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Fractions: Teaching and Learning
Activities
Penny Lane
Conceptual understanding vs procedural knowledge
The most important thing we have to bear in mind when teaching fractions is that
our students need to develop conceptual understanding of fractions, not
procedural knowledge. That is, we have to ensure students really understand
fraction concepts, rather than simply follow procedures for working with fractions.
Classroom snapshots
Let’s take a look at what some of our students think about fractions.
Year 6 students were shown a shoe and asked how many pairs of
shoes they could see. Of 30 students asked, four gave the answer
half a pair. Twenty five other students said none. One student said
What a cool shoe; where did you get it?
Year 4 students were shown this circle, and asked what
fractions they could see.
The whole class agreed with what one student said:
You have to have equal parts to have fractions, so there
aren’t any fractions in this circle.
1
Year 1 students were shown a third of a circle, and asked what it
was.
It’s a fat quarter.
If you put four together you’ll get a fat circle.
Year 4 students were shown a circle divided into thirds.
They’re quarters.
That’s three quarters.
Year 4/5 students were shown a third of a circle.
It’s a big quarter.
It’s three quarters.
Year 4 students were shown a quarter of a circle divided into three
equal parts and asked what one of the parts was.
Three quarters.
Year 4 students were asked to suggest some fraction words.
Half
Quarters
… a long silence…
Fifths?
Year 4 students were asked what fifths are.
It’s like when they draw a circle divided into six parts and they tell
you to colour in five of the parts.
2
Two Year 3 students with a considerable amount of procedural
knowledge and very quick to respond with number facts when
called on to do so, were asked to divide a circle into five equal
parts.
They drew a line dividing the circle in half.
Then they drew two more diagonals to divide the
circle into six equal parts. They counted the parts
and erased all their lines.
Next they divided their circle in half with a line from
top to bottom, added two more diagonals, counted
the six parts and erased their lines.
It’s impossible.
After being unsuccessful on the above task, the same two Year 3
students asked the teacher to give them some tables questions.
They were asked what was 6 x 2.
12
Then they were asked what was 6 x 2½.
12½
3
A Year 6 student was asked how much pie each person would get
if two pies were shared among three people. He drew this.
They’d get two-sixths each, because it’s two parts out of six.
Year 2/3 students were asked to share two pizzas among three
people. They drew this.
They get two thirds each.
They were then asked what if one pizza had olives and one had
anchovies, and each person wanted one slice of each pizza.
They drew this.
They get two sixths each. They get two parts out of six.
The teacher asked them did they get the same amount of pizza as
before.
Yes, two thirds is the same as two sixths.
How can that be?
It’s two sixths because it’s two parts out of six, and it’s two thirds
because there are two slices and the slices are thirds.
4
How has this come about?
Perhaps teachers are doing too much scaffolding, too much modelling, too much
explaining.
We present each concept neatly and develop it sequentially according to the
knowledge we have about fractions, rather than the knowledge students have
about fractions; we model concepts and language that can lead to confusion,
such as ‘one part out of four’; we explain that fractions are related to equal parts
and we demonstrate equal parts with models and diagrams that make the equal
parts concrete, without understanding the complexity of this concept (it is not so
much the concept of concrete equal parts that is at the heart of understanding
unit fractions, it is the size of one part in relation to the whole); we focus on
specific fractions rather than on general fraction concepts; we make the
connections among fractions for students, for example when we begin with
halves and proceed to quarters then eighths, rather than allowing students to
make connections; we ask students to colour in three parts out of five parts that
have already been drawn for them, rather than supporting them to explore and
represent such fractions in their own ways; we base our teaching on specific
syllabus content items rather than on what is already in students’ minds.
What should we do instead?
We can give students a range of challenging tasks that elicit their prior
knowledge and give them opportunities to create new knowledge, note their
unscaffolded responses, and base further questions and tasks on these. In this
way we will build on what students already know and can express to work
towards the syllabus outcomes.
Students should be challenged and supported to devise their own strategies for
working with fractions, as this benefits the development of fraction concepts and
reveals to their teachers their strong understandings, limited understandings and
misunderstandings, and their effective use, limited use and misuse of language
in the context of fractions.
5
What content?
We need to support students to build general fraction concepts as well as
concepts of specific fractions, and sometimes we need to go beyond the syllabus
content for a particular stage in order to build strong concepts. These general
fraction concepts include the following.
Objects, collections and measurements can be divided into parts.
Unit fractions are named according to the number of parts of that size that make
up the whole.
There are relationships among different size parts of an object, collection or
measurement.
Non-unit fractions are named according to their relationship to unit fractions.
Specific fractions can be expressed in more than one way because of the
relationships among the different size parts.
We must keep returning to these general fraction concepts to ensure students
understand the concepts related to specific fractions. So, for example, if we are
building the concept of equivalent fractions by focusing on halves, quarters and
eighths, we must challenge students to transfer this concept to fractions
generally, rather than move onto equivalence among another set of specific
familiar fractions such as thirds, sixths and twelfths.
Fraction models
Students need to investigate a range of fraction models. Circles are often
preferable to rectangles for working with fractions of whole objects because once
a rectangle is cut it looks like two new wholes, whereas when a sector is cut from
a circle the original circle is still apparent. A circle is also a familiar shape in real
fraction contexts such as dividing pizzas, pies and cakes.
6
When using rectangles, it is a good idea to have a ‘base’ rectangle the same size
as the rectangle that is to be folded or cut, so students always have an image of
the whole to hand.
When using circles and naming them as pizzas, be aware that bought pizzas
have the added complexity of being delivered already cut into slices, usually
eight. This feature can be exploited at times, or students can be asked to imagine
the pizza before it is cut.
Students’ own problems
At all stages, students can pose fraction problems, including oral and written
word problems, and problems for investigation. In this way, students explore their
own interests, develop their understandings, and reveal their levels of
understanding of fraction concepts and language.
Also, students are highly interested in working on problems posed by their peers,
and like to challenge their classmates with increasingly more sophisticated
problems, thus building their own understanding and skills.
Suggested teaching and learning activities
Following are several suggested teaching and learning activities, many of which
have been trialled with students K-6 in the St George District.
This symbol appears in lessons that have been trialled and the
‘likely student responses’ are actual responses from those lessons.
The syllabus Fractions and Decimals content covered in each lesson is specified.
Relevant outcomes from other strands are recorded; fractions tasks have a lot of
potential to integrate with other strands.
7
Early Stage 1
Outcome: Describes halves, encountered in everyday contexts,
as two equal parts of an object
8
Sharing a birthday cake
Content
K&S: Recording fractions of objects using drawings
WM: Explain the reason for dividing an object in a particular way
Materials
Paper and pencils
Ask students to draw a birthday cake cut into slices and describe why they are
dividing it the way they are.
Likely student responses include the following.
Three people are eating my cake. That’s ten pieces.
The pieces are the same size.
9
What is a whole?
Content
WM: Using fraction language in everyday situations
Materials
None
Ask: What is ‘a whole’?
Likely students responses include the following.
A big round circle you dig with a spade
Wombats can dig a hole
Ask: What is ‘a whole cake’?
Likely student responses include the following.
A round shape
You’ve got all of it
A cake that hasn’t been eaten
A cake that’s not a half cake
Ask: What does ‘our whole class’ mean?
Likely student responses include the following.
Everyone
All of us
Students can participate in ‘collecting’ holes and wholes to develop an
understanding of the meanings of, and the difference between, the two words.
10
What do you do if 2 people want to share a cake?
Content
K&S: Sharing an object by dividing it into two equal parts
WM: Describe how to make equal parts
Materials
Paper and pencils
The following tasks focus on sharing food items that are different shapes.
Ask: If you have a cupcake and you want to share it with a friend, how could you
do it?
Students can describe what they could do.
Students can draw what they could do.
Ask: If you have a jelly snake and you want to share it with a friend, how could
you do it?
Students can describe what they could do.
Students can draw what they could do.
Ask: If you have a slice of bread and you want to share it with a friend, how could
you do it?
Students can describe what they could do.
Students can draw what they could do.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
11
What is a half?
Content
WM: Using fraction language in everyday situations
Materials
None
Constructing definitions is an important mathematical activity. Students explore
both the concept of a half and the associated language.
Ask: What is ‘a half’?
Likely student responses include the following.
It’s not a full thing, it’s a part of it.
It’s like a birthday cake with a cut down the middle.
12
Has this shape been divided into halves?
Content
K&S: Recognising that halves are two equal parts
K&S: Recognising when two parts are not halves of the one whole
Materials
Circles and rectangles with lines dividing them into two parts, some with equal parts and some
with unequal parts (see below)
Ask: Are these shapes divided into halves?
Students are likely to say the first and third rectangles are divided
into halves, and the middle one is not, explaining that the line must
be in the middle to make the two sides the same size.
Students are likely to say the first circle is divided into halves, and
the middle one is not, explaining that the line must be in the middle
to make the two sides the same size. They are likely to say the third
circle is not divided into halves because the line has to be straight
down.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
13
How many circles?
Content
K&S: Recognising that halves are two equal parts
Materials
12 semicircles, all the same size
With the students sitting in a circle on the floor, place the semicircles in a pile on
the middle, and ask: What are these?
Likely student responses include the following.
They’re like circles.
They’re half circles.
They’re circles cut in half.
They’re semicircles.
Ask: Can we use these shapes to make circles?
Then ask: How many circles do you think we can make with these shapes?
Because the semicircles are in a pile, this task requires students to make
estimates of the answer. Ask some students to explain how they worked out their
estimates.
Spread the semicircles out and ask the students again: How many circles can we
make? Ask some students to explain their strategies.
On another occasion, ask students to work out how many circles they could
make if they had say 14 half circles, and ask them to show on a sheet of paper
how they worked out their answer.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
14
How much is left?
