Fractions Workshop

Have a go at the Fraction Hunt or Letter Fractions

on your table while you are waiting!

4 out of 3 people have trouble with fractions

Proportions and Ratios Workshop

Session 1: 9:15-10:30Warm-up, Police Test, What is Proportional Thinking?- Key IdeasModel “Fraction Circles” lesson. Play Dotty pairs.

Session 2: 11:00-12:15Stage 7 & 8 Equivalent Fractions.Play games/become familiar with activities.

Session 3: 1:00-2:45Stage 6+ Decimal Fractions, Percentages, Ratios and RatesExplore lessons from Book 7.

“He has no sense of proportion”

Proportions and Ratios Workshop

Warm Up- I have… Who has…

Reviewing Multiplication and Division…(Stage 7 = whole numbers, Stage 8 = decimals)

7 x 0.297.2 ÷ 4

Tidy Numbers Place Value

Proportional Adjustment

Written form

A sample of numerical reasoning test questionsas used for the NZ Police

recruitment

½ is to 0.5 as 1/5 is to

a. 0.15

b. 0.1

c. 0.2

d. 0.5

1.24 is to 0.62 as 0.54 is to

a. 1.08b.1.8c. 0.27d.0.48

Travelling constantly at 20kmph, how long will it take to travel 50 kilometres?

a. 1 hour 30 minsb. 2 hoursc. 2 hours 30 minsd. 3 hours

If a man weighing 80kg increased his weight by 20%, what would his weight be now?

a. 96kgb. 89kgc. 88kgd. 100kg

Proportional Thinking

To be proportional thinker you need to be able to think multiplicatively

How do you describe the change from 2 to 10?

Additive Thinking: views the change as an addition of 8

Multiplicative Thinking:Describes the change as multiplying by 5

Developing Proportional thinking

Fewer than half the adult population can be viewed as proportional thinkers

And unfortunately…. We do not acquire the habits and skills of proportional reasoning simply by getting older.

What is proportional thinking?

Here are twelve jellybeans to spread out evenly on top of the cake. You eat one third. How many jellybeans do you eat?

What is ¾ of 28?

Stage 1 Stage 2-4 (AC)

Stage 5 (EA) Stage 6 (AA)

Unequal Sharing

Equal Sharing Using addition, 5+5+5=15

Using addition &

multiplication

Stage 4 Can write unit fraction symbols, e.g. ½

Stage 5 Can compare and order unit fractions, e.g. ¼, 1/3, ½

Stage 6 Describes the size of fractions with reference to both numerator and denominator,e.g which number is the same as eight sixths?

Revision of Early Proportional Thinking (Level 1 - 3)

Key Idea 1Introduce fractional language with care

Use words first then introduce symbols.

e.g. ‘1 fifth’ not 1/5

How do you explain the top and bottom numbers?

1

2

The number of parts chosen

The number of parts the whole has been divided into

+ = “I ate 1 out of my 2 sandwiches, Kate ate 2 out of her 3 sandwiches so together we ate 3 out of the 5 sandwiches”!!!!!

12

23

35

The problem with “out of”

86

x 24 = 2 out of 3 multiplied by 24!23

= 8 out of 6 parts!

one quarter

€

1

4

Key Idea 2

Connect different representations

words - symbols – regions - sets - numberlines

Shapes/Regions(Continuous

models)

Sets(Discrete Models)

Year 7 Student Responses

What is this fraction? 5/2

How do I write 3 halves?

2 fifths, five lots of halves, tenth, five twoths

3 1/2 1

/3

Key Idea 3Fractions can be more than 1Push over 1 whole to consolidate the meaning of fraction symbols and how 1 whole is created.

Sam had one half of a cake, Julie had one quarter of a cake, so Sam had most.

True or False or Maybe

Sam Julie

Key Idea 4Fractions are always relative to the whole. Halves are not always bigger than quarters, it

depends on what the whole is.

What is B?

A A

B B B B

C

D D D D D D D D

Communicating this idea to students

(what’s the part, what’s the whole?)

5 children share three chocolate bars evenly. How much chocolate does each child receive?

3 ÷ 5

1/2 1/

2 1/2 1/

2 1/2

What are these pieces called?

1/2 +

1/10 =

2/12 !!

What would have been an easier way to solve this?

(Wafers, Book 7 p.16)

Key Idea 5Division is the most common context for fractions i.e. quotients

3 ÷ 5

1/5+1/5+1/5 =3/5

Y7 response: “3 fifteenths!” Why?

