Fractions WorkshopMarie Hirst

Have a go at the Fraction Hunt on your table while you are

waiting!

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

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

The Number Frameworks

StrategiesKnowledge

Number Identification

Number Sequence and Order

Grouping and Place Value

Basic Facts

Written Recording

Addition & Subtraction

Multiplication & Division

Fractions and Proportions

Assess Your Fraction Strategies and Fraction Knowledge

Assigning a strategy stage for proportions and ratios

Fraction Snapshots

Here are 12 jelly beans to spread on the cake.

If you ate one third of the cake how many jelly beans will you eat?

Stage 1 Stage 2-4 (AC) Stage 5 (EA)

Unequal Sharing Equal Sharing Use of Addition and known facts e.g. 4 + 4 + 4 = 12

Fraction Snapshots (cont’d)

What is 3/4 of 80? Stage 6 (AA)Using multiplication

16 is four ninths of what number?

Stage 7 (AM)Using division

To make 8 aprons it takes 6 metres of cloth. How many metres would you need to make 20 aprons?

Stage 8 (AP)

What misconceptions may 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 to move to equal sharing?

Introduce the vocabulary of equal / fair shares with both regions and sets for halves and then quarters.

Draw two pictures of one quarter

Discrete and continuous modelsOne Quarter:

Continuous Discrete

(regions/lengths) (sets)

Label your drawings as discrete or continuous models.

Children need experience with both models from the very start.

Key Idea 1

Work with both shapes and sets of

fractions from early on.

Linking regions/shapes and setsFind one quarter

The Strategy Teaching Model

Using Number Properties

Using Imaging

Using Material

s

New Knowledge & Strategies

Existing Knowledge & Strategies

Using Materials

Using Materials - fraction regionsFind one quarter

Using Materials - fraction regions

Find one quarter of 12

The Strategy Teaching Model

Using Number Properties

Using Imaging

Using Material

s

New Knowledge & Strategies

Existing Knowledge & Strategies

Using Materials

Using ImagingFind one quarter of 12

Key idea: quarters means you need 4 equal groups. One quarter is the number in one of those groups.

The Strategy Teaching Model

Using Number Properties

Using Imaging

Using Material

s

New Knowledge & Strategies

Existing Knowledge & Strategies

Using Materials

Using Number Properties

Find one quarter of 40, 400, 4000

Develop early additive thinking by using addition facts

Find one quarter of 12

?

??

?3

3

3

3

Using Materials - cubes

Four birds found a worm in the ground 20 smarties long.

What proportion of the worm do they each get?

How many smarties will each bird get?

Key Idea 2

3 sevenths 3 out of 7 7/3 7 thirds

5 views of fractions

€

3

73 over 73 : 7

3 out of 7 3 ÷ 7

3 sevenths

+ =

“I ate 1 out of the 2 sandwiches in my lunchbox, Kate ate 2 out of the 3 sandwiches in her lunchbox, so together we ate 3 out of the 5 sandwiches”

12

23

35

The problem with “out of”

23

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

Fraction Language

Use words before and use symbols with care.

e.g. ‘one 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

Fractional vocabulary

One half

One third

One quarter

Don’t 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

1 - ?/20 = 3/2017

Key Idea 2

Fraction language is confusing. Emphasise the ‘ths’ code.

Use words before symbols. Introduce symbols with care. The bottom number tells how many parts the whole has been split into,the top number tells how many of those parts have been chosen.

6 is one third of what number?

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

Key Idea 3

18

Key Idea 3

Go from part-to-whole as well as whole-to-part with both shapes and sets.

Children need experience in both reconstructing the whole as well as dividing a whole.

Perception check on two key ideas

Where in the table does this question fit?Hemi got two thirds of the lollies. How many were there altogether?

Part-to-Whole Whole-to-Part

Continuous(region or

length)

Discrete(sets)

Write 3 more questions to fit the other parts of the table.

Model Part - to - Whole Whole - to - Part

Continuous(Region or

length)

Discrete(sets)

Hemi got two thirds of the lollies. How many were

there altogether?

Extending the idea of going from part-to-whole with non-unit fractions

Hemi got three fifths of the lollies and got 12. How many lollies were there altogether?i.e. 12 is three fifths of what number?

Draw a diagram/use equipment to help your thinking.

12 is three fifths of what number?

12

4 44 4 4

20

8

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

Discuss in groups what you think children would do and then how you would solve this problem.

3 ÷ 5

Key Idea 4

Division3 ÷ 5

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

Key Idea 4

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

e.g. 5 ÷ 3

Which letter shows 5 halves as a number?

0 1 2 3

A B C D E F

Key Idea 5

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

1 521/2

5 halves

Using fraction number lines to consolidate understanding of denominator and numerator

Push over 1

0 1 half 2 halves 3 halves 4 halves

0 1/2 2/2

3/2 4/2

0 1/2 1 11/2 2

Fraction Circles

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

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

1/2 is a number between 0 and 1 (number)

Find one half of 12 (operator)

Key Idea 6Fractions are numbers as well as operators

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 and double number line to find 3/5 of 100. (Fractions as an operator). How will you work it out?

Key Idea 7

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

True or False or Maybe

Key Idea 7

Fractions are always relative to the whole.

Halves are not always bigger than quarters, it depends on what the whole is.

What is the whole?

A A

B B B B

C

D D D D D D D D

Key Idea 8 - Ratios!

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 real life contexts when ratios are used.

1:1

3/7

Key Idea 8

There is a link between ratios and fractions.

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

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

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)

Exploring simple ratios at Stage 6

2 green beans : 3 red beans

How many green and red beans in 6 packets?

