New
New
Year 9
Problem solving with non-linear relationships
I can use my knowledge of non-linear relationships to solve problems.
New
New
Year 9
Problem solving with non-linear relationships
I can use my knowledge of non-linear relationships to solve problems.
Lesson details
Key learning points
- If you can spot a sequence, it is possible to predict behaviour.
- By predicting future behaviour, you can plan how to react.
- As with all predictions, it is not a guarantee.
Common misconception
If you combine two sequences of the same type the resulting sequence will still be that type.
Adding corresponding terms of two sequences will be good preparation for future units but it also a way to explore what sequences can be generated by combining other sequences.
Keywords
Geometric sequence - A geometric sequence is a sequence with a constant multiplicative relationship between successive terms.
Triangular number - A triangular number (or triangle number) is a number that can be represented by a pattern of dots arranged into an equilateral triangle.
Exploring the Collatz conjecture is great for getting students interested in sequences and in conjectures in maths. The slides for the final learning cycle could be used as prompts for the teacher and the students can explore how long it takes to get to 1 from different starting numbers.
Teacher tip
Licence
This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).
Video
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Starter quiz
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6 Questions
Q1.
Which of these are triangular numbers?
0
Correct answer: 1
1
5
Correct answer: 10
10
20
Q2.
Use a calculator to work out the value of $$3^{8}$$. .
Correct Answer: 6561, 6 561, 6,561
6561, 6 561, 6,561
Q3.
The next term in the geometric sequence 0.4, 2, 10, ... is .
Correct Answer: 50, fifty
50, fifty
Q4.
Which of these could be the first 4 terms in an arithmetic sequence?
28, 22, 16, 11, ...
136, 163, 190, 209, ...
Correct answer: 13, 49, 85, 121, ...
13, 49, 85, 121, ...
Correct answer: 17, 9, 1, -7, ...
17, 9, 1, -7, ...
-21, -8, 8, 21, ...
Q5.
What is the $$n^{\text{th}}$$ term rule for the linear sequence which starts 13, 8, 3, -2, ...?
Correct answer: $$18 - 5n$$
$$18 - 5n$$
$$13 - 5n$$
$$8 - 5n$$
$$5n + 8$$
$$5n + 13$$
Q6.
Which of these could be the first 5 terms of a sequence with a common second difference? (These are called quadratic sequences).
Correct answer: 2, 3, 5, 8, 12, ...
2, 3, 5, 8, 12, ...
8, 12, 18, 28, 40, ...
9, 10, 12, 16, 24, ...
Correct answer: 12, 17, 27, 42, 62, ...
12, 17, 27, 42, 62, ...
Correct answer: 15, 21, 28, 36, 45, ...
15, 21, 28, 36, 45, ...
Exit quiz
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6 Questions
Q1.
This table shows the value of a car two years after it was made. If the values form a geometric sequence, what is the common multiplier between terms? (You may use your calculator.)
0.2
0.4
0.6
Correct answer: 0.8
0.8
1.25
Q2.
This table shows the value of a car 2 years after it was made. If the values continue to follow the same geometric sequence, what is the car's value after 3 years? (You may use your calculator.) £
Correct Answer: 10 240, £10 240, £10240, 10240, £10,240
10 240, £10 240, £10240, 10240, £10,240
Q3.
This table shows the value of a car two years after it was made. Why might this sequence not continue as the same geometric sequence forever?
If the geometric sequence continues the value will drop below zero pounds.
Sequences cannot go on forever, they will eventually stop.
Correct answer: The value of the car may stop decreasing, it could reach minimum value.
The value of the car may stop decreasing, it could reach minimum value.
Correct answer: The value may decrease sharply in the first 2 years, then the pattern may change
The value may decrease sharply in the first 2 years, then the pattern may change
The car loses the most value in the first year, not the same amount each year.
Q4.
Which two sequences can be added to make the first 5 terms of a linear sequence?
Correct answer: 3, 7, 11, 15, 19, ...
3, 7, 11, 15, 19, ...
5, 10, 20, 40, 80, ...
6, 7, 9, 12, 16, ...
Correct answer: 14, 11, 8, 5, 2, ..
14, 11, 8, 5, 2, ..
16, 24, 36, 54, 81, ...
Q5.
Order these terms so they form part of a Collatz sequence: "If the term is even, the next term is half the previous term. If the term is odd, the next term is 3 times the previous term add 1".
1 - 33
1
- 33
2 - 100
2
- 100
3 - 50
3
- 50
4 - 25
4
- 25
5 - 76
5
- 76
6 - 38
6
- 38
Q6.
Which two sequences can be added to make the first 5 terms of a linear sequence?
Correct answer: 1, 3, 8, 16, 27, ...
1, 3, 8, 16, 27, ...
8, 11, 13, 14, 14, ...
Correct answer: 10, 19, 25, 28, 28, ...
10, 19, 25, 28, 28, ...
15, 16, 19, 24, 31, ...
20, 17, 14, 11, 8, ...