© boardworks ltd 2005 1 of 54 n3 fractions ks4 mathematics
TRANSCRIPT
© Boardworks Ltd 2005 1 of 54
N3 Fractions
KS4 Mathematics
© Boardworks Ltd 2005 2 of 54
N3.1 Equivalent fractions
N3 Fractions
Contents
N3.2 Finding fractions of quantities
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.5 Multiplying and dividing fractions
© Boardworks Ltd 2005 3 of 54
Look at this diagram:
3
4=
6
8
×2
×2
=18
24
×3
×3
Equivalent fractions
© Boardworks Ltd 2005 4 of 54
Look at this diagram:
2
3=
6
9
×3
×3
=24
36
×4
×4
Equivalent fractions
© Boardworks Ltd 2005 5 of 54
Look at this diagram:
18
30=
6
10
÷3
÷3
=3
5
÷2
÷2
Equivalent fractions
© Boardworks Ltd 2005 6 of 54
Equivalent fractions spider diagram
© Boardworks Ltd 2005 7 of 54
Cancelling fractions to their lowest terms
A fraction is said to be expressed in its lowest terms if the numerator and the denominator have no common factors.
Which of these fractions are expressed in their lowest terms?
14
16
20
27
3
13
15
21
14
35
32
15
Fractions which are not shown in their lowest terms can be simplified by cancelling.
7
8
5
7
2
5
© Boardworks Ltd 2005 8 of 54
Mixed numbers and improper fractions
When the numerator of a fraction is larger than the denominator it is called an improper fraction.
For example,
15
4is an improper fraction.
We can write improper fractions as mixed numbers.
15
4can be shown as
15
4= 3
3
4
© Boardworks Ltd 2005 9 of 54
Improper fraction to mixed numbers
Convert to a mixed number.378
378
=88
+ + +88
88
88
+58
581 + 1 + 1 += 1 +
= 4 5
8437 ÷ 8 = 4 remainder 5 37
8= 4
5
84This is the number of times 8 divides into 37.
4
This number is the remainder.
5
© Boardworks Ltd 2005 10 of 54
227733
Mixed numbers to improper fractions
Convert to a mixed number.273
273 =
271 + 1 + 1 +
=77
+ + +77
77
27
=237
To do this in one step,
=
Multiply these numbers together …
… and add this number …
… to get the numerator.237
© Boardworks Ltd 2005 11 of 54
Find the missing number
© Boardworks Ltd 2005 12 of 54
N3.2 Finding fractions of quantities
Contents
N3 Fractions
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.1 Equivalent fractions
N3.5 Multiplying and dividing fractions
© Boardworks Ltd 2005 13 of 54
Finding a fraction of an amount
We can see this in a diagram:
23
of £18 = £18 ÷ 3 × 2 = £12
23
of £18?What is
© Boardworks Ltd 2005 14 of 54
Let’s look at this in a diagram again:
710
of £20 = £20 ÷ 10 × 7 = £14
710
of £20?What is
Finding a fraction of an amount
© Boardworks Ltd 2005 15 of 54
56
of £24 =16
of £24 × 5
= £24 ÷ 6 × 5
= £4 × 5
= £20
56
of £24?What is
Finding a fraction of an amount
© Boardworks Ltd 2005 16 of 54
What is of 9 kg?4
7
To find of an amount we can multiply by 4 and divide by 7.4
7
We could also divide by 7 and then multiply by 4.
4 × 9 kg = 36 kg
36 kg ÷ 7 = =36
7kg 5
1
7kg
Finding a fraction of an amount
© Boardworks Ltd 2005 17 of 54
When we work out a fraction of an amount we
multiply by the numerator
and
divide by the denominator
For example,
23
of 18 litres = 18 litres ÷ 3 × 2
= 6 litres × 2
= 12 litres
Finding a fraction of an amount
© Boardworks Ltd 2005 18 of 54
To find of an amount we need to add 1 times the amount to two fifths of the amount.
