Fractions Domain
Fractions, perhaps with the exception of a half, are much more difficult for students to understand than
whole numbers because they are essentially very different. Therefore teachers should expect student
understandings of fractions to emerge very slowly.
Complicating the teaching of fractions is the fact that many students have been taught to view fractions in a
variety of five unrelated ways. This Fractions Domain will avoid four of these entirely since all problems can
be solved by one single unified view. What the four incorrect ways of regarding fractions are is mentioned
in this document at appropriate times.
Beginning Teaching Fractions
There is no urgency to introduce any fractions to young children. However, when the teaching of fractions
begins teachers should be aware that they need to create a sound basis for later very challenging fraction
ideas, so initially great care is needed.
Early on it is suggested that students have multiple opportunities to model a half of a
region. The equality of halves needs to be emphasised. Young children will understand
the injustice that “my brother got the big half of the cake” without realising that a “big
half” is a contradiction. So students need experiences with equal parts. At this stage
Figure 1
students will typically see “halves” as two identical regions. So, for example, while in
Figure 1 the shaded portion of the geoboard is actually a half, it is too much to
expect students to understand the other unshaded pieces also make a half.
At later stages halves, or indeed any unit fraction, will not have to appear identical
in all ways but merely equal in some measure. For example, Figure 2 shows halves
on a geoboard; they are halves because the areas are equal even though the
shapes are not identical.
Frequent exposure to problems involving drawing and colouring in halves is
important. Cutting to produce halves would be good, but young children may lack
Figure 2
the fine motor skills for this.
A more challenging, but worthwhile, view of
halves, is to give students a region
representing a half, and ask them to make a
whole. For example Figure 3 shows a given
Figure 3
half being turned into a possible whole.
Another problem for very young students is they may not understand that cutting two
pieces or “halves” out of a cake requires the whole cake to be used. For example,
Figure 4 shows how students might think they are creating halves by cutting two equal
pieces of cake, although they have left most of the cake behind.
Finally, when introducing fractions, they should be written in words not the symbols
Figure 4
initially, and the whole object should be explicitly mentioned. For example, students
should be encouraged to say or write “one half of an orange” rather than simply “one half” or “ 12 ”. This
does two things:
• it encourages student to always think about what the “whole” is and
• it avoids the confusions that arise from reading, say, 12 as “one out of two” or “one over two’, or “one
divided by two” or “the ratio one to two”.
So it is not a good idea to introduce these fraction symbols too early: 12 , 13 , 32 , 14 , 42 and 34 .
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Fractions of Sets
To find fractions of numbers students should be encouraged first to solve division problems then attempt
to link the answers to fractions. For example, to solve a third of twelve students should be encouraged to
solve the problem:
• 12 sweets are shared among three students. How many sweets does each students receive?
Students need to create three piles each with four sweets by some process, which is likely to be by sharing
one-by-one. The next question is then:
• What fraction of the sweets did each person receive?
The next question is then:
• What does 1 third of 12 equal?
The connections that students must make in such problems are not trivial; teachers should expect many
students to take some time to make the required connection between division and finding a fraction of a
number.
Fraction Symbols
So far the use of the common fraction symbols has been avoided in the essential introductory work on
fractions. This has been very deliberate; it helps students avoid incorrectly constructing inadequate and
wrong concepts about fractions. When the standard fraction symbols are introduced it is important
teachers and students to read the symbol aloud by using the familiar fraction words. For example, students
seeing ¾ should be saying three quarters, not three out of four, or three divided by four. Mixing writing
the fraction words and the symbols is a good idea in problems. This encourages students to read fraction
words and symbols in one way only. For example, a written problem might be add 73 and 4 sevenths. This
is easy to solve provided students regard 73 as a compact way of writing 3 sevenths.
Eventually the dominant form will be the symbol rather than fraction words. However, it is worth noting
that, when introducing any new fraction idea, reversion to the word forms
€ in a context is very desirable.
3
For example, to solve 1− 11 when
€ it is a new kind of problem it would be better to make up a story and
use fraction words:
• At a party three elevenths of a cake is eaten. What fraction of the cake remains?
