1. A10 Generating sequences
Boardworks KS3 Maths 2009
A10 Generating sequences
A10 Generating sequences
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For more detailed instructions, see the Getting Started presentation.

2. A10.1 Generating sequences
Boardworks KS3 Maths 2009
A10 Generating sequences
A10.1 Generating sequences

3. Boardworks KS3 Maths 2009 A10 Generating sequences
Sequence grid
Start the activity by asking pupils to work out the number that is hidden by the orange counter.
Initially, do not tell pupils how the numbers in the grid have been generated.
Ask pupils if they can see any patterns.
The numbers in the rows, columns and diagonals form arithmetic sequences. Pupils will need to determine the rule for two of these sequences by subtracting numbers in adjacent squares (either horizontally, vertically or diagonally) and counting on.
You may click on any counter at any time to reveal the number beneath it.
As an extension or homework activity ask pupils to fill in their own sequence grids. They can generate the table by choosing one rule for the rows and another for the columns. They can then investigate how the sequences in the diagonals are related to the sequences in the rows and columns.

4. Generating sequences from flow charts
Boardworks KS3 Maths 2009
A10 Generating sequences
Generating sequences from flow charts
A sequence can be given by a flow chart. For example:
START
Write down 3.
This flow chart
generates
the sequence:
3, 4.5, 6, 6.5, 9, 10.5.
Add on 1.5.
Write down the answer.
This sequence has only six terms.
Click to reveal each section of the flow chart.
Follow the instructions with the class until you get to 10.5 and then click to reveal the yes arrow and the instruction to stop.
Is the answer
more than 10? No
It is finite.
Yes
STOP

5. Generating sequences from flow charts
Boardworks KS3 Maths 2009
A10 Generating sequences
Generating sequences from flow charts
START
Write down 5.
This flow chart generates the sequence
5, 2.9, 0.8, –1.3, –3.4.
Subtract 2.1.
Write down the answer.
Is the answer
less than –3? No
Go through this flow chart in the same way.
The sequence generated is 5, 2.9, 0.8, –1.3, –3.4.
Yes
STOP

6. Generating sequences from flow charts
Boardworks KS3 Maths 2009
A10 Generating sequences
Generating sequences from flow charts
START
Write down 200.
This flow chart generates the sequence
200, 100, 50, 25, 12.5, 6.25.
Divide by 2.
Write down the answer.
Is the answer
less than 7? No
Go through this flow chart in the same way.
The sequence generated is 200, 100, 50, 25, 12.5, 6.25.
Yes
STOP

7. Generating sequences from flow charts
Boardworks KS3 Maths 2009
A10 Generating sequences
Generating sequences from flow charts
START
Write down 3 and 4.
This flow chart generates the sequence
3, 4, 7, 11, 18, 29, 47, 76, 123.
Add together the two
previous numbers.
Write down the answer.
Is the answer
more than 100?
Go through this flow chart in the same way.
The sequence generated is 3, 4, 7, 11, 18, 29, 47, 76, 123. No
Yes
STOP

8. Predicting terms in a sequence
Boardworks KS3 Maths 2009
A10 Generating sequences
Predicting terms in a sequence
Usually, we can predict how a sequence will continue by looking for patterns.
For example:
87, 84, 81, 78, ...
We can predict that this sequence continues by subtracting 3 each time.
However, sequences do not always continue as we would expect.
Discuss ways in which the sequence 1, 2, 4 might continue. For example, by doubling the previous term or by adding 1, adding 2, adding 3 etc..
For example:
A sequence starts with the numbers 1, 2, 4, ...
How could this sequence continue?

9. Boardworks KS3 Maths 2009 A10 Generating sequences
Continuing sequences
Here are some different ways in which the sequence might continue: 1 2 4 7 11 16 22 +1 +2 +3 +4 +5 +6 1 2 4 8 16 32 64 ×2 ×2 ×2 ×2 ×2 ×2
We can never be certain how a sequence will continue unless we are given a rule or we can justify a rule from a practical context.

10. Boardworks KS3 Maths 2009 A10 Generating sequences
Continuing sequences
This sequence continues by adding 3 each time. 1 4 7 10 13 16 19 +3 +3 +3 +3 +3 +3
We can say that rule for getting from one term to the next term is add 3.
This is called the term-to-term rule.
The term-to-term rule for this sequence is +3.

11. Using a term-to-term rule
Boardworks KS3 Maths 2009
A10 Generating sequences
Using a term-to-term rule
Does the rule +3 always produce the same sequence?
No, it depends on the starting number.
If we start with 2 and add on 3 each time we have, 2, 5, 8,
11,
14,
17,
20,
23, ...
If we start with 0.4 and add on 3 each time we have,
There are infinitely many sequences that follow the rule ‘add three’.
Encourage pupils to think of different possible starting numbers. The starting number could be a fraction, a negative number or a very large number.
Is it possible to find a sequence for which:
a) all the numbers are multiples of 3?
b) all the numbers are odd?
c) all the numbers are multiples of 9?
d) none of the numbers is a whole number?
0.4,
3.4,
6.4,
9.4,
12.4,
15.4,
18.4,
21.4, ...