Choose your share of chocolate!

Which letter represents 5 halves?

0 1 2 3

A B C D E F

Key Idea 6

Fractions are numbers as well as operators

e.g. 3/5 is a number between 0 and 1 (number)

e.g. Find 3/5 of 100 (operator)

Communicating fractions as operators to students using double number lines

Put a peg on where you think 3/5 will be. (Fractions as a number). How will you work it out?

35

0 1

0 100

15

20 60

Use a bead string / double number line to find 3/5 of 100. (Fractions as an operator).

How will you work it out?

x3

80

20 2020

The distance between Masterton and Wellington is 80 kilometres. Hemi has travelled 3/4 of the trip. How many kilometres is that?

Operating with fractions - Result Unknown

20

0 80x3 6020

4/9 of ? = 16, 16 is four ninths of what number?

Draw a diagram to help to solve this problem.

Tino has travelled 16km, which is four ninths of his journey. How much further does he need to travel?What is the number sentence for this problem?

16

4 4 4 4

Operating with fractions - Change Unknown

0 36

÷ 4

416 4

90 1

X 9

Key Idea

Go from part-to-whole (change unknown) as well as whole-to-part (result unknown) with shapes and sets.

Your turn….

If this is one quarter of a shape.

What does the whole look like?

12 is two thirds of what number?

Lesson Modelling Fraction Circles (Book 7, p20)

Play “Dotty Pairs” game

Session 2: Stages 7 & 8

Finding Fractions

Throw 2 dice and make a fraction,

e.g. 4 and 5 could be 4 fifths of 5 quarters.

Try and make a true statement each time the dice is thrown.

Throw dice 10 times, Miss a go if you cannot place a fraction.

12 is 2/3 of a number what is the number? Stage 7

There are 21 boys and 14 girls in Anna’s class. What percentage of Ana’s class are boys? Stage

8 It takes 10 balls of wool to make 15 beanies, how many balls of wool does it take to make 6 beanies?

Moving to Stage 7 and 8 (Level 4-5)

Which of these numbers is the same as two thirds? Stage 7Which of these numbers is the smallest?

0.478 0.8 0.39

Which of these fractions is the smallest? Stage 8Write 137.5% as a decimal

What is 3/8 as a decimal?

Equivalent Fractions

ObjectivesTeaching Fractional equivalence

In order to apply understanding to:•Adding and subtracting with fractions•Multiplying and dividing with fractions

What do students need to know about fractions before Stage 7?

Any fraction equivalence??

Stage 7 (AM) Key Ideas (level 4)Fractions and Decimals

• Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3

• Find equivalent fractions using multiplicative thinking, and order fractions using equivalence and benchmarks. e.g. 2/5 < 11/16

• Convert common fractions, to decimals and percentages and vice versa.

• Add and subtract related fractions, e.g. 2/4 + 5/8

• Add and subtract decimals, e.g. 3.6 + 2.89

• Find fractions of whole numbers using multiplication and division e.g.2/3 of 36 and 2/3 of ? = 24

• Multiply fractions by other factions e.g.2/3 x ¼

• Solve measurement problems with related fractions, e.g. 1½ ÷ 1/6 = 9/6 ÷ 1/6 =9

• Solve division problems expressing remainders as fractions or decimals e.g. 8 ÷ 3 = 22/3 or 2.66

Percentages

• Estimate and solve percentage type problems such as ‘What % is 35 out of 60?’, and ‘What is 46% of 90?’ using benchmark amounts like 10% and 5%

Ratios and Rates

• Find equivalent ratios using multiplication and express them as equivalent fractions, e.g. 16:8 as 8:4 as 4:2 as 2:1 = 2/3

• Begin to compare ratios by finding equivalent fractions, building equivalent ratios or mapping onto 1).

• Solve simple rate problems using multiplication, e.g. Picking 7 boxes of apples in ½ hour is equivalent to 21 boxes in 1½ hours.

Stage 7 (AM) Key Ideas (level 4)Fractions and Decimals

• Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3

• Find equivalent fractions using multiplicative thinking, and order fractions using equivalence and benchmarks. e.g. 2/5 < 11/16

• Convert common fractions, to decimals and percentages and vice versa.

