Year 3 - 4
MATHS NOTEBOOK
…………………………………………………………….
Compiled
by
E. C. Wood
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Number
Some Important Basics
Number Bonds
Number bonds are pairs of numbers that combine to make a third. Number bonds help show that
every whole number larger than one is made up of other whole numbers. 1 & 4, 2 & 3 both make
five. They are the number bonds of five.
Number Bonds means addition sums that make particular totals and their corresponding
subtraction facts (see example below).
Number Bonds of 10
10 – 10 = 0
10 – 9 = 1
10 – 8 = 2
10 – 7 = 3
10 – 6 = 4
10 – 5 = 5
10 – 4 = 6
10 – 3 = 7
10 – 2 = 8
10 – 1 = 9
10 – 0 = 10
0 + 10 = 10
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
7 + 3 = 10
8 + 2 = 10
9 + 1 = 10
10 + 0 = 10
Number bonds are vital for mental arithmetic. When you ‘partition’, or ‘decompose’ a
number you get number bonds..
For instance, knowing that can be 15 can be partitioned to 10 and 5 helps you perform
calculations with 15. You can calculate with the ’10′ part, and then with the ’5′ part, and
combine their answers. 5 and 10 are number bonds of 15.
Knowing and remembering the number bonds of 100 makes it easier to
handle money, or give change. If I know that 70 and 30 are number bonds
of 100, then I know instantly how much change to expect from a pound
when I buy a 70p packet of sweets.
Number
bonds can help you add. Knowing that 5 and 2 are number bonds of 7 makes
Place
Value
adding
7
much faster.
five to five
get 10,
andathen
add indicates
the remaining
two
Pupils needtotofive
understand
thatAdd
the position
of to
a digit
within
number
its value.
to example,
get 12! And
this
is just
scratching
the 4surface.
For
they
need
to know
that the
in 24 is 4 units but the 4 in 47 is 4 tens or 40.
1
Place Value
It is important to understand that the position of a digit within a number indicates its value.
For example, the 4 in 24 is 4 units but the 4 in 47 is 4 tens or 40.
Partitioning
A skill vital to many of the calculations in this booklet is that of partitioning, which is
related to the concept of place value. Partitioning means splitting a number into its digits
and giving each its place value, as in the examples below.
27
20
638
7
600
30
45·6
8 40
5
0·6
Relationship between Calculation Types
It is important to understand the relationship between addition, subtraction, multiplication
and division. They will learn that:
Addition and subtraction are inverse operations – that is, the opposite of one
another.
Adding 3 is the opposite of subtracting 3, so that if 6 + 3 = 9 then 9 – 3 = 6.
Multiplication and division are also inverse operations.
One way of viewing multiplication is that it is repeated addition: so, for example 5 x
3 (we can say ‘five threes’) is the same as adding five threes together:
5 x 3 = 3 + 3 + 3 + 3 + 3 = 15.
Division can also be viewed as repeated addition. If I divide 16 by 5, I can simply start
from zero and count up in groups of 5 until I have less than 5 left, therefore 16 ÷ 5 =
3 r 1.
Commutative Law
The commutative law says that when adding or multiplying numbers together, the order of
the numbers does not matter. So:
4 + 6 is the same as 6 + 4
20 x 3 is the same as 3 x 20
4 + 9 + 16 + 11 is the same as 16 + 4 + 11 + 9
From this we also find that:
If 9 – 6 = 3 then 9 – 3 = 6
If 20 ÷ 5 = 4 then 20 ÷ 4 = 5.
Distributive Law
The distributive law says that when multiplying or dividing a number, it is possible to split
the number, multiply or divide each part, then recombine it. For example:
4 x 26 is the same as (4 x 20) + (4 x 6)
28 ÷ 2 is the same as (20 ÷ 2) + (8 ÷ 2)
2
Addition
increase
add
more
altogether
sum
total
make
addition
greater
Partitioning - splitting numbers down into hundreds, tens and units
47
40
+
1. Split 47 and 76 into tens and
units.
76
7
70
6
2. Add the tens together and then
the units together.
40 + 70 = 110
7 + 6 = 13
3. Finally add the unit and tens
together
110 + 13 = 123
With partitioning, some people will find it easier to only split one of the numbers up.
47 + 76 = (47 + 70) + 6 = 123
Rearranging the numbers so as to add from the largest number
47 + 76 = 76 + 40 + 7
Using a number line
Using a combination of partitioning
and a number line can aid addition.
