proportions and ratios workshop have a go at the pirate problem or fraction hunt on your table while...

Proportions and Ratios Workshop

Have a go at the Pirate Problem or Fraction Hunt on your table while you are waiting!

4 out of 3 people have trouble with fractions

Objectives• Understand the progressive strategy stages of

proportions and ratios

• Understand common misconceptions and key ideas when teaching fractions and decimals.

• Explore equipment and activities used to teach fraction knowledge and strategy

Overview• Key Teaching Ideas (Stages 2-6)• Number Framework Progressions

Morning Tea• Decimals and Stage 7• Getting into Book 7 and the Planning Units Lunch• Modelling Sessions• Year Overiews – Putting it all together

Play Which Mystery Letter Am I?

1/3

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

recruitment

½ is to 0.5 as 1/5 is to

a. 0.15b. 0.1c. 0.2d. 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

Developing Proportional thinkingFewer 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 misconceptions might young children have when beginning fractions?

Misconceptions about finding one half when beginning fractions:• Share without any attention to equality• Share appropriate to their perception of size, age etc.• Measure once halved but ignore any remainderSo what do we need to teach? Introduce the vocabulary of equal / fair shares with both regions and sets for halves and then quarters.

Bev Dunbar: ‘Exploring Fractions’

Key Teaching Ideas Stages 2- 6

Draw two pictures of one quarter

Fractions Key Teaching Ideas1. Use sets as well as regions and lengths

from early on and connect different representations

Shapes/Regions Sets

€

14

1 quarter Lengths

3 out of 7 7/3 3 sevenths 7 thirds

Fractions Key Teaching Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first then introduce symbols with care.

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!

Fraction SymbolsIn 2001 42% of year 7 & 8 students who sat the initial NUMPA could not name these symbols

41

31

21

Fractional vocabulary

One half

One third

One quarter

Don’t know

Emphasise the ‘ths’ code1 dog + 2 dogs = 3 dogs1 fifth + 2 fifths = 3 fifths1/5 + 2/5 =

3/5

3 fifths + ?/5 = 11 - ?/5 = 3/5

6 is one third of what number?

This is one quarter of a shape. What does the whole look like?


representations.2. Use words first & introduce symbols with care.3. Go from part-to-whole as well as whole-to-part

with both shapes and sets.

Which letter shows 5 halves as a number?

0 1 2 3

A B C D E F


representations.2. Use words first & introduce symbols with care.3. Go from part-to-whole as well as whole-to-part with both shapes

and sets.

4. Fractions are not always less than 1. Push over 1 early to consolidate the understanding of the top and bottom numbers.

What is this fraction? 5/2

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

How do I write 3 halves? 3 1/2

1/3

Y7 student responses decile 10

Fraction number lines and counting in fractions

0 1 half 2 halves 3 halves 4 halves0 1/2

2/2 3/2

4/2

0 1/2 1 11/2 2

Teaching Ideas Fraction Circles

Understanding Fraction Representations

How could these activities be adapted? e.g. decimal identification

A. Spin a WholeB. Fraction DotsC. Happy FamiliesD. Fraction CirclesE. I have…, Who Has…F. Number MatG. Numerators and DenominatorsH. Fraction BingoI. Dominoes

• Form groups of 3.• Explore your game.• Number yourselves 1 – 3 • Number 1’s get together…• Share your game

5 children share three chocolate bars evenly.

How much chocolate does each child receive?

3 ÷ 5



and sets.4. Fractions are not always less than 1. Push over 1 early to

consolidate understanding.

5. Division is the most common context for fractions when units of one are not accurate enough for measuring and sharing problems.

Initially this is done by halving and halving again.

e.g. 3 ÷ 5

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 do you think they have done?

½

A more sophisticated method for 3 ÷ 5

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

Y7 response: “3 fifteenths!” Why?

Put a peg where you think 3/5 will be.

Put a peg where you think 3/5 of 100 will be.

0 1




consolidate understanding. 5. Division is the most common context for fractions, e.g. 3 ÷ 5

6. Fractions are numbers as well as operators

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

Three fifths of 100 is 60 (operator)

350 1

0 100

15

20 60x3

Teaching IdeasUsing double number lines

Connecting sets with regions and lengths¼ ¼ of 12

Sam had one half of a cake, Julie had one quarter of a cake, so Sam had most. True or False or Maybe

SamJulie




consolidate understanding. 5. Division is the most common context for fractions, e.g. 3 ÷ 5. 6. Fractions are numbers as well as operators.