12. Writing sequences from term-to-term-rules
Boardworks KS3 Maths 2009
A10 Generating sequences
Writing sequences from term-to-term-rules
A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms.
To generate a sequence from a term-to-term rule we must also be given the first number in the sequence.
For example:
1st term
Term-to-term rule 5
Add consecutive even numbers starting with 2.
This gives us the sequence, 5 7 11 17 25 35 47
... +2 +4 +6 +8
+10
+12

13. Sequences from a term-to-term rule
Boardworks KS3 Maths 2009
A10 Generating sequences
Sequences from a term-to-term rule
Write the first five terms of each sequence given the first term and the term-to-term rule.
1st term
Term-to-term rule 10
Add 3
10,
13,
16,
19, 21
100
Subtract 5
100,
95,
90,
85, 80 3
Double 3, 6,
12,
24, 48
Edit the numbers in this slide to produce more or less challenging examples. 5
Multiply by 10 5,
50,
500,
5000,
50000 7
Subtract 2 7, 5, 3, 1, –1
0.8
Add 0.1
0.8,
0.9,
1.0,
1.1,
1.2

14. Sequences from position-to-term rules
Boardworks KS3 Maths 2009
A10 Generating sequences
Sequences from position-to-term rules
Sometimes sequences are arranged in a table like this:
Position
1st
2nd
3rd
4th
5th
6th …
nth
Term 3 6 9 12 15 18 … 3n
We can say that each term can be found by multiplying the position of the term by 3.
This is called a position-to-term rule.
Point out that once we know a position-to-term rule we can find any term in the sequence given its position in the sequence.
Ask pupils to give other terms in the sequence with the position-to-term rule 3n.
For example, what is the 15th term in the sequence?
You could also ask pupils to give you the position of a given term in the sequence using inverse operations. For example,
42 is a term is this sequence. What position is it in?
For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence.
What is the 100th term in this sequence?
3 × 100 = 300

15. Sequences from position-to-term rules
Boardworks KS3 Maths 2009
A10 Generating sequences
Sequences from position-to-term rules
Start by revealing the value of the nth term and ask pupils to find the value of each term using substitution. You may choose from linear and quadratic sequences.
Alternatively, give pupils the six terms and ask then to find an expression for the nth term.

16. Writing sequences from position-to-term rules
Boardworks KS3 Maths 2009
A10 Generating sequences
Writing sequences from position-to-term rules
The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms.
We can use algebraic shorthand to do this.
We call the first term T(1), for Term number 1,
we call the second term T(2),
we call the third term T(3), ...
Explain the algebraic notation. Reassure pupils that using letters is a good way to save us lots of writing!
The T stands for term and the number in the bracket is the position of the term in the sequence. n can be any whole number.
Ask:
What do we call the 10th term of a sequence? (T(10))
What do we call the 450th term? And so on.
Explain again that we find the value of a term by substituting its position number into the rule for the nth term.
and we call the nth term T(n).
T(n) is called the the nth term or the general term.

17. Writing sequences from position-to-term rules
Boardworks KS3 Maths 2009
A10 Generating sequences
Writing sequences from position-to-term rules
For example, suppose the nth term of a sequence is 4n + 1.
We can write this rule as:
T(n) = 4n + 1
Find the first 5 terms.
T(1) =
4 × = 5
T(2) =
4 × = 9
T(3) =
4 × = 13
Photo credit: © Shutterstock 2009, Dmitrijs Dmitrijevs
Repeat that T(1) is a short way of writing the first term. To find the value of T(1) substitute 1 into the rule 4n + 1. To find the value of T(2) substitute 2 into the rule 4n + 1 etc.
What do you notice about this sequence? (It goes up 4 each time. An even better answer is: it is the numbers from the 4 times table with 1 added on each time.)
Ask pupils to use the rule to work out the value of the 10th term, the 50th term, the 243rd term, etc.
T(4) =
4 × = 17
T(5) =
4 × = 21
The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.

18. Writing sequences from position-to-term rules
Boardworks KS3 Maths 2009
A10 Generating sequences
Writing sequences from position-to-term rules
If the nth term of a sequence is 2n2 + 3.
We can write this rule as:
T(n) = 2n2 + 3
Find the first 4 terms.
T(1) =
2 × = 5
T(2) =
2 × = 11
T(3) =
2 × = 21
Photo credit: © Shutterstock 2009, Rui Vale de Sousa
Emphasize that in a linear sequence the n can only be raised to the power of 1 (though this is not usually written).
In a quadratic sequence n can still be raised to the power of 1 but there also needs to be a term with n raised to the power of 2.
T(4) =
2 × = 35
The first 4 terms in the sequence are: 5, 11, 21, and 35.
This sequence is a quadratic sequence.

19. Sequence generator – linear sequences
Boardworks KS3 Maths 2009
A10 Generating sequences
Sequence generator – linear sequences
Use this activity to review finding position-to-term rules and term-to-term rules for simple arithmetic sequences.
Click on the position-to-term rule to edit it.

20. Sequence generator – non-linear sequences
Boardworks KS3 Maths 2009
A10 Generating sequences
Sequence generator – non-linear sequences
Use this activity to generate non-linear sequences.
Click on the position-to-term rule to edit it.

21. Boardworks KS3 Maths 2009 A10 Generating sequences
Sequences and rules
Which rule is best?
The term-to-term rule?
The position-to-term rule?
Photo credit: © Shutterstock 2009, Kiselev Andrey Valerevich
Discuss the difference between the term-to-term rule and the position-to-term rule for a sequence.
Some may say that the term-to-term rule is better because it doesn’t use algebra.
Conclude that the position-to-term rule is usually more useful because usually allows you to find the value of any term in the sequence given its position. For example, if you want to know the 100th term in a sequence you don’t have to calculate the first 99 terms to find it.
The position-to-term rule is usually written algebraically. We call it the rule for the ‘nth term’.
Explain that the ‘nth term’ of a sequence can also be called the ‘general term’ of the sequence. The general term gives us a rule which is true for every number in the sequence using algebra. For example, the general term of the sequence 5, 10, 15, 20, 25, ... is 5n.
Point out that for some sequences that increase in unequal steps, the position-to-term rule can be very complicated to find and use, for example, in The Fibonacci Sequence where each term is the sum of the previous two terms. For such sequences the term-to-term rule is best.