Introducing Fractions:
Initial ideas:
Young children come to school with an intuitive
sense of proportion based on ‘fair shares’ and a
working knowledge of what is meant by, “half”
and “quarter”.
In Prep to Year 3, children need to be exposed to
the language and concepts of fractions through
‘real-world’ examples. These occur in two forms:
•
“You’ve got more than me, that’s not fair!”
•
half of the apple, the glass is half full
•
a quarter of the orange,
•
3 quarters of the pizza
This is a useful starting point, but much more is
needed before children can be expected to work
with fractions formally
Use real-world examples AND non-examples to
ensure students understand that EQUAL parts
are required.
CONTINUOUS
3 quarters of the pie
2 thirds of the netball court
5 eighths of the chocolate
bar left
Continuous models are
infinitely divisible
DISCRETE
Half a dozen eggs
2 thirds of the marbles
Discrete models are
collections of whole
Note: language only, no symbols
The consequences of not appreciating the need for
equal parts.
They know how to ‘play the game’ but what do they really know?
Cut plasticene ‘rolls’
and ‘pies’ into equal
and unequal parts –
discuss ‘fair shares’
Share jelly-beans or
smarties equally and
unequally – discuss
‘fair shares’
Work Sample from SNMY Project 2003-2006 [Male, Year 5]
Explore paper folding, what do you notice as
the number of parts increases?
Fold a sheet of newspaper in
half. Repeat until it can’t be
folded in half again – discuss
what happens to the number
of parts and the size of the
parts
Halve paper strips of
different lengths,
compare halves – how
are they the same?
How are they different?
The size of the part depends upon the
whole and the number of parts
Partitioning:
Counting and colouring parts of someone else’s
model is next to useless - students need to be
actively involved in making and naming their own
fraction models.
Formalising Fraction Knowledge:
1. Prior knowledge and experience - informal
experiences, fraction language, key ideas
Equal parts
As the number of
parts increases, the
2. Partitioning – the missing link in building
size of the part
fraction knowledge and confidence, strategies decreases
for making, naming and representing
The number of
fractions
parts names
the part
3. Recording common fractions and decimal
fractions – problems with recording, the
The numerator tells
fraction symbol, decimal numeration (to
‘how many’, the
tenths)
denominator tells
4. Consolidating fraction knowledge –
comparing, ordering/sequencing, counting,
and renaming.
‘how much’
Links to
multiplication and
division
Explore partitioning informally through paper
folding, cutting and sharing activities based on
halving using a range of materials, eg,
plasticene rolls and icy-pole sticks
Partitioning (making equal parts) is the key to this:
paper streamers
• develop strategies for halving, thirding and
fifthing;
rope and pegs
• generalise to create diagrams and number lines;
• use to make, name, compare, order, and rename
mixed and proper fractions including decimals.
Kindergarten Squares
Smarties
The ‘halving’ strategy
For example,
Both shapes
are 1 half
Explore paper folding with coloured paper
squares, paper streamers and newspaper.
Explore: make and name as many fractions in the
‘halving family’ as you can
8 equal parts,
eighths
How are they different?
How are they the same?
For example, make a poster
2 and 3 quarters
Write down as many things as you can about your
fraction. How many different ways can you find to name
your fraction?
How many different designs can you make
which are 3 quarters red and 1 quarter yellow?
It’s bigger than 2 and a half ... Smaller than 3 .... It’s 11
quarters ... It’s 5 halves and 1 quarter ... It could be 2
and 3 quarter slices of bread ...
Extend partitioning to diagrams:
Ask: What did the
second fold do?
Ask: What did the first
fold do?
It cut the top and
bottom edges in half
again
It cut the top and
bottom edges in half
Estimate 1 half
The ‘thirding’ strategy:
Ask: What did the
third fold do? It cut the
side edges in half.
Think: 3 equal parts
... 2 equal parts …
1 third is less than 1
half ... estimate
Halve the
remaining part
How would you describe this strategy
using paper streamers?