0

0 2

3

12

18

4

6

6

9

10

15

8

12

I have 22 green beans, how many red will I have?

green

red

33

Summary of Fractions Key Ideas(Stages 2 - 6)

1. Use sets as well as shapes/regions from early on

2. Fraction Language - use words first and introduce symbols carefully

3. Go from Part-to-Whole as well as Whole-to-Part

4. Division is the most common context for fractions.

5. Fractions are not always less than 1, push over 1 early.

6. Fractions are numbers as well as operators.

7. Fractions are always relative to the whole.

8. Be careful of the relationship between ratios and fractions

9. Fractions are a context for add/sub and mult/div strategies

Choose your share of chocolate!

Getting into book 7

• Explore an activity in book 7.• Focus on the key ideas we have

discussed whilst exploring the activity.

Fractions, Ratios and Decimals

"My life is all arithmetic”the young businesswomanexplains. "I try to add to my

income, subtract from my weight, divide my time, and avoid multiplying..."

Little League Video Clip

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.

Numerical Reasoning Testas 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

Objectives

• Consolidate understanding of key ideas when teaching fractions, decimals and percentages

• Understand common misconceptions with ratios and decimals.

• Explore equipment and activities used to teach key ideas within these higher stages.

Decimals

At what stage are decimals introduced?

(knowledge and strategy)

Teaching Decimal Knowledge using Book 4

• Decimal Number Lines (Bk 4: 15) MM 4-31

• Squeeze / Number Line Flips: Bk 4 (15)

• Using Decimats: (Bk 4: 8,9), MM 4-21

What did these activities practice?

How are your decimals?•Order these decimals from smallest to largest:

. 3.48 3.6 3.067

•Write one eighth as a decimal

•What is the answer to 5 ÷ 4

•What is the answer to 3 - 1.95

•What is 0.3 x 0.4

•Order these fractions decimals and percentages . 2/3

7/16 30% 0.61 2/5 75%

0.38

Stacey’s Homework

Continue these sequences:

a) 0.7, 0.8, 0.9, 0.10, 0.11, 0.12

b) 2.97, 2.98, 2.99 2.100, 2.101, 2.102

Write down which is the smallest number:

a) 0.8, 0.5, 0.1 0.1

b) 2.3, 2.191, 2.161 2.3

c) 3.856, 3.29, 3.4 3.4

What do you think Stacey is doing?

Hemi’s Homework

Write down which is the smallest number:

a) 0.8 0.5 0.1 0.1

b) 2.3 2.191 2.16 2.191

c) 3.856 3.29 3.4 3.856

What do you think Hemi is doing?

Discuss what other common misconceptions you think children may have about decimals.

Decimal Misconceptions Decimals are two independent sets of whole numbers

separated by a decimal point, e.g. 3.71 is bigger than 3.8 and 1.8 + 2.4 = 3.12

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

Decimals are negative numbers.

1/2 is 0.2 and 1/4 is 0.4, e.g. 0.4 is smaller than 0.2

When you multiply decimals the number always gets bigger.

When you multiply a decimal number by 10, just add a zero, e.g. 4.5 x 10 = 4.50

Equivalent Fractions

You 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.

Converting Fractions to Decimals

Using Decipipes Bk 7 p.22 (or Decimats)

Start with tenths, fifths, halves, quarters, and then eighths,

Operating with decimals

Using Candy Bars (book 5) Understanding tenths and hundredths using candy bars:

Pose division problems using the equipment to find the number of wholes, tenths and hundredths;

e.g. 6 ÷ 5, 4 ÷ 5, then 5 ÷ 4, 3 ÷ 4, 13÷ 4

Operate with the decimals using addition/subtraction and multiplication to consolidate understanding requiring exchanging across the decimal point, e.g.

3.6 - 1.95, 3.4 + 1.8, 4.3 - 2.7, 7 x 0.4, 1.25 x 6

Using Advanced Additive strategies for decimals

Solve 3.6 - 2.98

Ww

w

Multiplying Decimals0.3 x 0.4

0 1

1

Ww

w

Multiplying Decimals0.3 x 0.4 = 0.12

0 1

1

0.3

0.4

How are your decimals?•Order these decimals from smallest to largest:

. 3.48 3.6 3.067

•Write one eighth as a decimal

•What is the answer to 5 ÷ 4

•What is the answer to 3 - 1.95

•What is 0.3 x 0.4

•Order these fractions decimals and 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

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

It is a method of comparing fractions by giving both fractions a common

denominator - hundredths. So it is useful to view percentages as

hundredths.

Why calculate percentages?

=

Percentages

At what stage are percentages introduced?

(knowledge and strategy)

Percentages

AM (Stage 7: NC Level 4)• Solve fraction decimal percentage conversions for

common fractions e.g. halves, thirds, quarters, fifths, and tenths

AP(Stage 8: NC Level 5)• Estimate and solve problems using a variety of strategies

including using common factors, re-unitising of fractions, decimals and percentages, and finding relationships between and within ratios and simple rates.

Applying PercentagesTypes of Percentage Calculations

• Finding percentages of amounts, e.g. 25% of $80

• Expressing quantities as a percentage (for easy comparison), e.g. 18 out of 24 = ?%

• 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?

Estimate and find percentages of whole number amounts.

E.g. Find 25% of $80 (easy!) 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)

Mini Teaching Session 1

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

What now? Use fraction snapshots if you think it would be useful

to regroup children.(On wikispace)

Review fraction long term planning units.

Teach fraction knowledge and proportions & ratios strategies in your classroom with your groups.

This is our last pick up session -Thank you all for coming.

Thought for the dayThere are three things to remember when teaching;

Know your stuff,

Know whom you are stuffing,

And stuff them elegantly.

Lola May