Download Multiples - Pearson Schools and FE Colleges

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Key words
Multiples
multiples
Find multiples of whole numbers
The multiples of 6 are found by multiplying any whole number by 6:
166
2 6 12
3 6 18
4 6 24
The numbers 6, 12, 18 and 24 are all multiples of 6.
There are lots of other multiples of 6:
33 6 198
198 is also a multiple of 6.
Example 1
Find the first five multiples of 7.
177
2 7 14
The first five multiples of 7 are found by
multiplying 7 by 1, 2, 3, 4 and 5.
3 7 21
4 7 28
5 7 35
The first five multiples of 7 are 7, 14, 21, 28 and 35.
Example 2
Write down the numbers in the cloud that are:
a) multiples of 3
b) multiples of both 4 and 5
15
5 3 15
7 3 21
11 3 33
a) 15, 21 and 33 are multiples of 3
b) 40 is a multiple of both 4 and 5
8 5 40
10 4 40
Exercise 1.1
Write down the first six multiples of the following numbers:
2
a) 3
b) 10
c) 5
d) 4
e) 9
f) 8
g) 100
h) 25
Maths Connect 1G
33
14 21 16
40
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Find the first four multiples of the following numbers:
a) 16
b) 49
c) 81
d) 112
Write down the numbers in the cloud that are multiples of:
a) 5
b) 3
c) 11
d) 7
e) 4
f) 6
18
Which of the numbers in the cloud are multiples of:
55 21
12
31
44
25
Look at Example 2.
21
16
12
55
100
24
a) both 3 and 7
b) both 4 and 6
c) both 5 and 11?
Write down a number which is a multiple of both the following numbers:
a) 2 and 4
b) 3 and 7
c) 5 and 10
d) 8 and 6
e) 10 and 11
Choco Bars come in boxes of 20. Mr Bruce can only order whole
boxes for his sweet shop.
a) What is the least number of Choco Bars he can order?
b) List five different quantities of Choco Bars that he can order.
c) If he needed 95 bars, how many boxes would he have to order?
a) Write down the first twelve multiples of 3.
b) Which of these numbers are also multiples of 6?
a) Write down the first five multiples of 1, 3, 5 and 7.
b) What do you notice about the multiples of odd numbers?
Investigation
Copy this hundred square onto cm
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
39
30
31
32
33
34
35
36
37
38
39
40
c) What can you say about the
multiples of 6?
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
d) Draw a new hundred square
and investigate the patterns
made by shading in multiples
of different numbers.
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
squared paper:
a) Shade in all the multiples of 2.
b) Are all the multiples of 6
shaded in?
Multiples
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Key words
Special numbers
multiples
square numbers
triangular numbers
Recognise and find square numbers
Decide whether numbers are multiples of particular numbers by
drawing them out as rows of dots
Multiples of a number are that number multiplied by any other whole number.
We can represent numbers by drawing them out as patterns of dots. For example, to
represent the number 9, we can draw out 9 dots in lots of different arrangements.
By drawing out 9
dots in columns of
2 dots we can see 9
not a multiple of 2
since there is a dot
left over.
We can also see
that 9 is an odd
number.
By drawing out
9 dots in
columns of 3
dots we can
see 9 is a
multiple of 3.
By drawing out
9 dots in
columns of 4
dots we can
see 9 is not a
multiple of 4
since there are
2 dots left over.
By drawing out
9 dots in
columns of 5
dots we can
see 9 is not a
multiple of 5
since there are
4 dots left
over.
Square numbers are numbers which when drawn out as patterns of dots can be
arranged into squares. Square numbers are found by multiplying a number by itself.
For example, 25 is a
square number.
5 5 25
Example
By drawing out the number 10 as patterns of dots find out whether it is:
a) An even number
a)
10 is an even number since
we can draw out dots in pairs
with none left over.
b) A square number
b)
c) A multiple of 4.
c)
10 is not a square number
since we cannot arrange 10
dots into a square.
10 is not a multiple of 4 since
when we draw out 10 dots in
columns of 4 dots, 2 dots are
left over.
Exercise 1.2
By drawing out the number 11 as patterns of dots, find out whether it is:
a) An even number
b) A multiple of 3.
By drawing out the number 16 as patterns of dots, find out whether it is:
a) A multiple of 5
4
Maths Connect 1G
b) A square number
c) An even number.
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In this diagram, the numbers 3, 6, 9 and 12 are drawn out as patterns as dots:
We can see from the diagram that these numbers are multiples of 3 because they can be
arranged in columns of 3 dots. Design patterns to show that:
a) 4, 8, 12 and 16 are multiples of 4
b) 5, 10, 15 and 20 are multiples of 5
c) 3, 5, 7 and 9 are odd numbers.
Copy and complete the table:
11
22
33
1
4
9
55
Pattern
Number of dots
16
The table in Q4 shows the first five square numbers.
a) Draw diagrams of the next two square numbers.
b) Look at your diagrams for part a). How many dots are there in the sixth and seventh
square numbers?
c) Predict how many dots there will be in the eighth square number.
