$
* Divide by 5: $n = 10$
* Yes, because 10 is a whole number. It is the 10th term.

5) Arrows/Hexagon halves Sequence
* Pattern: These shapes share a vertical side.
* Term 1: 6 sticks
* Term 2: 11 sticks (add 5)
* Term 3: 16 sticks (add 5)
* Rule: $5n + 1$.
* Question: How many sticks for the 10th term?
* Calculation: $5 \times 10 + 1 = 50 + 1 = 51$.

6) Crosses with Tiles Sequence
* Pattern: Look at the white tiles only.
* Term 1: 4 white tiles
* Term 2: 7 white tiles (add 3)
* Term 3: 10 white tiles (add 3)
* Rule: $3n + 1$.
* Question: Can a term be made using 19 white tiles?
* Equation: $3n + 1 = 19$
* Subtract 1: $3n = 18$
* Divide by 3: $n = 6$
* Yes, it is the 6th term.

7) Dots Sequence
* Pattern: Count the total dots.
* Term 1: 3 dots
* Term 2: 6 dots (add 3)
* Term 3: 9 dots (add 3)
* Rule: $3n$ (multiples of 3).
* Question: How many dots for the 12th term?
* Calculation: $3 \times 12 = 36$.

8) Rectangles Sequence
* Pattern: The rectangles connect corner-to-corner (diagonally).
* Term 1: 1 rectangle
* Term 2: 2 rectangles
* Term 3: 3 rectangles
* Rule: The number of rectangles equals the term number ($n$).
* Question: Explain how this pattern is different.
* In all the previous patterns (1–7), the shapes were joined by sharing a full side (an edge). In this pattern, the shapes only touch at a single corner point. This means no sides are shared or saved; every new shape requires all 4 of its own sticks.

──────────────────────────────────────

Final Answer:

1) 19 sticks
2) 26 sticks
3) None (No term uses exactly 35 sticks; Term 4 uses 31 and Term 5 uses 38).
4) Yes, it is the 10th term.
5) 51 sticks
6) Yes, it is the 6th term.
7) 36 dots
8) The shapes touch at corners (points) instead of sharing full sides.