018. Spatial Patterns and Rules

Learning Intentions

• To know that a spatial pattern begins with a starting design and has a repeating design
• To understand how spatial patterns are related to number patterns
• continue a spatial pattern given the first few shapes
• describe and use a rule relating the number of shapes and the number of objects required to make them

Pre-requisite Summary

• Understand that a pattern follows a rule and repeats or grows in a predictable way
• Be able to identify the first term or starting design in a pattern
• Be able to count objects accurately in a diagram or shape
• Understand that number patterns can increase or decrease by a constant amount
• Recognise that each new shape in a spatial pattern may add the same number of objects
• Be able to compare one design with the next to describe what changes
• Understand that a rule can connect the shape number to the number of objects used
• Be familiar with simple growing patterns made from matchsticks, tiles or dots

Worked Examples

Worked Example 1

A spatial pattern is made from squares in a row.
Shape $1$ uses $4$ matchsticks.
Shape $2$ uses $7$ matchsticks.
Shape $3$ uses $10$ matchsticks.
a) Describe the repeating change from one shape to the next.
b) State the related number pattern.
c) Draw or describe Shape $4$.

Worked Example 2

A pattern of triangles is growing.
Shape $1$ uses $3$ counters.
Shape $2$ uses $6$ counters.
Shape $3$ uses $9$ counters.
a) Describe how the pattern grows.
b) State the number pattern.
c) Find the number of counters in Shape $5$.

Worked Example 3

A pattern begins with one square tile. Each new shape adds $2$ more tiles than the previous shape.
Shape $1$: $1$ tile
Shape $2$: $3$ tiles
Shape $3$: $5$ tiles
a) Continue the pattern for Shapes $4$ and $5$.
b) State the number pattern.
c) Describe the rule in words.

Worked Example 4

A row of joined pentagons is made with matchsticks.
Shape $1$ uses $5$ matchsticks.
Shape $2$ uses $9$ matchsticks.
Shape $3$ uses $13$ matchsticks.
a) Find the increase each time.
b) Write a rule relating shape number and number of matchsticks.
c) Find how many matchsticks are needed for Shape $6$.

Worked Example 5

A dot pattern has:
Shape $1$: $2$ dots
Shape $2$: $5$ dots
Shape $3$: $8$ dots
Shape $4$: $11$ dots
a) Describe the number pattern.
b) Explain how this relates to the spatial pattern.
c) Find the number of dots in Shape $8$.

Worked Example 6

A growing pattern of connected hexagons has:
Shape $1$: $6$ matchsticks
Shape $2$: $11$ matchsticks
Shape $3$: $16$ matchsticks
a) State the number pattern.
b) Describe the growth of the spatial pattern.
c) Find a rule for the number of matchsticks in Shape $n$.

Problems

Problem 1

A spatial pattern is made from squares in a row.
Shape $1$ uses $4$ matchsticks.
Shape $2$ uses $7$ matchsticks.
Shape $3$ uses $10$ matchsticks.
a) Describe the repeating change.
b) State the related number pattern.
c) Draw or describe Shape $4$.

Problem 2

A pattern of triangles is growing.
Shape $1$ uses $4$ counters.
Shape $2$ uses $8$ counters.
Shape $3$ uses $12$ counters.
a) Describe how the pattern grows.
b) State the number pattern.
c) Find the number of counters in Shape $5$.

Problem 3

A pattern begins with one square tile. Each new shape adds $3$ more tiles than the previous shape.
Shape $1$: $2$ tiles
Shape $2$: $5$ tiles
Shape $3$: $8$ tiles
a) Continue the pattern for Shapes $4$ and $5$.
b) State the number pattern.
c) Describe the rule in words.

Problem 4

A row of joined pentagons is made with matchsticks.
Shape $1$ uses $5$ matchsticks.
Shape $2$ uses $9$ matchsticks.
Shape $3$ uses $13$ matchsticks.
a) Find the increase each time.
b) Write a rule relating shape number and number of matchsticks.
c) Find how many matchsticks are needed for Shape $7$.

Problem 5

A dot pattern has:
Shape $1$: $3$ dots
Shape $2$: $7$ dots
Shape $3$: $11$ dots
Shape $4$: $15$ dots
a) Describe the number pattern.
b) Explain how this relates to the spatial pattern.
c) Find the number of dots in Shape $6$.

