Teaching Fractions, Percentages and Proportions Presenter: Anna MacDougall [email protected].

Teaching Fractions, Percentages and Proportions

Presenter: Anna MacDougall

[email protected]

mailto:[email protected]

Goals for today:

• Understand the importance of multiplicative thinking in the development of proportional reasoning.

• Understand the relationship between fractions and percentages

• Understand the difference between ratio and proportion

• Have strategies and ideas for teaching percentages and proportional reasoning and extending Stage 6+ students.

Fractional LanguageDenominator: bottom digit, number of parts in a whole, what is being counted (Page 11)

Numerator: the top digit, shows the count, how many parts we have (Page 11)

Unit Fraction: has a numerator of 1 eg ¼ (Page 11)

Non-unit Fraction: has a numerator greater than 1 eg ¾ (Page 11)

Improper Fraction: numerator greater than denominator eg 5/4 (3)

Mixed Fraction: whole number and fraction eg 1¾ (Page 33)

Stage 6-7 Ordering and rounding of decimals. Ordering non-unit fractions for halves, quarters, thirds, fifths

and tenths. Use common factors to simplify fractions and find equivalent

fractions Fractions, decimals and percentages conversions for halves,

quarters, thirds, fifths and tenths.

Use multiplication and addition facts to find answers to fraction problems eg.

7/8 of 64 = 56. 1/8 = 8 so 7/8 = 7x8

Use mental strategies to solve word problems Including Place Value and Compensating from tidy numbers

3

15

3

16

Mystery Stars: Part to Whole

¼ of a number is 6.What is the number?If this is ½, what could the whole look like?

Mystery Stars:Whole to Part

There are 24 sweets on the cake and you eat ¾ of the cake. How many sweets do you eat?

Equation to solve

My Strategy/Explanation of solution

Holly started with a whole basket of chocolates. She ate 5/8 of them. That left 21 chocolates for her friends to eat. How many chocolates were there to start with?

My

grou

p’s

mos

t effi

cien

t str

ateg

yEquipm

ent/diagram

Finding Common Denominators:• 2/4 + 3/4 = 5/4 = 1 1/4• Use the fraction strips to solve this problem

⅝ + ¾ = ?

⅛ ⅛ ⅛ ⅛ ⅛ ¼¼¼

¼¼¼ ⅛ ⅛ ⅛ ⅛ ⅛

1

Multiplying Fractions• This is ⅓ of one whole strip.

• If it is cut into quarters, four equivalent pieces, what will each new piece be called?

⅓

121

1/4 x 1/3 = 1/12

http://www2.nzmaths.co.nz/LearningObjects/fractions/index.html

Fractions as Operators ContinuedScaffolding problems can help students work

through to difficult number properties:⅓ of 1/5 = ?

So what is ⅔ of 1/5 = ?

1/51/15 1/51/15 1/15

Decimats for DecimalsDecimals are special cases of equivalent

fractions. The decimal system says that fractions must be expressed as tenths, hundredths, etc.

Use your decimats to solve this problem:7 ÷ 4 = ?How does your answer compare with what the

calculator gives as the answer?

Stage 6-7• Use the Deci Pipes to first develop the concept

of decimals at Stage 5.• Use at Stage 6 for adding and subtracting of

decimals.• Double number line to develop links between

fractions, decimals and percentages.

12/40

75%50% 100%25%

1/4 3/4

Teaching Percentages

• Why do we have Percentages?It is a method of comparing fractions by giving

both fractions a common denominator, namely hundredths.

So it is useful to view percentages as hundredths.

In order to rename a common fraction as a percentage what do you need to know?

=And in order to find equivalent fractions you need to be multiplicative thinking.

1/10 10%

1/5 20%

1/2 50%

1/4 25%

1/8 12.5%

Estimate and find percentages of whole number amounts.

E.g. Find 25% of $80 25% = 1/4 so 25% = 1/4 of 80 = $20 E.g. Find 35% of $80 (harder!)

