N
2.1
Properties of numbers
Previous learning
Objectives based on NC levels 3 and 4 (mainly level 4)
Before they start, pupils should
be able to:
In this unit, pupils learn to:
recall addition and
subtraction facts for each
number to 20
recognise odd and even
numbers
recognise multiples of 2, 5
and 10.
represent problems using words or diagrams
look for and visualise patterns
manipulate numbers
record, explain and compare methods
engage in mathematical discussion of results
begin to generalise
and to:
identify squares of numbers to 12 3 12
recognise multiples and use simple tests of divisibility
order, add and subtract positive and negative numbers in context.
Lessons
1 Square numbers
2 Multiples and divisibility
3 Positive and negative integers
About this unit
Pupils’ confidence in responding to numbers in everyday situations is
strengthened by having a good ‘feel for number’. This means being aware of
significant relationships between numbers and knowing at a glance which
properties they possess and which they do not.
In this unit pupils learn to use patterns to help them to recall number facts
and recognise number properties.
Assessment
Common errors and
misconception
| N4.1
Properties of numbers
This unit includes:
an optional mental test that could replace part of a lesson (p. 00);
a self-assessment section (N2.1 How well are you doing? class book p. 00);
a set of questions to replace or supplement questions in the exercises
or homework tasks, or to use as an informal test (N2.1 Check up,
CD-ROM).
Look out for pupils who:
have difficulty in remembering number facts, such as addition and
subtraction facts to 20, or multiplication facts to 10 3 10;
confuse squaring and doubling;
lack confidence in working in the negative part of the number line, and
who think that 23 1 5 5 8, or that 23 2 5 5 8.
Key terms and notation
Practical resources
Exploring maths
Useful websites
problem, solution, method, pattern, relationship, order, solve, explain,
represent
calculate, calculation, calculator, add, subtract, multiply, divide, divide
exactly, divisible, sum, total, difference, product, greater than (.), less than
(,), value
positive, negative, integer, odd, even, multiple, square, perfect square,
square root, digit sum
temperature, degrees Celsius (°C)
calculators for pupils
individual whiteboards
Tier 2 teacher’s book
N2.1 Mental test, p. 00
Answers for N2.1, pp. 00200
Tier 2 CD-ROM
PowerPoint files
N2.1 Slides for lessons 1 to 3
Prepared toolsheets
N2.1 Toolsheets 3.1 to 3.3
Tier 2 programs and tools
Calculator tool
Number grids
Number and shape sorter
Tier 2 class book
N2.1, pp. 00200
N2.1 How well are you doing?, p. 00
Tier 2 home book
N2.1, pp. 00200
Tier 2 CD-ROM
N4.1 Check up, p. 00
Multiples an NRICH package of problems and puzzles
nrich.maths.org/public/viewer.php?obj_id=5530
Grid game
www.bbc.co.uk/education/mathsfile/gameswheel.html
Multi sequencer
www.amblesideprimary.com/ambleweb/mentalmaths/supersequencer.html
N4.1 Properties of numbers | 1 Square numbers
Learning points
When a number is multiplied by itself the result is a square number.
81 is the square of 9. It can be written as 92.
A square number can be represented by dots arranged in the shape of a square.
Starter
Show slide 1.1. to discuss the objectives for this unit. Say that this lesson is about
square numbers.
Show slide 1.2. Point to the 3 by 3 pattern of dots.
How many dots are there? How are they arranged?
Write 9 in the box below the dots and 3 3 3 in the box below. Continue up to
6 3 6.
Point to the row of numbers 1, 4, 9, 16, 25, 36. Say that these are called square
numbers – each is the result of multiplying a number by itself and can be
represented by dots arranged in a square shape.
Say that there is a special way of reading and writing square numbers. Point to 12
and 22 saying ‘one squared, two squared’. Ask pupils to write on their whiteboards
32, 42, 52 and 62. Enter these in the table.
Launch Number grids. Choose a multiplication grid with 10 rows and columns, a
start number of 0 and a step of 1. Say that you will highlight the first six numbers
in the sequence of square numbers. Click to highlight 1, 4, 9, 16, 25, 36. Point out
that 16 or 4 3 4 lies in the fourth column and the fourth row.
SIM
What are the next numbers in the sequence? [49, 64, 81, 100]
How would we write seven squared?
What number do you square to get 81?
What is the square of ten?