• Add and subtract related fractions, e.g. 2/4 + 5/8

• Add and subtract decimals, e.g. 3.6 + 2.89

• Find fractions of whole numbers using multiplication and division e.g.2/3 of 36 and 2/3 of ? = 24

• Multiply fractions by other factions e.g.2/3 x ¼

• Solve measurement problems with related fractions, e.g. 1½ ÷ 1/6 = 9/6 ÷ 1/6 =9

• Solve division problems expressing remainders as fractions or decimals e.g. 8 ÷ 3 = 22/3 or 2.66

Percentages

• Estimate and solve percentage type problems such as ‘What % is 35 out of 60?’, and ‘What is 46% of 90?’ using benchmark amounts like 10% and 5%

Ratios and Rates

• Find equivalent ratios using multiplication and express them as equivalent fractions, e.g. 16:8 as 8:4 as 4:2 as 2:1 = 2/3

• Begin to compare ratios by finding equivalent fractions, building equivalent ratios or mapping onto 1).

• Solve simple rate problems using multiplication, e.g. Picking 7 boxes of apples in ½ hour is equivalent to 21 boxes in 1½ hours.

Stage 8 (AP) Key Ideas (level 5)Fractions and Decimals

• Add and subtract fractions and mixed numbers with uncommon denominators, 2/3 + 14/8

• Multiply fractions, and divide whole numbers by fractions, recognising that division can result in a larger answer, e.g. 4 ÷ 2/3 = 12/3 ÷ 2/3 = 6

• Solve measurement problems with fractions like ¾ ÷ 2/3 by using equivalence and reunitising the whole

• Multiply and divide decimals using place value estimation and conversion to known fractions, e.g. 0.4 × 2.8 = 1.12 (0.4< ½ ) , 8.1 ÷ 0.3 = 27 (81÷ 3 in tenths)

• Find fractions between two given fractions using equivalence, conversion to decimals or percentages

Percentages

• Solve percentage change problems, e.g. The house price rises from $240,000 to $270,000. What percentage increase is this?

• Estimate and find percentages of whole number and decimal amounts and calculate percentages from given amounts e.g. Liam gets 35 out of 56 shots in. What percentage is that?

Ratios

• Combine and partition ratios, and express the resulting ratio using fractions and percentages, e.g. Tina has twice as many marbles as Ben. She has a ratio of 2 red to 5 blue. Ben’s ratio is 3:4. If they combine their collections what will the ratio be? i.e. 2:5 + 2:5 + 3:4 = 7:14 = 1:2 ,

• Find equivalent ratios by identifying common whole number factors and express them as fractions and percentages, e.g. 16:48 is equivalent to 2:6 or 1:3 or ¼ or 25%

Rates:

• Solve rate problems using common whole number factors and convertion to unit rates, e.g. 490 km in 14 hours is an average speed of 35 k/h (dividing by 7 then 2).

• Solve inverse rate problems, e.g. 4 people can paint a house in 9 days. How long will 3 people take to do it?

Stage 8 (AP) Key Ideas (level 5)Fractions and Decimals

• Add and subtract fractions and mixed numbers with uncommon denominators, 2/3 + 14/8

• Multiply fractions, and divide whole numbers by fractions, recognising that division can result in a larger answer, e.g. 4 ÷ 2/3 = 12/3 ÷ 2/3 = 6

• Solve measurement problems with fractions like ¾ ÷ 2/3 by using equivalence and reunitising the whole

• Multiply and divide decimals using place value estimation and conversion to known fractions, e.g. 0.4 × 2.8 = 1.12 (0.4< ½ ) , 8.1 ÷ 0.3 = 27 (81÷ 3 in tenths)

• Find fractions between two given fractions using equivalence, conversion to decimals or percentages

Percentages

• Solve percentage change problems, e.g. The house price rises from $240,000 to $270,000. What percentage increase is this?

• Estimate and find percentages of whole number and decimal amounts and calculate percentages from given amounts e.g. Liam gets 35 out of 56 shots in. What percentage is that?

Ratios

• Combine and partition ratios, and express the resulting ratio using fractions and percentages, e.g. Tina has twice as many marbles as Ben. She has a ratio of 2 red to 5 blue. Ben’s ratio is 3:4. If they combine their collections what will the ratio be? i.e. 2:5 + 2:5 + 3:4 = 7:14 = 1:2 ,

• Find equivalent ratios by identifying common whole number factors and express them as fractions and percentages, e.g. 16:48 is equivalent to 2:6 or 1:3 or ¼ or 25%

Rates:

• Solve rate problems using common whole number factors and convertion to unit rates, e.g. 490 km in 14 hours is an average speed of 35 k/h (dividing by 7 then 2).

• Solve inverse rate problems, e.g. 4 people can paint a house in 9 days. How long will 3 people take to do it?

How could you communicate this idea of equivalence to students?