Comparing €
Fractions
Students who do not understand fractions can often be diagnosed by asking them to compare fractions. For
example students might be asked to pick the largest fraction from:
2 2 2
2
• 7, 9, 11, 13,
Students who view fractions as “out of” often pick
€
2
13
because 13 is the largest denominator.
Another factor to consider here is that the comparison of fractions can strictly speaking only proceed if the
fractions all refer to the same wholes. This diagnostic question helps detect whether students realise this:
• Jack spends ½ his pocket money on sweets, and Jill spends ¼ of her money on sweets. Does Jill buy
€
more sweets than Jack?
The answer is yes – provided Jill started with at least twice as much pocket money as Jack.
2 2
So the comparison question above – which is the largest of 7,2 9,2 11,
13, would be better asked as:
• Four students all buy the same-sized cake. Jill eats 27 of her cake, Jack eats
her cake, and Lee eats 132 of his cake. Who ate the most?
2
9
of his cake, Mere eats
€
€
€
€
€
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2
11
of
Fractions of Sets
Up to this point the fractions have mainly been about fractions of regions, except for halves, thirds and
quarters of numbers. Now a fraction of a set may be introduced where students solve division problems
first, then link them to finding a unit fraction of a number. (A unit fraction always has 1 in the numerator, so
1 is a unit fraction, but 2 is not.) For example, the students may attempt to solve problems like:
11
11
€
€
• Mary eats 1 fifth of her fifteen sweets. How many does she eat?
Here they need to connect working out 1 fifth of her fifteen with 15 ÷ 5.
Unit fractions are
€ used at this time because finding a non-unit fraction of a number needs a key extra step,
and is consequently sharply more difficult for students than unit fractions. For example:
• Mary eats 3 fifths of her fifteen sweets. How many does she eat?
They need to connect working out 1 fifth of her fifteen with 15 ÷ 5; this as was needs for unit fractions of a
number, but now need to co-ordinate then multiply by three.
Such non-unit fraction problems need to be delayed.
Notice problems like 27 of 23 are another step harder; in this case, because 7 is not a factor of 23 the
calculation requires understanding that 27 of 23 nees to be transformed into (2 x 23 ) ÷ 7. It is not at all
obvious why this is true.
Fractions €
Arising from Division
Up to this point fractions like€¾ have arisen from creating four equal parts of a whole and taking three of
these parts. This is why young students can read ¾ as “three out of four” without any immediate ill-effect.
But definition of a fraction is not adequate to cope with harder fraction problems; a new extended
definition of a fraction is introduced:
•
A fraction is a number that is the answer to a division of whole numbers.
This definition of fraction needs considerable care when introduced if for no other reason than teachers,
especially secondary, regard the line in a fraction as a division sign - so, for example, such teachers
incorrectly read ¾ as three divided by four.
Most teachers, and therefore the students that they teach, find it hard to understand why 3 ÷ 4 equals 3
quarters is not a tautology, that is, it is not merely a repetition of the same idea in different words or
symbols. Secondary teachers will often claim, understandably but incorrectly, that the line in a fraction – it
is called a vinculum - is in fact a division sign. The cause is that in secondary school algebra the vinculum
does mean division, and very usefully it acts as bracket without the messiness. For example, it is much
2x + 3y −11
neater to write
than (2x + 3y −11) ÷ (x + y) . The power of this notation becomes even more
x+y
evident as the complexity of expression rises: Consider whether
2x + 3y −11 11x + 3y −11
−
x+
6x + 7y€
€y
or ((2x + 3y −11)/(x + y) − (11x + 3y −11)/(6x + 7y))/(6 /(x − 4))
6
x−4
is more desirable. Clearly the left hand expression, while still complex, is much the easier to understand
€
than the right hand expression.
But the everyday usage of the vinculum in algebra meaning division in
secondary school mathematics is no argument for confusing this with its use in fractions.
When introducing division problems with fractional answers it is very desirable to fall back to the division
story problems. Firstly we should revise the meaning of division with whole number answers:
• Freddy has 6 bananas to share out among 3 people. How many bananas does each person get?