What is the next square number after 100? And after that?
Ask the class to chant the sequence of square numbers. Point to the numbers as
pupils say them. Click on ‘Hide products’ and chant again.
If you prefer, use slide 1.3 instead of Number grids. You can white out the slide
for the last chant by pressing W on the keyboard, or by right-clicking then
choosing Screen > White screen. Press W or right-click again to restore the slide.
TO
Main activity
Use the Calculator tool to show pupils how to use the x2 . key on their
calculators. Tell them that they are going to investigate sums of two square
numbers.
What is six squared? What is two squared?
What is the sum of six squared and two squared? [40]
Demonstrate the key sequence 6 x2 1 2 x2 
| N2.1
Properties of numbers
Explain that 40 is an interesting number because it is the sum of two square
numbers. Say that 13 is also the sum of two squares. Ask pupils to discuss in pairs
what the two squares might be [2 and 3].
Which other numbers up to 30 are the sum of two different squares?
Ask pupils to investigate this in pairs for a few minutes. Discuss how to work
systematically, e.g. add 12 to each of 22, 32, 42 and 52. Since 12 added to 62 is too
big, now add 22 to each of 32, 42 and 52.
Why don’t we need to try adding two squared to one squared?
[same as 12 1 22]
Leave the pairs to work for a few more minutes, then gather the complete set of
results: 5, 10, 13, 17, 20, 25, 26, 29.
How do we know that we have found all the possibilities?
Establish that because they have worked systematically through all the possible
pairs they must have checked them all.
Select individual work from N2.1 Exercise 1 in the class book (p. 00).
Review
Say that a mystery number, when squared, has the answer 225.
Is the mystery number less than 10?
[No – its square would be less than 100.]
Is the mystery number more than 20?
Confirm with the Calculator tool that 202 5 400, so the number is less than 20.
Could the mystery number be even?
TO
Establish that the product of two even numbers is always even, so the mystery
number is odd (and lies between 10 and 20). Write 11, 13, 15, 17 and 19 on the
board.
Which of these numbers could it be? Explain why.
Draw out that 15 is the only number which, when multiplied by itself, will result
in a number with a units digit of 5. Confirm using the Calculator tool.
Sum up with the points on slide 1.4.
Homework
Ask pupils to do N2.1 Task 1 in the home book (p. 00).
N2.1 Properties of numbers | 2 Multiples and divisibility
Learning points
A multiple of 5 is a number that divides exactly by 5.
A number is a multiple of 2 if its last digit is even.
A number is a multiple of 3 if its digit sum is a multiple of 3.
A number is a multiple of 4 if half of it is even.
A number is a multiple of 5 if its last digit is 0 or 5.
A number is a multiple of 10 if its last digit is 0.
Starter
Say that this lesson is about multiples. Remind pupils that a multiple of a number
divides exactly by the number.
Draw a large square box on the board. Ask pupils
to suggest some numbers below 60. If they
are multiples of 5, write them in the box. If not,
write them outside.
Once there are at least three numbers
in the box, ask:
27
55
59
20
1
35
23
What is my rule for putting numbers in the box?
Continue asking for numbers until pupils recognise the rule. Repeat with
multiples of 7, multiples of 2 and multiples of 11. Invite pupils who think that
they know the rule to the board to write another number in the box.
Main activity
Write 96 on the board.
Is this number odd or even? How do you know?
Point out that the last digit is even, so the whole number is even. An even
number is divisible by 2, i.e. divides exactly by 2 with no remainder. It is also a
multiple of 2. Say a few numbers and ask pupils to say if the number is divisible
by 2.
Return to 96.
Is this number divisible by 3?
Tell the class that there is a quick way to find out by adding up all the digits, i.e.
9 1 6 5 15. Explain that 15 is called the digit sum. Because it is a multiple of 3, the
number 96 is a multiple of 3. Test a few more numbers for divisibility by 3.
Return to 96.
Is this number divisible by 4?
Slide 2.1
| N2.1
Properties of numbers
Say that there is an easy way to find out. We know that 96 is even so it divides
exactly by 2, so find half of 96. Point out that because 48 is even, 96 can be
divided exactly by 2 and then exactly by 2 again. So 96 is divisible by 4. Test a few
more numbers for divisibility by 4.
Return to 96.
Is this number divisible by 5 or by 10?
Confirm that it is not, since it does not end in 5 or 0.