7. Fractions are always relative to the whole. Continually ask “What is 1?”

What is B?

A A

B B B B

C

D D D D D D D D

What is the whole? (Trains Book 7, p32)

Fractions Key Teaching Ideas 1. Use sets as well as regions from early on and connect different


and sets.4. Fractions are not always less than 1. Go over 15. Division is the most common context for fractions, e.g. 3 ÷ 5. 6. Fractions are numbers as well as operators7. Fractions are always relative to the whole.

8. Fractions are really a context for applying

add/sub and mult/div strategies

Connect their division strategies to finding a fraction of a number, i.e. finding 1 third of a number is the same as dividing a number by 3.

Framework Practice Match the strategy stages

to their definitions and assessment task(s) from GloSS.

Stage 1 Stage 2-4 (AC) Stage 5 (EA)Unequal Sharing Equal Sharing Use of Addition and

known facts e.g. 5 + 5 + 5 = 15

Stage 6 (AA)Using multiplication

Stage 7 (AM)Using division

4/9 of ? = 1616 is four ninths of what number?

16

4 44 4 4 4

36

8

At Stage 7, students should be using a range of multiplication and division strategies to solve problems with fractions, proportions and ratios.

What strategy would be used to find 1 third of 27 when the division fact is unknown?

Stage 2- 4: Equal sharing by ones

Stage 5: Anticipate the result of equal sharing using repeated addition or skip counting,e.g. 9 + 9 + 9 or 9, 18, 27

Stage 6:Use multiplication, e.g. 3 x ? = 27 Can easily extend to finding 2 thirds of 27.

Stage 2- 4:

Stage 5:

Stage 6:

Ratios (Introduced at Stage 6)

Write 1/2 as a ratio

3: 4 is the ratio of red to blue beans. What fraction of the beans are red?

Think of some contexts when ratios are used.

1:1

3/7

RatiosHow are ratios and fractions connected?

Ratios describe a part-to-part relationship e.g. 2 parts blue paint : 3 parts red paint

But fractions compare the relationships of one of the parts with the whole, e.g.The paint mixture above is 2/5 blue

Perception Check

What have you remembered about these important key teaching ideas?

Choose your share of chocolate!

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.

Stage 7 Decimals

Decimals are special cases of equivalent fractions where the denominator is always a

power of ten.

Stage 7 (AM) Level 4 Key IdeasFractions• Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3 • Find equivalent fractions using multiplicative thinking,, e.g. 2/6 = how many twelfths?• Order fractions using equivalence and benchmarks like 1 half , e.g. 2/5 < 11/16

• Add and subtract related fractions, e.g. 2/4 + 5/8 • Find fractions of whole numbers using mult’n and div’n 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 =9Decimals• Order decimals to 3dp• Round whole numbers and decimals to the nearest whole or tenth• Solve division problems expressing remainders as decimals, e.g. 8 ÷ 3 = 22/3 or 2.66 • Convert common fractions, i.e. halves, quarters etc. to decimals and percentages• Add and subtract decimals, e.g. 3.6 + 2.89 Percentages• Estimate and solve percentage type problems like ‘What % is 35 out of 60?’, and ‘What is 46% of 90?’ using benchmark amounts like 10% & 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.

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

Developing understanding of decimal place value

The CANON law in our place value system is that 1 unit must be split into TEN of the next smallest

unit AND NO OTHER!

Developing Decimal Place Value Understanding

1. Use decipipes, candy bars, or decimats to understand how tenths and hundredths arise and what decimal numbers ‘look like’

2. Make and compare decimal numbers, e.g. Which is bigger? 0.6 and 0.47

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

1. establish the whole, half, quarter rods then tenths 2. 1 half = ? Tenths, why is it 0.5 as a decimal?3. 1 quarter = ? tenths + 4. 1 eighth = ? tenths? +

Using Decipipes

View children’s response to this task

Now compare:0.4 0.38 0.275

3 chocolate bars shared between 5 children.