Example 1
76 + 47 =
+10
76
+10
86
+10
96
+10
106
1. Split the smaller of the two
numbers into hundreds tens units
etc. 47 = 4 tens and 7 units
+7
116 123
47 + 76 = 123
Example 2
3
2. Starting from the larger number
use the number line and count up in
tens (either as individual tens or all
together) and then the units.
47 + 76 =
+40
47
In this example the same method is
used. However, instead of counting
up steps of ten, the tens are just
added in one whole step.
+7
116
123
47 + 76 = 123
Adding the nearest 10 and Adjusting (for example when adding 19, 29 etc)
36 + 29
36 + 30 – 1
This method works when adding numbers which are one away
from the nearest ten, for example 19, 29 etc.
36 + 30 = 66
In this example view 29 as 30 - 1
66 – 1 = 65
Add 36 and 30 = 66
Finally subtract 1 from 66 = 65
Standard written method
Example 1
135 + 54 =
H T U
1 3 5+
5 4
1 8 9
Start by rewriting the sum down the page so that the
units, tens, hundreds etc are lined up in the same
column.
Add up each column, starting with the units, writing in
the answers as you go.
Units: 5 + 4 =9
Tens: 3 + 5 = 8
Hundreds: 1 + 0 = 1
Example 2
147 + 7 =
14 7+
76
223
1 1
In this example the columns add up to 10 or more, so it is
necessary to carry across into the next column.
Units: 7 + 6 = 13; write down the 3 and carry the one to the
tens column.
Tens: 4 + 7 + 1 = 12; write down the 2 and carry the one to
the hundreds column.
Hundreds: 1 + 1 = 2; write down the 2
4
Subtraction
minus
less
difference
subtract
between
leave
takeaway
Using a number line
Using a Number Line
Draw a line from 0 to the higher
number (the number you are
subtracting from).
Partition the number to be taken away 48 = 40 + 8.
Take away first the ten, then the units.
Example 1
86 - 48 =
-2 -6
0 38 40
-10
46
-10
56
-10
66
-10
76
86
86 - 48 = 38
Example 2
86 - 48 =
-2
In this example the same method is
used. However, instead of counting
down in steps of ten, the tens are
subtracted in just one step.
-40
-6
38 40
46
86
86 - 48 = 38
Example 3
103 - 92 = 21
+8
92
+10
90
In this example the number line is
used to find the difference by
counting up from the smaller of the
two numbers.
+3
100
103
Subtracting the nearest 10 and Adjusting (for example when subtracting 19, 29 etc)
5
86 - 29
This method works when subtracting numbers which are one
away from the nearest ten, for example 19, 29 etc.
86 – 30 + 1
In this example view 29 as 30 =+ 1
86 - 30 = 56
Subtract 30 from 86 = 56
56 + 1 = 57
Finally 56 + 1 = 57
Expanded Methods
These involve partitioning both numbers before subtracting. Partitioning means separating
the digits of the numbers e.g. 43 is 40 plus 3.
Example 1 - with no decomposition.
Split both of the numbers into tens and
units.
Subtract the ten( 40 -20) and the units (6-2)
Add the two answers together (20 +4) to
give you a final answer.
46 - 22 = 24
46 - 22
40 + 6
- 20 + 2
20 + 4 = 24
Example 2 - with decomposition.
73 - 47
60 13
70 + 3
40 + 7
20 + 6 = 26
-
In this example 7 units cannot be taken away from
3 units so decompositionis used.
The number 73 is changed from 70 + 3 to 60
+ 13 by moving ten from the tens column to
the units column.
Subtract the ten( 60 - 40) and the units (137).
Add the two answers together (20 +6) to
give you a final answer.
73 - 47 = 26
Example 3 - in this example decomposition is used twice.
714 - 286
100
600
-
110
14
700 + 10 + 4
200 + 80 + 6
400 + 20 + 8 + 428
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Standard written method
Example 1
987 – 76 =
9 8 776
911
Start by rewriting the sum down the page so that the units, tens,
hundreds etc are lined up in the same column.
Remember to put the bigger number on the top.
Subtract the bottom number from the top number in each
column, starting with the units, writing in the answers as you go.
Units: 7 - 6 = 1
Tens: 8 - 7 = 1
Hundreds: 9 - 0 = 9
Example 2.
138 – 29 =
2
1
1 3 82 9
1 0 9
In this example the number on the bottom of a column is larger than the
number on the top, so it is necessary to use decomposition (to borrow
from the next column to the left).
Units: because 9 is bigger than 8 a ten is taken from the tens column of
the top number to make it 18 - 9 = 9
Tens: because a ten has been moved to the units column the 3 is
crossed out and replaced by a 2. Therefore 2 -2 = 0.
