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)
• find 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 $4,7,10,13,\dots$
c) Find the next two terms.

Worked Example 2

a) Describe the rule for the pattern $36,31,26,21,\dots$
b) State the common difference.
c) Find the next two terms.

Worked Example 3

a) Describe the rule for the pattern $3,6,12,24,\dots$
b) State the common ratio.
c) Find the next two terms.

Worked Example 4

a) Describe the rule for the pattern $160,80,40,20,\dots$
b) State the common ratio.
c) Find the next two terms.

Worked Example 5

a) Find the $6$th term of the pattern $5,9,13,17,\dots$
b) Find the $8$th term of the pattern $2,6,18,54,\dots$

Worked Example 6

a) Decide whether $7,14,21,28,\dots$ has a common difference or a common ratio as its simplest description.
b) Decide whether $81,27,9,3,\dots$ 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 $6,11,16,21,\dots$
c) Find the next two terms.

Problem 2

a) Describe the rule for the pattern $50,43,36,29,\dots$
b) State the common difference.
c) Find the next two terms.

Problem 3

a) Describe the rule for the pattern $4,12,36,108,\dots$
b) State the common ratio.
c) Find the next two terms.

Problem 4

a) Describe the rule for the pattern $243,81,27,9,\dots$
b) State the common ratio.
c) Find the next two terms.

Problem 5

a) Find the $6$th term of the pattern $8,12,16,20,\dots$
b) Find the $7$th term of the pattern $5,10,20,40,\dots$

Problem 6

a) Decide whether $9,18,27,36,\dots$ has a common difference or a common ratio as its simplest description.
b) Decide whether $64,32,16,8,\dots$ has a common difference or a common ratio.
c) Justify each choice.

Exercises

Understanding and Fluency

1. State the rule and find the next two terms:
   a) $2,5,8,11,\dots$
   b) $15,20,25,30,\dots$
   c) $41,37,33,29,\dots$

2. State the common difference and find the next two terms:
   a) $9,14,19,24,\dots$
   b) $100,90,80,70,\dots$
   c) $3,1,-1,-3,\dots$

3. State the rule and find the next two terms:
   a) $2,4,8,16,\dots$
   b) $5,15,45,135,\dots$
   c) $96,48,24,12,\dots$

4. State the common ratio and find the next two terms:
   a) $7,21,63,189,\dots$
   b) $128,64,32,16,\dots$
   c) $6,12,24,48,\dots$

5. Find the missing terms:
   a) $4,9,\mathrm{\_},19,\mathrm{\_},29$
   b) $81,27,\mathrm{\_},3,\mathrm{\_}$
   c) $12,18,\mathrm{\_},30,\mathrm{\_}$

6. Find the required term:
   a) the $7$th term of $3,8,13,18,\dots$
   b) the $6$th term of $2,10,50,250,\dots$
   c) the $8$th term of $40,35,30,25,\dots$

7. Continue each pattern:
   a) $11,22,44,\dots$
   b) $30,27,24,\dots$
   c) $1,4,7,\dots$

8. Decide whether each pattern is additive or multiplicative:
   a) $14,18,22,26,\dots$
   b) $3,9,27,81,\dots$
   c) $200,100,50,25,\dots$

Reasoning

9. Explain why $5,10,15,20,\dots$ is best described using a common difference rather than a common ratio.

10. A student says the pattern $2,6,10,14,\dots$ has a common ratio of $4$. Explain the error.

11. Explain why $64,32,16,8,\dots$ has no common difference but does have a common ratio.

12. Compare the patterns $3,6,9,12,\dots$ and $3,6,12,24,\dots$. Explain how their rules are different.

Problem-solving

13. A staircase has $4$ tiles in the first step, $7$ in the second, $10$ in the third, and $13$ in the fourth. If the pattern continues, how many tiles are in the $8$th step?

14. A bacteria culture doubles each hour. If it starts with $5$ bacteria, how many are there after the first $6$ terms: $5,10,20,\dots$?

15. A machine prints $120$ pages in the first minute, then $110$, then $100$, and so on. How many pages will it print in the $7$th minute?

16. A ball bounces to half its previous height each time. If the bounce heights are $96,48,24,12,\dots$, what is the $6$th bounce height?

17. A saving pattern is $20,25,30,35,\dots$. How much will be saved in the $10$th term?

18. A game score triples each round: $2,6,18,54,\dots$. What is the score in round $6$?

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