“Pondering Percentages” NS&AT 3-4.1(12-13)

Find 35% of 80

Let us make this into a word problem for our students to solve…..

We can use: •Square paper•A double number line•Find 10% strategy

Equation to solve


Wiremu wanted to buy a scooter, so he waited until it was on sale. It cost $80 and the discount was 35%. How much did Wiremu save? How much did he pay for the scooter? M

y gr

oup’

s m

ost

effici

ent s

trat

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Equipment/draw

ing

• Double number line to develop links between fractions, decimals and percentages.

10%

80400

75%50% 100%25%

20 60

Calculating PercentagesPercentage strips help students to see that

calculating percentages is like mapping a fraction onto a base of 100.

Leonne got 18 out of her 24 shots in.What was her shooting percentage?

10 20 240

Leonne’s Percentage

So Leonne’s 18 out of 24 maps onto 75 out of 100 (75%).

How does this relate to how you would calculate 18/24 as a percentage?

10 20 240

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0%

Using Percentage StripsUse your percentage strips to work out these

percentage problems:

Keith got 16 out of 25 for his spelling test.What percentage was that?

Bubble Puzzles for quick learners. These are great as they have to create problems instead of just solving them.

Renee paid $360 000 for her house.

Using What I Know to Calculate Percentages

Mind map what I know about 240•What is 10%•So what is 5%•What else can you work out?

•Try this with 360 – so if an I pod cost $360.00 and there was a 36% discount offered, how much would the ipod cost now?•Percentage games…..

Equation to solve


Room 13 decided to have a shared sushi lunch. There are 48 pieces of teryaki sushi and 16 pieces of avocado sushi. What percentage of the sushi was avocado sushi?

My

grou

p’s

mos

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cien

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yEquipment/

diagram/table

Ratios and RatesWhat is the difference between a ratio and a rate?Both are multiplicative relationships.

• A ratio is a relationship between to things that are measured by the same unit,

e.g. 4 shovels of sand to 1 shovel of cement (shovel vs shovel)

• A rate involves different measurement units,e.g. 60 kilometres in 1 hour (km vs hr)

A rate involves different measurement units,)

Ratio and proportionRATIO compares part to part,

‘two to every three’ or ‘two for every three’

What is the ratio of red to white?

What is the ratio of white to red?




one red to every two white


two white to every one red




1 to 2


2 to 1

RatiosMake up this paint recipe using unifix cubes.24 yellow cubes with 18 blue cubes (24:18).

If this recipe makes a big pot of paint, what might go into smaller pots with the same colour?

What is the key multiplication idea here?

Equivalent RatiosEqual subdivisions

Double Number Lines

This could be shown on a double number line:

0

0

24

18

12

90

0

24

18

12

9

4

3

Paint MixturesMake up these paint recipes.For each recipe show how it might be shown on

the circular colour chart.18 yellow

6 blue

14 yellow21 blue

Paint Mixture Recipe

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8 9

Number of parts (Blue)

Nu

mb

er o

f p

arts

(Y

ello

w)

Ratio and proportionPROPORTION compares a part to the whole,

‘two in every five’ (this is like a fraction)

What proportion of this shape is red?

What proportion of this shape is white?




four reds out of twelve squares in total


eight whites out of twelve squares in total




four out of twelve


eight out of twelve




four / twelve


eight / twelve





4

12

8

12





4

12

8

12

1

3

2

3

=

=

Contexts for proportional reasoning:

• Scaling, enlarging and shrinking• Duplication, e.g. recipes• Repeating, sequential patterns• Calculating best deals• Predicting probabilistic outcomes• Fair sharing• Measure, including unit conversion• Speed• Pulleys and gears• Density, e.g. mass per volume or flavour of mixtures• Consumption

How long is the short side of the grey rectangle?How long is the short side of the grey rectangle?long is the short side of the grey rectangle?

18 cm

24 cm

?