Fold kindergarten squares or paper
streamers into 3 equal parts
The ‘fifthing’ strategy
Use to draw diagrams, for example,
Apply thirding strategy to
top and bottom edge,
halving strategy to side
edges to get sixths
Think: 5 equal parts ...
4 equal parts …
1 fifth is less than 1
quarter ... estimate
Then halve and
halve again
Notice:
Use to draw diagrams, for example,
Apply fifthing
strategy to top and
bottom edge, halving
strategy to side
edges to get tenths
Halving
family
Thirding
family
Fifthing
family
Apply to number line
4
5
Fold kindergarten squares or paper
streamers into 5 equal parts
No. of parts
Name
1
whole
2
halves
3
thirds
4
quarters (fourths)
5
fifths
6
sixths
8
eighths
9
ninths
10
tenths
12
twelfths
15
fifteenths
Halving
and
Thirding
Halving
and Fifthing
Thirding
and Fifthing
As the number of parts increases, the size of the parts
gets smaller – the number of parts, names the part
Explore strategy combinations to recognise
that:
Thirds by quarters
give twelfths
quarters
thirds
thirds
Thirds by fifths
give fifteenths
fifths
What other fractions can
be generated by halving
and thirding or by
fifthing and thirding?
tenths
What other
fractions can be
generated by
fifthing and
tenths
halving?
Tenths by tenths
give hundredths
Recording common fractions:
Use real-world examples to explore the
difference between ‘how many’ and ‘how much’
Young children
expect numbers
to be used to
say ‘how many’
This tells
‘how many’
tens
34
This tells
‘how many’
ones
Informally describe and compare:
Is it a big share or a little share? Would you rather have 2
thirds of the pizza or 3 quarters of the pizza? Why? How
could you convince me?
Construct fraction diagrams to compare more
formally
Introduce the fraction symbol:
Introduce recording once key ideas have been
established through practical activities and
partitioning:
•
•
equal shares - equal parts Explore non-examples
fraction names are related to the total number
of parts (denominator idea – the more parts
there are, the smaller they are)
This tells how much
•
the number of parts required tells how many
(numerator idea – the only counting number)
This tells how many
2 fifths
2
5
2
out of
5
2
5
This number tells how many
This number names the parts and tells
how much
Make and name mixed common fractions
Recognise:
Introducing Decimals:
third
3rd
• different meanings for ordinal number names,
eg, ‘third’ can mean third in line, the 3rd of April
or 1 out of 3 equal parts
• that the ‘out of’ idea only works for proper
fractions and recognised wholes, eg,
3 ‘out of’ 4
Note: this idea does not work for
improper fractions, eg, “10 out of 3”
is meaningless!
But “10 thirds” does make
sense, as does “10 divided into
3 equal parts”
Recognise tenths as a new place-value part:
Recognise decimals as fractions – use halving
and fifthing partitioning strategies to make and
represent tenths
Halves by fifths are tenths
fifths
7 out of ten parts, 7 tenths
halves
Fifth then halve each part or halve then fifth each part, 2 and 4 tenths
2
2.4
3
Name decimals in terms of their place-value parts,
eg, “two and four tenths” NOT “two point four”
Why is this important?
Extend decimal place-value:
1. Introduce the new unit: 1 one is 10 tenths
Recognise hundredths as a new place-value part:
2. Make, name and record ones and tenths
1. Introduce the new unit: 1 tenth is 10 hundredths
The decimal point shows
where ones begin
3. Consolidate: compare, order, count forwards
and backwards in ones and tenths, and
rename
Note: Money and MAB do not work – Why?
5.0
5.30
5.3 5.4
6.0
5.37
5.40
hundredths
1!3
ones
ones tenths
one and 3 tenths
tenths
via partitioning
2. Show, name and record ones, tenths &
hundredths
5!3 7
3. Consolidate: compare, order, count forwards
and backwards, and rename
Establish links between tenths and
hundredths, and hundredths and per cent:
Consolidating decimal place-value:
1. Compare decimals – which is larger, which is
smaller, why? Which is longer, 4.5 metres or 4.34 metres?
Which is heavier, 0.75 kg or 0.8 kg?