Paper Folding

Fraction Tiles / Strips

1/4 = ?/8 x2

x2

Fraction Circles Multiplicative thinking

Equivalence Games/Activities

• Fraction Frenzy, FIO Number Level 3, Book 3• www.maths-games.org Click on “Fraction

Games” (Fraction Booster, Fraction Monkeys, Melvin’s Make-a-Match, Fresh Baked Fractions)

• Fraction circles/wall and dice game• Fraction bingo, pictures then words• The Equivalence Game: PR3-4+ p.18-19

Once you understand equivalence you can……

1.Compare and order fractions

2.Add and Subtract fractions

3.Understand decimals, as decimals are special cases of equivalent fractions where the denominator is always a power of ten.

Key IdeaOrdering using equivalence and benchmarks

A½ or ¼1/5 or 1/9

5/9 or 2/9

Circle the bigger fraction of each pair.

B6/4 or 3/5

7/8 or 9/7

7/3 or 4/6

D7/10 or 6/8

7/8 or 6/9

5/7 or 7/9

Example of Stage 8 fraction knowledge

2/3 3/4

2/5 5/8 3/8

C7/16 or 3/8

2/3 or 5/9

5/4 or 3/2

unit fractions

More or less than 1

related fractions

unrelated fractions

What did you do to order them?

4/5 or 2/3

Which is bigger?

(Order/compare fractions: Stage 7)

12/1510/15

Find fractions between two fractions, using equivalence: Stage 8

Feeding Pets

3/4 2/3

What fractions come between these two??

When is one method easier than another?

When the fractions are easier to convert to decimals fifths, tenths, halves, quarters, eighths or commonly known ones- eg. thirds).

Usefulness of decimal conversion and equivalent fraction methods?

Both are equivalent fraction methods.

Tri Fractions

Game for comparing and ordering fractions

FIO PR 3-4+

Add and Subtract related fractions

(Stage 7)e. g ¼ + 5/8

Fraction circles / fraction wall tiles*Play create 3 (MM 7-9)

•halves, quarters, eighths•halves, fifths, tenths•halves, thirds, sixths

What could you use to help students understand this idea?

Add and Subtract fractions with uncommon denominators

(Stage 8)e.g. 2/3 + 9/4

Using fraction circles / fraction wall tiles*Play “Fractis”

•How??•Find common

denominators/equivalent fractions

Create 3 (MM7-9) Each player chooses a fraction to place their counter on

Take turns to move your counter along the lines to another fraction

Add the new fraction to your total.

The first player to make exactly three wins.

Go over three and you lose.

Multiplying Fractions (Stage 7)

Using fraction circles / fraction wall tiles • 6 x ¼

Using paper folding / wall tiles / OHT fractions/drawing

•½ x ¼

Using multiplicative thinking, not additive

Pirate Problem• Three pirates have some treasure to share. They

decide to sleep and share it equally in the morning. • One pirate got up at at 1.00am and took 1 third of

the treasure.• The second pirate woke at 3.00am and took 1 third

of the treasure.• The last pirate got up at 7.00am and took the rest

of the treasure.

Do they each get an equal share of the treasure? If not, how much do they each get?

1st pirate = 1 third

2nd pirate =1/3 x 2/3 = 2 ninths

3rd pirate = the rest = 1 - 5 ninths = 4 ninths

Pirate Problem• One pirate got up at at 1.00am and took 1 third of the treasure.• The second pirate woke at 3.00am and took 1 third of the treasure.• The last pirate got up at 7.00am and took the rest of the treasure.

€

3

4×

5

6

Multiplying fractions

Jo ate 1/6 of a box of chocolates she had for Mother’s Day. Her greedy husband ate ¾ of what she left. What fraction of the whole box is left?

How might you help student understand this idea?

€

3

4×

5

6

€

5

6

€

3

4

€

15

24

€

15

24

Multiplying fractions

Digital Learning Objects “Fractions of Fractions” tool

Multiplying fractions – your turn!

• What is a word problem / context for:

Play:Fraction Multiplication grid game

3 x 58 6

Draw a picture, or use the Fraction OHTs to represent the problem

Dividing by fractionsStage 7: Solve measurement problems with related fractions, (recognise than division can lead to a larger answer)

You observe the following equation in Bill’s work:

Consider…..• Is Bill correct?• What is the possible reasoning behind his answer?• What, if any, is the key understanding he needs to

develop in order to solve this problem?

No he is not correct. The correct equation is

1/2 of 2 1/2 is 1 1/4.