Content
K&S: Recording fractions of objects using drawings
WM: Explain the reason for dividing an object in a particular way
Materials
Paper and pencils
Ask: If I have a cupcake and I eat half of it, can you draw a picture showing how
much of the cupcake is left?
When students have drawn their pictures, ask: Why did you draw it this way?
Likely student responses include the following.
The straight line is where you cut the cupcake and the curvy line is
the edge of the cake.
It’s half the cake.
It’s a semicircle.
On other occasions the students could be asked to draw other objects that can
be represented with half circles such as half a pie or a half moon, or they could
be asked to draw what is left after half a pizza has been eaten.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
15
Drawing half circles
Content
K&S: Recording fractions of objects using drawings
WM: Explain the reason for dividing an object in a particular way
Materials
Paper and pencils
This is a follow-up task for students who did not draw semicircles in response to
the previous task, How much is left? See the work samples from Ilina, Sophie
and Rana. In some cases, the students might not be able to conceptualise the
solution (eg possibly Rana), and in other cases the students may be limited by
their inability to draw a semicircle (eg possibly Ilina).
Ask the students to draw circles. Can they do this successfully?
Give each student a circle cut from paper and ask them to show where they
would draw a line to divide their circle in half, and explain why they would draw it
there. If they are successful, ask them to draw the line.
Next they can cut along the line. Ask them to describe the shapes they now
have.
Finally, get them to draw one of the shapes.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
16
Investigating folding and cutting shapes in half
Content
K&S: Sharing an object by dividing it into two equal parts
K&S: Recognising that halves are two equal parts
Materials
Paper shapes – rectangles (oblongs and squares), circles, diamonds; scissors
Set students the task of investigating what happens when you fold and cut a
shape in half.
If they don’t include folding and cutting a rectangle along a diagonal, lead them to
doing so by giving them a square and asking them if there is a way to fold it other
than from one edge to the opposite edge. Then give them other rectangles to try.
They might now fold these along a diagonal or from one corner to the opposite
corner, both of which produce interesting halves.
Folding allows students to see the shape divided while remaining whole, and
cutting allows them the opportunity to deconstruct the shape, match the two
halves by overlapping, and reconstruct the original shape. It is particularly useful
to be able to cut out and match the two halves in the case of an oblong halved
along a diagonal because the two halves don’t match when folded.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
17
Investigating folding and cutting triangles in half
Content
K&S: Sharing an object by dividing it into two equal parts
K&S: Recognising that halves are two equal parts
K&S: Recognising when two parts are not halves of the one whole
Materials
Paper triangles (equilateral, isosceles and scalene), scissors
Set students the task of investigating what happens when they try to fold and cut
a triangle in half.
This activity will help students extend their understanding of what is a triangle,
because they will be working with different types of triangle.
They should find that only equilateral and isosceles triangles can be folded and
cut into matching halves. Ask them to talk about the shape of each half.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
18
What is the shape?
Content
K&S: Recognising that halves are two equal parts
Materials
Shapes (oblongs, squares, circles, triangles) cut from paper and folded in half
This activity helps students visualise shapes and sizes.
Show each folded shape in turn and ask: If this is half a shape, what is the whole
shape?
When students have given their suggestions and explanations, open up the
shapes.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
19
Covering half the area of a rectangle
Content
K&S: Recognising that halves are two equal parts
Materials
Large rectangles drawn on paper; assorted coloured cardboard rectangles, each one being half
the area of one of the large drawn rectangles
Give students a large rectangle drawn on paper, and ask them to find a
cardboard rectangle that covers half its area.
Other outcomes covered
MES1.2: Describes area using everyday language and compares areas using direct comparison
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
20
Filling and half-filling containers
Content
K&S: Recognising that halves are two equal parts
Materials
Clear plastic containers such as cylindrical jugs; plastic cups; water
Ask students to estimate how many cupfuls of water it will take to fill a specific
container. Then ask them to estimate how many cupfuls will half-fill the same
container. They should explain their strategies for making their estimates.
They can investigate filling and half-filling containers with water and record (in
drawing and writing) what they find out.
They can discuss what happens when they half-fill a cylindrical jug and a tapered
jug.
Other outcomes covered
MES1.3: Compares the capacities of containers and the volumes of objects or substances using
direct comparison
21
Sharing a collection of objects
Content
K&S: Recognising that halves are two equal parts
Materials
Collections of up to 20 objects (eg oranges, marbles, blocks, counters)
Give students a collection of up to 20 objects and ask: If you share these so you
and a friend get half each, how would you do it? Allow them to use the objects to
demonstrate their strategies.
This task gives students the opportunity to transfer their understanding of the
concept of ‘half’ to a collection of objects. They can also be given sharing tasks
to work out mentally, such as the following.
Show students a small number of objects (up to 10) and ask: How many objects
you would get if you were given half of these? They should explain their
strategies.
Show up to 30 counters in a pile and say: Look and think about how many
counters you would get if you were given half of these. Again, they should
explain their strategies. Estimation tasks such as this help build students’
reasoning skills and concepts of number relationships.
Other outcomes covered
NES1.3: Groups, shares and counts collections of objects, describes using everyday language
and records using informal methods
22
Stage 1
Outcome: Describes and models halves and quarters, of objects
and collections, occurring in everyday situations
23
Which piece would you like?
Content
WM: Explain why the parts are equal
Materials
Two identical oblong chocolate slices; a knife; two identical oblong pieces of cardboard that
measure 4 counters by 6 counters
Show students the two chocolate slices and establish they are the same size.
Cut one slice lengthwise and the other one crosswise and put one of each on a
plate.
Ask: Which one would you prefer to have or doesn’t it matter? Why?
Likely students responses include the following.
I want this one because it’s longer.
I want this one because it’s fatter.
Show the two cardboard oblongs, establish they are identical, then cut one
lengthwise and the other crosswise. Ask: Which one is bigger, or are they the
same size?
Pick up a crosswise half and ask: How many counters do you think it will take to
cover this shape? Place counters on the shape until it is covered. Repeat with a
lengthwise half. Guide students to discuss why 12 counters cover each of the
halves. Likely students responses include the following.
They’re the same size. Hey, they must be because they’re both
halves.
One’s fatter and one’s longer but they’re both halves.
Other outcomes covered
MS1.2: Estimates, measures, compares and records areas using informal units
24
What is a half?
Content
WM: Explain why the parts are equal
Materials
None
Ask: What is ‘a half’?
Likely student responses include the following.
It’s something that’s got a line down the middle and there’s a part
on this side of the line (gesturing to the right) and a part on this side
of the line (gesturing to the left) and both parts are the same size.
25
Black and white
Content
WM: Explain why the parts are equal
Materials
Cards with rectangles which are half black and half white (see below)
Show each card in turn and ask: How much of this shape is black and how much
is white?
This task should help students understand that ‘a half’ doesn’t have to be just
one region of a whole, nor is it a standard rectangular shape.
Students can be set the task of making a rectangle and colouring it half one
colour and half another colour, in an interesting way. They should justify their
pattern, explaining how they know it is half one colour and half the second colour.
At another time, they can be given a square of paper marked out in squares (say,
16 or 36 squares) and asked to colour the paper to make a design that is half
one colour and half another colour, using the square grid as a guide. They could
be asked to make the design symmetrical, as another challenge.
Other outcomes covered
SGS1.2 Manipulates, sorts, represents, describes and explores various two-dimensional shapes
26
Comparing lengths
Content
WM: Visualise fractions that are equal parts of a whole
Materials
2 lines drawn on cardboard – one line 6 counters long and the other 12 counters long
With students sitting in a circle, put out a cardboard sheet with the shorter line
drawn on it and ask: Look and think… how many counters will fit along this line?
Once the students have made their estimates, begin putting counters along the
line, and allow the students to change their estimates as the counters are placed.
Pause after three counters are on the line and ask if they can now work out how
many will fit along the line. Likely responses include the following.
Three more, because the bit of line that’s left is the same as the
covered bit.
Six, because you’ve covered half the line.
Cover the line. Next, put out the longer line and ask students to think about how
many counters will fit along it. Likely responses include the following.
10, because it’s longer than the other line.
12, because it’s double the other line.
12, because the other line is half the long line.
Other outcomes covered
MS1.1: Estimates, measures, compares and records lengths and distances using informal units,
metres and centimetres
27
Cutting holes
Content
WM: Visualise fractions that are equal parts of a whole
Materials
Sheets of paper; scissors
Show the students that you are folding a sheet of paper, then cut a rectangular
hole on the fold.
Ask: What shape will the hole be when I unfold the sheet of paper?
Ask students to fold a sheet of paper and cut a hole that will be a specific shape
(eg circle, square, triangle) when they unfold the paper. Ask them to explain why
they think their hole will be the shape specified.
Likely student responses include the following.
Mine will be a circle because I’ve cut out a curve.
It will be a circle because I cut a semicircle.
Mine is a square because I cut out half a square.
I think mine will be a triangle because I cut out a triangle…it’s not,
it’s a diamond.
(The triangle is quite a challenge.)
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
28
Is it a half?
Content
K&S: Describing parts of an object or collection as ‘about a half’, ‘more than a half’ or ‘less than a
half’
Materials
Counters
Cards showing black and white circles
Put about 20 blue counters and about 20 yellow counters in a clear plastic jar or
in a pile on the floor. Ask: Are about half the counters blue, or more than half, or
fewer than half? Why do you think that?
Vary the proportions of the two colours and ask the same questions.
Show cards such as the following and ask similar questions.
29
Sharing Problems
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Problems written on sheets of paper and put up on display (Masters follow)
Share 9 marbles between 2 people.
Share 9 cupcakes between 2 people.
If 9 people are driving to the beach in 2 cars, how many people should travel in each car?
These are all problems involving division of 9 by 2, but each one is answered differently. The first
answer is 4 marbles each with one left over; the second answer is 4½; the third answer is 5 in
one car and 4 in the other.