Problem 6

A growing pattern of connected hexagons has:
Shape $1$: $6$ matchsticks
Shape $2$: $11$ matchsticks
Shape $3$: $16$ matchsticks
a) State the number pattern.
b) Describe the growth of the spatial pattern.
c) Find the number of matchsticks in Shape $5$.

Exercises

Understanding and Fluency

1. A matchstick pattern has:
   Shape $1:3$ sticks
   Shape $2:5$ sticks
   Shape $3:7$ sticks
   a) Write the next two terms in the number pattern.
   b) State how many sticks are added each time.

2. A tile pattern has:
   Shape $1:2$ tiles
   Shape $2:4$ tiles
   Shape $3:6$ tiles
   a) Write the next three terms.
   b) Describe the rule in words.

3. A dot pattern has:
   Shape $1:4$ dots
   Shape $2:7$ dots
   Shape $3:10$ dots
   a) Find the common difference.
   b) Find the number of dots in Shape $6$.

4. A pattern of joined squares uses matchsticks:
   Shape $1:4$
   Shape $2:7$
   Shape $3:10$
   a) Find the number of matchsticks in Shape $4$.
   b) Find the number of matchsticks in Shape $8$.

5. A pattern grows by adding $5$ counters each time.
   Shape $1:6$ counters
   a) Write the first five terms.
   b) Find the number of counters in Shape $7$.

6. A spatial pattern has the number pattern $1,4,7,10,\dots$
   a) State the common difference.
   b) Find the $6$th term.
   c) Find the $10$th term.

7. A pattern of triangles uses counters:
   Shape $1:3$
   Shape $2:6$
   Shape $3:9$
   a) Find the number of counters in Shape $4$.
   b) Find the number of counters in Shape $8$.

8. A row of connected pentagons uses matchsticks:
   Shape $1:5$
   Shape $2:9$
   Shape $3:13$
   a) Find the increase each time.
   b) Find the number of matchsticks in Shape $5$.

9. A dot pattern has:
   Shape $1:2$
   Shape $2:5$
   Shape $3:8$
   a) Continue the pattern to Shape $6$.
   b) Describe the rule.

Reasoning

10. Explain how the spatial pattern of joined squares is related to the number pattern $4,7,10,13,\dots$

11. A student says the pattern $5,8,11,14,\dots$ grows by $2$. Explain the mistake.

12. Why does a row of joined squares not increase by $4$ matchsticks each time, even though one square has $4$ sides?

13. A student says Shape $5$ in the pattern $3,6,9,12,\dots$ has $14$ counters. Explain why this is incorrect.

Problem-solving

14. A builder uses tiles to make a pattern. Shape $1$ uses $6$ tiles, Shape $2$ uses $10$ tiles, and Shape $3$ uses $14$ tiles. How many tiles are needed for Shape $8$?

15. A matchstick pattern starts at $4$ and increases by $3$ each time. How many matchsticks are needed for Shape $12$?

16. A pattern of dots has Shape $1=7$, Shape $2=10$, Shape $3=13$. How many dots are in Shape $20$?

17. A pattern of connected hexagons uses $6$ matchsticks for Shape $1$, $11$ for Shape $2$, and $16$ for Shape $3$. How many matchsticks are needed for Shape $10$?

18. A student draws a growing pattern where each new shape adds $2$ tiles. If Shape $1$ has $5$ tiles, how many tiles will Shape $15$ have?

Potential Misunderstandings

• Students may think any picture pattern is repeating, when some spatial patterns are growing patterns
• Students may confuse the starting design with the repeating or growing part of the pattern
• Students may count all objects from scratch each time rather than noticing the constant increase
• Students may not connect the visual growth in the pattern to the related number pattern
• Students may describe the total number of objects instead of the change from one shape to the next
• Students may think a joined-shape pattern adds the full number of sides each time, ignoring shared sides
• Students may identify the wrong common difference in the number pattern
• Students may apply an incorrect rule when finding a later term in the pattern
• Students may confuse the shape number with the number of objects in the shape

Next: 019. Rules and Input-Output Tables