He is dividing by 2. He is multiplying by 1/2. He reasons that “division makes smaller” therefore the answer must be smaller than 2 1/2.

Possible reasoning behind his answer:

Key Idea: To divide the number A by the number B is to find out how many lots of B are in A

For example:

There are 4 lots of 2 in 8

There are 5 lots of 1/2 in 2 1/2

To communicate this idea to students you could…• Use meaningful representations for the

problem. For example:I am making hats. If each hat takes 1/2 a metre of

material, how many hats can I make from 2 1/2 metres?

• Use materials or diagrams to show there are 5 lots of 1/2 in 2 1/2 :

Key Idea:

Division is the opposite of multiplication.The relationship between multiplication and division can be used to help simplify the solution to problems involving the division of fractions.

Use contexts that make use of the inverse operation:

To communicate this idea to students you could…

Your turn!

4 ½ ÷ 1 1/8 is

Use materials or diagramsUse contexts that make use of the inverse operation:

Remember the key idea is to think about how many lots of B are in A, or use the inverse operation…

Stage 8 Advanced Proportional

Solve measurement problems with fractions by using equivalence and reunitising the whole.

Example

€

3

4÷

2

3

€

→9

12÷

8

12

€

→9

8

Why not 9/8 twelfths?

(Or 1 1/8 )

Ref: Book 8 : p21, Dividing Fractions p22, Harder Division of Fractions

Book 7: p68, Brmmm! Brmmm!

€

3

4÷

2

3

€

→9

12÷

8

12

€

→9

8

Why not 9/8 twelfths?

(Or 1 1/8 )

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

112

How many times will 8/12

go into 9/12?

1 lot of 8/12 1/8 more again

€

3

4÷

2

3

€

→9

12÷

8

12

€

→9

8

Why not 9/8 twelfths?

(Or 1 1/8 )How many times

will 8/somethings go into 9/somethings?

1 lot of 8 1/8 more again

ExampleMalcolm has ¾ of a cake left.He gives his guests 1/8 of a cake each.How many guests get a piece of cake?

¾ ÷ 1/8

ExampleMalcolm has ¾ of a cake left.He gives his guests 1/8 of a cake each.How many guests get a piece of cake?

¾ ÷ 1/8

Or, 6/8 ÷ 1/8

How many one eighths in six eighths?...Answer 6

Brmmm! Brmmm!Book 7, p68

Trev has just filled his car.He drives to and from work each day. Each trip takes three eighths of a tank. How many trips can he take before he runs out of petrol?

1 ÷ 3/81

18

18

18

18

18

18

18

18

18

“How many three-eighths measure one whole?”

1 lot 2/3 lot1 lot

2 2/3

Harder Division of FractionsBook 8, p22

Malcolm has 7/8 of a cake left.He cuts 2/9 in size to put in packets for his guests.How many packets of cake will he make?

7/8 ÷ 2/9

Why is this hard to compare?

Harder Division of FractionsBook 8, p22

Malcolm has 7/8 of a cake left.He cuts 2/9 in size to put in packets for his guests.How many packets of cake will he make?

63/72 ÷ 16/72 63÷ 16 63/16 or 315/16

Rewrite them as equivalent fractions

Example

€

3

4÷

2

3

€

7

8÷

1

4€

→9

12÷

8

12

€

→9

8

Your turn:Make a word story/context for each problem.

Use pictures/diagrams to model

3 ÷ 25 8

Chocoholic

You have three-quartersof a chocolate block left.

You usually eat one-third of a block each sitting for the good of your health.

How many sittings will the chocolate last?

Fractions Revision sheet… enjoy!!

Equivalence Games/Activities- have a go!• Fraction Frenzy, FIO Number Level 3, Book 3• www.maths-games.org Click on “Fraction

Games” (Fraction Booster, Fraction Monkeys, Melvin’s Make-a-Match, Fresh Baked Fractions)

• Fraction circles/wall and dice game• Fraction bingo, pictures then words• The Equivalence Game: PR3-4+ p.18-19

Decimal FractionsYou need to understand equivalent fractions before understanding decimals, as decimals are special cases of equivalent fractions where the denominator is always a power of ten.

What is 3/8 as a decimal?

Session 3 Stages 7 & 8

Misconceptions with Decimal Place Value:

How do these children view decimals?