Launch Number and shape sorter. Choose a two-way Carroll diagram and
then ‘Is a multiple of 3 / Is a multiple of 4’. Involve pupils in dragging the
numbers to the correct part of the diagram. Ask questions such as:
SIM
How do you know that 42 is a multiple of 3?
How do you know that 42 is not a multiple of 4?
How do you know that 24 is divisible by 3 and divisible by 4?
Select individual work from N2.1 Exercise 2 in the class book (p. 00).
Review
Write 96 on the board again.
Is 96 divisible by 6?
Explain that all multiples of 6 divide exactly by 2 and also by 3. We know that 96
divides exactly by 2 because it is even. We also know that it divides exactly by 3
because its digit sum of 15 is a multiple of 3. So 96 is divisible by 6.
Launch Number and shape sorter again. This time choose a two-way Venn
diagram, then ‘Is a multiple of 3 / Is a multiple of 6’. Involve pupils in dragging
the numbers to the correct region. Ask, for example:
SIM
How do you know that 39 is a multiple of 3?
How do you know that 39 is not a multiple of 6?
Sum up with the points on slides 2.1 and 2.2.
Homework
Ask pupils to do N2.1 Task 2 in the home book (p. 00).
N2.1 Properties of numbers | 3 Positive and negative integers
Learning points
The negative number 6 is called ‘negative 6’ and written as 26.
Numbers get less as you count back along the number line beyond nought or zero, so 210 is
less than 25.
Six degrees below zero is minus six degrees Celsius (26°C).
210°C is a lower temperature than 25°C.
Always include the units when you write a temperature.
Starter
Say that this lesson is about positive and negative numbers.
Show the first number line on toolsheet 3.1 Point at random to divisions on the
line and ask pupils to say the number. As they do so, write it on the line.
TO
Explain that the line shows the integers from 25 to 5. Integers are positive or
negative whole numbers and zero. The negative integers are called ‘negative one’,
‘negative two’, and so on. Say that we usually don’t write 12 for ‘positive two’ but
write 2.
Show the number line on toolsheet 3.2. Again, point at random to divisions on
the line and ask pupils to say the number. Write each number on the line, then
ask questions like:
TO
Tell me a number that is less than 220.
That is more than 230.
That lies between 220 and 10.
Record answers on the board using the < and > signs, e.g.
TO
Main activity
260 < 220
29 > 230
220 < 0 < 10
Show the number line on toolsheet 3.3. Explain that, just like a number line with
only positive numbers, it is possible to add and subtract by counting steps along
the line.
What is 5 more than 22? [record as22 1 5 5 3]
What is 6 less than 4? [record as 4 2 6 5 23]
What is the difference between 5 and23?
Stress that a difference is measured by the
number of steps or the distance between
the numbers, and can be recorded as
5 2 (23) 5 8.
3
-3
5
0
+5
Which pairs of numbers have a difference of 4?
Show slide 3.1. Say that the thermometers show the temperatures in Leeds
and Barcelona on the same day in winter. Point out the °C (degrees Celsius)
abbreviation. Say that as you move down the scale the temperature is falling.
| N2.1
Properties of numbers
What is the temperature in Leeds? In Barcelona?
How much colder is it in Leeds than Barcelona?
Point out 25°C on each scale. Invite a pupil to show 27°C. Explain that this is
read as ‘minus seven degrees Celsius’ not ‘negative seven degrees Celsius’ and
that it means that the temperature is seven degrees Celsius below zero.
The temperature falls by 5 degrees in each city.
What are the temperatures now?
On another day, Leeds is 3°C. Barcelona is 7 degrees colder.
What is the temperature in Barcelona?
Remind pupils that they should include the units when they write a temperature.
Show slide 3.2 and work through the questions.
Select individual work from N2.1 Exercise 3 in the class book (p. 00).
Review
Show slide 3.3. Ask pupils to write the temperatures in order on their
whiteboards from hottest to coldest. Use the temperatures to ask questions
such as:
What is the difference between 12°C and 37°C?
Between 12°C and –12°C?
Between 212°C and –2°C?
The temperature is 22°C.
How many degrees must it rise to reach 12°C?
The temperature falls from 37°C to 212°C.
How many degrees has it fallen?
After each question, ask pupils how they worked out the answer.
Sum up the lesson with the points on slide 3.4.