Some numbers can be drawn out as triangular patterns of dots. These numbers are called
triangular numbers.
Pattern
Number of dots
1
3
6
10
15
a) Draw diagrams of the next two triangular numbers.
b) Look at your diagrams for part a). How many dots are there in the sixth and seventh
triangular numbers?
c) Predict how many dots there will be in the eighth triangular number.
Special numbers
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Key words
Sequences
Predict the next term in a sequence of numbers or shapes
Describe sequences of numbers or shapes
sequence
term
consecutive
A number sequence is a set of numbers in a given order. You can find number
sequences everywhere:
Your telephone number is a sequence
of numbers.
The door numbers on houses in a
street is a sequence of numbers.
Each number in a sequence is called a term . You write sequences with commas
between each of the terms: 4, 8, 12, 16, …
Consecutive terms are terms which are next to each other.
The multiples of 4 produce this sequence:
4 and 8 are consecutive terms.
8 and 12 are consecutive terms.
12 and 16 are consecutive terms.
Example 1
4, 8, 12, 16, …
Here is a sequence of diagrams.
Spot the pattern and draw the next two terms in the
sequence.
Each time two more dots are added:
This is the 5th term.
Example 2
This is the 6th term.
Write down the next four terms in each of the following sequences:
a) 5, 10, 15, 20, …
b) 66, 55, 44, …
a) The next four terms in the sequence are:
25, 30, 35, 40
To find the next term in the sequence you add 5.
b) The next four terms in the sequence are:
33, 22, 11, 0
6
Maths Connect 1G
To find the next term in the sequence you
subtract 11.
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Exercise 1.3
Spot the pattern and draw the next two terms in each sequence:
a)
b)
c) * * * *
****
**
****
**
**
****
**
**
**
d)
Spot the pattern for each of these sequences:
a) 3, 6, 9, 12, …
d) 6, 8, 10, 12, …
b) 7, 14, 21, 28, …
e) 10, 15, 20, 25, …
c) 11, 22, 33, 44, …
f) 16, 24, 32, 40, …
Each of these sequences
is going up or ascending.
Spot the pattern and draw the next two terms in each sequence:
a)
b)
c) * * * * * * *
*******
*******
d)
*******
*******
****
*******
*******
*
*******
*****
Find the next three terms in each of the following sequences:
a) 48, 42, 36, 30, …
b) 20, 15, 10, 5, …
Each of these sequences is
going down or descending.
c) 12, 10, 8, 6, …
The house numbers on a road go from 1 to 100.
The even numbers are on one side of the street and the odd
numbers are on the other.
a) If you are walking down the side of the street with odd
numbers, write down the numbers on the first five doors
you walk by.
b) What is the number on the last house you walk by?
The table below shows the average number of hours of daylight per day each month:
Jan
Feb
Mar
Apr
May
Jun
9
10
11
12
13
14
Jul
Aug
Sep
Oct
Nov
Dec
a) If the sequence continues like this how many hours of daylight will there be in October?
b) Do you think the sequence will continue like this? Explain your answer.
Investigation
Number sequences can be found in real life. Describe a real-life number sequence which is:
a) ascending
b) descending
c) neither ascending nor descending.
Sequences
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Key words
Function machines
function machine
input
operation
output
Find the missing output in a function machine
A function machine looks like this:
An operation
is performed
on the number.
The input is the
number you put
into the machine.
Input
Output
Operation
The output is
the result.
Here is an example of a function machine:
The operation
is 11.
The input is 5.
Example 1
5
16
The output is 16:
5 11 16.
Find the missing output of this function machine:
7
3
Output
To find the output of the function machine we
subtract 3 from the input: 7 3 4
The output is 4.
Example 2
11
Raashad’s parents calculate the amount of pocket money he gets by
multiplying his age by 10.
Draw a function machine where the input is his age and the output is the
amount of pocket money he receives.
Age
10
Amount of
pocket money
Raashad’s age is the input of the function machine
because this is the value that we start with.
The operation is 10 since the amount of pocket
money Raashad gets is found by multiplying his age
by 10.
The output is the result of multiplying the input
(Raashad’s age) by 10.
8
Maths Connect 1G
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Exercise 1.4
What is the output of this function machine if the input is:
a) 17
b) 0
c) 112
d) 89
Input
5
Output
Input
12
Output
Input
6
Output
Input
8
Output
What is the output of this function machine if the input is:
a) 36
b) 49
c) 100
d) 122
What is the output of this function machine if the input is:
a) 6
b) 12
c) 1000
d) 25
What is the output of this function machine if the input is:
a) 24
b) 40
c) 888
d) 8000
Florence is saving up to buy a mobile phone. Her parents say they will give her £50.
She draws a function machine to calculate how much money she saves:
Amount
saved
£50
Total
a) How much money will Florence have in total if she saves £33?
b) How much money will Florence have in total if she saves £21.50?
Function machines can have two operations. Look at this function machine:
2
Input
7
Output
a) What is the output if the input is 12?
b) What is the output if the input is 30?
