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Key words
Sequences
Find terms of a sequence and say whether it is ascending or descending,
finite or infinite
Find the next term in a sequence of numbers or shapes
sequence
term
consecutive
infinite
finite
A number sequence is a set of numbers in a given order, such as 1, 4, 7, 10, …
Each number in a sequence is called a term .
Terms next to each other are called consecutive terms.
Sequences may be ascending (e.g. 2, 4, 6, 8, …) or descending (e.g. 18, 15, 12, 9, …).
The sequence 1, 2, 3, 4, 5 ... is infinite . We could go on counting forever.
The sequence 10, 12, 14 ... 98 is finite . The dots mean that there are missing terms and
that the sequence continues in the same way until the final value, 98, is reached.
Example 1
Here is a sequence of diagrams.
a) Spot the pattern and draw the next two terms in
the sequence.
b) Is the sequence finite or infinite?
a) Each time two more dots are added.
This is the
fifth term.
This is the
sixth term.
b) The sequence is infinite.
Example 2
The first term of a sequence is 12; each term in the sequence is found by
subtracting 4 from the previous term.
a) Write down the first five terms of the sequence.
b) Is this sequence ascending or descending?
First term 12
Second term 12 4 8
Third term 8 4 4
Fourth term 4 4 0
Fifth term 0 4 4
a) 12, 8, 4, 0, 4
b) The sequence is descending.
Exercise 1.1 ..........................................................................................
Spot the pattern and draw the next two terms in each sequence.
1st term
a)
b)
c)
d)
2
Maths Connect 1R
2nd term
3rd term
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Decide whether each of the sequences in Q1 are finite or infinite.
Find the next two terms of each of the following sequences and decide whether they are
ascending or descending:
a) 6, 9, 12, 15, …
c) 5, 50, 500, 5000, …
b) 0.5, 1, 1.5, 2, 2.5, 3, …
d) 128, 64, 32, 16, …
e) 11, 9, 7 …
Write down the first five terms of each of the following sequences:
a)
b)
c)
d)
The first term is 12 and each term is found by adding 4 to the previous term.
The first term is 81 and each term is found by dividing the last term by 3.
The second term is 12 and each term is found by subtracting 1 from the previous term.
The fifth term is 17 and each term is found by adding 4 to the previous term.
Copy and complete these sequences of numbers and describe the pattern you are using
to move from term-to-term:
a) 2, …, 10, …, 18, …, 26
b) 15, …, …, 6, …, …, 3
c) 1, …, 100, …, 10000
1
1
1
1
d) 2, …, 8, …, 32, …, 128
Look at the pattern for moving from the
1st to the 3rd term, the 3rd to the 5th and
so on. How can you adapt this pattern for
moving from term-to-term?
To make one beach-hut out of matchsticks we used six
matchsticks.
To make two beach-huts we used an extra five (eleven
matchsticks in total).
a) Copy and complete the table below:
b)
c)
d)
e)
Number of beach-huts
1 2 3 4
Number of matchsticks
6 11
5 6
7
Predict how many matchsticks you would need for ten beach-huts.
How many lots of five matchsticks did you need for ten beach-huts?
How many matchsticks would you need for 70 beach-huts?
Describe how you calculated the number of matchsticks needed for
70 beach-huts.
How many lots
of five
matchsticks do
you need?
The first few terms of a sequence are: 3, 6, 12, 24.
How many terms are less than 200?
Each term of a sequence is found by adding 5 to the previous term.
a) i) If the first term is 5 what is the sixth term?
b) i) If the first term is 6 what is the seventh term?
ii) How many lots of 5 is this?
ii) How many lots of 5 is this?
Investigation
The first term of a sequence is 100. To find the next term you divide by 2.
a) Write down the first few terms of the sequence.
b) Replace ‘divide by 2’ with ‘add 2’, ‘subtract 2’ or ‘multiply by 2’ and find the first
few terms of each sequence.
c) When the pattern was:
i) divide by 2
ii) multiply by 2
iii) add 2
iv) subtract 2
was the sequence ascending or descending?
d) Will this also be true if you replace 100 and 2 with other numbers?
Sequences
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Key words
Generating sequences
generate
term
term-to-term rule
Generate a sequence given a starting point and a rule to go from
term-to-term
Use the rule to find a term in a sequence without finding all the values in between.