Round off the unit by referring again to the objectives. Suggest that pupils find
time to try the self-assessment problems in N2.1 How well are you doing? in
the class book (p. 00).
Homework
Ask pupils to do N2.1 Task 3 in the home book (p. 00).
N2.1 Properties of numbers | N2.1 Mental test
Read each question aloud twice.
Allow a suitable pause for pupils to write answers.
1 Write the next odd number after twenty-nine.
2 Which is the lowest of these temperatures?
[Write on board:
2°C
25°C
5°C
0°C
21°C]
2005 KS2
3 What is three times three added to four times four?
2003 KS2
4 Write three even numbers that add to twenty.
2004 KS3
5 What temperature is ten degrees lower than seven degrees Celsius?
2006 KS2
6 What number is nine squared?
1997 KS3
7 Write down an even number that is a multiple of seven.
2005 KS3
8 What is the smallest whole number that is divisible by five and by three?
2004 KS3
9 The temperature on Monday was minus eight degrees Celsius.
[Write on the board 28°C.]
On Tuesday, it was ten degrees higher.
What was the temperature on Tuesday?
2005 PT
10 What is the next square number after thirty-six?
2005 PT
11 Write a number that is a multiple of ten and also a multiple of twelve.
2006 PT
12 What number multiplied by eight equals forty-eight?
Key:
KS3 Key Stage 3 Mental test PT Progress test KS2 Key Stage 2 Mental test
Questions 1 to 3 are at level 3. Questions 4 to 12 are at level 4.
Answers
  1 31  2 25°C
  3 25  4 One of these sets of three numbers:
2, 2, 16; 2, 4, 14; 2, 6, 12; 2, 8, 10;
4, 4, 12; 4, 6, 10; 4, 8, 8; 6, 6, 8
  5 23°C  6 81
  7 e.g. 14, 28  8 15
10 | N2.1
1998 KS2
  9 2°C
10 49
11 e.g. 60, 120
12 6
Properties of numbers
2005 KS3
N2.1 Check up and resource sheets
Check up
N2.1
Answer these questions by writing in your book.
Check up [continued]
Properties of numbers (no calculator)
Properties of numbers (calculator allowed)
1
5
1995 level 3
Ali drew a picture to show what there is above and below the sea at Aber.
2003 Progress Test level 4
The 4th square number is 16.
What is the 5th square number?
hotel
+20 m
6
bird
boat
Here is a grid with some numbers shaded.
0m
The grid continues.
Will the number 35 be shaded?
Write Yes or No.
Explain your answer.
diver
fish
2006 Progress Test level 3
20 m
7
1
2
3
5
6
7
4
8
9
10
11
12
2006 Progress Test level 4
The diagram shows what is above and below sea level.
anchor
40 m
chest
The anchor is at about �40 m.
30
bird
20
kite
10
butterfly
a What is at about �10 m?
b What is at about �10 m?
c What is about 30 m higher than the chest?
2
2005 KS2 level 3
11 12 13 14 15 16 17 18 19
3
0
boat
10
diver
20
fish
30
eel
metres
Which three of these numbers add to make a multiple of 10?
1997 KS2 level 3
One of these numbers when multiplied by itself gives the answer 49.
Which number is it?
2 3 4 5 6 7 8 9
4
2002 KS2 level 4
Write all the multiples of 8 in this list of numbers.
18 32 56 68 72
Pearson Education 2008
a What is about 50 m lower than the bird?
b An octopus is at about �40 m.
About how many metres higher is the diver than the octopus?
Tier 2 resource sheets | N2.1 Properties of numbers | N2.1
Pearson Education 2008
Tier 2 resource sheets | N2.1 Properties of numbers | N2.1
N2.1 Properties of numbers | 11
N2.1 Answers
Class book
EXERCISE 1
EXERCISE 2
1 a 10, 20, 30
c 6, 12, 18
e 21, 42, 63
b 3, 6, 9
d 9, 18, 27
2
b
d
f
h
1 a 100
c 400
b 225
d 1225
2 a 8  8 5 64
c 13 13 5 169
e 14  14 5 196
b 9  9 5 81
d 12  12 5 144
f 22  22 5 484
3
a
c
e
g
b
d
f
h
4
Other solutions may be possible.