251
1 × 3.5 m =3.5 m and2
5of 3.5 m = 1.4 m
so, of 3.5 m =2
51 3.5 m + 1.4 m =4.9 m
What is of 3.5m?2
51
Finding a fraction of an amount
We could also multiply by 7
5
© Boardworks Ltd 2005 19 of 54
MathsBlox
© Boardworks Ltd 2005 20 of 54
N3.3 Comparing and ordering fractions
Contents
N3 Fractions
N3.4 Adding and subtracting fractions
N3.2 Finding fractions of quantities
N3.1 Equivalent fractions
N3.5 Multiplying and dividing fractions
© Boardworks Ltd 2005 21 of 54
Using decimals to compare fractions
Which is bigger or ?38
720
We can compare two fractions by converting them to decimals.
38
= 3 ÷ 8 = 0.375
= 7 ÷ 20 = 0.35720
0.375 > 0.35
so 38
>720
© Boardworks Ltd 2005 22 of 54
Which is bigger or ?38
512
Another way to compare two fractions is to convert them to equivalent fractions.
First we need to find the lowest common multiple of 8 and 12.
The lowest common multiple of 8 and 12 is 24.
Now, write and as equivalent fractions over 24. 38
512
38
=24
×3
×3
9and
512
=24
×2
×2
10so,
38
512
<
Using equivalent fractions
© Boardworks Ltd 2005 23 of 54
Ordering fractions
© Boardworks Ltd 2005 24 of 54
Mid-points
© Boardworks Ltd 2005 25 of 54
N3.4 Adding and subtracting fractions
Contents
N3 Fractions
N3.3 Comparing and ordering fractions
N3.2 Finding fractions of quantities
N3.1 Equivalent fractions
N3.5 Multiplying and dividing fractions
© Boardworks Ltd 2005 26 of 54
When fractions have the same denominator it is quite easy to add them together and to subtract them.
For example,
3
5+
1
5=
3 + 1
5=
4
5
We can show this calculation in a diagram:
+ =
Adding and subtracting fractions
© Boardworks Ltd 2005 27 of 54
7
8–
3
8=
7 – 3
8=
4
8
Fractions should always be cancelled down to their lowest terms.
1
2 =1
2
We can show this calculation in a diagram:
– =
Adding and subtracting fractions
© Boardworks Ltd 2005 28 of 54
1
9+
7
9+
4
9=
1 + 7 + 4
9=
12
9
Top-heavy or improper fractions should be written as mixed numbers.
= 1 3
9
1
3 = 1 1
3
Again, we can show this calculation in a diagram:
+ + =
Adding and subtracting fractions
© Boardworks Ltd 2005 29 of 54
Fractions with common denominators
Fractions are said to have a common denominator if they have the same denominator.
For example,
1112
,412
and512
all have a common denominator of 12.
We can add them together:
1112
+412
+512
=11 + 4 + 5
12=
2012
= 1 812
= 1 23
© Boardworks Ltd 2005 30 of 54
Fractions with different denominators
Fractions with different denominators are more difficult to add and subtract.
For example,
We can show this calculation using diagrams:
What is 56
–29
?
–
1518
–418
=
=15 – 4
18=
1118
© Boardworks Ltd 2005 31 of 54
+
1220
+1520
=
=12 + 15
20=
2720
= 1 720
What is 35
+34
?
Using diagrams
© Boardworks Ltd 2005 32 of 54
–
2520
–1420
=
=25 – 14
20=
1120
What is –710
?1 14
Using diagrams
© Boardworks Ltd 2005 33 of 54
Using a common denominator
1) Write any mixed numbers as improper fractions.
1 34
= 74
2) Find the lowest common multiple of 4, 9 and 12.
The multiples of 12 are: 12, 24, 36 . . .
36 is the lowest common denominator.
What is +19
?134
+512
© Boardworks Ltd 2005 34 of 54
3) Write each fraction over the lowest common denominator.
74
=36
×9
×9
63 19
=36
×4
×4
4 512
=36
×3
×3
15
4) Add the fractions together.