Clearly the answer is 6 ÷ 2 = 3. Here division is a process than leads to an answer.
Now apply this argument to a problem whose answer is not a whole number:
• Freddy has 3 bananas to share out among 4 people. What fraction of a banana does each person
receive?
Here 3 ÷ 4 is a process that ends up producing a fractional answer 3 quarters. Unfortunately teachers and
students often find this answer difficult to obtain. Consider this problem with its range of possible answers
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• We have 3 whole oranges and we cut each orange into four equal pieces, then share them out. What
does 3 ÷ 4 equal?
Suppose four children, Jo, James, Anu and Hoani each receive 1 quarter from each orange as indicated in
Figure 4.
Jo
James
Jo
James
Jo
James
Hoani
u
Anu
Hoani
u
Anu
Hoani
u
Anu
Figure 4
Each person receives three pieces, or 3 quarters of one orange. This demonstrates that 3 ÷ 4, a process
leads to the answer 3 quarters of one whole.
Unfortunately it is common to see 3 ÷ 4 = 123 . Typically people do this because they read 123 as three out
of twelve not three quarters. This is another important reason why teachers should never say “out of” for
fractions.
Occasionally teachers read 123 as €
three twelfths. While this is correct we need €
to realise that it is three
twelfths of three oranges not three twelfths of one orange. This illustrates the extreme importance teachers
should attach to fractions being related to one unit whatever that unit may be. This is not altogether
straightforward for€students.
A further complication in the example above, where each student got three quarters of an orange, the
quarters came from three different oranges, so their “three quarters” relates to an imaginary one orange.
This does cause problem for some teachers and many students.
Numbers Greater Than One
Students often believe that fractions are always part of a whole, so they cannot be more than 1. So for
them symbols like 67 are meaningless. Yet using with the extended definition of fractions arising from whole
number divisions we can have fractions that are greater than one unit; for example, 7 ÷ 6 = 67 .
Numbers like 67 are well named as improper fractions because they are not the way we express the answer
to the division
7 ÷ 6 in the adult world. Here it is sensible to do a whole number division first, which leads
€
to 7 ÷ 6 = 1 16 .
€
Sometimes numbers like 1 16 are called mixed fractions, but more correctly, because it is a mixture of a
€
whole number and fractional part, it should be described as a mixed number; this is the kind of number that
the adult world generally uses, involving as it does decimal fractions almost always.
€
€
Equivalent Fractions,
and Addition and Subtraction
Unlike whole numbers, which come in one way, fractions have an infinite number of equivalent forms. This
factor alone explains why fractions are inherently more difficult for students to learn than whole numbers.
Unfortunately being able to move between various forms of fractions is absolutely essential; the equivalence
of fractions is needed to perform operations with fractional numbers. For example, what underpins
working out
2 1
+ ? Firstly students need to realise that this problem corresponds to a story such as:
3 4
• Jake eats 2 thirds of a cake then he eats 1 quarter of another cake that is the same size. What fraction of
a cake is eaten altogether?
!
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A vital idea that students need to understand about addition of fractions here is that the cakes must be the
same size. It is significant to note that many students will be confused by this problem because there is no
cake that has had 11 twelfths of it eaten.
(It would be inappropriate to solve this problem:
• Jules eats 2 thirds of a small cake and Jake eats 1 quarter of a big cake. What fraction of a cake is eaten
altogether?
This problem is impossible to work out as the cakes are of unknown different sizes.)
Returning to the original problem 2 thirds and 1 quarter ( of the same unit) cannot be added since they
refer to different fractions. But knowing 2 thirds equals 8 twelfths and 1 quarter equals 3 twelfths then:
• 2 thirds = 4 sixths = 6 ninths = 8 twelfths = 10 fifteenths = 12 eighteenths = 14 twenty-firsts = 16
twenty-fourths = … And
• 1 quarter = 2 eighths = 3 twelfths = 4 sixteenths = 5 twentieths-= 6 twenty-fourths = ….
