Fractions
4 out of 3 people have
trouble with fractions
23 August
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
Connect different representations
words - symbols - drawings - number lines
1
4
Sets
(Discrete Models)

Shapes/Regions(Co
ntinous 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÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
The problem with “out of”
1
2 +
2
3
8
6
2
3
3 = 5
x 24 =
“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”!!!!!
2 out of 3 multiplied by 24!
= 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?
The number of parts chosen
1
2
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
3
7
= three
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.
Key Idea 3
6
is one third of what number?
This is one quarter of a
shape. What does the
whole look like?
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
Continuous
(region or
length)
Discrete
(sets)
Whole-to-Part
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.
3
3
6
6
3
3
Developing the use of addition facts
(to find one quarter of 12)
? + ? + ? + ? = 12
3?
?3
?3
?3
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
1/
5+
1/
5+
1/
5
= 3 /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.
6 quarters
1
12/4
5
Using fraction number lines to consolidate
understanding of denominator and numerator
Push over 1
0
1 half
0
1/
2
0
1/
2
2 halves
2/
2
1
3 halves
3/
2
11/2
4 halves
4/
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?
A
0
B
C
1
D
E
2
F
3
Key Idea 6 (Stage 6)
Fractions are numbers as well as operators
3/ is a number between 0 and 1 (number)
4
Find three quarters of 80
(operator)
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?
3/4 of 80
80
20
20
20
3/4 of 80 = 3 x 20 = 60
Using Double Number Lines
0
20
60
100
0
1
5
3
5
1
Put a peg on where you think 3/5 will be.
(Fractions as a number). How will you work it out?
Use a bead string and double number line to find
3/ of 100. (Fractions as an operator). How will you
5
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
Julie
Sam
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
1:1
3: 4 is the ratio of red to blue beans. 3/
7
What fraction of the beans are red?
Think of some real life 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
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