Fractions

23 August

4 out of 3 people have trouble with fractions

Body Fractions Game

• Arm Span = 1• One arm = half• What is a quarter?

• Make one half, three quarters, one, etc• With a partner three halves• In a group of four…

Objectives• Understand the strategy stages progressions for

proportions and ratios

• Understand common misconceptions and key ideas when teaching fractions.

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

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.

Key Idea 1

Draw two pictures of one quarter

one quarter

€

1

4Connect different representations

words - symbols - drawings - number lines

Shapes/Regions(Continous

models)

Sets(Discrete Models)

Recapping Key Idea 1

Work with shapes, lengths and sets of

fractions from early on.

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 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 Language

Use words first then introduce 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

Top and bottom numbers

• The top number counts

• The bottom number tells what is being counted.

Fractional vocabulary

One half

One third

One quarter

Don’t know

Use multilink wrapped as wholes to demonstrate the “th” code.

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

Fraction Symbols and fraction words

= three

€

3

7

Three thirteenths = 13

Provide plenty of opportunities for children to work between words and symbols

Recapping 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

Recapping 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, he has 12 lollies. How many were there altogether?

Part-to-Whole Whole-to-Part

Continuous(region or

length)

Discrete(sets)

Key Idea 4 :

Anticipate the result of equal sharing by grouping (using addition or skip counting) to solve fraction problems rather than equal sharing by ones(Early Additive Stage 5)

Find one quarter of this shape in different ways

Find one quarter of this shape

Find one quarter of this shape

Now find one quarter of 12 Finding quarters: Introduce the idea of halving and halving again.

66

33

33

Developing the use of addition facts (to find one quarter of 12)

?

??

?3

3

3

3

? + ? + ? + ? = 12

Find one quarter of 12 (imaging)Key idea: quarters means you need 4 equal groups. One quarter is the number in one of those groups.

3

Recapping Key Idea 4 :

Anticipate the result of equal sharing by using addition or skip counting (Stage 5 EA: Animals Book 7, p.18)

Linking the use of regions and sets helps to do this.

Hungry Birds (Book 7, page 22)

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?

5 + 5 + 5 + 5 = 20

Key Idea 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 but harder examples require more sophisticated methods

e.g. 3 ÷ 5 = 3 fifths

Division 3 ÷ 5

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

Key Idea 6

Order these fractions: ¼ 6 quarters, 3/4 nine quarters, 2/4

Key Idea 6

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

1 512/4

6 quarters

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

Perception Check – Discuss these key ideas. (Stage 4 and 5).

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

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

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

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

5. Division is the main context for using fractions

6. Fractions are not always less than 1, push over 1 early to consolidate meaning of fraction symbols.

Key Idea 6.

Which letter shows 5 halves as a number?

0 1 2 3

A B C D E F

3/4 is a number between 0 and 1 (number)

Find three quarters of 80 (operator)

Key Idea 6 (Stage 6)

Fractions are numbers as well as operators

80

20 2020

3/4 of 80 = 3 x 20 = 60

3/4 of 80

Key Idea 7 (Stage 6 AA)

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

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 8 (stage 6)

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

Recapping Key Idea 8 (stage 6)

Fractions are always relative to the whole.

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

Key Idea 9 (stage 6) - 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

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

Summary of Fractions Key Ideas

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. Consider the relationship between ratios and fractions

9. Use addition/skip counting to find fractions of sets then develop and apply multiplicative thinking –

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