24cm

16cm

Ratio tables

× ⅔ × ⅔

Small Rectangle

Large Rectangle

18

?

24

1612

Ratio tables

× ¾

Small Rectangle

Large Rectangle

18

?

24

1612 × ¾

Ratio and proportion

Here is a recipe for raspberry ice cream.

This recipe is for 8 people. Josie makes enough raspberry ice cream for 12 people. How much cream does she use?

Fred makes raspberry ice cream in the same way. He uses 2½ kg of raspberries. How much sugar does he use?

Within measures

M1 M2Weight Cooking Time.5 102.5 ?

x5x5

Cooking a chicken takes 10 minutes for every 500g. How long to cook a 2.5kg chook.


Double Number Lines

It takes Cara 8 hours to groom 12 cars. Each car takes the same time. How long will it take her to groom 20 cars?

8

12

?

20

0

0

Hours

Cars


Equation to solve


Juice mix for camp consists of 2 cups of concentrate for every 3 cups of water. If there are 240 campers and each camper has ½ cup of juice, how much concentrate and water will be required? M

y gr

oup’

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effici

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Equipment/draw

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I have eight pizzas to share among three people.How much pizza does each person get?

Five girls share three pizzas equally and three boys share two pizzas equally. Each pizza is the same size.Who gets more pizza, a boy or a girl?How much more?

Problem One

The elephants are in proportion.What is the height of the little elephant?

Problem Two

Making bread!40 loaves of bread can be made from 6 kg of flour. How many loaves can be made from 15 kg of flour?


14 bags of meal can feed 56 calves

How many bags can feed 20 calves

Try this!

Extra for ExpertsAt Kiwi School 40% of the children are girls.At Tui School 70% of the children are girls.If the schools meet for sports day, what percentage of

all the children are girls?Kiwi school has twice as many children as Tui School.

In SummaryChildren must be multiplicative to solve percentage and

proportional problemsThey must be able to recognise common factorsGive children a range of contexts in order to solve

proportional problems

Figure It Out – eg. Problem Solving, Financial Literacy, Special Topics, Proportional Reasoning

‘High Achieving Students’ including Bright Sparks and Problem Solving– on www.nzmaths.co.nz

Level 5 units on www.nzmaths.co.nz

mathsplayground.com Problem solving, videos to show how to work it out, games, logic and reasoning

Learning Objects – www.nzmaths.co.nz Schools need to register for access (free to schools)

Otago Problem Solving see http://www.maths.otago.ac.nz/pc/PCdetails.php

Census at School http://www.censusatschool.org.nz/

Mathletics http://www.mathletics.co.nz/ there is a charge for this site

Problem Solving websites, eg http://www.transum.org/software/SW/Starter_of_the_day/index.htm

http://www.figurethis.org/index.html

http://illuminations.nctm.org/ActivitySearch.aspx

http://nzamt.org.nz/tasks

Websites to investigate for games and activities:

http://www.aplusmath.com/Games/index.html

http://www.ictgames.com/resources.html

http://www.bbc.co.uk/schools/websites/4_11/site/numeracy.shtml

http://nrich.maths.org/public/index.php

http://www.wicked.org.nz/Interactives/Maths

http://www.visualfractions.com/

http://www.abc.net.au/countusin/

http://www.transum.org/software/

http://www.woodlands-junior.kent.sch.uk/maths/

http://www.oswego.org/ocsd-web/games/Mathmagician/cathymath.html

http://www.studyladder.co.nz/learn/mathematics?lc_set=

http://tutpup.com/

http://nz.ixl.com/

http://www.sparklebox.co.uk/maths/#.UD_59KBwxFk

http://www.sheppardsoftware.com/math.htm

http://nrich.maths.org/public/

http://www.nrich.maths.org/public/

http://www.nzmaths.co.nz/

Teaching Fractions, Percentages and Proportions Presenter: Anna MacDougall [email protected].

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Transcript of Teaching Fractions, Percentages and Proportions Presenter: Anna MacDougall [email protected].