1. Bernie says that 0.657 is bigger than 0.7

2. Sam thinks that 0.27 is bigger than 0.395

3. James thinks that 0 is bigger than 0.5

4. Adey thinks that 0.2 is bigger than 0.4

5. Claire thinks that 10 x 4.5 is 4.50

Decimal Misconceptions Summary

1. Decimals are two independent sets of whole numbers separated by a decimal point, e.g. 3.71 is bigger than 3.8

2. The more decimal places a number has, the smaller the number is because the last place value digit is very small. E.g. 2.765 is smaller than 2.4

3. Decimals are negative numbers.

4. 1/2 is 0.2 and 1/4 is 0.4, so therefore 0.4 is smaller than 0.2.

5. When you multiply a decimal number by 10, just add a zero, e.g. 1.5 x 10 = 1.50

6. When you multiply decimals the number always gets bigger

Developing conceptual decimal understanding

• Use candy bars, decimats or decipipes to understand how tenths and hundredths arise and what they ‘look like’ (PV understanding)

• Compare decimal numbers, e.g. 0.6 and 0.47

• How much more make.. e.g. 0.47 + ? = 0.6

3 ÷ 5 5 children share three chocolate bars evenly. How much chocolate does each child receive?

30 tenths ÷ 5 =

0 wholes + 6 tenths each = 0.6

Key Idea: Stage 7 AMSolve division problems - express remainders as decimals.

Add and Subtract decimals

Using candy bars, decimats, decipipes

• 0.6 + 0.23 (mixed tenths and hundredths)• 0.4 + 1.8 (Exchanging ten for one)• 3.6 – 2.98 (Choose from efficient strategies)

Add and subtract decimals

3.6 - 2.98

Tidy Numbers Place Value

Equal Additions Reversibility

Standard written form (algorithm)

‘Target Time’ (from FIO Number L3 Book 2 page 16)

Target Number is 6

+ =

• Roll a dice and place the number thrown.

• Try and make the number sentence as close to the target number as possible.

• Score = the difference between your total and the target number.

Ww

w

Multiplying Decimals by decimals (Stage 8)e.g. 0.4 x 0.3

Why does the answer get smaller when multiplying?

0 1

1

Ww

w

0 1

1

0.3

0.4

Multiplying Decimals by decimals (Stage 8)e.g. 0.4 x 0.3 = 0.12

It is a method of comparing fractions by giving both fractions a common denominator i.e. hundredths. So it is useful to view percentages as hundredths.

Why calculate percentages?

=

Applying PercentagesTypes of Percentage Calculations at Level 4 (stage 7)

• Estimate and find percentages of amounts,

e.g. 25% of $80

• Expressing quantities as a percentage

e.g. What percent is 18 out of 24?

Estimate and find percentages of whole number amounts.

25% of $80

35% of $80

Using benchmarks like 10%, and ratio tablesFIO: Pondering Percentages NS&AT 3-4.1(p12-13)

Teaching common conversions halves, thirds, quarters, fifths, tenths

Book 8:21 (MM4-28) , Decimats. Bead strings, slavonic abacus

Practising instant recall of conversionsBingo, Memory, I have, Who has, Dominoes,

100%

Find __________ (using benchmarks and ratio tables)

100%

$80

Find 35% of $80

$80

100%

$80

Find 35% of $80

$80

100%

$80

Find 35% of $80

100%

$80

Find 35% of $80

$8

10%

$8

35%

$28

$4

5%

$4

$8$8

30%

$24

Now try this…

46% of $90

46% of $90100% 10% 40% 5% 1% 6% 46%

$90 $9 $36 $4.50 $0.90 $5.40 $41.40

Is there an easier way to find 46%?

46% of 90

Estimating Percentages

16% of 3961 TVs are found to be faulty at the factory and need repairs before they are sent for sale. About how many sets is that?

(Book 8 p.26 - Number Sense)

Using Number Properties:

Explain how you would estimate 61% of a number?

About 600

Dice: 20% 25% 40% 50% 60% 75%

Roll then move to chosen spot60% MM (7-5)

Applying PercentagesPercentage Calculations at Stage 8

Increase and decrease quantities by given percentages, including mark up, discount and GST e.g. A watch cost $20 after a 33% discount. - What was it’s original price? (Stage 8)

Rates and Ratios

Ratios

Write 1/2 as a ratio

3: 4 is the ratio of red to blue beans.

What fraction of the beans are red?

When are ratios used?

1:1

3/7

Key Idea :

Ratios describe a part-to-part relationship e.g. 3 parts red paint : 4 parts blue paint

But fractions compare the relationships of one of the parts with the whole, e.g.The paint mixture above is 3/7 red or 4/7 blue

Smarties

Mini boxes have 6 red and 3 blue smarties.