a 25 5 16 1 9
b 50 5 25 1 25
c 17 5 16 1 1
d 29 5 25 1 4
e 85 5 49 1 36
f 52 5 36 1 16
g 61 5 25 1 36
h 125 5 100 1 25
i 16 5 25 2 9
j 20 5 36 2 16
k 15 5 16 2 1
l 40 5 49 2 9
m 77 5 81 2 4
n 21 5 25 2 4
o 64 5 100 2 36 p 35 5 36 2 1
5 a No
c No
e Yes
5
1
5 12
1 1 3
5 22
1 1 3 1 5
5 32
1 1 3 1 5 1 7
5 42
1 1 3 1 5 1 7 1 9 5 52
1 1 3 1 5 1 … 1 19 5 102 5 100
Extension problem
8
72
98
34
4
50
45
21
100
a
c
e
g
True
False
False
False
3 18, 56, 72
4 30, 45, 60
b Yes
d Yes
f No
6 54 1 36 or 34 1 56
7 a 15
c 63
e 22
b 36
d 25
f 36 or 72
Numbers from 30 to 60
41
Multiples of 5
35
12 | N2.1
Properties of numbers
49
47
43
33
55
30
60
44
38
51
36
42
34
31
58
52
54
57
39
48
56
46
Multiples of 2
EXERCISE 3
1 a 22°C, 27°C, 21°C, 25°C
b 21°C
c 27°C 25°C 22°C 21°C 3°C
2 The temperature rose 10 degrees
3 5 degrees colder
37
Multiples of 3
45
32
Extension problem
6 There are 8 ways to write 150 as the sum
of four squares:
144 1 4 1 1 1 1
121 1 16 1 9 1 4
100 1 25 1 16 1 9
81 1 64 1 4 1 1
81 1 49 1 16 1 4
64 1 49 1 36 1 1
64 1 36 1 25 1 25
49 1 49 1 36 1 16
True
True
True
True
53
59
4
City
Temperature
difference (degrees)
Belfast
9
Liverpool
10
Cardiff
10
Newcastle
9
London
10
Plymouth
9
York
9
5 11 cm
6 a 6 degrees
c 12 degrees
7
a
c
e
g
b 6 degrees
b
d
f
h
23
0
4
24
8 a 24 1 2 5 22
c 5 2 7 5 22
e 2 2 4 5 22
1
21
25
25
b 23 1 4 5 1
d 22 2 4 5 26
f 21 1 3 5 2
5 a 11
b 36
6 a 5°C
b 29°C, 23°C, 0°C, 6°C
7 a
b
Any multiple of 10 that does not divide
exactly by 20, e.g. 10, 30, 50, 70, 90,
110, …
Any multiple of 20 must also be a
multiple of 10, so it is not possible to
put a number in section B
Home book
TASK 1
1
Numbers
3
6
7
5
4
9
11
15
12
Squares
9
36
49
25
16
81
121
225
144
2 a 32 1 42 5 52
c 92 1 122 5 152
b 62 1 82 5 102
d 52 1 122 5 132
3 150 5 1 1 49 1 100
150 5 4 1 25 1 121
150 5 25 1 25 1 100
TASK 2
9
23
4
21
2
0
–2
1
24
3
How well are you doing?
1 28 and 35
1 a 5, 10, 25
c 8, 16, 24
b 7, 14, 21
d 31, 62, 93
2 a 40, 50, 60
c 45, 63, 72, 81
e 24, 60, 72
b 40, 45, 50, 60, 75
d 16, 24, 32, 40, 72
3
41
Numbers from 30 to 50
2 A 219°C
B 16 degrees colder
C 222°C
3 25
43
46
47
49
50
Multiples of 4
Multiples of 3
36
30
33
39
45
42
44
32
40
48
31
34
35
37
38
4 A 4, 16, 36 or 64 B 1, 9, 25, 49 or 81
C Any even number that is not a
square number
D Any odd number that is not a
square number
N2.1 Properties of numbers | 13
TASK 3
1 a 22°C,
2 A fall of 9 degrees
3 a 26°C, 24°C, 2°C, 4°C
b i 10 degrees ii 2 degrees
iii 8 degrees
iv 8 degrees
4 50 degrees
Tier 2 CD-ROM
CHECK UP
1 a Bird
c Fish
b Diver
2Any three numbers from 11 to 19 inclusive
that sum to a multiple of 10
3 7
4 32, 56, 72
5 25
6No. 35 will not be shaded because all the
shaded numbers are even
7 a Fish
14 | N2.1
Properties of numbers
b About 30 metres