Generating a sequence means writing down the terms of the sequence. To do this
you need to know the pattern that the sequence follows.
To generate a sequence you may be given a starting point and a rule that connects one
term to the next. This is called the term-to-term rule .
For example, for this sequence:
5, 11, 17, 23, 29, 35, …
the first term of the sequence is 5 and the term-to-term rule is ‘add 6’.
Since we add 6 each time, we can compare the sequence to the multiples of 6.
ⴙ6 ⴙ6 ⴙ6 ⴙ6 ⴙ6
Sequence:
5, 11, 17, 23, 29, 35, …
Multiples of 6:
6, 12, 18, 24, 30, 36, …
We can see that this sequence is the multiples of 6 minus 1.
We can write this sequence like this:
1st term
2nd term
3rd term
4th term
5th term
1615
2 6 1 11
3 6 1 17
4 6 1 23
5 6 1 29
By following the pattern it is easy to find any term in the sequence.
For example, the 65th term is 65 6 1 389
Example
A sequence starts with 2 and the term-to-term rule is ‘add 4’.
a) Find the first five terms of the sequence.
First Term 2
b) Find the tenth term of the sequence without
Second Term 2 4 6
Third Term 6 4 10
working out all the terms in between.
Fourth Term 10 4 14
Fifth Term 14 4 18
a) 2, 6, 10, 14, 18
b) Each term is 2 less than the multiples of 4.
So the tenth term will be 10 4 2 38
Since we add 4 each time we can compare
this sequence to the multiples of 4:
Sequence
2, 6, 10, 14, 18, ...
Multiples of 4 4, 8, 12, 16, 20, ...
Exercise 1.2 ..........................................................................................
Write down the first five terms of a sequence given the following first terms and rules.
a)
b)
c)
d)
e)
4
First term
7
3
8
100
74
Maths Connect 1R
Term-to-term rule
Add 4
Add 2
Add 3
Subtract 6
Subtract 12
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Look at Q1. Fill in the gaps in the following sentences:
The sequence in Q1a is the multiples of
The sequence in Q1b is the multiples of
The sequence in Q1c is the multiples of
add
add
add
.
.
.
Write down the ninth term of the sequences in Q1a, b and c.
The first term of a sequence is 30. The term-to-term rule is ‘subtract 2’.
a) Write down the first five terms.
b) How many more times will you have to subtract 2 to get from the fifth term to the
seventh term?
c) Write down the tenth term without continuing the sequence.
Look at the sequence in Q1d.
a) How many times will you have to subtract 6 to get from the fifth term to the seventh
term?
b) Write down the tenth term without continuing the sequence.
The second term of a sequence is 5. The term-to-term rule is ‘add 4’.
a) Find the first term.
b) Find the 100th term.
The term-to-term rule is ‘multiply by 2’. Copy and complete the sequence:
__, __, __, __, 88
Write down the 15th and 21st terms of the following sequences:
a) The first term is 36, to get to the next term add 12
b) The first term is 0, to get to the next term add 5
c) The first term is 8, to get to the next term add 0.5.
Investigation
The first two terms in a sequence are: 1, 2
a) Find and describe a term-to-term rule connecting these terms and then write the
next three terms in the sequence.
b) There is another rule that could connect the first two terms of this sequence, what
is it? Write down the first five terms of the sequence using the second term-toterm rule.
c) Randomly pick two numbers between 1 and 10 as the first two terms of a
sequence. List as many term-to-term rules as you can for the sequence, and write
the next three terms for each sequence.
d) If you are only given the first two terms
If you think this is true explain why it is
of a sequence can you always find more
true, if you think this is false show an
example where it doesn’t work – this is
than one rule?
called a counter example
e) If you are given the first three terms of a
sequence, can there be more than one rule
which connects the terms?
Generating sequences
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Key words
The general term
term
term number
general term
Generate a sequence given a general term
Find the general term of a sequence
We can describe a sequence of numbers using the first term and a term-to-term rule.
For example, the sequence whose first term is 4 and term-to-term rule is ‘add 3’ is
4, 7, 10, 13, …
Another way of describing a sequence is by giving a rule that connects the
term number and the term. We call this the general term .
Look at the sequence 8, 10, 12, 14, 16 … . The difference between terms is 2 so we
compare it to the multiples of 2. The sequence is the multiples of 2 add 6.