Another set of problems could be as follows.
Share 21 marbles among 4 people.
Share 21 jelly snakes among 4 people.
If 21 people have 4 tables to sit at, how many people should sit at each table?
Display the problems on a classroom wall. When students have worked out
answers, using concrete materials, drawings and mental strategies, ask them to
explain what they did.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
30
Dividing a cake into equal slices
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
Form the students into different size groups of between 3 and 6, and ask each
group to draw a round birthday cake (top view) and divide it into enough slices for
each person in the group to have the same size slice, with no cake left over.
Likely student responses include the following.
It’s quarters
It’s four halves
It’s one four half
We started with a dot in the middle and
we drew the lines. We just looked and
thought where the lines would go.
We drew the line to cut it in half then we
worked out there would be three slices
in each half of the cake.
They don’t look right.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
31
Dividing a pie into three equal slices
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
It is important that students explore fractions other than halves and quarters in
order to build the understanding that halves and quarters are not the only
fractions. If we limit students’ experiences to halves and quarters, some students
might think that any fraction that is not a half is called a quarter.
Ask students to draw a circle on a sheet of paper, imagine it is the top view of a
pie, and draw lines to divide the pie into 3 equal slices with none left over.
Encourage them to work out where their lines will go before they draw them; they
could use pencils or popsticks to show where the lines could go.
Likely student responses are shown in the photographs overleaf.
The students can also explore pies that are in the shape of a square, an oblong
and an equilateral triangle.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
32
Three Year 1 boys worked together to divide a ‘pie’ into three equal slices. The
following four photos show the progress of their thinking.
33
How old is the person having the birthday?
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
‘Birthday cake slices’ (one being a cardboard 3D model of a slice of cake with a candle in it; the
others being cardboard segments of circles, with a mark on each showing where a candle would
go); paper and pencils
Show the 3D model of the birthday cake slice and explain that the mark on the
cake is where a candle was, and the candles were equally spaced around the
cake. Ask: How old is the person having the birthday? How do you know?
Give pairs of students 2D ‘slices’ and ask them to work out the age of the person
having the birthday. Have paper and pencils available, because students often
use the strategy of drawing around the slice enough times to create a cake.
This person is 24. We did 4 slices and that’s 12 candles because
it’s 3, 6, 9, 12, and it’s half the cake so we doubled it.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
34
In 20 seconds
Content
WM: Use fraction language in a variety of everyday contexts
Materials
None
Tell the students: In 20 seconds I can write the word ‘happy’ 8 times. Ask: How
many times would I write it in 10 seconds? Why? Do they use the word ‘half’ in
their explanations?
Ask: How many times can I write ‘happy’ in 15 seconds?
The students can time themselves writing words or numbers, or doing other
actions such as clapping. They can pose problems for their classmates based on
their results.
Other outcomes covered
MS1.5: Compares the duration of events using informal methods and reads clocks on the half-
hour
35
Fat Cat and Little Kitten Problems
Content
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
Problems, perhaps recorded in booklets, based on two characters, a fat cat and a little kitten. The
fat cat eats twice as much as the little kitten.
My little kitten ate 3 pieces of fish. My fat cat ate twice as many pieces as my little kitten. How
many pieces of fish did my fat cat eat?
My fat cat ate 10 pieces of meat. My little kitten ate half as many pieces as my fat cat. How many
pieces of meat did my little kitten eat?
My fat cat ate 5 fish. My little kitten ate half as many fish as my fat cat. How many fish did my little
kitten eat?
My little kitten ate 1½ cans of cat food. My fat cat ate twice as many cans as my little kitten. How
many cans of cat food did my fat cat eat?
My fat cat ate 11 sardines. My little kitten ate half as many sardines as my fat cat. How sardines
did my little kitten eat?
Read the problems to the students and ask them to explain their strategies for
working out the answers.
They could work with materials, use drawings or work mentally to work out the
answers.
Students could write their own Fat Cat and Little Kitten problems.
36
Measuring
Content
WM: Use fraction language in a variety of everyday contexts
Materials
Large paperclips, classroom objects
Have a pile of paperclips on show. Hold up something like a pencil that is about
3½ paperclips long and ask students how many paperclips they think would fit
along it. Line up paperclips alongside the pencil and ask how many paperclips
long is the pencil.
Note that some students might say it is 4½ paperclips long, counting the fourth
paperclip as a whole and adding the half. Ask them to explain their answers.
Students can be set tasks involving half-unit measurements, such as
constructing a cylinder that is 5½ paperclips long.
Other outcomes covered
MS1.1: Estimates, measures, compares and records lengths and distances using informal units,
metres and centimetres
37
Halfway and quarter way
Content
WM: Use fraction language in a variety of everyday contexts
Materials
Pictures of paths, lines, hills
Show a picture of a hill, a path or a line and ask questions such as the following.
Where would a person be if they walked halfway along this path?
Halfway up this hill?
Run your finger along the line and stop when you get to the halfway point.
Where would a person be if they walked a quarter of the way along this path?
A quarter of the way up this hill?
Run your finger along the line and stop when you get a quarter of the way.
Other outcomes covered
MS1.1: Estimates, measures, compares and records lengths and distances using informal units,
metres and centimetres
38
A one-handed clock
Content
WM: Use fraction language in a variety of everyday contexts
Materials
A model of a clock made with only the hour hand
Show the one-handed clock with the hand pointing to the four, ask the students
to work out what the time would be, and ask them to explain their thinking.
Likely student responses include the following.
It’s four o’clock because the hand is pointing to the four.
Move the hand, with the students watching its clockwise movement, and stop it
when it points to another number. Ask the students to work out what the time
would be. Repeat several times.
Move the hand and stop it midway between two numbers and ask the students to
work out the time and explain their thinking.
Likely student responses include the following.
It’s half-past seven because it’s halfway between the seven and the
eight.
Half-past seven because it’s halfway past the seven.
It’s seven thirty because it’s halfway past the seven.
Other outcomes covered
MS1.5: Compares the duration of events using informal methods and reads clocks on the half-
hour
39
How many circles?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
24 quarter circles, all the same size
With the students sitting in a circle on the floor, place the quarter circles in a pile
on the middle, and ask: What are these?
Likely student responses include the following.
They’re shapes.
They’re cake shapes.
They’re semicircles.
They’re kind of like a puzzle; you put shapes together.
They’re quarter circles.
Ask: Can we use these shapes to make circles?
Then ask: How many circles do you think we can make with these shapes?
Because the quarter circles are in a pile, this task requires students to make
estimates of the answer. Ask some students to explain how they worked out their
estimates.
Count the quarter circles, then put them away and ask the students to work out
how many circles they could make with 24 quarter circles. Ask them to show on
paper how they worked it out. Likely students responses are shown overleaf.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
40
Likely student responses to the task of working out how many circles can be
made with 24 quarter circles.
41
Has this shape been divided into quarters?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Circles and rectangles with lines dividing them into four parts, some with equal parts and some
with unequal parts (see below)
Ask: Are these shapes divided into quarters?
Students often say the third rectangle above is divided into quarters without
questioning the fact that the four triangles are not identical. While each triangle is
actually a quarter of the area of the rectangle, this needs to be explored.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
42
Are these quarters?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
An oblong divided by two diagonals
Show students an oblong divided by two diagonals.
Ask: Has this rectangle been divided into quarters? How do you know?
Discuss the two different triangles they can see.
Give pairs of students photocopies of the rectangle and ask them to work out if
the two different triangles have the same area.
Other outcomes covered
MS1.2: Estimates, measures, compares and records areas using informal units
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
43
What is a quarter?
Content
K&S: Modelling and describing a half or a quarter of a whole object
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
None
Constructing definitions is an important mathematical activity. Students explore
both the concept of a quarter and the associated language.
Ask: What is ‘a quarter’?
Work with students to produce a definition of a quarter that reflects their current
level of understanding and use of language, write it on a poster and display it on
a classroom wall. Return to it from time to time to refine it.
44
What shape can a quarter of a square be?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Paper and pencils
Students can investigate the different ways that a square can be divided into
quarters.
If they produce a limited range of responses, ask them to recall the different ways
a square can be divided into halves, and base their divisions into quarters on
those. Then they might produce solutions such as the following.
This is an appropriate context to introduce the terms trapezium and quadrilateral.
Students could investigate the following question: Does every quarter of a square
have a square corner?
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
45
Has this rectangle been divided into quarters?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Card with the diagram below. The rectangle should be made so that each quarter can be
measured by counters eg the card could be 12 counters by 8 counters, so that the longer
quarters measure 3 by 8 counters and the wider quarters measure 6 by 4 counters.
Show the following card and ask: Has this rectangle been divided into quarters?
This task focuses on the concepts of matching and non-matching quarters.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
46
Halving regular polygons
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Templates of regular polygons (pentagon, hexagon, heptagon, octagon, nonagon, decagon) for
students to trace around
Students can investigate how regular polygons can be halved. They should find
that regular polygons with an even number of sides can be halved by a line
drawn from corner to corner or from the a point on one side to a point on the
opposite side. Regular polygons with an odd number of sides can only be halved
by a line joining a corner to the midpoint of the opposite side.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
47
Quartering regular polygons
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Templates of regular polygons (pentagon, hexagon, heptagon, octagon, nonagon, decagon) for
students to trace around
Students can investigate if regular polygons can be divided into matching
quarters.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
48
Quarters game
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Dice marked with ¼, 2/4, ¾, 4/4 and two blank faces; paper and pencils
Students play in pairs, taking turns to throw the dice and attempting to be the first
to complete five circles. They draw what they throw, so if they throw ¾, they draw
three-quarters of a circle.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
49
Quarter hours on a one-handed clock
Content
WM: Use fraction language in a variety of everyday contexts
Materials
A model of a clock made with only the hour hand
Show the one-handed clock and move the hand until it is a quarter of the way
past a number. Ask the students what time they think it is, and why they think so.