So 2 thirds + 1 quarter equals 11 twelfths by noting there is a the common denominator, 12. But there are
other common denominators in these lists such as 24, so 2 thirds + 1 quarter also equals 22 twentyfourths. In fact all are an infinite number of correct answers: 2 1 11 22 33 44
+ = =
=
=
= ...
3 4 12 24 36 48
So a huge complicating factor in addition and subtraction of fractions is not only the need to spot a
common denominator but also know how to determine the Lowest Common Denominator
As already indicated adding or subtraction of fractions
! are difficult to understand. Students who look for
procedures without ever seeking understanding, or who regard fractions as “out of” commonly make
3 2
5
errors like 4 + 3 = 7 where numerators and denominators have been added.
It is useful to introduce all new fraction problems by writing the fractions in words initially; this will show
just how complex the new idea is.
Multiplication of Fractions
When introducing a new idea it is a powerful teaching idea to try to link this new concept to previous
understandings. In the case of multiplication the keyidea with whole numbers is repeated addition. It is
possible to use this idea for some multiplications involving fractions. Unfortunately it is not possible in all
cases; this will require construction of new meanings for multiplication.
A Whole Number Times a Fraction
A whole number of fractional pieces can be connected easily to multiplication as repeated addition. For
example, 4 × 83 = 83 + 83 + 83 + 83
Finding the answer to these multiplications is relatively straight-forward: Consider working out
12 x 43 . Starting with a problem this could be:
€
ۥ 12 students each receive 3 quarters of an apple from their teacher. How many apples did the teacher
need to buy?
Obviously the teacher needs 36 quarters because 12 x 3 = 36. But each apple creates 4 quarters so the
teacher needs to buy 36 ÷ 4 = 9 apples.
This problem is like a whole multiplication problem in which “times” for whole numbers is applied to
fractions.
A Fraction of a Whole Number
Now consider now working out 43 x 12. Firstly, this cannot mean 12 repeatedly added to “three quarters”
times. So it is not immediately evident what this 43 x 12 might mean. Again a problem helps:
• There are 12 oranges left at the supermarket. Harry buys 3 quarters of the oranges. How many oranges
does Harry have?
€
Here it naturally suggests that 1 quarter of 12 is 3, and 3 lots of 3 is 9. So reading 43 x 12 as three quarters
€
of twelve is useful. This implies it is more natural to write 43 x 12 as 43 of 12; in fact many students find 43
of 12 easier to work out than 43 x 12
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€
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€
€
€
€
Remarkably, although 12 x 43 and 43 of 12 are totally different problems, their answers are the same. It
cannot be overemphasised that it is not obvious to students that the commutative law applies to a whole
number times a fraction.
€
€
Fraction of a Fraction
Whole number notions of multiplication of restricted use when finding fraction of a fraction. Teachers
should note that students must construct new mental schema to cope with this new idea – and all such
new constructions are always difficult. Consider working out 13 of 14 :
Finding a quarter of 4, 8, 12, 16, 20 etcetera is easy, leading to answers 1, 2, 3, 4, 5 etcetera. Finding a third
of answers 3, 6, 9 etcetera are easy. So a good choice is a grid that is three by four, i.e. the grid has 12
squares. this will make finding a 13 of 14 of the grid easy.
€
Discard 3 quarters
Discard 2 thirds
€
€
A Fraction Times a Fraction
A fraction of a fraction is a different idea from multiplication of fractions.
What confuses the issue for students this that their “rule” is the same.
Before understanding this students need a new schema for multiplication
based on the multiplication as area of a rectangle.
Firstly we need to connect multiplication of whole numbers to area of
rectangles. For example, Figure 5 illustrates why the area of a 4 metre by 5
metre rectangle is 20 square metres; this shows, in effect, that 4 metres x 5
metres = 20 square metres.
The important idea here is to see that the counting unit is a “tile” that is 1
metre by 1 metre with area 1 square metre, and there are 20 such tiles
because there are five tiles in each row and four rows.