1st term 1 2 6
2nd term 2 2 6
3rd term 3 2 6
50th term 50 2 6
General term term number 2 6
You can use the general term to find the value of any term in the sequence. You
substitute the term number into the general term.
For example, the 20th term of this sequence is 20 2 6 46
Example
a) Find the general term of the sequence 5, 7, 9, 11, 13, …
b) Find the 16th term of the sequence.
a) 1st term 5 1 2 3
Look for a connection
between the term number
and the term.
2nd term 7 2 2 3
5th term 13 5 2 3
We can see that this sequence is the multiples of 2 add 3.
The general term is: term number 2 3
Substitute the term
number into the
general term.
b) The 16th term is 16 2 3 35
Exercise 1.3 ..........................................................................................
For each of these sequences:
a) Find the general term.
b) Find the 100th term.
6
Maths Connect 1R
Term Number
1
2
3
4
Sequence A
11
12
13
14
Sequence B
0
1
2
3
Sequence C
6
7
8
9
Sequence D
2
4
6
8
Sequence E
3
6
9
12
Sequence F
3
5
7
9
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Write down a general term for each of the following sequences. Then work out the tenth
term in each sequence.
a) 3, 8, 13, 18, …
b) 99, 198, 297, 396, …
c) 76, 151, 226, 301, …
d) 0.1, 0.2, 0.3, 0.4, …
e) 40, 75, 110, 145, …
The first five patterns of some sequences are shown in the table.
For each sequence:
a) How many extra dots are added each time?
b) Write down the general term for the pattern.
Term
Number
1
2
3
4
5
Pattern A
Pattern B
Pattern C
Pattern D
Two rival taxi firms have the following rates:
Rovers Rides £3.00 basic charge 50p per mile
Candice Cars £2.50 basic charge 60p per mile.
a) Find the cost of each firm for journeys of 1, 2, 3, 4 and 5 miles.
b) Which company would you use for a journey of 1 mile?
c) Which company would you use for a journey of 10 miles?
d) Find the general term for each taxi firm’s rates.
The bill for a mobile phone is calculated using the table below:
Duration of call (minutes)
Cost (£)
1
2
3
4
£5.50
£6.00
£6.50
£7.00
a) Find the general term for the cost of the phone bill.
b) Calculate the phone bill for a call lasting 20 minutes.
Investigation
Look at the differences between consecutive terms in each of the sequences in Q1,
Q2 and Q3.
a) What is the connection between the difference and the general term for the
sequence?
b) Write a set of instructions for finding the general term of a sequence.
The general term
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Key words
Function machines
function machine
input
output
inverse
unknown
mapping diagram
Find unknown numbers and operations in function machines
Draw simple mapping diagrams
A function machine changes one number (the input ) to another number (the
output ) according to a rule.
For example:
⫹4
Input
Output
Here the rule is ‘add 4’. The operation 4 is performed on any input.
You can use any of these operations in a function machine: addition, subtraction,
multiplication or division. You can use the inverse operation to work backwards
through the function machine to find an unknown input.
We can use a mapping diagram to show how one number moves to another number
using a rule.
For the function machine above, the mapping diagram is:
0
1
2
3
4
5
6
7
8
9 10
0
1
2
3
4
5
6
7
8
9 10
Example 1
Find the unknown number in these function machines:
⫻3
a)
a)
27 3, so
27
b) 20
4
Always check
your answer.
Check: 4 5 20 ✓
Example 2
Input
5
You need to work
backwards. The inverse of
multiplication is division.
9
Check: 9 3 27 ✓
b) 20 4 5, so
⫼
⫻7
⫹2
Output
Function machines can
have more than one step.
a) What is the output if the input is 3?
b) What is the input if the output is 37?
a) Output 3 7 2 23
b) 37 2 35
35 7 5
So Input 5
8
Maths Connect 1R
Work backwards and use
inverse operations.
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Exercise 1.4 .............................................................................................
Draw mapping diagrams for parts a) to d) and find the rules that connect the pairs of
numbers.
a) 1 → 8
4 → 11
6 → 13
b) 3 → 12
4 → 16
5 → 20
c) 12 → 7
10 → 5
8→3
d) 30 → 5
24 → 4
18 → 3
Draw function machines for the rules in Q1.
Copy and complete the function machines below:
a) 8
⫻2
⫼5
c)
7
b) 17
⫺
12
d) 42
⫼
7
Draw function machines to represent the following statements.