Likely student responses include the following.
It’s not half-past four because it’s not halfway after it.
It’s quarter-past I think.
It might be ten minutes past four or 20 minutes past four or
something like that.
Is it four and a quarter?
The students should discuss the different responses.
Other outcomes covered
MS1.5: Compares the duration of events using informal methods and reads clocks on the
half-hour
50
Collecting data
Content
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
Paper and pencils
Form students into groups of 8 or 12. Direct them to find a food that only half the
group likes and a food that only a quarter of the group likes, and to show the
information in graphs.
Other outcomes covered
DS1.1: Gathers and organises data, displays data using column and picture graphs, and
interprets the results
51
How much of the shape is black?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Cards showing circles or rectangles partly coloured black (see below)
Show a card and ask: How much of this shape is black? Students should explain
their thinking.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
52
How many circles?
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
18 thirds of circles, all the same size
With the students sitting in a circle on the floor, place the circle thirds in a pile in
the middle, and ask: What are these?
Likely student responses include the following.
They’re quarters.
They’re big quarters.
Ask: Can we use these shapes to make circles?
Then ask: How many circles do you think we can make with these shapes?
Because the circle thirds are in a pile, this task requires students to make
estimates of the answer. Ask some students to explain how they worked out their
estimates.
Count the thirds, then ask the students if they can work out how many circles can
be made, and explain their strategies.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
53
Which is bigger – a quarter or a third?
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Several quarters and thirds of circles of different sizes; use different coloured cardboard for
different size circles from which to make the circle sectors, so that, say, a green third and a green
quarter are from the same circle and a red third an a red quarter are from a larger circle, thus
allowing comparisons among different size quarters or between a third and a quarter from the
same circle
Ask students: Which is bigger – a quarter or a third? They should give reasons
for their answers.
Put out the quarters and thirds of circles and ask them to discuss the sizes of the
pieces.
Likely student responses include the following.
Thirds can be bigger and quarters can be bigger.
It depends on how big the circle is.
A red third is bigger than a red quarter and a green third is bigger
than a green quarter.
Ask questions such as: Would you rather have a third of this finger bun or a
quarter of the bun? Why? When would I cut it into thirds and when would I cut it
into quarters? What if I shared it among five people; would the pieces be larger
or smaller?
54
Sharing a blackbird pie
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
Students often find it easier to divide an object (such as a pie) into fractions than
a collection, so this activity is valuable because it links finding fractions of a
collection to fractions of an object.
Students begin by dividing a pie into halves and quarters. Discuss the blackbird
pie in ‘Sing a Song of Sixpence’ which has 24 blackbirds.
Ask: If there are 24 blackbirds in a pie, how many would be in each half of the
pie, if the blackbirds are shared equally between the two halves?
How many blackbirds would there be in a quarter of the pie, if the blackbirds are
shared equally among the four quarters?
Students can investigate how many different ways the blackbird pie can be
divided so each slice has the same number of blackbirds.
Likely student responses are overleaf.
This type of activity helps build the understanding that halves and quarters are
not the only fractions.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
55
Likely student responses to the task of sharing a blackbird pie.
56
Sharing a plum pie
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
This activity is similar to ‘Sharing a blackbird pie’. It allows a focus on any
number as the number of plums in the plum pie can be varied.
Students can find halves and quarters of pies with 20 plums (or any other
number divisible by 4), with 18 plums (or any other number which will require 2
plums being halved), with 21 plums (or any other number requiring a plum to be
quartered), and perhaps with 23 plums.
Students can be set the task of investigating how many ways they can divide a
pie with, say, 30 plums, with a whole number of plums in each slice. This type of
task leads to an understanding later of factors.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
57
What fraction is red?
Content
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
Counters
Give each pair of students eight counters - four red and four of a second colour.
Ask: What fraction of your collection of counters is red?
Add eight more counters of the second colour to each collection so each pair of
students has four red counters and twelve counters of a second colour.
Ask: What fraction of your new collection is red?
Ask students to explain the strategies they used.
58
How many oranges?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Some oranges cut into halves; picture of several half oranges; card with ‘½’ written several times
Show students the oranges already cut into halves and ask: How many oranges
are there?
They should discuss their answers (because some students are likely to count
each half as a whole orange) and their strategies.
Show the picture and ask the same question.
Show the card with ½ written several times and ask: What would the total be if
we added all these halves? Some students may think the question is ‘How many
halves are there?’ so it is a good idea to ask students to explain what the
question is asking.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
59
How many half oranges?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
None
Ask students questions such as: How many half oranges would there be if I cut
six oranges into halves?
This type of question can be recorded as 6 – ½, which reads as ‘6, how many
halves?’
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
60
Stage 2
NS2.4 Models, compares and represents commonly used
fractions and decimals, adds and subtracts decimals to two
decimal places, and interprets everyday percentages
61
Dividing a circle into equal parts
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
Large circle drawn on butchers’ paper; lengths of tape secured at the centre of the circle
Have students in a circle on the floor or around a table, and put out a large circle
drawn on paper, with its centre marked. Secure some lengths of tape at the
centre of the circle and ask the class: Which students should hold the end of the
four lengths of tape to divide the circle into equal parts?
Encourage the students to discuss what is happening.
Change the number of lengths of tape and repeat.
62
A Year 4 class used tape to divide a circle into equal parts. The students stood in
a circle around the table and worked out who should hold the end of each length
of tape in order to make the parts equal.
63
Dividing a cherry cake into equal slices
Content
K&S: Modelling, comparing and representing fractions
Materials
A circle drawn on paper with 30 ‘cherries’ (red counters) placed around the perimeter;
photocopies of the ‘cherry cake’ (Master follows)
Show students the model of the cherry cake and ask them to work out the
number of cherries on the edge. Ask: How could we use the cherries to help us
divide the cake in half?
Give pairs of students photocopies of the cherry cake and ask them to divide the
cake into more than two equal slices, with none of the cake left over. Do they use
the cherries as a guide?
Some work samples follow to show likely student responses.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
64
These students investigated the different ways they could divide the cake into
equal slices, with a whole number of cherries on each slice. This type of work
builds the concept of factors.
65
Dividing a cake into fifths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
A circle drawn on paper with 30 ‘cherries’ (red counters) placed around the perimeter;
photocopies of the ‘cherry cake’; paper and pencils
Ask students to draw a circle representing the top view of a cake and ask them to
divide the cake into five equal slices, with no cake left over. Ask students to
discuss their strategies for dividing their cakes.
Give each student a photocopy of a cherry cake and ask them if the cherries on
the edge of the pie would help them divide the cake into five equal slices. Again,
ask students to discuss their strategies for dividing the cakes.
66
Strategies for dividing cakes into equal slices
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
Card showing a circle divided into halves
When students discuss their strategies for dividing cakes in activities such as
‘dividing a cake into fifths’, they need their unsuccessful strategies to be
challenged.
Some students sometimes begin by drawing a line through the middle of the
circle, thus dividing it in half. They should discuss why this was unsuccessful for
dividing the cake into fifths. Show a drawing of a cake divided in half, and ask: If I
start by cutting the cake into halves, how many slices do you think I will be able
to divide the cake into if I make some more cuts?
Some students do not visualise, or cannot represent, the slices as lines radiating
from the centre of the circle.
They need to be asked to describe what they can see as a real cake is being cut,
and when lines are drawn from the centre of a circle to the perimeter.
67
Dividing a circle into thirds
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
Cards showing circles divided into three parts with a ‘Y’,
one with equal parts and some with unequal parts eg
When one student divided a ‘cake’ into fifths, he began by dividing it into thirds,
then he added two more radii. He said I did a Y first. He thought he had divided
the cake into five equal slices. There are three aspects here (equal parts, from
thirds to fifths, the ‘Y’) that need challenging at an individual level and at a whole
class level.
The class can discuss strategies for checking that the parts are equal. Possible
student responses include the following.
You could cut out one of the parts out and put it on top of the other
parts to see if they are the same size.
You could cut out all the parts and put them on top of each other.
You can just tell by looking carefully.
The students could work in pairs to discuss whether or not a circle that has been
divided into thirds could then be divided into fifths.
It’s not possible, because if you start with thirds and then divide
the thirds all in half, you’d get sixths.
Students often discuss the fact that a circle divided into thirds looks like a peace
sign, a Mercedes logo or a Y inside a circle. They need to be challenged to build
the understanding that the three lines are equally spaced around the circle. Show
them some circles with different ‘Y’ shapes drawn inside and ask them if each is
divided into thirds.
68
Dividing a cake into quarters
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
A circle drawn on paper with 30 ‘cherries’ (red counters) placed around the perimeter;
photocopies of the ‘cherry cake’; paper and pencils
Give each student a ‘cherry cake’ and ask them to divide it into quarters.
As they work, ask them how many cherries there will be on each quarter.
If they have unequal numbers of cherries on their slices, ask them if the cake is
divided into fair shares. Then ask them to try again to divide the cake into
quarters.
69
What shape is it?
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
About 15 cardboard isosceles triangles with sides 5cm, 5cm and 6cm and the angles being 72
degrees and 54 degrees (two) – mark the 72 degree angle in some way, eg with a coloured dot
Paper and pencils
Give each pair of students a cardboard triangle and ask: If this is a fifth of a
shape, what could the shape be?
The mark on the 72 degree angle will help students orient the triangle if they try
rotating, flipping or sliding it.
A related activity would be to give each pair of students an equilateral triangle
and ask: If this is a sixth of a shape, what could the shape be?
Other outcomes covered
SGS2.2a: Manipulates, compares, sketches and name two-dimensional shapes and describes
their features
70
How many marbles?
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
WM: Pose questions about a collection of items
Materials
None
Ask students questions such as: If 10 marbles is one-fifth of my collection of
marbles, how many marbles do I have?
