Fractions, decimals, percentage, ratio & proportion

decimal Fractions, decimals, percentage, ratio & proportion CPD Course 04/05 Nigel Davies half 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? 2

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” 3

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” 4

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 5

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. 6

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 7

Diagrammatic representations of numbers or measurements 8

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. 9

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’ … 10

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. 11

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 12

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. 13

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 14

Fractions Decimals Percentages 0 2 0 2 0 200% Making connections Mark each of these on each line : ½ 0.2 40% 150% 1.75 15

Fractions Decimals Percentages 0 2 0 2 0 200% Making connections Mark each of these on each line : ½ 0.2 40% 150% 1.75 ½ 0.5 50% 16

Fractions Decimals Percentages 0 2 0 2 0 200% Making connections Mark each of these on each line : ½ 0.2 40% 150% 1.75 1/5 0.2 20% 17

Fractions Decimals Percentages Identifying equivalents Use the empty number lines to identify the equivalents for a variety of quarters, fifths, tenths & hundredths: 21

Addressing the misconceptions 1 Fractions are always parts of one, never bigger than 1 22

Max. Waiting Time : 21/2 hours Addressing the misconceptions 2 Fractions are parts of shapes, not numbers in their own right 23

Addressing the misconceptions 3 The bigger the bottom number of the fraction, the bigger the value 24

Addressing the misconceptions 4 Decimals with more digits are bigger 25

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 26

Addressing the misconceptions 6 2:3 is the same as 2/3 27

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? 28

1 2 3 4 5 6 7 8 9 10 5 6 7 9 8 11 10 14 13 12 15 16 Changing the number line 29

100-square 30

Using a 100-square What proportion of the numbers in the chart : Are odd? Lie between 33 & 54 (exclusive)? Have at least one 3 as a digit? Are prime numbers? 1 in every 2, ½ , or 50% 20 out of the 100, 1/5 , or 20% 19 out of the 100, 19/100 , or 19% 25 out of the 100, 1/4 , or 25% 31

Using a 100-square What is the ratio of : Odd numbers to even numbers? Multiples of five to multiples of four? 50 to 50, or 1:1 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%? 32

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 …” 33

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. 34

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 35

“What else do I know?” 5% = 2% = 20% = 100% = 80 50% = 1% = 10% = 40% = 25% = 21/2% = 36

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? 10% is the same as 1/10 or divide by 10 10% of £120 = £12  2  2 5% of £120 = £ 6  2  2 21/2% of £120 = £ 3 171/2% = £21 37

FDPRP Webs 6  10 6 10 60% 30 50 3  5 3 5 “3 out of every 5” 0.6 1/5 of 3 60 100 0.2 x 3 38

Proportional change Change/convert 45miles to km 5 miles  8km 45 miles  x 9 39

Proportional change Change/convert 45miles to km 5 miles  8km 45 miles  x 9 x 9 40

Proportional change Change/convert 45miles to km 5 miles  8km 45 miles  72 km x 9 x 9 41

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” 42

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? 43

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 44

Useful resources Maths Pack 1 45

Useful resources Maths Pack 2 46

Useful resources Primary Games 1 47

Useful resources Fractions ITP 48

Useful resources Logotron : Visual Fractions 49

Another possible resource 50

Fractions, decimals, percentage, ratio & proportion