Now this idea can be extended to multiplication of fractions:
2 2
Suppose we want to find × . We could imagine tiles that are each 1
3 5
third of a metre by 1 fifth of a metre; such tiles are shown in grey in Figure
2 2
6. Then × would represent 2 rows of such tiles with 2 tiles in each
3 5 €
row. Now 15 tiles that are each 1 fifth of a metre by 1 third of a metre
1
cover 1 square metre. Therefore the area of one tile is
of a square
15
€
metre. So 2 fifths of a metre times 2 thirds of a metre equals 4 fifteenths of
a square metre
2 2 4
€
i.e. × =
3 5 15
€
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5 metres
4 metres
€
Beginning with the 12 squares cut and discarding three quarters of the twelve squares leaves one quarter of
the original grid i.e. three squares. Cutting the three squares and discarding two thirds of the squares of
this
new
grid
leaves
one
third
i.e.
one
square.
1
1
1
So 3 of 4 of the original whole equals 12 of the original whole.
1
1
1
We may generalise this:
where a and b are any whole numbers.
of =
a
b a×b
€ we can show, with some difficulty, why, say, 3 of 5 = 15 or in general:
Extending the previous thinking
4
7 28
c
d c×d
€where a, b, c and d are any whole numbers.
of =
a
b a×b
Figure 5
1
5
1
5
1
5
1
5
1
5
1
m
3
1
m
3
m
m
m
m
m
Figure 6
©Mathematical Literacy for Lower-Achieving Students Page 6 of 14
1
m
3
In general:
c d c×d
where a, b, c and d are any whole numbers.
× =
a b a×b
Comparing the Operations “Of” and “Times” for Fractions
3
5 15
3 5 15
Knowing
that both
and × =
it is tempting to think that “of” and “times “ for fractions
of =
€
4
7 28
4 7 28
are the identical. And, because they have the same rule, swapping “of” and “times “ this cannot lead to
errors in calculation. However, conceptually students need to understand that “of” and “times” mean two
quite different
€ things.
€
Fraction Division
Division of fractions is one of the hardest concepts for student to understand in all work with numbers.
With whole numbers a division always leads to an answer that is smaller than the original number. For
example, for 12 ÷ 4 the 12 is reduced to 3. So it is no surprise that students often think wrongly that
8 ÷ ½ = 4.
So a new schema need to be constructed to cope with division by a fraction – this is an example of what
Piaget means by accommodation. And any accommodation is difficult and may take a long time to make a
successful construction. So teachers should expect that teaching division of fractions is likely to be very
difficult.
Whole Numbers Divided by a Unit Fraction
The fact that, anti-intuitively, 8 ÷ ½ does not equal 4 requires a reconstruction of the meaning of division
for whole numbers before moving on to apply new ideas to fraction division.
What students need to understand is that every multiplication problem leads to two conceptually different
division problems. Imagine this multiplication problem:
• 4 children each eat 3 apples, so altogether they eat a total of 12 apples.
The division problems associated with this are:
• 12 apples are shared among 4 children, so each child gets 3 apples
This is the sharing meaning of division, and it is the one that is overwhelmingly common in schools. And:
• 12 apples are shared out. Each child gets 3 apples so 4 children get bags of apples.
Here the meaning of division is to find the number of groups rather than the number of items each person
in a group gets.
So, how might we attempt to work out 8 ÷ ½? Firstly we could attempt to construct a sharing problem:
• 8 apples are shared among half a person.
This is nonsense so we should abandon this path. However, asking a problem involves creating groups does
work:
• 8 apples are cut into halves. How many people can be given a half an apple?
The answer is obviously 18.
The gravest risk for teachers here is to provide a meaningless rule; here, for example, it could be “change
the division to a multiplication and flip the fraction”. For example:
2 16
•
8 × = = 16 .
1 1
We may generalise how to find a whole number divided by a unit fraction:
1
•
a ÷ = a × b where a and b are any whole numbers.
b
€
A Whole Number Divided by Any Fraction
A problem like 12 ÷ 2 thirds is more challenging than 12 ÷ 1 third. Again, if we want to work out, say,
€ 12 ÷ 2 we should first construct a word problem:
3
• Yvonne has 12 bars of chocolate; she gives two thirds of a bar to each person in the class. How many
children are there in the class?