Midi boxes have 16 red and 8 blue smarties.

Maxi boxes have 40 red and 20 blue smarties.

What do you notice about these proportions?

Key Idea

They all have the same ratio of 2:1 and therefore the same proportion of blue (1 third)

Finding simple equivalent ratios and expressing them as equivalent fractions

If the ratio of green paint to red paint is 2:3 and I have 18 litres of red paint, how much green paint should I

need? What fraction is red?

0

0 2

3

12

18

4

6

6

9

10

15

8

12

green

red

(Stage 6: Seed Packets, Book 7 p.30)

x6

Which recipe gives the darkest green paint? Blue : Yellow

A 1 : 3 B 4 : 8 C 3 : 5

• Method 1: Convert to equivalent fractions

• Method 2: Find equivalent ratios by ‘equalising’

• Method 3: Find equivalent ratios by ‘mapping onto 1’

Key IdeaCompare ratios by finding equivalent fractions or equivalent ratios (Mixing Colours, Book 7 page 50)

Method 1. Convert to equivalent fractions

Which recipe gives the darkest green paint?

Blue : Yellow

A 1 : 3

B 4 : 8

C 3 : 5

Blue = 1/4

Blue = 4/12 = 1/3

Blue = 3/8

Method 2. Find equivalent ratios by equalising the total parts in the ratios.

Which recipe gives the darkest green paint?

Blue : Yellow

A 1 : 3

B 4 : 8

C 3 : 5

Total of parts

A = 4

B = 12

C = 8

24 is the common

multiple so…

(x6)

(x2)

(x3)

Method 2. Find equivalent ratios by equalising the total parts in the ratios.

Which recipe gives the darkest green paint?

Blue : Yellow

A 1 : 3

B 4 : 8

C 3 : 5

6 blue

8 blue

9 blue

Method 3. Mapping onto 1 unit part

Which recipe gives the darkest green paint?

Blue : Yellow

A 1 : 3

B 4 : 8

C 3 : 5

Map onto 1 part blue

A = 1:3

B = 1:2

C = 1:12/3

Your turn! - Using Imaging Which recipe gives the darkest green paint?

Blue : Yellow

A 6 : 14

B 4 : 6

C 7 : 8

• Convert to equivalent fractions

• Equalising the total (building up the ratio)

• Mapping onto 1

Your turn - Using Number Properties

Which ratio gives the darker shade of green?(Blue: Yellow)

4 : 7 or 9 : 18

• Convert to equivalent fractions

• Equalising the total (building up the ratio)

• Mapping onto 1

Ratios- Using the Framework

What is the difference between “ratios” in Stages 6, 7 & 8?

These boxes of smarties have two colours, yellow and green.

If 6 smarties are green, 2/5 are yellow.

What is the ratio of green to yellow?

How many are yellow?

What percentage are green?

4

6:4 or 3:2

60%

Stage 8 – Advanced ProportionalFind equivalent ratios by identifying common whole number factors and express them as fractions and percentages, e.g. 16:48 is equivalent to 2:6 or 1:3 or ¼ or 25%

Ratios and Rates

What is the difference between a ratio and a rate?

Both are multiplicative relationships.

A ratio is a relationship between two things that are measured by the same unit,e.g. 4 shovels of sand to 1 shovel of cement.

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

Stage 8 (AP) : Combine and partition ratios,

Video Clip from “Little League”

Hours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Joe 1 2 3 4 5Sam 1 2 3

ClueIf both painters started work at the same time, how many hours would pass before they finished painting a house at exactly the same time?

So, their average rate, in houses per hours, would be…?

So, in general, if Painter A takes a hours to paint a house, and Painter B takes b hours to paint a house, how long does it take them to paint a house together?

Why does a x b work? a + b

Joe takes 3hrs/houseSam takes 5hrs/house How long does it take to paint 1 house together?

Houses : Hours 8 : 15

House : Hours 1 : 15/8 1 7/8

Stage 8 Advanced Proportional

Nikki jogs 2.4km (6 laps) in 12 minutes

What other questions could be asked from this statement?

• Solve rate problems using common whole number factors and conversion to unit rates, e.g. 490 km in 14 hours is an average speed of 35 km/h (dividing by 7 then 2).

• Solve inverse rate problems, e.g. 4 people can paint a house in 9 days. How long will 3 people take to do it?

Nikki jogs 24 laps in 12 minutes

How long did it take to run 1 lap?

How far can she run in 1 minute

How long will it take to run 10 laps?

