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 uses matchsticks.

Shape uses matchsticks.

Shape uses matchsticks.

a) Describe the repeating change from one shape to the next.

b) State the related number pattern.

c) Draw or describe Shape .

Worked Example 2

A pattern of triangles is growing.

Shape uses counters.

Shape uses counters.

Shape uses counters.

a) Describe how the pattern grows.

b) State the number pattern.

c) Solve the number of counters in Shape .

Worked Example 3

A pattern begins with one square tile. Each new shape adds more tiles than the previous shape.

Shape : tile

Shape : tiles

Shape : tiles

a) Continue the pattern for Shapes and .

b) State the number pattern.

c) Describe the rule in words.

Worked Example 4

A row of joined pentagons is made with matchsticks.

Shape uses matchsticks.

Shape uses matchsticks.

Shape uses 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 .

Worked Example 5

A dot pattern has:

Shape : dots

Shape : dots

Shape : dots

Shape : dots

a) Describe the number pattern.

b) Explain how this relates to the spatial pattern.

c) Find the number of dots in Shape .

Worked Example 6

A growing pattern of connected hexagons has:

Shape : matchsticks

Shape : matchsticks

Shape : 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 .

Problems

Problem 1

A spatial pattern is made from squares in a row.

Shape uses matchsticks.

Shape uses matchsticks.

Shape uses matchsticks.

a) Describe the repeating change.

b) State the related number pattern.

c) Draw or describe Shape .

Problem 2

A pattern of triangles is growing.

Shape uses counters.

Shape uses counters.

Shape uses counters.

a) Describe how the pattern grows.

b) State the number pattern.

c) Find the number of counters in Shape .

Problem 3

A pattern begins with one square tile. Each new shape adds more tiles than the previous shape.

Shape : tiles

Shape : tiles

Shape : tiles

a) Continue the pattern for Shapes and .

b) State the number pattern.

c) Describe the rule in words.

Problem 4

A row of joined pentagons is made with matchsticks.

Shape uses matchsticks.

Shape uses matchsticks.

Shape uses 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 .

Problem 5

A dot pattern has:

Shape : dots

Shape : dots

Shape : dots

Shape : dots

a) Describe the number pattern.

b) Explain how this relates to the spatial pattern.

c) Find the number of dots in Shape .

Problem 6

A growing pattern of connected hexagons has:

Shape : matchsticks

Shape : matchsticks

Shape : matchsticks

a) State the number pattern.

b) Describe the growth of the spatial pattern.

c) Find the number of matchsticks in Shape .

Exercises

Understanding and Fluency

Exercise 1.

A matchstick pattern has:

Shape sticks

Shape sticks

Shape sticks

a) Write the next two terms in the number pattern.

b) State how many sticks are added each time.

Exercise 2.

A tile pattern has:

Shape tiles

Shape tiles

Shape tiles

a) Write the next three terms.

b) Describe the rule in words.

Exercise 3.

A dot pattern has:

Shape dots

Shape dots

Shape dots

a) Find the common difference.

b) Find the number of dots in Shape .

Exercise 4.

A pattern of joined squares uses matchsticks:

Shape

Shape

Shape

a) Find the number of matchsticks in Shape .

b) Find the number of matchsticks in Shape .

Exercise 5.

A pattern grows by adding counters each time.

Shape counters

a) Write the first five terms.

b) Find the number of counters in Shape .

Exercise 6.

A spatial pattern has the number pattern

a) State the common difference.

b) Find the th term.

c) Find the th term.

Exercise 7.

A pattern of triangles uses counters:

Shape

Shape

Shape

a) Find the number of counters in Shape .

b) Find the number of counters in Shape .

Exercise 8.

A row of connected pentagons uses matchsticks:

Shape

Shape

Shape

a) Find the increase each time.

b) Find the number of matchsticks in Shape .

Exercise 9.

A dot pattern has:

Shape

Shape

Shape

a) Continue the pattern to Shape .

b) Describe the rule.

Reasoning

Exercise 10.

Explain how the spatial pattern of joined squares is related to the number pattern

Exercise 11.

A student says the pattern grows by . Explain the mistake.

Exercise 12.

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

Exercise 13.

A student says Shape in the pattern has counters. Explain why this is incorrect.

Problem-solving

Exercise 14.

A builder uses tiles to make a pattern. Shape uses tiles, Shape uses tiles, and Shape uses tiles. How many tiles are needed for Shape ?

Exercise 15.

A matchstick pattern starts at and increases by each time. How many matchsticks are needed for Shape ?

Exercise 16.

A pattern of dots has Shape , Shape , Shape . How many dots are in Shape ?

Exercise 17.

A pattern of connected hexagons uses matchsticks for Shape , for Shape , and for Shape . How many matchsticks are needed for Shape ?

Exercise 18.

A student draws a growing pattern where each new shape adds tiles. If Shape has tiles, how many tiles will Shape 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