Fractions, percentages, ratio and proportion
Year 6 Autumn 4
Revise finding fractions of shapes
Previous learning
Core for Year 6
Extension
Use, read and begin to write these words:
Use, read and write these words:
Use, read and write these words:
fraction, part, whole, equivalent, cancel, ….
numerator, denominator, recurring…
proper fraction, improper fraction, mixed number,…
fraction, part, whole, equivalent, cancel, ….
numerator, denominator, recurring, …
proper fraction, improper fraction, mixed number,…
Revise finding fractions of shapes, e.g.
Revise finding fractions of shapes, e.g.
Revise finding fractions of shapes, e.g.
• What fraction of this shape is shaded?
• What fraction of this shape is shaded?
Write your answer as simply as possible.
• This rectangle has 13 identical shaded squares inside it.
fraction, part, whole, equivalent, numerator, denominator, ….
proper fraction (e.g.
mixed number (e.g.
• Shade
1
5
3
), improper
4
3
1 )…
4
fraction (e.g.
7
4
),
What fraction of the rectangle is shaded?
of this shape.
• Shade one third of the diagram.
• Shade more triangles so that
2
3
of the hexagon is shaded.
• How many more small squares need to be shaded so that
4
of the shape is shaded?
5
• If
3
8
• Shade
5
8
of this shape.
Express one quantity as a fraction of another, e.g.
• What fraction of an hour has passed between 2:10 pm and
2:30 pm?
• What fraction of 1 metre is 35 centimetres?
What fraction of 1 kilogram is 24 grams?
of a shape is shaded, what fraction is not shaded?
• What fraction of a whole turn is 90°, 36°, 120°, 450°?
Change an improper fraction to a mixed number, e.g.
33
8
1
Previous learning
Core for Year 6
to 4 8
Extension
Count along a counting stick from 0:
• in steps of one half, then one quarter, then one third.
Each time, label the divisions on the stick with equivalent
proper and improper fractions and mixed numbers.
© 1 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Extension
Change an improper fraction to a mixed number, e.g.
Change an improper fraction to a mixed number, e.g.
Convert between improper fractions and mixed numbers.
• Change
23
10
to a mixed number.
• Change
33
8
33
8
Use base 10 materials to represent a mixed number such
as 2 3 , showing that this is equivalent to 23 tenths.
10
Respond to questions such as:
to a mixed number.
= 33 ÷ 8 = 4
• How many fifths are there in 7 1 ?
5
1
8
• Harry runs 5 km each day. The track he runs is
Change a mixed number to an improper fraction, e.g.
Use diagrams to illustrate the mixed number
2 3 and recognise that this represents 23 tenths or
= 23 ÷ 10 = 2 3
10
10
3 =
2 10
2 × 10 + 3
10
=
km long.
How many times round the track does he run each day?
• Change 2 3 to an improper fraction.
10
23
10
1
4
23
.
10
23
10
Recognise the equivalence between fractions, e.g. between sixteenths, eighths, quarters and halves, and between hundredths, tenths and halves
Previous learning
Core for Year 6
Extension
Recognise the equivalence between quarters and eights,
thirds and sixths, and fifth and tenths.
Recognise the equivalence between sixteenths, eighths,
quarters and halves, and hundredths, tenths and halves.
Calculate equivalent fractions, e.g.
Use a fraction wall to identify equivalent fractions, e.g.
Use a fraction wall with 16 intervals to identify equivalences
between sixteenths, eighths, quarters and halves.
Recognise that any fraction can be: changed to an equivalent
fraction by multiplying both numerator and denominator by
the same number, e.g.
Use a number line with 100 intervals to identify equivalences
between hundredths, tenths, quarters and halves.
7
15
=
7×4
15 × 4
=
28
60
Recognise that :
Identify fractions with a total of 1, e.g.
• Think about the fraction
1
8
.
How many of them add to make 1?
10
100
=
1
10
50
100
=
5
10
© 2 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
=
1
2
20
100
=
2
10
25
100
=
1
4
= 0.2, etc.
75
100
=
3
4
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Reduce a fraction to its simplest form
Previous learning
Core for Year 6
Extension
Recognise simple equivalent fractions.
Reduce a fraction to its simplest form.
Reduce a fraction to its simplest form.