Hundreds: 1 - 0 = 1; write down the 2
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Multiplication
multiply
lots of
multiples of
groups of
times
repeated addition
product
of
multiplied by
Mental Methods
Multiplication Tables. There’s no escaping them – these are still vital! By the end of
Year4 all facts up to 10 x 10 should be secure.
However, knowing tables facts is not just a matter of chanting tables: it is
important to know that 6x8 is 48 or that 48 ÷ 8 is 6 without having to recite the
rest of the 8 times table in order to reach the answer.
A good knowledge of tables facts enables pupils to concentrate more fully on
grasping new methods while still producing accurate work.
Using memorised tables facts
Combining knowledge of tables with multiplying by powers of 10
so, for example:
4 x 8 = 32
so 4 x 80 = 320
Repeated addition.
Multiplication can be seen as repeated addition.
2+ 2 + 2 + 2 + 2 = 10 is the same as 5 lots of 2 = 10 or 5 x 2 = 10.
Number lines
0
4
A number line can be used to
support repeated addition.
8
4 x 3 can be done by counting up
groups of 4.
12
Arrays.
These are a systematic arrangement of objects, usually in rows and columns.
5x3=
X
X
X
X
X
X
X
X
X
X
= 15
X
X
X
X
X
8
Partitioning numbers and multiplying each part of the number
So, for example: 64 x 2=
64
60 +
x2
120
x
1. Split 64 into tens and units:
60 + 4
2
2. Multiply both of the numbers by 2:
60 x 2 = 120
4x2=8
4
x2
+
8
= 128
3. Finally add the unit and tens together:
120 + 8 = 128
64 x 2 = 128
Written Method
1) The Grid method.
The grid method builds on the use of partitioning to multiply two-digit by one-digit
numbers and then three-digit by one-digit numbers.
Example 1
In example 1 the 45 has been partitioned
into 40 and 5
Each part is then multiplied by 3.
40 x 3 =120 and 5 x 3 =15
The products are then added back
together to give the answer.
120 + 15 = 135
45 x 3 =
X
40
5
3
120
15
135
45 x 3 = 135
Example 2
In example 2 a three-digit number
multiplied in the same way.
The 263 has been partitioned into 200,
60 and 3.
Each part is then multiplied by 4.
200 x 4 =800, 60 x 4 =240 and 3 x4 =12
The products are then added back
together to give the answer.
800 + 240 + 12 = 1052
263 x 4 =
X
200
60
3
4
800
240
12
1052
263 x 4 = 1052
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2) The Standard Compact Method
In example 1 a two digit number is
multiplied a single digit
Starting with the units, multiply each
of the numbers on the top row by 4.
9 x 4 = 36 - write in the 6 (unit) and
carry the 3 (tens) across to the tens
column.
1x4=4+3=7
Example 1
19 x 4 =
1 9x
4
7 6
3
19 x 4 = 76
Example 2
In example 2 a three digit number is
multiplied a single digit
Starting with the units, multiply each
of the numbers on the top row by 4.
5 x 4 = 20 - write in the 0 (unit) and
carry the 2 (tens) across to the tens
column.
4 x 4 = 16 + 2 = 18 - write in the 8
(tens) and carry the 1 (tens) across to
the hundreds column.
3 x 4 = 12 + 1 = 13 - write in the 3
(hundreds) and carry the 1 (thousand)
across to the thousands column.
345 x 4 =
3 4 5x
4
13 8 0
1 1 2
345 x 4 = 1380
Both the Grid Method and the Standard Compact Method can be used in later years for
long multiplication.
e.g. 345 x 23
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Division
divided by
share equally
share
divide
division
equal groups of
divided into
Mental Methods
Using memorised tables facts: from knowing that 6 x 4 = 24 you can deduce that 24
÷ 4 = 6 and 24 ÷ 6 = 4
Grouping
Using jottings to support mental calculation. Drawing pictures to represent the
calculation can be useful.
15 ÷ 5 means that 15 needs to be
split into 5 equal groups.
15 ÷ 5 = 3
15 ÷ 5 = 3
Number lines
+5
0
+5
5
The use of the number line depends on understanding
that division is related to repeated addition; so to divide
15 by 5 you can add in groups of 5 repeatedly until you
reach 15. Count how many groups of 5 you have to find
the answer.
+5
10
15
15 ÷ 5 = 3
Chunking
52 ÷ 4 = 13
In this example repeated addition is still used, however,
the first ten groups have been chunked together.
There are 13 groups of 4 in 52, therefore 52 ÷ 4 =13
10 lots of 4
+4
0
40
+4
44
+4
48
52
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Expanded Methods
Short division
As you look at this division sum, ask: ‘How many
twos in twenty four?’ or ‘What is 24 divided by 2?