Find the original number in each case.
a) Twice this number is 102.
c) 12 less than this number is 79.
b) Three more than this number is 20.
d) Half this number is 35.
Find the missing inputs in these two-step function machines:
a) Input
⫻5
⫹6
16
b) Input
⫼2
⫺3
7
c) Input
⫹7
⫼10
2
d) Input
⫺8
⫻12
48
What number am I thinking of?
a) I think of a number, divide by 4, add 3 and get 28.
b) I think of a number, subtract 6, multiply by 5 and get 0.
c) I think of a number, divide by 2, multiply by 3 and get 30.
Draw function
machines to help you.
Tom’s calculator isn’t working properly. When he enters a number it multiplies it by 10
and subtracts 11.
a) If he enters the number 30, what will the calculator show?
b) If the calculator shows the number 9, what number did Tom enter?
c) If Tom enters 7 8, what will the calculator show on its screen?
d) If the calculator shows the number 1989, what number did Tom enter?
Copy and complete the following function machines using the numbers written next to
them. The first one is done for you.
⫻ 9
⫺ 7
1, 3, 5, 8
⫼
⫹
c) 2, 5, 6, 7
⫼
⫹
2
a) 2, 7, 9, 11
b
Are these two function machines identical?
Explain your answer.
Input
⫻5
⫻9
Output
11
Experiment with different inputs.
Input
⫻9
⫻5
Output
Replace ‘5’ and ‘9’ with other operations (e.g. 2, 12, 6) and decide whether the
order in which the operations are carried out affects the output.
Function machines
9
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Key words
More function machines
algebra
unknown
expression
equation
Write simple expressions and equations using letters to stand in for
unknown numbers
Solve simple equations by converting them into function machines
In maths we often use a letter to stand in for a number we don’t know.
This is called algebra .
If you chose a number and added 5, you could represent this by writing: x 5
It doesn’t matter which letter you use. It is just a method of writing an unknown
number.
We call x 5 an algebraic expression .
The sentence ’I choose a number, subtract 10 and get 12’ can be written:
x 10 12
This is called an algebraic equation since it contains an unknown (x) and an sign.
The same rules and conventions that you use in arithmetic also apply in algebra.
Example 1
Write the following function machines as algebraic equations and find the
unknowns:
a) Input
a) m 12 14
⫺12
14
b) Input
⫻3
⫹10
37
We can choose any letter
to represent an unknown.
m 14 12
26
Check: 26 12 14 ✓
b) 3x 10 37
3x 27
x 27 3
Use the inverse operation
to find the unknown.
3x is the same as 3 x or
x 3.
9
Example 2
Write each statement as an algebraic equation and find the unknown:
a) I think of a number, multiply it by 5, add 2 and get 27.
b) I think of a number, divide it by 4 and get 12.
a) 5b 2 27
5b 25
b5
x
b) 12
4
x 12 4
48
10
Maths Connect 1R
Represent the
unknown number with
a letter, in this case b.
In algebra we write
x
x 4 as 4
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Exercise 1.5 .............................................................................................
Georgina, Raashad and Emily are all given the same amount of money.
Use the letter b to represent the number of pence they are given.
a) Georgina spends 45p of hers. How much money does she have now?
b) Raashad shares his with his brother. How much money does he
have now?
c) Emily is given £1 more. How much money does she have now?
Write the following statements as algebraic expressions:
a)
b)
c)
d)
I choose a number and subtract 5.
I choose a number and triple it.
I choose a number, multiply it by 5 and add 14.
I choose a number divide it by 7 and subtract 10.
Choose a letter to stand in
for the unknown number.
Write the following as algebraic equations and calculate the number I choose.
a)
b)
c)
d)
e)
f)
I choose a number, add 12 and get 17.
I choose a number, divide it by 5 and get 12.
I choose a number, multiply it by 12 and get 48.
I choose a number, subtract 0.5 and get 1.5.
I choose a number, multiply it by 100, subtract 12 and get 1388.
I choose a number, divide it by 8, subtract 10 and get 1.
Write the following function machines as algebraic equations and find the missing inputs:
a) Input
⫻5
⫹3
8
b) Input
⫼10
⫺12
19
c) Input
⫼7
⫹3
11
d) Input
⫻6
⫺5
37
A set of numbers is mapped to another set of numbers using the following function
machine:
Input
⫻3
⫹0.5
Output
a) Find the output if the input is 5.
b) Find the input if the output is 5.
c) Write an algebraic expression
Use one letter for the input and another for the output.
connecting the input to the output.