If 21 marbles is three-fifths of my collection, how many marbles do I have?
(This question is appropriate for students who are very familiar with the multiples
of 7.)
Students can pose problems of this type for their peers.
These types of questions can be rephrased as clues to the number of items in a
collection when students are briefly shown the collection and asked to estimate,
then calculate, how many there are. For example, show briefly a bag of 40
marbles or a card with 40 marbles pictured and ask: How many marbles do you
think I have in this bag? (Allow students to make an estimate.) I’ll give you a clue:
five marbles would be an eighth of my collection.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
71
Equivalent fractions
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by finding
equivalence among halves, quarters and eighths
Materials
Diagrams of a circle divided into quarters, a circle divided into eighths
Hold up the card showing a circle divided into quarters and ask: What can you
see?
After students have described the circle as being divided into quarters, ask: If I
cut the circle in half, what would you see? Cut the circle in half and hold up one
half.
Likely student responses include the following.
I can see a half of a circle.
A semicircle.
Two quarters.
Do the same with the circle divided into eighths.
Summarise what students are saying: So half a circle is the same as two
quarters of a circle and the same as four eighths of a circle. Is that right? … We
say that one-half and two-quarters and four-eighths are equivalent fractions.
What do you think ‘equivalent’ means? … Can you find any other fractions that
are equivalent to one-half?
72
Equivalent fractions - fifths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Diagram of a circle divided into fifths
Hold up the card showing a circle divided into fifths and ask: What can you see?
After students have described the circle as being divided into fifths, ask: If I cut
the circle in half, what would you see? Cut the circle in half and hold up one half.
Likely student responses include the following.
Half a circle.
Two fifths and another fifth cut in half.
Two and a half fifths?
Direct students to work in pairs to discuss what fractions equivalent to a half they
can see or imagine in the half circle.
Likely student responses include the following.
You can see there would be five tenths, because you can imagine a
line down the middle of each fifth to match this bit (pointing to the
tenth where the fifth has been cut in half).
Two and a half fifths is the same as five tenths.
73
Equivalent fractions game
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by finding
equivalence among halves, quarters and eighths
Materials
A game: Pizzas, using the following materials: 2 cardboard circles cut into halves, 2 cut into
1 1 2 1 2 3
quarters and 2 cut into eighths; a dice marked 2, 4, 4, 8, 8, 8
This is a game for two players, or teams of players. The players take turns to
throw the dice and collect the ‘pizza slices’ (circle sectors) matching the fraction
thrown. The first player or team to make two pizzas wins. Pizzas can be made
with non-matching slices.
A variation on the game is to complete two pizzas, each one having identical
slices.
Both variations of the game involve students in exchanging slices, so help
develop equivalent fraction concepts.
Similar games can be made with fifths, tenths and twentieths, and with thirds,
sixths and twelfths.
74
Unequal fractions
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by finding
equivalence among halves, quarters and eighths
Materials
Diagram of a circle divided as follows: (It is divided into a half, a quarter, a sixth
and a twelfth.)
About 15 photocopies of the diagram
Show students the circle divided into unequal fractions and ask: Is this circle
divided into fractions?
Likely student responses include the following.
No, because fractions have to be the same size.
You know, like quarters are four parts the same size, so this can’t
be fractions.
No, they’re all different.
You have to have equal parts to have fractions, so there aren’t any
fractions in this circle.
Give each pair of students a copy of the circle and say: Have a careful look at the
circle. You might recognise some fractions there. Talk about what you can see.
Likely student responses include the following.
We can see a half over here, and a quarter.
They’re different sizes but they’re still fractions.
We know this is a half and this is a quarter and we think these ones
are a sixth and a twelfth.
75
Folding a paper strip
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
A4 paper cut into strips about 4cm wide
Give each student some paper strips and ask them to work out how they can fold
them to make halves, quarters and eighths. They should discuss the strategies
they are using.
Some students will be able to fold into halves but not into the other fractions. If
they do understand that there are four quarters in a whole and eight eighths, the
problem is with the folding and visualising. Allow these students to cut a strip into
halves, then ask them how they could make quarters and allow them to cut
again. If they are successful, help them line up the four quarters to reconstruct
the strip and ask them to describe where the cuts are. Give them another strip
and ask them to fold it into quarters, explaining at each step what they are
planning to do.
If there are students who do not understand there are four quarters in a whole
and eight eighths, they can be asked to ‘fold the strip into four equal parts’ and, if
they are successful with this task, into ‘eight equal parts’.
76
Cutting a paper strip into three equal pieces
Content
K&S: Modelling, comparing and representing fractions
Materials
A4 paper cut into strips about 4cm wide; scissors
Hold up one strip, fold it in half end to end, hold a pair of scissors as though
about to cut it in the centre between the ends and the fold, and ask: What would I
have if I cut this strip across here and opened it up?
Likely student responses include the following.
You’d have a long piece and two short pieces, because where the
its folded you open it up and it makes a long piece.
You’d have a half and two quarters.
Cut the strip and open it up to show the results. Then fold a second strip the
same way and ask students where it should be cut to get three strips the same
length. Give each student a strip of paper and some scissors, direct them to fold
the paper strip and not to open it again until they had made their cut. As they
work, ask them to explain their strategies for working out where to cut the strip.
They can measure, but without opening up the strip.
Likely student responses are overleaf.
77
```
78
Cutting a paper strip into fifths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
A4 paper cut into strips about 4cm wide; scissors
This is the same as the previous task, but students are set the task of working
out where to cut the folded strip to divide the strip into fifths.
Likely student responses follow.
79
80
Measuring area
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Cardboard rectangles whose areas can be measured with counters, including half-counters (7 x
1½, 3 x 3½, 2½ x 4, 2½ x 2½); counters
Show students a rectangle (that is nine counters long by two counters wide) and
ask them to estimate how many counters it would take to cover it. Ask them to
demonstrate how they would work it out. Show another rectangle (6 counters
long by 3 counters wide) and repeat. Ask them if there are any other rectangles
that would hold 18 counters. They might suggest a rectangle measuring 18
counters by one counter. Ask if there are any others. They are likely to say no.
Give students the cardboard rectangles and direct them to work out how many
counters would fit on each one.
Ask them to show on paper how they worked it out.
Likely student responses are overleaf.
At the end of the lesson, return to the question of whether there are any other
rectangles that would fit 18 counters; some students might suggest a rectangle
that measures 36 counters by half a counter.
81
A student works out how many counters fit on this rectangle.
82
83
As an extension task, a pair of students was asked to draw a rectangle that
would fit 20¼ counters.
84
Half-filling a jug
Content
K&S: Modelling, comparing and representing fractions
Materials
Clear cylindrical jug, clear tapered jug the same height
Show the students the two jugs and ask them to describe what is different about
them. They should notice that one is a cylinder and the other is almost a cone.
Set the students the task of drawing the two jugs and showing them half-filled
with water.
Likely student responses include the following.
Most students are likely to draw the top of the water halfway up the vertical
height of the tapered jug, so draw this on the board and ask the students if it
looks correct. They will be likely to say it does. Leave the diagram there.
With the students watching, fill the cylindrical jug with cupfuls of water, then ask
the students to work out how to half-fill the jug. They might suggest pouring out
half the number of cupfuls.
85
Repeat with the tapered jug, and encourage the students to discuss why the jug
doesn’t look half-full when half the water has been poured out.
Return to the diagram of the tapered jug on the blackboard and ask the students
to suggest where the water will come up to when the jug is half-full.
Other outcomes covered
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
milliliters and cubic centimetres
86
Dividing rectangles into quarters
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Paper rectangles; 3 cardboard rectangles that measure 12 by 8 counters, divided as follows:
Give each student some paper rectangles and ask them to divide them into
quarters in different ways. They can use both sides of the paper. Ask them to
record on each quarter how many counters they estimate would fit on it. Remind
them they can think of using half-counters so the whole surface is covered.
Likely student responses include the following.
11 22
16 16 12 12 12 12 11
22 22
11
16 16
22
11
22 21
26 26 27 27 27 27 22
21 21
22
26 26
21
22
Some students think there will be an odd number of counters in the quarters of
the last rectangle because ‘one counter will fit in the pointy bit’ of the triangle,
indicating the angle at the midpoint of the rectangle.
20 30
15 20 20 15 25 30 24
20 20
22
15 25
30
23
87
At the conclusion of the activity, when students are gathered together and
discussing what they did, put out the three cardboard rectangles divided into
quarters with the undivided sides showing and establish with the students that
they are all the same size. Turn them over and ask the students what they notice.
They should realise that the rectangles are divided into quarters in different ways.
Ask them to estimate how many counters will fit on each quarter. Then begin to
cover them with counters. The students will find that each quarter will fit the same
number of counters, and should discuss reasons for this.
In the photo below, note that instead of individual counters being used, there are
strips of photocopied counters, which makes the task quicker and supports the
development of multiplication and division concepts.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
88
How much of this pie has been eaten?
Content
K&S: Modelling, comparing and representing fractions
Materials
Pictures of divided ‘pies’ with one slice missing, for example:
Pictures of undivided ‘pies’ with one piece missing, for example:
Hold up each divided ‘pie’ in turn and ask: How much of this pie has been
eaten? Students are likely to say in each case that one piece has been eaten,
except when shown a pie with half missing when they will say that half has been
eaten.
Spread out the pies to which the students responded ‘one piece has been eaten’
and ask: Have the people who have eaten the slices of pie had the same amount
from each pie? Is there a way we could say how much of each pie has been
eaten that will show how big each slice is? What did you say when I showed you
this pie (show the pie with half missing)?
These questions should generate discussion of fractions generally.
Likely student responses include the following.
The bits that are missing are fractions.
You know how many slices there are so you can tell what the
fraction is.
The slices in each pie are the same size so it’s fractions.
Show the undivided pies with a piece missing to generate further discussion.
Likely student responses include the following.