Recognise from diagrams that:
Recognise patterns in equivalent fractions, e.g. for one half,
one third, one quarter, one fifth and one tenth.
Know that if the numerator and the denominator have no
common factors, the fraction is expressed in its lowest terms
and is in its simplest form.
–
–
–
–
1
2
1
4
1
3
2
3
2
4
2
8
2
6
4
6
is equivalent to
is equivalent to
is equivalent to
is equivalent to
or
3
6
and
or
or
4
8
or
3
4
is equivalent to
Recognise that a fraction can be:
6
8
• reduced to an equivalent fraction by dividing both
numerator and denominator by the same number, which is
called cancelling, e.g.
4
12
8
12
5
20
Recognise patterns in equivalent fractions, e.g.
1
2
=
2
4
=
3
6
=
4
8
5
10
=
=
6
12
=
7
14
3
10
Respond to questions such as:
3
4
1
2
? To
. To
2
3
1
3
? To
1
4
?
.
4
2
4
=
1
4
=
3 × 10
10 × 10
=
30
100
8
3
4
=
• Write the fraction
5
20
as simply as possible.
• Write a different fraction that is equivalent to
• Fill in the missing numbers in the boxes.
=
=
Respond to questions such as:
• What fractions are equivalent to
1
2
5 ÷5
20 ÷ 5
• changed to an equivalent fraction by multiplying both
numerator and denominator by the same number, e.g.
…
and similar patterns for one fifth and one third.
• Find a fraction equivalent to
=
4
5
.
• Fill in the missing numbers in the boxes.
1
2
8
=
2
12
6
=
6
1
2
=
1
24
=
6
24
Relate finding fractions to division and use them as operators to find fractions of quantities, including several tenths and hundredths
Previous learning
Core for Year 6
Extension
Relate finding fractions to division and use division to find
simple fractions of quantities, e.g.
Relate finding fractions to division and find fractions,
including several tenths and hundredths, of quantities, e.g.
Calculate fraction of numbers and quantities, e.g.
Understand that:
•
12
3
• Find
is another way of writing 12 ÷ 3;
1
5
2
5
• when 3 whole cakes are divided equally into 4, each
person gets three quarters, or 3 ÷ 4 =
3
4
2
5
;
• Find
• finding one fifth is equivalent to dividing by 5, so find
1
of 30 by working out 30 ÷ 5.
5
© 3 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
• What is half of
of 20
of 20
of 20
20 ÷ 5 = 4
4×2=8
• What is
• What is
3
4
1
3
2
3
?
of 500?
of
3
4
of 100?
3
of 90
10
1
10
7
10
of 90
90 ÷ 10 = 9
of 90
9 × 7 = 63
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Recognise from diagrams and use relationships such as:
1
1
is half of
4
2
1
is one tenth
100
of
1
is
8
1
10
half of
1
4
• What is
1
4
Recognise from diagrams and use relationships such as:
1
4
1
10
Use halving to find halves, quarters and eighths, e.g.
of 64?
is half
• What is
is double
1
10
1
1
is half of
6
3
1
is one tenth
100
of
1
10
3
100
of £650?
£650 ÷ 10 = £65
£65 ÷ 10 = £6.50
£6.50 × 3 = £19.50
Without using a calculator, respond to questions such as:
1
5
of the pages.
How many pages have I read?
1
3
3
4
1
4
is half of
1
of £650
10
1
of £650
100
3
of £650
100
• One eighth of a number is 2. What is the number?
• What is
1
8
1
5
Find hundredths by finding one tenth of one tenth, e.g.
• What is one eighth of 64? Of 120?
• A book has 60 pages. I have read
1
2
1
of
5
is half of
• What is
Solve problems such as:
Extension
of 27?
of 24? Of 200?
• Joe spent three-quarters of his pocket money.
He has 50p left. How much pocket money did he have?
• Work out
• What is
2
3
• Calculate
• Find
3
10
3
5
• Calculate
of £40.
of 66?
3
4
Without using a calculator, respond to questions such as:
1
4
of 148.
• Li and Anil each have 45 books.
4
2
of Li’s books and
of Anil’s books are novels.
of £15.
5
3
How many more novels does Li have than Anil?
of 80 metres.
• James had £360. He spent
Using a calculator, respond to questions such as:
• Calculate
3
8
7
9
of it.
How much money did he have left?
of 980.