2. Order decimal fractions on a number line, eg,
Order from smallest to largest and place on a 0 to 2 number line (rope):
3.27, 2.09, 4.9, 0.45, 2.8
0.7 is 7 tenths or 7
10
Recognise per cent ‘benchmarks’:
50% is a half, 25% is a quarter,
10% is a tenth, …
33 % is 1 third …
0.75 is 7 tenths, 5 hundredths
75 hundredths
75 per cent, 75%, or
75
100
Extending Fraction & Decimal Ideas:
By the end of primary school, students are
expected to be able to:
Requires:
partitioning
• rename, compare and
strategies, fraction
order fractions with unlike
as division idea
denominators
and ‘region’ idea
• recognise decimal fractions
for multiplication
to thousandths
Requires: partitioning strategies, place-value
idea that 1 tenth of these is 1 of those, and
the ‘for each’ idea for multiplication
3. Count forwards and backwards in place-value
parts, eg,
… 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, …
…5.23, 5.43, 5.63, 5.83, …
4. Rename in as many different ways as possible, eg,
4.23 is 4 ones, 2 tenths, 3 hundredths
4 ones, 23 hundredths
42 tenths, 3 hundredths
423 hundredths …
Renaming Common Fractions:
1
3
3 parts
4 parts
3
4
Use paper
folding &
student
generated
diagrams
to arrive at the
generalisation:
2
6
9
12
9 parts
12 parts
If the total number of parts increase by a
certain factor, the number of parts required
increase by the same factor
Comparing common fractions:
Comparing common fractions:
Which is larger 3 fifths or 2 thirds?
Which is larger 3 fifths or 2 thirds?
3
9
=
5
15
But how do you know? ... Partition
fifths
2
10
=
3
15
thirds
THINK: thirds by fifths ... fifteenths
THINK: thirds by fifths ... fifteenths
Extend decimal place-value:
Compare, order and rename decimal fractions:
•
The more digits the larger the number (eg, 5.346 said
to be larger than 5.6)
2. Show, name and record ones, tenths, hundredths
and thousandths
•
The less digits the larger the number (eg, 0.4
considered to be larger than 0.52)
•
If ones, tens hundreds etc live to the right of 0, then
tenths, hundredths etc live to the left of 0 (eg, 0.612
considered smaller than 0.216)
•
Zero does not count (eg, 3.01 seen to be the same as
3.1)
•
A percentage is a whole number (eg, do not see that
67% is 67 hundredths or 0.67)
5.370
6.0
5.37 5.38
5.376
5.40
5.380
tenths
5.30
5.3 5.4
ones
5.0
thousandths
Some common misconceptions:
hundredths
Recognise hundredths as a new place-value part:
1. Introduce the new unit: 1 hundredth is 10
thousandths
via partitioning
5!3 7 6
3. Consolidate: compare, order, count forwards
and backwards, and rename
Compare, order and rename decimal fractions:
a) Is 4.57 km longer/shorter than 4.075 km?
b) Order the the long-jump distances: 2.45m,
1.78m, 2.08m, 1.75m, 3.02m, 1.96m and 2.8m
d) Express 7! % as a decimal
ones
9
tenths
0
hundredths
1. Compare mixed common fractions and
decimals – which is bigger, which is smaller,
why?
2. Order common fractions and decimal
fractions on a number line
c) 3780 grams, how many kilograms?
2
Consolidating fraction knowledge:
3. Count forwards and backwards in
recognised parts
7
thousandths
1
Use Number Expanders to rename decimals
4. Rename in as many different ways as
possible.
Which is bigger? Why?
2/3 or 6 tenths ... 11/2 or 18/16
For example,
(Gillian Large, Year 5/6, 2002)
(Gillian Large, Year 5/6, 2002)
Make a Whole:
Games:
For example,
• Make a Whole
• Target Practice
• Fraction Concentration
(Make a Whole Game Board, Vicki Nally, 2002)
(Vicki Nally, 2002)
Make a Model, eg, a Think Board
(Vicki Nally, 2002)
(Gillian Large, 2002)