You can sort of tell what the fraction is because you can imagine
where the other slices would be.
89
Sorting circle fractions
Content
K&S: Modelling, comparing and representing fractions
Materials
Several quarters, fifths and thirds of circles of different sizes
Ask students to sort the shapes and explain how they have sorted them.
Ask questions such as: Which is larger – a quarter or a third? A quarter or a fifth?
A fifth or a sixth? The last question will challenge students to extend their thinking
to fractions other the ones they can see.
90
Visualising thirds
Content
K&S: Modelling, comparing and representing fractions
Materials
3 sectors of circles, one being a third of a circle and the others being fractions that look similar to
a third (eg five-twelfths and seven-twentyfourths)
Hold up the three sectors and ask: Which one of these is a third of a circle?
This is a useful task because some students may think of a third of a circle in
terms of its shape rather than in terms of its size in relation to the circle.
It is also useful because it helps students build familiarity with thirds, which will
lead to visualising and conceptualising the fraction two-thirds.
As an extension task, the other fractions could be displayed in the classroom and
students set the challenge of working out what fractions of a circle they are.
91
Visualising tenths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Various sectors of a circle, including a tenth
Display the sectors and ask: Which one of these is a third of a circle? Why do
you think it is that one?
Give each group of students one of the sectors and set them the task of showing
on paper what fraction of a circle it is. They will probably draw around the sector
to make a circle, or half a circle.
Ask questions such as: Which is larger – a tenth or an eighth? Why? Which
fraction is twice as big as a tenth? What fraction do you think would be half as
large as a tenth? (Do not have a twentieth available – this question is to extend
their thinking beyond the fractions available.)
92
Investigating shapes that can be divided into
tenths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Templates of various shapes including circles, rectangles, pentagons, pentagrams and decagons,
and others that cannot easily be divided into tenths
Set students the task of investigating which two-dimensional shapes can be
divided into tenths.
Other outcomes covered
SGS2.2a: Manipulates, compares, sketches and names two-dimensional shapes and describes
their features
93
What fraction is black?
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
WM: Pose questions
Materials
Cards showing rectangles divided into tenths, twentieths, thirtieths and so on, with some of the
squares black and some white, as follow:
The last three cards have two sides,
with the grid not shown on one of the
sides.
Show one card at a time (showing the side without the grid) and ask: What
fraction of this rectangle is black?
Students should explain their thinking. Turn the cards over to show the grids to
help students check their ideas.
Ask further questions to elicit discussion about tenths, and about the equivalence
of five-tenths and one-half.
Set students the task of drawing their own shapes, colouring part and asking
their peers fraction questions about them.
94
Chocolate hundredths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
A ‘giant block of chocolate’ – a brown cardboard square, say 20cm by 20cm, marked into 100
squares in a 10 by 10 grid; several brown cardboard parts of a ‘block of chocolate’ such as half a
block, 99-hundredths and 30-hundredths
Show students the ‘block of chocolate’ with the grid showing, without calling it
that, and ask them what it is. Several students are likely to suggest it looks like a
block of chocolate.
Display the block of chocolate so the grid is not seen, hold up the half-block, and
say: This is what is left after some of the chocolate has been eaten. How much is
left? Students usually say that half the block is left. Turn the block of chocolate so
the grid can be seen and use it to check how much is left.
Show the 99-hundredths-block and ask: How much of this block has been eaten?
Students are likely to say that one piece has been eaten. Ask them: What
fraction of the block of chocolate has been eaten?
Show other part-blocks, asking what fraction of the block has been eaten or how
much is left, turning over the block so the grid is seen and using it to check the
students’ responses.
Likely student responses include the following.
(Being shown a five piece by five piece section of a block)
A quarter of the block.
Twenty five hundredths, because it’s five rows of five I think.
Teacher: So do we all agree it’s a quarter of the block?
(She turns it over to show the grid.) And is it twenty five
hundredths?
Twenty five hundredths is the same as a quarter.
95
96
100 echidnas
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
100 five-cent coins or large replicas of them
Put out the hundred coins or coin replicas and ask students to estimate how
many there are. Begin to count them into groups of ten and let the students
change their estimates as you do so.
Ask the students how they could be arranged in a pattern to see easily that there
are 100 coins.
Arrange them in a ten by ten array. If using enlarged photocopies of coin
pictures, perhaps glue them to a sheet of cardboard for display; to make it easier,
make photocopies of a line of ten coins, thereby being able to glue ten coins at a
time.
Ask: How much money is here altogether? How did you work it out?
When it is established that there is $5, ask: What is a tenth of $5? How do you
know? What is a hundredth of $5? How do you know? How many 50c items
could you buy for $5? How many 5c items could you buy for $5?
The same activities can be done with ten-cent coins.
Then students could be asked: What if the coins were 20c coins, how much
money would there be altogether? What if they were 50c coins?
Later questions could be of the type: What is a tenth of $8? What is a hundredth
of $8?
97
Measurement relationships
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Metric measurement equipment
Set students in small groups the task of investigating the relationships between
centimetres and metres, including making models to show how long a tenth of a
metre is and how long a hundredth of a metre is. They can use materials such as
paper streamers, tape, metre rulers and cardboard strips. One group could be
set the task of making a visual representation for display.
Repeat this task for area, asking students to investigate and model a tenth of a
square metre and a hundredth of a square metre. They need a large space for
this, so could go outside and use materials such as string to peg out a square
metre. One group could be set the task of making a visual representation for
display.
Repeat the task for volume and capacity, using materials such as water, cubic
centimetre blocks, and cardboard to make models, to investigate tenths and
hundredths of a litre and relationships between MAB blocks measuring one cubic
centimetre and 1 000 cubic centimetres.
Other outcomes covered
MS2.1: Estimates, measures, compares and records lengths, distances and perimeters in metres,
centimetres and millimeters
MES2.2: Estimates, measures, compares and records the areas of surfaces in square
centimetres and square metres
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
millilitres and cubic centimetres
98
Measurement tenths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Metric measurement equipment
Set students in pairs to investigate the relationship between tenths of a litre and
tenths of a kilogram. (There are 100g in a tenth of a kilogram and 100mL in a
tenth of a litre.)
Some students will benefit from using measurement equipment to investigate.
For example, they might pour a litre of water into ten cups, putting the same
amount in each cup, then working out, with support, how much is in each cup
and noticing it is about 100mL in each case. They might measure a kilogram in
smaller masses, say 10g masses, and divide the smaller masses into ten equal
groups to work out how much is in each tenth.
Other students will be able to work without equipment, simply using their
knowledge of the number of millilitres in a litre and the number of grams in a
kilogram.
Other outcomes covered
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
millilitres and cubic centimetres
MS2.4: Estimates, measures, compares and records masses using kilograms and grams
99
Farms
Content
K&S: Modelling, comparing and representing fractions
Materials
‘Farms’, A5 sheets of card divided into sections, for example:
Hold up one card at a time and explain that the card is a farm, with the lines
being fences breaking the farms up into fields. Ask: How many cows will there be
in each field if there are 36 cows altogether and they need to have the same
amount of grass each?
Ask them to explain their strategies for working out their answers.
Likely student responses include the following.
You can work out halves first because one fence goes
across in the middle, so that’s eighteen up the top,
then it’s half again so it’s nine in the quarter one, then
you can work out how many go in the smaller bits, it’s
three and three and three.
Well, see how’s there’s kind of like six the same size
but a line’s missing, so you divide six into 36 and then
it’s six in each one and twelve in the big one.
You can do thirds and that’s twelve in the first field,
and six in the two fields at the end because six is half
of twelve, and six in the top middle field because it’s
the same as the one next to it, and three in the bottom
middle fields because three and three is six.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
100
101
Party
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Record the following sequence of numbers on the board and ask the students
what they notice: 1, 5, 9, 13, 17, 21 …
Likely student responses include the following.
You’re adding four each time.
It’s plus one to the multiples of four.
You started at one and then counted by fours after that.
Say: Imagine a party with four people sharing 1 pizza, 5 cupcakes, 9 sausages
and 13 jelly snakes. How much would each person have if they shared
everything fairly and there was nothing left over?
Likely student responses are overleaf.
Students should discuss what they have found out when they have completed
the task.
Likely student responses include the following.
The numbers are all in the sequence you wrote on the board.
Every answer is something and a quarter.
Because you have to divide the last thing into quarters, that’s why.
Related tasks are to share 1, 4, 7, 10 and 13 items (multiples of 3, plus 1) among
3 people, and so on.
102
103
Football fractions
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Paper and pencils
Set each student the task of writing ‘a fraction problem’. Possibly the only
fractions represented in their problems will be a half, a quarter and three-
quarters, and this can result in a discussion of why this is so.
Ask students about fractions they know about outside the classroom. They
usually mention half-hour shows on television, clock times, pizza slices, cutting
toast. Most of the fractions they mention will be unit fractions, particularly a half
and a quarter.
Set the students to work in groups to draw a rugby league field, and ask them
what fractions they used when drawing. They will probably mention the half way
line. Ask them if there are any other ‘football fractions’, for example in the field
positions. They should be able to name the ‘half’, ‘three-quarters’ and ‘five-
eighth’. Ask them to talk in their groups about why these positions have these
names, and if it has anything to do with the field layout. They can use their
drawings of the field to think about and demonstrate what they know.
104
Mixing cordial
Content
K&S: Modelling, comparing and representing fractions
Materials
Clear cylindrical jug
Mark the jug into sixths up the side.
Pour concentrated cordial into the jug, up to the first mark. Add water up to the
fifth mark.
Ask: What fraction of the mixture is cordial and what fraction is water?
The amount of mixture can be measured in millilitres, and the amounts of cordial
and water worked out from that measurement.
Other outcomes covered
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
milliliters and cubic centimetres
105
Pizza sharing problems
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Pose the following problems and allow students to work with pencil and paper.