Understand percentage as the number of parts in every 100 and express halves, quarters, tenths and hundredths as percentages
Previous learning
Core for Year 6
Extension
Use, read and begin to write:
Use, read and write:
Use, read and write:
percentage, per cent (%), …
percentage, per cent (%), discount, increase, decrease, …
percentage, per cent (%), discount, increase, decrease, …
Understand percentage as the number of parts in every 100.
Understand percentage as the number of parts in every 100.
Understand percentage as the number of parts in every 100.
• Recognise the percentage of 100 Multilink cubes that are
red, yellow, blue, green, …
• Work out the percentage of the numbers 1 to 100 that are
even, are multiples of 5, have a digit 3, are greater than 40,
lie between 60 and 70, …
• Work out the percentage of pupils in the class who are
girls… are aged 11… have brown eyes…
Solve problems such as:
Solve problems such as:
Solve problems such as:
• 40% of a class of children are girls.
What percentage of the class are boys?
• 97% per cent of the Earth’s water is salt water.
The rest is fresh water.
What percentage of the Earth’s water is fresh water?
• A group of girls, boys and adults is at the Zoo.
52% of the group are girls. 10% are adults.
What percentage of the group are boys?
© 4 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Extension
Understand the relationship between simple fractions and
percentages, and know that::
Understand the relationship between fractions and
percentages, and know that:
Understand the relationship between fractions, decimals and
percentages, and know that:
one whole = 100%
one half = 50%
1
10
1
20% =
5
1
1% =
100
23
23% =
100
one quarter = 25%
one tenth = 10%
10% =
25% =
50% =
75% =
1
4
1
2
3
4
1
10
1
20% = 0.2 =
5
1
1% = 0.01 =
100
23
23% = 0.23 =
100
10% = 0.1 =
25% = 0.25 =
50% = 0.5 =
75% = 0.75 =
1
4
1
2
3
4
In addition know that:
1
8
= 12.5%
1
3
Respond to questions such as:
Respond to questions such as:
• Write in its lowest terms the fraction that is equivalent
to 40%.
• Write the fraction that is equivalent to 5%.
• What percentage is equivalent to
3
5
?
1
3
= 33 %
• Write the fraction which is equivalent to 0.8.
• Write
7
100
as a decimal.
Find simple percentages of shapes and of whole number quantities, e.g. 10%, 20%, 40% and 80 % by doubling, and 25% by finding a quarter
Previous learning
Core for Year 6
Extension
Find simple percentages of shapes, e.g.
Find percentages of shapes, e.g.
Find percentages of shapes, e.g.
• What percentage of each shape is shaded?
• What percentage of this grid is shaded?
• What percentage of this diagram is shaded?
• Shade 37.5% of this diagram.
• What percentage of this diagram is shaded?
Find percentages of quantities by using fractions, e.g.
Find percentages of quantities by using fractions, e.g.
Find percentages of quantities (in addition to those on the
left) by using fractions, e.g.
• To find 50%, find one half
• To find 50%, find one half.
• To find 90%, find one tenth and multiply by 9.
• To find 25%, find one quarter (half of one half)
• To find 25%, find one quarter (half of one half)
• To find 15%, find 5% and add it to 10%.
• To find 75%, find one quarter and add it to one half
• To find 75%, find 25% and add it to 50%.
• To find 1%, find one hundredth by dividing by 100.
• To find 30%, find one tenth and multiply by 3.
• To find 40%, find 10% then multiply by 4.
• To find 7%, find 1% and multiply by 7.
• To find 5%, find 10% then halve it.
• To find 12.5%, find one eighth.
© 5 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Extension
Without a calculator, find percentages of quantities such as:
Without a calculator, find percentages of quantities such as:
Without a calculator, e.g.
25% of £40
50% of £30
10% of £5
25% of £800
60% of £3
30% of 3 m
• 20% of a number is 6. What is the number?
10% of 2 kg
25% of 1 kg
20% of 80 cm
75% of £30
40% of 5 kg
75% of 2 litres
• 5% of a number is 16. What is 25% of the number?
Using a calculator, find percentages of quantities such as:
Using a calculator, e.g.
• Calculate 35% of 320.
• Calculate 17% of £3275.
Find
1
100
of 320, then multiply by 35.
3 2 0 ÷ 1 0 0 × 3 5 =
Find
1
100
of 3275, then multiply by 17.
