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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. 0.15
- b. 0.1
- c. 0.2
- d. 0.5

1.24 is to 0.62 as 0.54 is to

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

- Travelling constantly at 20kmph, how long will

it take to travel 50 kilometres? - a. 1 hour 30 mins
- b. 2 hours
- c. 2 hours 30 mins
- d. 3 hours

- If a man weighing 80kg increased his weight by

20, what would his weight be now? - a. 96kg
- b. 89kg
- c. 88kg
- d. 100kg

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 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 remainder
- So 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 Ideas

- Use sets as well as regions and lengths from

early on and connect different representations

1 quarter

Lengths

Shapes/Regions

- 3 out of 7 7/3 3 sevenths

7 thirds

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - 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

The problem with out of

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!!!!!

x 24

2 out of 3 multiplied by 24!

8 out of 6 parts!

Fraction Symbols

In 2001 42 of year 7 8 students who sat the

initial NUMPA could not name these symbols

Fractional vocabulary

- One half
- One third
- One quarter
- Dont know

- Emphasise the ths code
- 1 dog 2 dogs 3 dogs
- 1 fifth 2 fifths 3 fifths
- 1/5 2/5 3/5
- 3 fifths ?/5 1
- 1 - ?/5 3/5

- 6 is one third of what number?

This is one quarter of a shape. What does the

whole look like?

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - Use words first introduce symbols with care.
- Go from part-to-whole as well as whole-to-part

with both shapes and sets.

Which letter shows 5 halves as a number?

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - Use words first introduce symbols with care.
- Go from part-to-whole as well as whole-to-part

with both shapes and sets. - 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

Teaching Ideas Fraction Circles

0 1/2 2/2

3/2 4/2

0 1/2 1 11/2

2

Understanding Fraction Representations

- Spin a Whole
- Fraction Dots
- Happy Families
- Fraction Circles
- I have, Who Has
- Number Mat
- Numerators and Denominators
- Fraction Bingo
- Dominoes

- Form groups of 3.
- Explore your game.
- Number yourselves 1 3
- Number 1s get together
- Share your game

How could these activities be adapted? e.g.

decimal identification

5 children share three chocolate bars evenly.

How much chocolate does each child receive?

3 5

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - Use words first introduce symbols with care.
- Go from part-to-whole as well as whole-to-part

with both shapes and sets. - Fractions are not always less than 1. Push over 1

early to consolidate understanding. - 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

What are these pieces called?

2/12 !! What do you think they have done?

½

1/2 1/10

A more sophisticated method for 3 5

1/51/51/5 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.

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - Use words first introduce symbols with care.
- Go from part-to-whole as well as whole-to-part

with both shapes and sets. - Fractions are not always less than 1. Push over 1

early to consolidate understanding. - Division is the most common context for

fractions, e.g. 3 5 - Fractions are numbers as well as operators

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

Three fifths of 100 is 60

(operator)

Teaching Ideas Using double number lines

x3

Connecting sets with regions and lengths

Sam had one half of a cake, Julie had one quarter

of a cake, so Sam had most. True or False

or Maybe

Julie

Sam

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - Use words first introduce symbols with care.
- Go from part-to-whole as well as whole-to-part

with both shapes and sets. - Fractions are not always less than 1. Push over 1

early to consolidate understanding. - Division is the most common context for

fractions, e.g. 3 5. - Fractions are numbers as well as operators.
- Fractions are always relative to the whole.

Continually ask What is 1?

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

What is B?

A A A A A A A A

B B B B B B B B

C C C C C C C C

D D D D D D D D

Fractions Key Teaching Ideas

- Use sets as well as regions from early on and

connect different representations. - Use words first introduce symbols with care.
- Go from part-to-whole as well as whole-to-part

with both shapes and sets. - Fractions are not always less than 1. Go over 1
- Division is the most common context for

fractions, e.g. 3 5. - Fractions are numbers as well as operators
- 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?

36

16

8

4 4

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)

11

Write 1/2 as a ratio 3 4 is the ratio of red to

blue beans. What fraction of the beans are red?

3/7

Think of some contexts when ratios are used.

Ratios

- How 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?

- 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.

1st pirate 1 third

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

3rd pirate the rest 1 - 5

ninths 4 ninths

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 Ideas

- Fractions
- 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 lt 11/16 - Add and subtract related fractions, e.g. 2/4

5/8 - Find fractions of whole numbers using multn and

divn 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 - Decimals
- 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. 168

as 84 as 42 as 21 2/3 - Begin to compare ratios by finding equivalent

fractions, building equivalent ratios or mapping

onto 1).