How far will she have run in 1 hour (if the same pace is kept)

Key Idea: The key to proportional thinking is being able to see combinations of factors within numbers.

How long will it take to run 10 laps? 24 laps:12 min so 10 laps:? min

Using ratio tables to identify the multiplicative relationships between the numbers involved.

Minutes Laps

Rate 1 24 12

Rate 2 10

24 laps:12 min so 10 laps:? min

X 5/12

÷ 2

24 laps:12 min so 10 laps:? min

Find the unit rate (there are always two unit rates)

1 lap: 0.5 min so multiply by 10 or

2 laps : 1 min so multiply by 5

Relationship within the rates 24 : 12 2:1

the minutes taken are half the laps 2 : 1

Relationship between the rates 24 : 12

10/24 = 5/12

10 : ?

How is your content knowledge?

• Name an equivalent fraction for 5/6

• Order these decimals from smallest to largest: 3.48 3.6 3.067

• Write one eighth as a decimal

• 5 ÷ 4

• 3 - 1.95

• 0.3 x 0.4

• What is 59% of 80?

• Order these fractions decimals & percentages . 2/3

7/16 30% 0.61 2/5 75% 0.38

3.067 3.48 3.6

0.125

1.25

1.05

0.12

47.2

30% 0.38 2/5 7/16 0.61 2/3 75%

Plan a lesson from Book 7

Focus on the key ideas we have discussed today

Plan a lesson from Book 7

Stage 4-5Wafers p16Animals p18Hungry Birds p22Stage 5-6Birthday Cakes p26Trains p32Stage 6-7How can two decimals so ugly make one so beautiful?

p45

• Required Knowledge for Stage 6 : Keeping Clever Box • Identify symbols for halves, quarters, thirds, fifths and tenths, including fractions greater than 1

• Order unit fractions (¼ ½,) and fractions with like denominators (3/4, 1/4)

I have, Who Has

Memory/ Happy FamiliesJigsaw

Dominoes

Snakes and chancesSnatch

Bingo

Knowledge being developed at Stage 6 : Keeping Clever Box(Required Knowledge for moving to Stage 7)

•Multiplication facts for all times tables.•Order unit fractions•Count forwards & backwards in halves, thirds, quarters, fifths…•Read any fraction including improper fractions•Reading decimals and rounding decimals to nearest 1•Count forward and backwards in tenths and hundredths•Know tenths and hundredths in decimals to two places e.g., tenths in 7.2 is 72

I have, Who Has

Memory/ Happy FamiliesJigsaw

Dominoes

Snakes and chancesSnatch

Bingo

Knowledge being developed at Stage 7 : Keeping Clever Box•Order decimals to 3 decimal places•Round decimals to the nearest whole number or tenth•Order related fractions, halves, thirds, quarters, fifths, tenths…•Recall decimal – fraction – percentage conversions for halves, thirds, quarters, fifths, tenths•Recall equivalent fractions for halves, thirds, quarters, fifths, and tenths

I have, Who Has

Memory/ Happy FamiliesJigsaw

Dominoes

Snakes and chancesSnatch

Bingo

What now?Use your data from IKAN and GloSS (Re-GloSS fractions if necessary) to identify class needs.

Use long-term planning units for Fractions

Teach fraction knowledge and proportions & ratios strategies with your groups/whole class.

In-Class Modelling visit??

FeedbackPlease complete an “Analysis” sheet for me!

-one colour for pre workshops, one for post.

Please add a comment: What was most useful from these sessions? What would you like more of?

A Fractional Thought for the day

Smart people believe only half of what they hear.

Smarter people know which half to believe.

Workshop Resources• Warm Up – I Have.. who has• Fraction Circle lesson• Paper Folding / Multiplying fractions OHT• Freddy Frog Chocolate• Dice and counters• Bead string / rope / pegs• Decipipes/candy bars/decimats• Percents Game / Create 3 Game• Blue/yellow cubes + practice sheet• AA packs: Seed Packets / Birthday cakes• AM Packs: Candy Bars / Hot Shots / Dividing fractions• AP Packs: Comparing Apples / Brrmmm! / Reverse % Problems• Books 1, 4, 7, 8• Fractis• Link to “Fractions of Fractions” tool• Fraction/decimal grid game• Fraction revision sheet• FIO book activities• Keeping clever blank template games

HandoutsAlso include:• Fraction Squares OHTs• Percents MM• Create 3 MM• I have, Who Has and C• Stage 7 & 8 Key ideas