3 2 7 5 ÷ 1 0 0 × 1 7 =
112.
556.75
Revise using ratio and proportion to describe the relationship between quantities, e.g. 3 red beads for every 2 blue beads, 3 out of every 5 beads are red
Previous learning
Core for Year 6
Extension
Use, read and begin to write these words:
Use, read and write these words:
Use, read and write these words:
for every, to every, in every, out of every, …
one/two/three times as many as, … proportion, …
for every, to every, in every, out of every, …
one/two/three times as many as, … ratio, proportion, …
for every, to every, in every, out of every, …
one/two/three times as many as, … ratio, proportion, …
Describe the relationship between two quantities using
statements such as:
Revise describing the relationship between two quantities
using statements such as:
• In this bead pattern:
• In this bead pattern:
Understand that a ratio can be simplified in the same way as
a fraction, by dividing each side by the same number, e.g.
the ratio of 5 : 10 is equivalent to a ratio of 1 : 2.
Solve problems such as:
1 bead in every 3 beads is red,
2 beads in every 6 beads are red, …
• 2 out of every 5 beads is red, and
3 out of every 5 beads are blue, so:
the proportion of red beads is 2⁄5;
2 beads in every 3 beads are blue,
4 beads in every 6 beads are blue, …
One third of all the beads are red.
Two thirds of all the beads are blue.
There are half as many red beads as blue beads.
There are twice as many blue beads as red beads.
• The ratio of fruit to cereal in a packet of cereal is 40 : 60.
Write this ratio in its simplest form.
the proportion of blue beads is 3⁄5.
• There are:
2 red beads to/for every 3 blue beads,
4 red beads to/for every 6 blue beads,…
There is:
1 red bead to/for every 2 blue beads,
2 red beads to every 4 blue beads, …
© 6 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
so the ratio of red beads to blue beads is 2 : 3.
Divide a quantity into two parts in a given ratio, e.g.
• Jamie makes a fruit salad using bananas, oranges and
apples.
For every one banana, he uses 2 oranges and 3 apples.
Jamie uses 18 fruits altogether.
How many oranges does he use?
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Solve simple problems involving direct proportion by scaling quantities up or down
Previous learning
Core for Year 6
Extension
Solve problems involving ratio or proportion, e.g.
Solve problems direct proportion by scaling quantities up or
down, e.g.
Solve problems involving ratio or direct proportion, e.g.
• 1 in every 4 of these squares is red.
• 1 in every 4 of these squares is red.
• Here is a recipe for carrot soup for 4 people.
Complete these statements.
Complete these statements.
For every red square, there are … blue squares.
in every 8 squares are red.
The number of blue squares is … times the number of red
squares.
in every 16 squares are red.
200 g carrots
2 onions
40 g butter
300 ml stock
• Write the recipe for carrot soup for 6 people.
3 in every … squares are red.
• This rectangle has six identical shaded squares inside it.
10 in every … squares are red.
• Deirdre is going to make some lemonade.
1
2
The finished drink should be ⁄3 lemon juice and ⁄3 water.
Jenny puts 100 ml of lemon juice in a glass.
How much water should she put with it?
• Peanuts cost 60p for 100 grams.
What is the cost of 350 grams of peanuts?
7.2cm
• Here is a recipe for pasta sauce.
300 g
120 g
75 g
• This is what you need to make 4 pancakes.
100 g flour
150 ml of milk
2 small eggs
What do you need to make 12 pancakes?
tomatoes
onions
mushrooms
Jamie makes the pasta sauce using 900g of tomatoes.
What weight of onions should he use?
• Chicken must be cooked for 50 minutes for every kg.
How long does it take to cook a 3 kg chicken?
length
The width of the rectangle is 7.2 centimetres.
Calculate the length of the rectangle.
• The distance from A to B is three times as far as from B to
C. The distance from A to C is 60 centimetres.
A
B
C
60cm
• This map has a scale of 1 cm to 6 km.
Calculate the distance from A to B.
Altburn
Ridlington
Carborough
The road from Ridlington to Carborough measured on the
map is 6.6 cm long.
What is the length of the road in kilometres?
• There are 30 children, in a class.
There are 3 boys for every 2 girls.
How many boys are there?
© 7 | Year 6 | Autumn TS4 | Fractions, percentages, ratio and proportion
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999