3663
+364
+3615
=36
63 + 4 + 15=
3682
= 2 3610
= 2 185
What is +19
?134
+512
Using a common denominator
© Boardworks Ltd 2005 35 of 54
Adding and subtracting fractions
© Boardworks Ltd 2005 36 of 54
Using a calculator
It is also possible to add and subtract fractions using the
key on a calculator.abc
For example, to enter 84
we can key in abc4 8
The calculator displays this as:
Pressing the = key converts this to:
© Boardworks Ltd 2005 37 of 54
To calculate: 23
+45
using a calculator, we key in:
abc2 3 + abc4 5 =
The calculator will display the answer as:
We write this as 1 157
Using a calculator
© Boardworks Ltd 2005 38 of 54
Fraction cards
© Boardworks Ltd 2005 39 of 54
N3.5 Multiplying and dividing fractions
Contents
N3 Fractions
N3.4 Adding and subtracting fractions
N3.3 Comparing and ordering fractions
N3.2 Finding fractions of quantities
N3.1 Equivalent fractions
© Boardworks Ltd 2005 40 of 54
When we multiply a fraction by an integer we:
multiply by the numerator
and
divide by the denominator
For example,
49
54 × = 54 ÷ 9 × 4
= 6 × 4
= 24
Multiplying fractions by integers
This is equivalent to of 54.4
9
© Boardworks Ltd 2005 41 of 54
57
12 × ?What is
57
12 × = 12 × 5 ÷ 7
= 60 ÷ 7
=607
= 8 47
Multiplying fractions by integers
© Boardworks Ltd 2005 42 of 54
Using cancellation to simplify calculations
712
What is 16 × ?
We can write 16 × as:712
161
×712
4
3=
283
=139
© Boardworks Ltd 2005 43 of 54
825
What is × 40?
We can write × 40 as:825
825
×401
8
5=
645
=4512
Using cancellation to simplify calculations
© Boardworks Ltd 2005 44 of 54
Multiplying a fraction by a fraction
To multiply two fractions together, multiply the numerators together and multiply the denominators together:
38
What is × ?25
38
45
× =1240
3
10
=310
We could also cancel at
this step.
© Boardworks Ltd 2005 45 of 54
56
What is × ?12255
Start by writing the calculation with any mixed numbers as improper fractions.
To make the calculation easier, cancel any numerators with any denominators.
1225
356
× =
7
5
2
1
145
= 2 45
Multiplying a fraction by a fraction
© Boardworks Ltd 2005 46 of 54
Multiplying fractions
© Boardworks Ltd 2005 47 of 54
Dividing an integer by a fraction
13
What is 4 ÷ ?
13
4 ÷ means, “How many thirds are there in 4?”
Here are 4 rectangles:
Let’s divide them into thirds.
4 ÷ = 1213
© Boardworks Ltd 2005 48 of 54
25
What is 4 ÷ ?
25
4 ÷ means, “How many two fifths are there in 4?”
Here are 4 rectangles:
Let’s divide them into fifths, and count the number of two fifths.
4 ÷ = 1025
Dividing an integer by a fraction
© Boardworks Ltd 2005 49 of 54
34
6 ÷ means, ‘How many three quarters are there in six?’
6 ÷ = 6 × 414
= 24
So,
6 ÷ = 24 ÷ 334
= 8
We can check this by multiplying.
8 × = 8 ÷ 4 × 334
= 6
34
What is 6 ÷ ?
Dividing an integer by a fraction
There are 4 quarters in
each whole.
© Boardworks Ltd 2005 50 of 54
Dividing a fraction by a fraction
18
What is ÷ ?12
means, ‘How many eighths are there in one half?’18
÷12
Here is of a rectangle:12
Now, let’s divide the shape into eighths.
= 418
÷12
© Boardworks Ltd 2005 51 of 54
45
What is ÷ ?23
To divide by a fraction we multiply by the denominator and divide by the numerator.
45
23
÷ can be written as54
23
×
Swap the numerator and the denominator and multiply.
54
23
× =1012
=56
Dividing a fraction by a fraction
This is the reciprocal of .45
© Boardworks Ltd 2005 52 of 54
67
What is ÷ ?353
Start by writing as an improper fraction. 353
185
353 =
185
÷67
=185
×76
3
1
=215
=154
Dividing a fraction by a fraction
© Boardworks Ltd 2005 53 of 54
Dividing fractions
© Boardworks Ltd 2005 54 of 54
Multiplying and dividing are inverse operations.
multiply by the numerator
and
divide by the denominator
When we multiply by a fraction we:
When we divide by a fraction we:
divide by the numerator
and
multiply by the denominator
Multiplying and dividing by fractions