One Pathway for Teaching Percentages

Where do Percentages sit in NZC?Level three Number and

AlgebraKnowledgeNA3-5• Know fractions and % in

everyday use

StrategiesNA3-1• Use a range of additive

and simple multiplicative strategies with whole numbers,fractions,decimals and %

Level four Number and Algebra

KnowledgeNA4-5• Know the equivalent

decimal and % for everyday fractions

Strategies• NA4-3Find

fractions,decimals and % of amounts expressed as whole numbers, simple fractions and decimals

Where do Percentages sit on the Number Framework?

KnowledgeStage 7• The student recalls

fraction,decimal,% conversions for halves,thirds,quarters,fifths and tenths.

Stage 8• The student recalls

fraction, decimal,% conversions for given fractions and decimals, eg 9/8 = 1.125 = 112.5%

StrategiesStage 7• The student can find simple

equivalent fractions and rename common fractions as decimals and %

Stage 8The student chooses from a

wide range of mental strategies to solve problems,

Eg 65% of 24 (50% = 12, + 10% = 2.4 ,+ 5%=

1.2) so the answer is 12 + 2.4+ 1.2 = 15.6)partitioning %

What do we mean by %

• Percentages are fractions with denominators hundredths

Some starters

Put an amount on the board eg $40

Students make that amount by as many different % as possible

Eg• 100% of 40 = 40• 50% of 80 = 40• 25% of 160 = 40• 200% of $20 = 40

100%$250

10% 5% 15% 10 2 3

100%$60

50% 25% 75%

100%$2

25% 5% 30%

100% 10%5.5kg

5% 1%

100% 10% 5%9

2.5%

100%$250

1% 2%

20%$250

10% 15%

100%90kg

80%

30%24kg

10% 11%

Box trails

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

This is a simple versionStudents draw up a 3 x 3 grid and pick 9 of these

0.5 0.15 0.7 0.01 0.10.9 0.2 2.1 1.5 0.1251.3 0.175 0.03 0.25 0.40.6 0.75 0.3 0.37 0.8

Call out % or the decimal and students pick the %

0.5(50%) 0.15(15%) 0.7(70%) 0.01(1%)0.1(10%)

0.9(90%) 0.2(20%) 2.1(210%) 1.5(150%)0.125(12.50%)

1.3(130%) 0.175(17.50%) 0.03(3%) 0.25(25%)0.4(40%)

0.6(60%) 0.75(75%) 0.3(30%) 0.37(37%)0.8(80%)

Teaching %

Where do we start?

• Knowledge is essential• Equipment very important at the start• Teach the strategy reasonably quickly and

then………• Application is crucial - students need lots of

opportunities to make the strategies work for them.

• Context• Revisit ideas frequently

Activities to build knowledge

2/10 0.2 20/100 20%

3/10

3/10 + 5/100

40%

65/100

0.7

7/100

Use a bead string to fill in the gaps

Fraction, percentage conversion

Use the 100 square to find the percentage to match the fraction

1/10 10% 1/20

1/5 1/25

1/4 2/10

1/8 3/5

3/4 1/100

Percentage Strips Imagine that the 100% is a row of elastic that when stretched stretches evenly. The 50 row is just the first 50 beads on the elastic stretched evenly to be the same length as the 100.

10 out of 50 maps on to 20% 35 out of 50 maps on to 70%

Can you use this double number line to map as many values out of 60 as you can onto a percentage.

0 30 60 0% 50% 100% Work through Hot Shots Book 7 – Teaching fractions, decimals and percentages

What we going to look at to day?

• How to use double number lines to answer % problems

• Using the teaching model• What our students need to know to do this work?

• Resources for practising and sustainability

Teaching progression

Materials

Images

Knowledge

Start by:•Using materials, diagrams to illustrate and solve the problemProgress to:•Developing mental images to help solve the problemExtend to:•Working abstractly with the number property

PercentagesWhat type of problem do we expect to meet in years 9/10 ?

•Finding one number as a % of another•Finding a % of a quantity•Finding the total given a % of the total•Increase/decrease by a %•Finding the original after an increase/decrease•GST and other problems

Using double number lines to solve %

problems

20% of 150 is 30

•20% of 150 is

•20% of is 30

• % of 150 is 30

0% 20% 100%

20% of 150 is

Question (in context)

The local dairy farmer is selling 20% of his herd of 150 cows. How many is he selling? Rewrite in

maths language

150

How do we use the lines to get the answer?

0% 20% 100%

150

20 x 5 = 100

150 divided by 5 = 30

0% 20% 100%

150Find 10% : 150 divided by 10

So 10% = 15

So 20% =30

0% 20% 100%

15015 x 10

10 x 1010 x 2

15 x 2

In a berry mix there are 30% strawberries and 20% raspberries and the rest are blackberries. In a 500gm punnet of berries what weight are the strawberries?

There are 30 students in 9CT and 40% are girls. How many girls are there in the class?

40% of 30 is

30% of 500gm is

Abigail is working on a set of 50 number problems and she has just finished question 28. What % of the questions has she finished.Mr Sharp spent the day at the races and his horses were placed in 8 out of 20 races. In what % of the races was he successful?

28 is % of 50

8 is % of 20

30% of the swimming team are girls. If there are 18 girls . How many are in the team altogether?

18 is 30% of

Activities to practice these skills

• Activity 1

• Activity 3 Note: these are not teaching activities

FIO’s• Fully grown Page 9• The Percentage Game page 14/1• Laser Blazer page 12/13

What do students need to be able to do before we use

this?

Have a sound knowledge of percentage

To know answers must be in context with correct

unitsCommon factors and lowest common

multiples

Recall of all multiplication and division facts

•AM/AP

Discounts, markups, inflation etc

Revision of % knowledge

Starter pack

Increasing/decreasing by a %

Decreasing by a %Sarah went shopping for a new bike which cost $350 When she got to town there was a sale and she got 20% off the price, What did she pay?

Did she pay more or less?

How much less?

So instead of paying 100% she only paid?

Show all this on the number lines

0% 80% 100%

$350

Increasing by a percentageThe value of a $400 antique vase has been increased by 20%. What is its value now? What questions do we ask?

120% of 400 is

Or divide 400 by 10 (to get 10%) and multiply by 12.

X 4 X 40% 120%100%

$400

Moving to number properties20% of 150 is ?

Now is time to link what they know about % with decimal fractions.

How else can we write this?

What does 20% actually mean?

How could we do this without the number line?

For some students this stage will be a long time coming! For others they will tell you.

Now might be the time to bring in a calculator and some more “awkward” q’s

Activity 5

Dominoes - using number properties

Finding the original amount after a % increase/decrease

Example: After an increase in his weekly wage of 20% Joe has $480.What was his wage before the increase?

Talk through and write the maths question

$480 is 120% of Complete a number line with this information

0% 100% 120%

$480

Revision (maintenance)

Some activities to use to give students constant revision.

Activities 2, 4, 6, 7 and 8