Misconceptions with Decimal Place ValueHow do

these children view decimals?

- Bernie says that 0.657 is bigger than 0.7
- Sam thinks that 0.27 is bigger than 0.395
- James thinks that 0 is bigger than 0.5
- Adey thinks that 0.2 is bigger than 0.4
- 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

- Use decipipes, candy bars, or decimats to

understand how tenths and hundredths arise and

what decimal numbers look like - Make and compare decimal numbers, e.g. Which is

bigger? 0.6 and 0.47 - How much more make.. e.g. 0.47 ? 0.6

Using Decipipes

- establish the whole, half, quarter rods then

tenths - 1 half ? Tenths, why is it 0.5 as a decimal?
- 1 quarter ? tenths
- 1 eighth ? tenths?

View childrens response to this task

Now compare 0.4 0.38 0.275

3 chocolate bars shared between 5 children.

Using candy bars (and expressing remainders as

decimals) 3 5

30 tenths 5 0 wholes 6 tenths each 0.6

Using decimats and arrow cards

- Read and Make (stage 6)
- Compare and Order (stage 6-7)
- Which is bigger 0.6 or 0.43?
- How much more make

- 3. Add and Subtract (stage 7)
- Rank these questions in order of difficulty.

- 0.8 0.3,
- 0.6 0.23
- 0.06 0.23,

Exchanging ten for 1 Mixed decimal

values Same decimal values

4. Multiply and Divide (stage 8)

Add and subtract decimals (Stage 7) using

decipipes or candy bars

Place Value

Tidy Numbers

1.6 - 0.98

Reversibility

Equal Additions

Standard written form (algorithm)

Decimal Games and Activities

- Digital learning Objects http//digistore.tki.org

.nz/ec/viewMetadata.action?idL1079 - Decimal Sort
- First to the Draw
- Four in a Row Decimals
- Beat the Basics
- Decimal Keyboard
- Target Time FIO N32,16

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

Long Term Planning Units

NZ Curriculum

knowledge

strategy

Plan a lesson using the Planning Units and Book 7

Stage 2- 5 Stage 6-7

Tanya, Jacinda, Fair Shares Alison and William Trains (Stage 6) or Hot Shots (Stage 7)

Amanda, Jessica, Nicole Hungry Birds Hamish and Marg Seed Packets (Stage 6) or Mixing Colours (Stage 7)

Emma and Ellie Fraction Circles Therie and Judy Birthday Cakes

Nikki and Cameron Animals or Wafers

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.

Modelling Stage 5

- Wafers
- Animals

Putting it all together

How Much Number? Y1-4 60-80 Y5-6 50-70

Y7-840-60

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 Journey

Organisation Orgnising routines, resources etc.

Focus on Content Familiarisation with books,

teaching model etc.

Focus on the Student Move away from what you are

doing to noticing what the student is doing

Reacting to the Student Interpret and respond to

what the student is doing

Evaluation

Thought for the day

- A DECIMAL POINT
- When you rearrange the letters becomes
- I'M A DOT IN PLACE

Additional Slides

Equivalent Fractions

Key IdeaOrdering using equivalence and benchmarks

What did you do to order them?

Circle the bigger fraction of each pair.

- A
- ½ or ¼
- 1/5 or 1/9
- 5/9 or 2/9

B 6/4 or 3/5 7/8 or 9/7 7/3 or 4/6

D 7/10 or 6/8 7/8 or 6/9 5/7 or 7/9

C 7/16 or 3/8 2/3 or 5/9 5/4 or 3/2

unit fractions

More or less than 1

related fractions

unrelated fractions

Example of Stage 8 fraction knowledge 2/3

3/4 2/5 5/8 3/8

How could you communicate this idea of

equivalence to students?

Multiplicative thinking

Fraction Circles

Paper Folding

Fraction Tiles / Strips

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

- Collect the chosen denominators
- Select how many denominators are needed.
- Make the fraction
- Compare the fraction (to ½ 1)
- Make another equivalent fraction

4

- Once you understand equivalence
- you can
- Compare and order fractions
- Add and Subtract fractions
- Understand decimals, as decimals are special

cases of equivalent fractions where the

denominator is always a power of ten.

Comparing Fractions - Which is bigger? (Bk 8)

4/5 or 2/3

12/15

10/15

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)

Which is bigger 3/4 or 9/8?

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.

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

Thinkboard Practicefive thirds or 7 4

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