017r. Number Patterns with Difference and Ratio

Learning Intentions

• To understand what a number pattern (or number sequence) is
• Describe a pattern where there is a common difference (added or subtracted)
• describe a pattern where there is a common ratio (multiplied or divided)
• Solve terms in a number pattern with a common difference or ratio

Pre-requisite Summary

• Understand that a number pattern is an ordered list of numbers that follows a rule
• Be able to compare consecutive terms to see how the pattern changes
• Recall addition and subtraction facts for identifying common differences
• Recall multiplication and division facts for identifying common ratios
• Understand that a common difference means the same number is added or subtracted each time
• Understand that a common ratio means the same number is multiplied or divided each time
• Be able to continue a simple pattern by applying the same rule repeatedly
• Be able to distinguish between additive change and multiplicative change

Worked Examples

Worked Example 1

a) Explain what a number pattern is.

b) Describe the rule for the pattern

c) Find the next two terms.

Worked Example 2

a) Describe the rule for the pattern

b) State the common difference.

c) Find the next two terms.

Worked Example 3

a) Describe the rule for the pattern

b) State the common ratio.

c) Find the next two terms.

Worked Example 4

a) Describe the rule for the pattern

b) State the common ratio.

c) Find the next two terms.

Worked Example 5

a) Find the th term of the pattern

b) Find the th term of the pattern

Worked Example 6

a) Decide whether has a common difference or a common ratio as its simplest description.

b) Decide whether has a common difference or a common ratio.

c) Justify each choice.

Problems

Problem 1

a) Explain what a number pattern is.

b) Describe the rule for the pattern

c) Find the next two terms.

Problem 2

a) Describe the rule for the pattern

b) State the common difference.

c) Find the next two terms.

Problem 3

a) Describe the rule for the pattern

b) State the common ratio.

c) Find the next two terms.

Problem 4

a) Describe the rule for the pattern

b) State the common ratio.

c) Find the next two terms.

Problem 5

a) Find the th term of the pattern

b) Find the th term of the pattern

Problem 6

a) Decide whether has a common difference or a common ratio as its simplest description.

b) Decide whether has a common difference or a common ratio.

c) Justify each choice.

Exercises

Understanding and Fluency

Exercise 1.

State the rule and find the next two terms:

a)

b)

c)

Exercise 2.

State the common difference and find the next two terms:

a)

b)

c)

Exercise 3.

State the rule and find the next two terms:

a)

b)

c)

Exercise 4.

State the common ratio and find the next two terms:

a)

b)

c)

Exercise 5.

Find the missing terms:

a)

b)

c)

Exercise 6.

Find the required term:

a) the th term of

b) the th term of

c) the th term of

Exercise 7.

Continue each pattern:

a)

b)

c)

Exercise 8.

Decide whether each pattern is additive or multiplicative:

a)

b)

c)

Reasoning

Exercise 9.

Explain why is best described Use a common difference rather than a common ratio.

Exercise 10.

A student says the pattern has a common ratio of . Explain the error.

Exercise 11.

Explain why has no common difference but does have a common ratio.

Exercise 12.

Compare the patterns and . Explain how their rules are different.

Problem-solving

Exercise 13.

A staircase has tiles in the first step, in the second, in the third, and in the fourth. If the pattern continues, how many tiles are in the th step?

Exercise 14.

A bacteria culture doubles each hour. If it starts with bacteria, how many are there after the first terms: ?

Exercise 15.

A machine prints pages in the first minute, then , then , and so on. How many pages will it print in the th minute?

Exercise 16.

A ball bounces to half its previous height each time. If the bounce heights are , what is the th bounce height?

Exercise 17.

A saving pattern is . How much will be saved in the th term?

Exercise 18.

A game score triples each round: . What is the score in round ?

Potential Misunderstandings

• Students may think any sequence with changing numbers has both a common difference and a common ratio
• Students may confuse a common difference with a common ratio
• Students may subtract consecutive terms incorrectly when the pattern is decreasing
• Students may divide consecutive terms in the wrong order when identifying a common ratio
• Students may assume repeated addition and repeated multiplication are the same kind of pattern
• Students may continue an additive pattern using multiplication, or a multiplicative pattern using addition
• Students may not Recognise that a decreasing multiplicative pattern can involve division
• Students may find one correct next term but then fail to continue the same rule consistently
• Students may think the largest change means the rule, rather than checking every pair of consecutive terms

Next: 018. Spatial Patterns and Rules