Transcript Fractions, decimals, percentage, ratio & proportion

Fractions, decimals,
percentage, ratio & proportion
half
CPD Course 04/05
Nigel Davies
decimal
fifth
Using FDPRP
Approximately, what is :
Your height in metres?
Your head circumference as a fraction of your height?
The ratio of your head circumference to your height?
The ratio of your height to your head circumference?
Your leg length as a proportion of your height?
Your leg length as a percentage of your height?
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Defining terms
There are 24 children in the class & 6 of them are boys.
Fraction
“One quarter (1/4) of the children in the class are boys”
Decimal
“0.25 of the children in the class are boys”
Percentage
“25% of the children in the class are boys”
Ratio
“The ratio of boys to girls in the class is 1 to 3, or 1:3, or there is 1
boy for every 3 girls”
Proportion
“One in every four of the children in the class is a boy”
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What about …
There are 40 teachers on a course & 5 of them are vegetarian.
Fraction
“One eighth (1/8) of the teachers are vegetarian”
Decimal
“0.125 of the teachers are vegetarian”
Percentage
“12.5% of the teachers are vegetarian”
Ratio
“The ratio of vegetarians to meat-eaters is 1 to 7, or 1:7, or there
is 1 vegetarian for every 7 meat-eaters”
Proportion
“One in every eight of the teachers is a vegetarian”
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Defining terms 2
Fractions, decimals, percentages, ratio & proportion are
different ways of expressing related ideas.
We might use them in one of the following ways :
Parts of a given group
A number of objects, a quantity or a measurement
Diagrammatic representations of numbers or measurements
A comparison of two parts, quantities or measurements
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Parts of a given group
Three quarters of the 60 cubes in a box are red.
0.25 of the 60 cubes in the same box are blue.
3 in every 5 of the people voting said ‘Yes’.
50% of the class of 24 children are girls.
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A number of objects, a quantity or a
measurement
Three and a half cakes
A 125% increase
£4.65
2.5 litres
3.25 metres of material
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Diagrammatic representations of
numbers or measurements
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A comparison of two parts,
quantities or measurements
Use 1 litre of red paint for every 2 litres of yellow paint.
In every 100g portion of a breakfast cereal, 80g is
carbohydrate.
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Progression
Reception :
Practical activities on grouping, sharing & comparing lead to
the idea of half full, half each …
Simple fractions (halves, quarters) appear in Key Stage 1 in
the context of time, shape & space and in doubling & halving.
Decimals are introduced in the context of money.
Fractions is a separate topic in Year 2.
Ratio appears in patterns that develop from ‘5 fingers on
every hand’, ‘four paws on every teddy’ …
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Years 3 & 4
Work on decimal place value extends work on whole number
place value.
Children are introduced to unit fractions, then fractions which
are several parts of a whole, mixed fractions & equivalent
fractions.
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Years 5 & 6
Work focuses on :
Relating fractions to division
Ordering numbers with up to 3 decimal places
Recognising the equivalence between fractions & decimals
Solving simple problems using ratio & proportion
Understanding percentage as the number of parts in every 100
Finding percentages of whole-number quantities
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Progression (contd.)
Calculations involving decimals are found in Key Stage 2 in
units of work on addition, subtraction, multiplication &
division and in solving word problems.
Fractions, decimals, percentages, ratio & proportion are linked
to problems involving ‘real life’, money & measurement.
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Children’s misconceptions
Fractions are always parts of one, never bigger than 1
Fractions are parts of shapes, not numbers in their own right
The bigger the bottom number of the fraction, the bigger the
value
Decimals with more digits are bigger
Percentages can never be bigger than 100%
2:3 is the same as 2/3
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Making connections
Mark each of these on each line :
½
0.2
40%
150% 1.75
Fractions
0
2
Decimals
0
2
Percentages
0
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200%
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Making connections
Mark each of these on each line :
½
0.2
40%
150% 1.75
Fractions
0
½
2
0.5
2
50%
200%
Decimals
0
Percentages
0
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Making connections
Mark each of these on each line :
½
0.2
40%
150% 1.75
Fractions
0
1/
5
2
0.2
2
Decimals
0
Percentages
0
200%
20%
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Making connections
Mark each of these on each line :
½
0.2
40%
150% 1.75
Fractions
0
2/
5
2
0.4
2
40%
200%
Decimals
0
Percentages
0
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Making connections
Mark each of these on each line :
½
0.2
40%
150% 1.75
Fractions
0
11/2
2
1.5
2
150%
200%
Decimals
0
Percentages
0
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Making connections
Mark each of these on each line :
½
0.2
40%
150% 1.75
Fractions
0
13 / 4
2
1.75
2
Decimals
0
Percentages
0
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175%
20
200%
Identifying equivalents
Use the empty number lines to identify the equivalents for a
variety of quarters, fifths, tenths & hundredths:
Fractions
Decimals
Percentages
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Addressing the misconceptions 1
Fractions are always parts of one, never bigger than 1
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Addressing the misconceptions 2
Fractions are parts of shapes, not numbers in their own right
Max. Waiting
Time :
21/2 hours
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Addressing the misconceptions 3
The bigger the bottom number of the fraction, the bigger the
value
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Addressing the misconceptions 4
Decimals with more digits are bigger
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Addressing the misconceptions 5
Percentages can never be bigger than 100%
I’ve got £50
I’ve got £100
I’ve got 50%
of your
amount
I’ve got £100
I’ve got £125
I’ve got 125%
of your
amount
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Addressing the misconceptions 6
2:3 is the same as 2/3
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Watching the video …
What do the children in Emma’s class already know about
fractions, decimals, percentages, ratio & proportion?