€
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The question becomes “how many two thirds are there in 12 wholes”? And since 12 wholes equals 36
thirds the question reduces to working out 36 ÷ 2 i.e. 18 children. The important connection students
need to make is that in fact division by a fraction is equivalent to multiplication by the reciprocal of the
fraction.
In general:
c
a ÷ = (a × b) ÷ c where a, b, and c are any whole numbers.
b
€
A Fraction Divided by A Fraction
Warning: this section is a threat to good mental health! It is difficult to understand properly.
17 2
A problem like
÷ is the most challenging idea to understand in all of fractions. Again we should make
13 9
up problem:
• Kelly has 17 thirteenths of a block of chocolate. She packs 2 ninths of a block into bags. How many bags
can€she pack?
The solution is hard! Here thirteenths and ninths cannot be directly compared but they can be converted
to a hundred and seventeenths by realising the Lowest Common Multiple of 13 and 9 is 117.
Because 17 x 9 = 153, and 2 x 13 = 26, 17 thirteenths ÷ 2 ninths = 153 a hundred and seventeenths
÷ 26 a hundred and seventeenths.
23 .
And 153 ÷ 26 = 5 26
23 bags of chocolate where each bag contains 2 ninths of a block of chocolate.
So Kelly can make 5 26
c d c×b
In general: ÷ =
where a, b, c and d are any whole numbers.
€ a b a×d
€
In Passing: Ratios are Not Fractions
In many textbooks “ratio” and “fraction” are seen as identical. This is particularly true of American
€
3
material. So the ratio 3 : 4 and the fraction
are regarded as the same thing. But they are not. Three
4
examples will help illustrate the point.
Suppose a mixture is made from 3 parts lemonade to 4 parts ginger ale, i.e. they are mixed in the ratio 3 :
4. Then the associated fractions are €
3/7 of the mixture is lemonade and 4/7 of the mixture is ginger ale – if
3
the ratio 3 to 4 were identical to
it would arise in the problem. But it does not.
4
A more subtle difference involves the similarity of equivalent ratios and equivalent fractions. Consider the
original ratio problem of mixing 3 litres of lemonade with 4 litres of ginger ale. Doubling the recipe we
would have a mixture of€6 litres of lemonade and 8 litres of ginger ale. So when we can write 3 : 4 = 6 : 8.
But here the equals sign means 3 to 4 is proportional to 6 to 8 but 3 : 4 is
not identical to 6 : 8.
3
Now consider
of a cake in Figure 7. Add two more vertical lines.
4
6
3
Then it is obvious that the fraction is now . But, unlike ratios, here
8
4
Figure 7
6
is identical
€ to .
8
So the mathematical rule that relates
is superficially the same rule as equivalent
€ similar ratios to each other
€
ratios. This is a major reason why many textbook authors don’t distinguish fractions from ratios – even
though
€ it is quite wrong conceptually.
And thirdly, if a ratio is a fraction, what fraction corresponds to the ratio for a fruit salad mix of 3 bananas,
5 apricots and 6 pears i.e. the ratio 3 : 5 : 6? The answer is clearly there isn’t one.
It is suggested that teachers never refer to fractions as ratios as this will only confuse students.