Investigation
There are 100 cm in a metre.
a) How many cm are there in 2 m?
b) How many cm are there in 5 m?
c) Copy and complete the function machine below.
Length
in m
?
Length
in cm
d) Use your function machine to work out how many cm there are in:
i) 3.5 m
ii) 10 m
iii) 0.2 m
iv) 9.7 m
e) Can you draw a function machine that will convert cm into m?
Function machines
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Key words
Finding the missing input
input
inverse
operation
Find the missing input in a function machine
You can find a missing input in a function machine by calculating the inverse of the
operation .
Input
11
16
Input
11
16
The inverse of: addition is subtraction
subtraction is addition
multiplication is division
division is multiplication
16 – 11 5 so the missing input is 5.
Example 1
Find the missing input of this function machine:
2
Input
Input
÷2
30
30
The inverse of multiplication is division.
30 2 15
The input is 15.
Check: 15 2 30
Example 2
✓
I choose a number and divide it by 3. The result is 7.
a) Draw a function machine to represent this information.
b) What number did I choose?
a)
Input
3
7
b) Input
3
7
By representing the problem as a function
machine it is much easier to find the original
number.
7 3 21
The input is 21. You chose the number 21.
The inverse of division is multiplication.
Exercise 1.5
Write down the inverse operation for each of the following:
a) ‘add 11’
10
Maths Connect 1G
b) ‘subtract 2’
c) ‘multiply by 10’
d) ‘divide by 12’
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Copy these function machines and find the missing inputs.
a) Input
12
22
b) Input
10
17
c)
4
5
d) Input
11
25
Input
Copy these function machines and find the missing inputs.
a) Input
3
27
b) Input
8
64
c)
2
26
d) Input
9
9
Input
I choose a number and add 5. The result is 16.
Look at Example 2.
a) Draw a function machine to represent this information.
b) What number did I choose?
I choose a number and divide it by 2. The result is 34.
a) Draw a function machine to represent this information.
b) What number did I choose?
Bob works in a burger bar.
He gets paid £5 per hour.
This function machine calculates his total pay:
Number of
hours worked
£5
Total pay
a) If he works for 40 hours how much does he get paid?
b) If his total pay in a week is £110 how many hours does he work?
c) If his total pay in a week is £65 how many hours does he work?
Lord Number loves his collection of china vegetables,
which he keeps locked in a safe. He works out the
combination for the safe by multiplying the number of
the month by 7.
For example:
February is the second month.
2 7 14
The combination for February is 14.
a) Draw a function machine to show how Lord
Number works out his combination.
b) What will his combination be in March?
c) In which month is his combination 42?
d) In which month is his combination 77?
e) Copy and complete this table:
Month
Combination
Jan
Feb
Mar
Apr
May
Jun
14
f) What pattern do you notice?
g) Create your own function machine to provide Lord Number with an alternative set of
combinations. Draw a table of your combinations.
Finding the missing input 11
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Key words
Finding the missing operation
function machine
input
output
operation
Find the missing operation in a function machine
Remember, a
function machine looks like this:
An operation is performed on
the number to give the output.
For example, 4 ⴙ 2 6
The input is the
number you put
into the machine.
Example
4
ⴙ2
The output is
the result.
6
These two function machines have the same operation. What is it?
5
Operation
15
12
Operation
22
To get from the number 5 to the number 15 you either:
a) multiply by 3
b) add 10
Looking at the second function machine, we can see that the
missing operation must be ‘add 10’ since 12 10 22
The missing operation is ‘10’.
Exercise 1.6
Find the missing operations for these function machines:
a)
3
2
6
24
12
2
c)
b)
17
8
63
101
9
d)
164
These two function machines have the same operation. What is it?
12
12
Operation
4
3
Operation
1
Maths Connect 1G
Thee four operations are:
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Asif and Jerome are playing a game. Asif chooses a
number and tells Jerome.
Jerome performs an operation on the number and tells
Asif the result.
Asif says:
Jerome says:
4
8
0.5
1
45
90
Look at the table. What operation is Jerome using?
Adrian receives £4 pocket money each week. He saves all his pocket money each week so
that he can buy a Skateboard that costs £49.99.
a) Copy and complete the function machine below:
Number of
weeks
Total saved
b) How much will he have after six weeks?
c) How many weeks will he have to save for?
This function machine has two operations:
5
Operation 1
11
Operation 2
There are many diffferent combinations of operation you could use to get from 5 to 11.
For example:
5
2
1
11
Find as many different combinations of operations as you can to get from 5 to 11.
Copy and complete these function machines using only the numbers written above them:
a) 2, 7, 9
11
7
6
b) 3, 5, 6
c) 2, 5, 7
Play the game in Q3 with a partner.
Player 1 chooses an operation.
If Player 2 guesses the operation correctly after one number he/she scores 3 points.
If Player 2 guesses the operation correctly after two numbers he/she scores 2 points.
If Player 2 guesses the operation correctly after three numbers he/she scores 1 point.
Player 1 and Player 2 take it in turns to choose an operation. The first one to score 10
points wins.
Find the missing operation 13