What do the children learn or consolidate in this lesson?
What were the successful features of the teaching in this
lesson?
Is there anything about the lesson that you would have done
differently?
How would you take forward the children’s learning?
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Changing the number line
1
5
6
2
7
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8
4
5
6
9
10
11
29
7
12
8
9
13
10
14
15
16
100-square
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Using a 100-square
What proportion of the numbers in the chart :
Are odd?
1 in every 2, ½ , or 50%
Lie between 33 & 54 (exclusive)?
20 out of the 100, 1/5 , or 20%
Have at least one 3 as a digit?
19 out of the 100, 19/100 , or 19%
Are prime numbers?
25 out of the 100, 1/4 , or 25%
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Using a 100-square
What is the ratio of :
Odd numbers to even numbers?
50 to 50, or 1:1
Multiples of five to multiples of four?
20 to 25 , or 4:5
What percentage of the numbers have only odd digits?
30%
What questions could you ask that have an answer of 10%?
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Fraction cubes
Ask the children to make a chain of 10 cubes, then break it
into two equal parts.
The children should record their work as :
“My chain is 10 cubes long.
There are … cubes in each half. ½ of 10 is …”
Make a chain of 12 cubes. Break it into 3 equal parts 
“My chain is 12 cubes long.
There are … cubes in each third. 1/3 of 12 is …”
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Fraction strips
Select five pupils, three of whom are girls.
“3/5 of these children are girls”
Explain that each girl will shade a part of the strip blue &
each boy will shade one part red.
How many different ways are there of shading 3/5 of the strip?
Choose strips with different numbers of parts & other
combinations of pupils.
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Decimal 3-in-a-row
A game for 2 players. Use number cards 1-8.
Each child in turn chooses two number cards, divides one by
the other (using a calculator, when required), & then marks
the answer on a 0-1 number line. Number cards can be used
more than once.
The first person to get three of their marks in a row wins.
1
0
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“What else do I know?”
5% =
2% =
20% =
1% =
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100% = 80
10% =
50% =
40% =
36
25% =
21/2% =
V.A.T. @ 171/2%
“How can we calculate VAT without a calculator?
Find the VAT on a meal costing £120
171/2% = 10% + 5% + 21/2% … how does that help?
2
2
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10% is the same as 1/10 or
divide by 10
10%
of £120 = £12
2
5%
of £120 = £ 6
21/
2%
of £120 = £ 3
2
171/2%
= £21
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FDPRP Webs
6  10
35
5
5
0.6
of 3
0.2 x 3
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60%
30
50
3
“3 out of
every 5”
1/
6
10
38
60
100
Proportional change
Change/convert 45miles to km
x9
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5 miles  8km
45 miles 
39
Proportional change
Change/convert 45miles to km
x9
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5 miles  8km
45 miles 
40
x9
Proportional change
Change/convert 45miles to km
x9
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5 miles  8km
45 miles  72 km
41
x9
Ideas for ratio
Use counting sticks & times tables to visualise equivalent
ratios.
Bead bracelets with differing ratios of colours :
“How many reds do I need if I the bracelet has 12 beads
altogether?
Recipes :
“Use 2 currants for every 3 raisins on top of the biscuits”
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Mixing paints
If the ratio of colours is the same, then the same shade
will be mixed :
Light blue is made from 4 parts blue & 1 part white.
“If I want 10 litres of light blue, how many litres of each
colour will I need?
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Useful resources
paper shapes
paper strips or strings
counting sticks
dominoes
packs of FDP cards
‘follow me’ cards
washing lines
money
squared paper
measuring equipment
card strips
base 10 apparatus
jam tart trays
an abacus
10 x 10 peg-boards
local paper advertisements
interlocking cubes
simple recipes
coloured tiles
calculators
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Useful resources
Maths Pack 1
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Useful resources
Maths Pack 2
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Useful resources
Primary Games 1
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Useful resources
Fractions ITP
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Useful resources
Logotron :
Visual Fractions
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Another possible resource
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Summary
When you teach fractions, decimals, percentages, ratio & proportion, aim
to :
Be aware of the progression from Reception to Year 6 & the aspects
children are likely to find difficult
Make totally clear to children the connections between the ideas involved
Ensure that children see FDPRP represented in different ways &
appreciate that the representations are of related ideas
Provide a wide range of opportunities to use FDPRP in different practical
contexts, including solving ‘real life’ problems
Help children to become confident users of the correct terms & notation
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