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Stage 2: Counting from One on Materials
Essential:
None
Optional:
•
•
•
•
Given wholes draw or cut halves of regions – halves are identical
Given identical halves of regions draw and construct wholes
Describe halves in words either orally or written, and always include reference to the whole
Find a half of a small number using materials – whole number answer
Stage 3: Counting from One by Imaging
Essential:
• Given wholes draw or cut halves of regions – halves are identical
• Given identical halves of regions draw and construct wholes
• Describe halves in words either orally or written, and always include reference to the whole
• Find a half of a small number using materials – whole number answer
Optional:
• Given wholes draw and cut identical shaped thirds and identical shaped quarters of regions. Include two
thirds of, say, an orange, two quarters of a cake, three quarters of a pie
• Describe thirds and quarters in words whether orally or written, and always include reference to the
whole
• Find a quarter of a small number using materials by first linking the problem to division - whole number
answer
• Find a third of a small number using materials by first linking the problem to division - whole number
answer
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Stage 4: Advanced Counting
Essential:
• Given wholes draw and cut identical shaped thirds and identical shaped quarters of regions. Include two
thirds of, say, an orange, two quarters of a cake, three quarters of a pie
• Describe thirds and quarters in words whether orally or written, and always include reference to the
whole
• Find a quarter of a small number using materials by first linking the problem to division - whole number
answer
• Find a third of a small number using materials by first linking the problem to division - whole number
answer
Optional:
• Introduction of the general pattern of naming fraction words as mostly ending in “-th”. For example if a
pie is cut into 13 pieces and 7 are eaten the amount eaten is 7 thirteenths of the pie and 6 thirteenths of
the pie remains. Identify the exceptional words: 1 half not 1 “twoth”, 1 third not 1 “threeth”, 1 quarter
not usually - though it is common in British North America - 1 fourth. There are others that are
somewhat problematic – for example logically “twenty-oneth” is a fraction but “a twenty-first “ probably
sounds better
• Introduce shorthand fraction symbols. E.g. 2 for two thirds. Mix them with use of fraction words
3
•
•
•
•
Use material to work out a unit fraction of a set. For example, model 1 fifth of 15
Given a fraction of whole draw the whole
Calculate a unit fraction of a whole number from materials
Calculate a unit fraction of a whole number from times tables
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Stage 5: Early Part-Whole Addition and Subtraction
Essential:
• Introduction of the general pattern of naming fraction words as mostly ending in “-th”- for example if a
pie is cut into 13 pieces and 7 are eaten the amount eaten is 7 thirteenths of the pie and 6 thirteenths of
the pie remains. Identify the exceptions: 1 half not 1 “twoth”, 1 third not 1 “threeth”, 1 quarter not 1
fourth in New Zealand - though it is common in British North America. There are others that are
somewhat problematic – for example it is logically “twenty-oneth” but “a twenty-first “ probably sounds
better
• Introduce shorthand fraction symbols. For example, two sevenths becomes 72 . Mix them with use of
fraction words
• Given wholes draw and cut thirds and quarters of regions – this time thirds and quarters are not
identical in all ways
• Given fraction like 73 looks like
show what a whole looks like
• Given fraction like
3
7
looks like
show what a whole looks like
• Use material to work out a unit fraction of a set. (A unit fraction has a numerator of 1.) For example,
model 1 fifth of 15. Notice non-unit fractions are much more challenging. For example, 3 fifths of 15
requires 15 ÷ 5 = 3, then 3 x 3 = 9, so it should be delayed.
• Do mental calculation involving converting 1 whole into fractions and vice-versa using fraction words
and
symbols.
For
example,
work
out
1
–
3
sevenths,
1
-
3
,
13
and
4 ninths + 89 - three ninths
• Comparison of fractions with the same wholes written in words and fraction symbols. For example:
Four students all buy the same-sized cake. Jill eats 72 of her cake, Jack eats 2 ninths of his cake, Mere
2
eats 11
of her cake, and Lee eats 2 thirteenths of his cake. Who ate the most?
• Comparison of fractions with equal denominators and different numerators expressed in a mixture of
fraction words and fraction symbols
• Comparison of fractions with different denominators and equal numerators expressed in a mixture of
fraction words and fraction symbols
• Use material to turn common fractions - halves, thirds and quarters - into mixed fractions and viceversa. For example, show why 7 halves of a pie equals 3 whole pies plus a half pie. And three and a third
cakes equals ten thirds of a cake. Mental conversions not expected yet
Optional:
• A big cake is twice as big as a smaller cake. Kerri eats 1 half of a small cake and 1 eight of a large cake.
Use pictures to show that she eats the equivalent of 3 quarters of a small cake
• Provided the students know the relevant times tables mentally, work out a unit fraction of a number
mentally. For example, Mandy bought a bag containing 12 apples. She ate a quarter of the apples. How
many did she eat? Here the students need to know 4 x 3 = 12 and connect this to fractions and division.