Ask the students: If you shared a pizza among four people, how much pizza
would each person get?
Likely student responses include the following.
Two pieces each because there are eight pieces.
A quarter.
Ask: Imagine I’ve made a pizza. I’m in the kitchen, about to slice the pizza. If I
share it among five people, how much would each person get?
Likely student responses include the following.
A fifth.
A sixth, if you’re having some as well as the other five people.
If you cut it into ten slices, they can have two slices each.
Ask: What if the pizza has to be shared between two people, but one of the
people is a very hungry adult and the other is a very young child?
Likely student responses include the following.
Cut it into thirds, two-thirds for the adult and one-third for the child.
Cut it into one-quarter for the child and three-quarters for the adult.
Ask: How could you share a pizza among two hungry adults and two very young
children?
Likely student responses include the following.
One-third for each adult and one-sixth for each child.
Three-eighths for each adult and one-eighth for each child.
106
Two students’ responses to the question: How could you share a pizza among
two hungry adults and two very young children?
107
Fraction sequences
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Begin by having in mind the number 2 048, and saying to the students: I’m
thinking of a four-digit number. Its digits are all even and they are all different. 6
is not one of the digits. The final digit is double the third digit, and the first two
digits add to 2. What number am I thinking of? The ‘clues’ can be written and put
on display.
When they have worked out the number, ask them to begin with 2 048, halve it,
halve the new number and keep repeating the process. They should generate
the sequence 2 048, 1 024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1 … Some
students may continue halving and add some fractions to the sequence. If not,
ask the students to keep on going once they reach 1. They will notice some
interesting patterns in the sequence and also learn about fraction relationships as
they investigate how to halve fractions.
Students can be set the task of creating other fraction sequences as an on-going
focus.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
PAS2.1: Generates, describes and records number patterns using a variety of strategies and
completes simple number sentences by calculating missing values
108
Yes/No game
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
None
Draw a vertical line on the board, creating two spaces, and write ‘Yes’ at the top
of one and ‘No’ at the top of the other.
Tell the students you are collecting certain numbers and they have to work out
which ones you are collecting. They take turns to suggest numbers, perhaps
within a specified range such as from 0 to 100, and you will record them in the
‘Yes’ space if they are being collected and in the ‘No’ space if they are not.
Numbers being collected might be even numbers, or multiples of 3, or two-digit
numbers, or single-digit odd numbers, or two-digit numbers ending in 6 and so
on.
In this case, specify the range from 1 to 20, and have in mind to collect mixed
numbers. The students will probably suggest all the whole numbers from 1 to 20
before someone realises mixed numbers as a possibility.
109
Adding mixed numbers
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Cards with three or four mixed numbers, two of which add to 10
For example:
8½ 6½ 5½
2½
1½ 4½ 2½
Hold up each card in turn and ask the students to work out which pair of numbers
add to 10, and explain how they know they are correct.
When students are familiar with the task and confident with mixed number
combinations of 10, ask them to work out the total of the numbers on the card.
What strategies do they use? Do they use their knowledge of the combinations of
mixed numbers of 10?
Likely student responses include the following.
8½ plus 1½ is 10, plus 2½ is 12½.
8½
2½ 8 and 2 is 10, plus 1 is 11, plus two halves is
1½ 12 and then one more half is 12½.
110
Multiplying mixed numbers
Content
K&S: Modelling, comparing and representing fractions
Materials
Cards with a mixed number repeated several times
For example:
3½ 3½
3½
3½
3½ 3½
3½
3½ 3½
3½
Hold up a card briefly and ask: If you added all the numbers, what would the total
be?
Students should explain and discuss their strategies.
Hold up the card again and allow students to work out the total. Again, they
should explain and discuss their strategies.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
111
Investigating multiplying mixed numbers
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils; coloured paper squares about 3cm squared
Begin to write the following sequence on the board, and ask students to
comment on what you are doing: 1½, 2½, 3½, 4½, 5½ …
Set them the task of investigating multiplying the numbers in the sequence by 5,
and ask them to use arrays to work out the sequence of answers. They can draw
arrays or use the paper squares to construct them, cutting them in half when
necessary and gluing them in place.
Ask the students to explain why they get the sequence of answers that they do.
Ask questions such as: What will be the hundredth number in the sequence of
answers? How do you know?
Other investigations of this type can be undertaken, such as investigating
multiplying 2½ by the numbers in the sequence of even numbers, investigating
multiplying numbers in the sequence 1½, 2½, 3½, 4½, 5½ … by 10.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
PAS2.1: Generates, describes and records number patterns using a variety of strategies and
completes simple number sentences by calculating missing values
112
Number lines
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by placing
halves, quarters and eighths on a number line
Materials
Paper and pencils
Set each student the task of making their own number line, beginning and ending
with numbers of their choice, and making regular markings on their line.
Likely student responses include the following. They usually work
with numbers with which they feel confident. Some of them will
fractions and/or include mixed numbers. Occasionally a student
will include decimals, such as a girl whose number line (1.5, 2, 2.5,
3 and so on), she explained, was like a number line showing depths
at her local swimming pool.
10 20 30 40 50 60 70 80 90 100
1 1½ 2 2½ 3 3½ 4 4½ 5 5½
No matter which numbers a student has placed on their number line, ask each
them to add one more number somewhere on their line between the first and last
numbers they already have. That is, they cannot simply extend their line to
continue the counting sequence they already have. So the student who
constructed the second number line above would have to add a number like 1¼
or 4¾ .
113
Fraction number lines
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by placing
halves, quarters and eighths on a number line
Materials
Paper and pencils
Draw a number line on the board, showing the numbers from 0 and 10 at each
end, and leaving large spaces between them. Ask: Are there any other numbers
we could place on this number line?
Students should be able to add some fractions and mixed numbers to the
number line.
Set students the task of investigating showing fractions on number lines. They
should discuss their own number lines in comparison with those made by others.
They are likely to draw lines showing mixed numbers rather than fractions on a
line from 0 to 1.
Later, set them the task of investigating showing fractions on number lines with
the stipulation that the number lines begin with 0 and end with 1.
114
Equivalent fraction number lines
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by placing
halves, quarters and eighths on a number line
Materials
Paper and pencils
Set students in pairs the task of making a set of number lines that show
equivalence among halves, quarters and eighths. Tell them one line should show
halves, one should show quarters and one eighths.
They should realise they will need to make their number lines the same length. It
is preferable they are not told this, nor given photocopied number lines of equal
length, as they then won’t have the opportunity to think the task through for
themselves and in so doing come to understand an important aspect of
equivalence.
Some students could be challenged to extend the task to other fractions.
115
Stage 3
NS3.4 Compares, orders and calculates with decimals, simple
fractions and simple percentages
116
Thirds
There are several activities outlined in Stage 2 that are useful in developing
concepts of thirds.
Dividing a circle into equal parts
Dividing a cherry cake into equal slices
Dividing a circle into thirds
How many marbles?
Equivalent fractions games
How much of this pie has been eaten?
Sorting circle fractions
Visualising thirds
Farms
Party
Mixing cordial
Pizza sharing problems
Fraction sequences
Yes/No game
Adding mixed numbers
Multiplying mixed numbers
Investigating multiplying mixed numbers
Number lines
Fraction number lines
117
From thirds to other fractions
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
Photocopies of a circle with lines dividing it into thirds
Give each pair of students photocopies of the circle and ask them to investigate
what other fractions the circle could be divided into, using the lines already there.
Likely student responses include the following.
You could draw lines halfway through each third and make sixths.
And twelfths if you do the same again.
Or you could do ninths, you would draw two lines in all the thirds.
Any multiple of three.
118
Cutting a paper strip into thirds
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
A4 paper cut into strips about 4cm wide; scissors
Hold up one strip, fold it in half end to end, hold a pair of scissors as though
about to cut it in the centre between the ends and the fold, and ask: What would I
have if I cut this strip across here and opened it up?
Likely student responses include the following.
You’d have a long piece and two short pieces, because where the
its folded you open it up and it makes a long piece.
You’d have a half and two quarters.
Cut the strip and open it up to show the results. Then fold a second strip the
same way and ask students where it should be cut to cut it into thirds. Give each
student a strip of paper and some scissors, direct them to fold the paper strip and
not to open it again until they had made their cut. As they work, ask them to
explain their strategies for working out where to cut the strip. They can measure,
but without opening up the strip.
119
Cutting a paper strip into thirds, sixths and
twelfths
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
A4 paper cut into strips about 4cm wide; scissors
Set students the task of investigating what fractions result when they fold a strip
of paper lengthwise, then lengthwise again, and, leaving it folded, making a cut
one-third from its end. The ends are different (one having two folds and the other
having one fold) so students should work with two strips, cutting one third in from
the end with one fold on one strip, and one-third in from the end with two folds on
the other strip.
They will find thirds, sixths and twelfths.
They can produce reports that show in diagrams what they found out.
120
Equivalent fraction number lines
Content
K&S: Placing thirds, sixths or twelfths on a number line between 0 and 1 to develop equivalence
WM:Eplain or demonstrate why two fractions are or are not equivalent
Materials
Paper and pencils
Set students the task of making a set of number lines that show equivalence
among thirds, sixths and twelfths. Tell them one line should show thirds, one
should show sixths and one twelfths.
They should realise they will need to make their number lines the same length. It
is preferable they are not told this, nor given photocopied number lines of equal
length, as they then won’t have the opportunity to think the task through for
themselves and in so doing come to understand an important aspect of
equivalence.
Some students could be challenged to extend the task to other fractions.
121
Investigating shapes that can be divided into
thirds, sixths or twelfths
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
Templates of various shapes, including regular dodecagons. It will be useful for this investigation
Set students the task of investigating which two-dimensional shapes can be
divided into identical thirds.
Repeat for which two-dimensional shapes can be divided into identical sixths.
Repeat for which two-dimensional shapes can be divided into identical twelfths.