Example 1.
24
2
2 )2 4
1 2
or
Rewrite the question, either way works.
1 2
2)2 4
Divide each digit of 24 by 2.
2÷2=1
4÷2=2
so 24 ÷ 2 = 12
Example 2.
665
7
7 )6 66 35
0 9 5
or
0 9 5 1
7) 6 6635 7
In example 2 it is necessary to use remainders
during the calculation
At the very start 7 does not fit into 6, so we say it
goes in zero times remainder 6. The remainder (6)
is placed next to the following number to make 66.
The process the process continues, but this time it
is how many times 7 goes into 66, which is 9 times
(because 7 goes into 63) remainder 3.
Continue this process until you reach the end of
the sum.
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Times Tables
There is no getting away from it, learning times table is very important.
Below are the times tables that you should try to know by the end of Year 4 and some
helpful hints that might make learning them easier.
The Best Trick
Remember that every multiplication
has a twin, which may be easier to
remember.
For example if you forget 8×5, you
might remember 5×8. This way, you
only have to remember half the
table.
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The 2 times table.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20.......
Remember, all numbers in the 2 times table are even.
All even numbers can be divided by 2.
The 3 times table.
3, 6, 9, 12, 15, 18, 21, 24, 27, 30.......
In the 3 times table, the digits will add up to a multiple of 3.
For example:
3 x 6 = 18.
The digits of 18 are 1 and 8.
1+8=9
9 is a multiple of 3. (3 x 3 = 9)
Another example:
3 x 7 = 21
The digits of 21 are 2 and 1.
2 + 1 = 3.
3 is a multiple of 3. (3 x 1 = 3)
This rule can help you with division too.
If you add the digits of a number together and the answer is a multiple of 3 (in the 3
times table), the number can be divided by 3.
E.g. 351
=3+5+1=9
So 351 can be divided by 3
The 4 times table.
4, 8, 12, 16, 20, 24, 28, 32, 36, 40......
Remember, all numbers in the 4 times table are even.
Double it and double again!
Look at 4 as 2 x 2, and since 2’s are doubles, just double the number then double
that answer.
For example
4x6
6 x 2 = 12
or
6 + 6 = 12
12 x 2 = 24
12 + 12 = 24
4 x 6 = 24
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The 5 times table.
5, 10, 15, 20, 25, 30, 35, 40, 45, 50......
Any number multiplied by 5 will always end 0 or 5
The 6 times table.
6, 12, 18, 24, 30, 36, 42, 48, 54, 60......
Every number in the 6 times table is even and if you add the digits together they will
always give you a multiple of 3.
6 x 7 = 42
4+2=6
6 is a multiple of 3 (in the 3 times table).
If you multiply 6 by an even number, they both end in the same digit.
Example: 6 × 2 = 12, 6 × 4 = 24, 6 × 6 = 36, etc
The 7 times table.
7, 14, 21, 28, 35, 42, 49, 56, 63, 70......
The 7 times table is one of the harder ones to learn, but this little trick below can help.
0
1
2
2
3
4
4
5
6
1. Start with a 3 x 3 grid.
2. Fill in the numbers as shown.
3. Starting from the top right hand
corner, put in the numbers 1 -9
going down the page.
4. You are now left with the 7 times
table.
07 14 21
28 35 42
49 56 63
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The 8 times table.
8, 16, 24, 32, 40, 48, 56, 64, 72, 80......
The numbers in the 8 × table are always even. This means they can be divided by 2 without
remainder. If it’s an odd number then it is not in the 8 × table!
Have a look at the 8 × table again. The unit digits
have a regular pattern - they go down in 2s.
8 × 4 is the same as 4 × 8 (= 32) so you can use
the 4 × table if you know it better.
The 9 times table.
9, 18, 27, 36, 45, 54, 63, 72, 81, 90......
Notice how the "units" place goes down: 9,8,7,6, ...? And at the same time, the
"tens" place goes up: 1,2,3,...?
If you add the answer's digits together, you get 9.
Example: 9×5=45 and 4+5=9. (But not with 9×11=99)
1. Hold your hands in front of you with your fingers spread out.
2. For 9 X 3 bend your third finger down. (9 X 4 would be the fourth finger etc.)
3. You have 2 fingers in front of the bent finger and 7 after the bent finger.
4. Thus the answer must be 27.
5. This technique works for the 9 times tables up to 10.
The 10 times table.
10, 20, 30, 40, 50, 60, 70, 80, 90, 100......
Any number multiplied by 10 will always end with a 0
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