So 14 of 12 = 12 ÷ 4 = 3
• Fractions arise from division using materials only. For example show why, if three apples are divided
among four people, each person gets three quarters of an apple
€
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©Mathematical Literacy for Lower-Achieving Students Page 11 of 14
Stage 6: Advanced Part-Whole Addition and Subtraction
Essential:
• Demonstrate that, to understand fractions students, must be aware of the relative sizes of the wholes
being referred to. For example, students may think that a half is always more than a quarter but in fact 1
half of 20 is less than 1 quarter of 4000
• A big cake is twice as big as a smaller cake. Kerri eats 1 half of a small cake and 1 eight of a large cake.
Use pictures to show that she eats the equivalent of 3 quarters of a small cake
• Provided the students know the relevant times tables mentally, work out a unit fraction of a number.
For example, Mandy bought a bag containing 12 apples. She ate a quarter of the apples. How many did
she eat? Here the students need to know 4 x 3 = 12 and connect this to fractions and division.
So 14 of 12 = 12 ÷ 4 = 3
• Fractions arise from division. Use material to explain why, if 3 apples are divided among four people,
each person gets three quarters of an apple. Generalise this to the use of symbols only. For example,
explain why 8 ÷ 9 = 8 ninths
€
• Mixed numbers arising from division. Use diagrams
• Mixed numbers arising from division. Use the two, three, and four times tables, and the result of dividing
by ten
• Use times tables to turn common fractions - halves, thirds and quarters - into mixed numbers and viceversa. For example, show why 15 quarters of a pie of a pie equals 3 43 pies. And three and a third cakes
equals ten thirds of a cake
€
3/11/2010
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©Mathematical Literacy for Lower-Achieving Students Page 12 of 14
Stage 7: Advanced Part-Whole Multiplication and Division
Essential:
• Use material to turn common fractions - halves, thirds and quarters - into mixed fractions and viceversa. For example, show why 7 halves of a pie equal 3 whole pies plus a half pie. And three and a third
cakes equals ten thirds of a cake
• Fraction of a fraction
• Fraction times a fraction applied to areas of rectangles
• Equivalent fractions, including reduction of fractions to their simplest form
• Addition and subtraction of fractions
• Addition and subtraction of mixed numbers
• Given a fraction of a set is a given number, find the size of the whole set by using times tables (whole
number answers only.) For example: Matiu started with a whole box and ate five ninths of the
chocolates. That left only 16 chocolates for Yves. How many were there in the fill box?
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©Mathematical Literacy for Lower-Achieving Students Page 13 of 14
Stage 8: Advanced Equivalent Fractions and Ratios
Essential
• A whole number divided by a unit fraction using times tables. For example, 7 ÷ 16 = 42
• Find mentally a whole number divided by a fraction using times tables, with the answer being a whole
number. For example, 9 ÷ 37 = 42
€ using times tables, with the answer
• Find by a written method a whole number divided by a fraction
3
being a whole number. For example, 11÷ 7 = (11× 7) ÷ 3 = 77 ÷ 3 = 25 23
• Find by a written
method a fraction divided by a fraction using times tables, with the answer being a
€
13
whole number. For example, 116 ÷ 113 = 116 × 113 = 121
= 6 18
18
• Use a calculator to express€an improper fraction with large numbers in the numerator and denominator
as
a
mixed
number,
and
check
the
answer.
For
example,
convert 319991
:
678
319991
653
319991 ÷ 678 = 47·9631268,
471 x 678 = 319338, 319991 – 319338 = 653, so 678 = 471 678 ,
€
Check: 471 x 678 + 653 = 319991
• Find fractions of a number that has been divided up in a given ratio, whole number answers
only. For
€
example, Janet and John and Hemi share 2350 lollies in the ratio 5 : 2 : 3. How
€ many lollies does each
child get?
• Make up a story that shows why ratios are equal, and what scale factor is involved. For example, for 3 :
7 : 4 = 30 : 70 : 40, a story could be:
Geraldine experiments with a new fruit salad recipe for a wedding reception. She uses 3 apples, 7
bananas, and 4 oranges. She likes the result so she scales up the production ten times for the wedding.
This needs 30 apples, 70 bananas, and 40 oranges
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©Mathematical Literacy for Lower-Achieving Students Page 14 of 14