Look at the difference between consecutive
d) Work out the outputs for inputs of 1, 2, 3, 4
terms in the sequence of answers.
and 5. Explain any pattern you see.
Ian’s pocket money (in pence) is calculated by doubling his age and adding 50p.
Gaby’s pocket money (in pence) is calculated by adding her age to 70p.
a) Calculate the amounts of pocket money Ian and Gaby receive when they are 12.
b) Calculate Ian’s age when he receives 78p pocket money.
c) Calculate Gaby’s age when she receives 78p pocket money.
Investigation
In Q6, Ian was born in 1990 and Gaby was born in 1992. Investigate the amount of
pocket money they will receive each year until they are 18. In which year will they
both receive exactly the same amount of pocket money? What will their ages be then?
More function machines 11
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Key words
Using letter symbols to stand
in for unknown numbers
unknown
relationship
algebra
Use letters to stand in for unknown numbers
Express relationships between unknowns
You already know that you can use a letter to stand in for an unknown number in an
equation, for example 5x 3 18.
Sometimes we want to express one unknown number in terms of its relationship
with another unknown.
For example, if Mary gets twice as much money as Nuala,
this can be expressed using algebra as M 2N
M is the amount of
money Mary receives, N
is the amount of money
Nuala receives.
We can use algebra to express a function machine:
⫼8
Input
⫺5
x
Using algebra, we write: 5 y
8
Example 1
Output
x represents the input and
y represents the output.
Express the relationships between the following pairs of letters using
algebra.
a) A is twice B.
b) D is four less than N.
c) M is five more than triple T.
d) Q is three times smaller than D.
a) A 2B
b) D N 4
c) M 3T 5
D
d) Q 3
Example 2
If Katy received 10p more pocket money and Asif received 50p less they
would receive the same amount. Use algebra to show the relationship
between the amounts of pocket money Katy and Asif receive.
p 10 q 50
12
We show things have
the same value by
using the ‘’ sign.
Maths Connect 1R
Choose a letter symbol to stand in for the
amounts of pocket money Katy and Asif receive.
Here p is the amount Katy receives and q is the
amount Asif receives.
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Exercise 1.6 .............................................................................................
Use algebra to show the relationships between the following pairs of letters:
a) M is three less than T.
c) Y is seven less than twice V.
e) X is four more than five times W.
b) X is four more than Z.
d) Q is four times smaller than R.
f) Z is seven times smaller than B.
Use algebra to show the relationship in each of the following situations:
a) James is four years younger than Andrea.
b) Georgina receives half the amount of pocket money Raashad receives.
c) If Angharad spent 10 minutes more on the phone she would spend half the amount of
time that Emily does.
Write the following algebraic expressions in another form:
a) 2x
b) x x x
We can write: 3 5 as 5 5 5
c) b b b b b
d) 7m
e) 3z
If possible, write the following function machines using only one ‘box’. If it is not
possible, say why not. When you have done this, express each function machine using
algebra and find the input for each one.
a) Input
⫹5
⫹4
18
b) Input
⫹16
⫺100
67
c) Input
⫻7
⫻9
126
d) Input
⫻4
⫼2
10
e) Input
⫼3
⫼2
4
f ) Input
⫻3
⫺5
10
Two whole numbers add together to make 15.
The two numbers multiplied together give 50.
a) Use algebra to express this information.
b) Find the numbers.
Try all the pairs of numbers that
multiply together to give 50.
In this Arithmagon the number in each square is the sum of the numbers in the two
circles on either side of it.
a) Find as many ways as you can to express
relationships between the letters using algebra.
b) Find the values of B, C and D if A 10.
c) Investigate other values for A, B, C and D.
You should find many sets of values that work.
A
17
19
C
B
19
21
D
Investigation
In this diagram the value in each cell is found by adding the values in the two cells
above it.
a) Copy and complete the diagram.
a
b
c
b) Does the value in the bottom cell change if you change the
a⫹b
order of a, b and c in the top row?
c) Draw a similar diagram with four unknowns a, b, c and d.
Investigate the value in the bottom cell for different arrangements of a, b, c and d
in the top row.
Using letter symbols to stand in for unknown numbers 13