30 tenths ÷ 5 = 0 wholes + 6 tenths each = 0.6

Using candy bars (and expressing remainders as decimals)

3 ÷ 5

Using decimats and arrow cards

3. Add and Subtract (stage 7) Rank these questions in order of difficulty. a)0.8 + 0.3, b)0.6 + 0.23 c)0.06 + 0.23,

Exchanging ten for 1

Mixed decimal valuesSame decimal values

1. Read and Make (stage 6)

2. Compare and Order (stage 6-7)• Which is bigger: 0.6 or 0.43?• How much more make…

4. Multiply and Divide (stage 8)

Add and subtract decimals (Stage 7) using decipipes or candy bars

1.6 - 0.98

Tidy Numbers Place Value

Equal Additions Reversibility

Standard written form (algorithm)

Decimal Games and Activities• Digital learning Objects: http://digistore.tki.org.nz/ec/viewMetadata.action?id=L1079

1. Decimal Sort2. First to the Draw3. Four in a Row Decimals4. Beat the Basics5. Decimal Keyboard6. Target Time FIO N3:2,16

‘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.

The Strategy Teaching Model

Using Number PropertiesUsing Imaging

Using Material

s

New Knowledge & Strategies

Existing Knowledge & Strategies

Using Materials

Long Term Planning Units

strategy knowledge

NZ Curriculum

Plan a lesson using the Planning Units and Book 7

Stage 2- 5 Stage 6-7Tanya, Jacinda,Fair Shares

Alison and William:Trains (Stage 6) orHot Shots (Stage 7)

Amanda, Jessica, NicoleHungry Birds

Hamish and MargSeed Packets (Stage 6) or Mixing Colours (Stage 7)

Emma and Ellie:Fraction Circles

Therie and JudyBirthday Cakes

Nikki and Cameron:Animals or Wafers

Finding FractionsThrow 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.

Modelling Stage 5• Wafers• Animals

Putting it all together

How Much Number?Y1-4: 60-80% Y5-6: 50-70% Y7-8:40-60%

Exploring www.nzmaths.co.nz

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

4 Stages of the PD JourneyOrganisation

Orgnising routines, resources etc.

Focus on ContentFamiliarisation with books, teaching model etc.

Focus on the StudentMove away from what you are doing to noticing what the

student is doing

Reacting to the StudentInterpret and respond to what the student is doing

Evaluation

Thought for the dayA DECIMAL POINT

When you rearrange the letters becomes

I'M A DOT IN PLACE

Additional Slides

Equivalent Fractions

Key IdeaOrdering using equivalence and benchmarks

A½ or ¼1/5 or 1/95/9 or 2/9

Circle the bigger fraction of each pair.

B6/4 or 3/57/8 or 9/77/3 or 4/6

D7/10 or 6/87/8 or 6/95/7 or 7/9

Example of Stage 8 fraction knowledge 2/3

3/4 2/5

5/8 3/8

C7/16 or 3/82/3 or 5/95/4 or 3/2

unit

fractionsMore or less

than 1related fractions

unrelated fractions

What did you do to order them?

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• Fraction circles and dice game• Fraction Wall Tile game (Norma)• Fraction domino pictures then words• The Equivalence Game: PR3+ p.18-19• Fraction Feud • Fraction Board

3

4

• Collect the chosen denominators

• Select how many denominators are needed.

• Make the fraction• Compare the fraction

(to ½ 1…)• Make another

equivalent fraction

Once you understand equivalence you can……

1.Compare and order fractions 2.Add and Subtract fractions3.Understand decimals, as decimals are special cases of

equivalent fractions where the denominator is always a power of ten.

4/5 or 2/3Comparing Fractions - Which is bigger? (Bk 8)

12/1510/15

Adding Related 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.

A Fractional Thought for the day

Smart people believe only half of what they hear.

Smarter people know which half to believe.

Fraction Circles (book 7 p.20)

Play the fraction circle game.Put the circle pieces in the “bank”. Take turns to roll the die and collect what ever you roll from the bank. You may need to swap and exchange as necessary. The winner is the person who has made the most ‘wholes’ when the bank has run out of fraction pieces.

Which is bigger 3/4 or 9/8?

Three in a row (use two dice or numeral cards)A game to practice using improper fractions as numbers

0 1 2 3 4 5 6

e.g. Roll a 3 and a 5

Mark a cross on either 3 fifths or 5 thirds.

The winner is the first person to get three crosses in a row.

X X

Thinkboard Practicefive thirds or 7 ÷ 4

proportions and ratios workshop have a go at the pirate problem or fraction hunt on your table while...

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