Other outcomes covered
SGS3.2a: Manipulates, classifies and draws two-dimensional shapes and describes side and
angle properties
122
Picnic
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Record the following sequence of numbers on the board and ask the students
what they notice: 1, 4, 7, 10, 13, 16 …
Likely student responses include the following.
19, 22, 25 … Just add threes.
You started at one and then plussed three every time.
Every number is a multiple of three plus one.
Say: Imagine a picnic with three people to share the food. How would they share
one bread stick, four sausages, seven square slices of cheese, ten slices of
tomato and thirteen carrot sticks so they all had the same amount to eat and
there was nothing left over?
Students should explain their strategies for sharing the items and discuss their
results. Why did they get the results they did?
A later task would be to share 2, 5, 8, 11 and 14 items (multiples of 3, plus 2)
among 3 people, and compare the results with the results from this task.
123
When would you need to find a third?
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
None
Ask students: When would you need to find a third of a collection of objects?
Likely student responses include the following.
If you have some marbles and your mum or dad makes you share
them with your brothers and if you’ve got two brothers.
When you and two friends want to share some food.
Ask: Which numbers of things are easy to find a third of?
Students should work in pairs on this question. Note that for some students it is
‘easy’ to find thirds of almost any whole number, whereas others will come up
with the answer that the multiples of three are ‘easy’ to find thirds of.
The same questions can be asked about sixths and twelfths, and later students
can be asked: Can you find a number that it is it easy to find thirds and sixths and
twelfths of? What other numbers is it easy to find thirds and sixths and twelfths
of?
124
Investigating shapes that can be divided into
sevenths
Content
Materials
Templates of various shapes, including regular heptagons. It will be useful for this investigation if
there are rectangles with linear dimensions that are multiples of 7.
Students have explored relationships among halves, quarters and eighths,
among fifths, tenths and hundredths, and among thirds, sixths and twelfths, so it
is useful for them to be challenged to transfer their conceptual understandings to
unfamiliar fractions.
Set students the task of investigating which two-dimensional shapes can be
divided into identical sevenths.
Ask: What other fractions can these shapes be divided into?
Other outcomes covered
SGS3.2a: Manipulates, classifies and draws two-dimensional shapes and describes side and
angle properties
125
Sevenths
Content
WM: Pose and solve problems
Materials
Paper and pencils
Ask: If you had enough food to last you a week, how much of the food would you
eat each day?
Ask: What if you had 28 slices of bread, how many would you eat each day of the
week if you had the same amount each day and had none left over at the end of
the week? 42 strawberries? 8 chocolate bars? 23 slices of cheese?
Students could pose some questions like this for their peers.
Ask: If you had an amount of money to buy your lunches for a week, how much
of the money would you need to keep for the weekend? If the amount you had
was $35, how much money would you spend on the weekend?
126
Lots of quarters
Content
K&S: Expressing mixed numerals as improper fractions
Materials
Three cardboard circles (not divided); cardboard quarter-circles
Perhaps begin by reciting the following rhyme, Quarters, written by some Stage 1
students.
One quarter
Two quarters
Three quarters
Four.
And that’s where we stop
Because there aren’t any more.
Ask: Is it correct that there aren’t any more than four quarters?
Show five cardboard quarter-circles and ask: How many quarters can you see?
Show a whole circle (not divided) and ask: How many quarters can you see?
Show three whole circles and ask: How many quarters can you see?
Ask: How could you write twelve quarters? Is twelve quarters the same as four?
12 20
Could we write 4 = 3 ? How would we finish this one: 4 = ?
10
And this one: 4 = ?
Set students in pairs the task of investigating other fractions like twelve-quarters.
Tell them that fractions like this are called improper fractions, and ask them to
suggest why they might be called this.
Perhaps students could suggest an alternative last two lines to the rhyme above.
127
Fraction posters
Content
K&S: Expressing mixed numerals as improper fractions
K&S: Developing a mental strategy for finding equivalent fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Poster-size paper and pencils
In the centre of a poster-size sheet of paper record the number 3 and record two
improper fractions equivalent to 3 as follows.
12
4
3
9
3
Ask students to comment on what they see and ask them to suggest some other
improper fractions equivalent to 3 that could be written on the poster.
Set students in groups the task of making a similar poster with a number of their
choice, other than 3, in the centre.
They should discuss their strategies for working out the improper fractions.
128
A dozen eggs
Content
K&S: Developing a mental strategy for finding equivalent fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Egg-cartons, one marked by different colours in halves, one in quarters, one in thirds, one in
sixths and one in twelfths; cardboard ‘eggs’; poster-size paper and pencils
Have the egg-cartons on display and ask students to talk about what they see.
Lead them to notice that an egg-carton can be divided into different fractions.
Ask: How many eggs are in a dozen? In a half-dozen? In a quarter of a dozen?
In a third of a dozen? In a sixth of a dozen?
Ask questions such as: How many eggs are there in three-quarters of a dozen?
How many eggs are there in five-twelfths of a dozen?
Ask: How many eggs are there in six-twelfths of the dozen? What other fractions
of the dozen are 6 eggs?
The students can help make a poster for fractions equivalent to a half, beginning
with the fractions found with the egg-cartons, as follows. They can add other
equivalent fractions.
6
12
1
2
2
4 3
6
129
More fraction posters
Content
K&S: Developing a mental strategy for finding equivalent fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Poster-size paper and pencils
Students can work in groups to make posters for fractions equivalent to the
following familiar fractions: a quarter, a third, two-thirds and three-quarters.
130
Puzzling posters
Content
K&S: Developing a mental strategy for finding equivalent fractions
K&S: Reducing a fraction to its lowest equivalent form
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Poster-size paper and pencils; equivalent fractions posters made previously
Provide posters such as the following and set students the task of adding some
more equivalent fractions, including the one that belongs in the centre. Explain
that there is only one possible fraction that can be placed in the centre and ask
them to work out what it should be, including by referring to the posters they have
previously made.
3 16
30 32
5
50 7
70 20
40
8
7 48
35
15
75 4 5
20 30
Students can make some of their own ‘puzzling posters’ for their peers to
complete.
131
Folding and cutting quarters
Content
K&S: Reducing a fraction to its lowest equivalent form
WM: Explain or demonstrate why two fractions are or are not equivalent
WM: Pose and solve problems
Materials
A4 paper cut into strips about 4cm wide; scissors
Ask students to fold a strip of paper lengthwise, then lengthwise again, and,
leaving it folded, making four small cuts at the edge of the strip at the one-
quarter, half and three-quarters spots.
Tell them to open up their strips and work out what fractions the small cuts and
the folds divide the strip into (sixteenths).
Set them the task of investigating what fractions result when they re-fold the
paper and make a cut one-quarter from an end. The ends are different (one
having two folds and the other having one fold) so students should work with two
strips, cutting one quarter in from the end with one fold on one strip, and one-
quarter in from the end with two folds on the other strip.
Ask students to name each fraction that results in its lowest form.
Students can pose their own paper folding and cutting problems.
132
Subtraction problems
Content
K&S: Using written, diagram and mental strategies to subtract a unit fraction from any whole
number
WM: Pose and solve problems
Materials
Paper and pencils
Pose problems such as the following.
If I had a bottle of drink and poured a third of it into a glass, how much would be
left in the bottle? To extend some students: If I poured half of what was left into
another glass, how much would be left in the bottle?
If I bought three chocolate bars and I ate half of one of them, how many bars
would I have left?
If I had eight socks and I lost one sock, how many pairs of socks would I still
have?
Show a picture of a shoe and ask: One shoe is missing; how many pairs of shoes
are left?
Students can pose similar problems
133
What can you see?
Content
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
A circles divided into sixths; photocopies of circles divided into fifths
Show the circle divided into sixths, cut off half so that students can see three-
sixths, and ask: What do you have if you cut off half of this circle?
They are most likely to tell you there is half left, or three-sixths.
Show a circle divided into fifths and repeat as above. Students might say there
are left, but it is likely they will not be sure they can call it ‘two-and-a-half fifths’.
Set them the task of investigating in pairs what fraction it could be called, other
than a half or two-and-a-half fifths.
Likely student responses are as follows, including overleaf.
134
135
Equivalent fraction problems
Content
WM: Explain or demonstrate why two fractions are or are not equivalent
WM: Pose and solve problems
Materials
Egg-cartons, one marked by different colours in halves, one in quarters, one in thirds, one in
sixths and one in twelfths; cardboard ‘eggs’; poster-size paper and pencils
Have the egg-carton and ‘eggs’ on display and pose problems such as the
following.
How many eggs would be left in a carton if three-twelfths of them have been
eaten? How do you know?
How many eggs would be left in a carton if a quarter of them have been eaten?
How do you know?
Why are the answers to the two questions the same?
And:
How many eggs have been eaten if four-sixths of the eggs are left?
How many eggs have been eaten if two-thirds of the eggs are left?
Why are your two answers the same?
Students can pose similar problems.
136
Reducing a fraction to its lowest equivalent form
Content
K&S: Reducing a fraction to its lowest equivalent form
Materials
None
Ask questions such as the following.
Why do we hear the word ‘half’ in everyday talk, rather than four-eighths or ten-
twentieths or six-twelfths?
If I have 30 counters, 15 red and 15 blue, is it sensible to say ‘fifteen-thirtieths of
my counters are red’ or is it sensible to say ‘half my counters are red’?
How do you know the most sensible way to say a fraction?
Students can be set the task of investigating fractions that can be reduced to
one-quarter, or fractions that can be reduced to two-thirds. At the conclusion of
the investigation, students can explain their strategies for reducing any fraction to
its simplest form.
137
Problems involving fractions with the same
denominator
Content
K&S: Adding and subtracting fractions with the same denominator
WM: Pose and solve problems
Materials
Paper and pencils
Pose problems such as the following.
If I had a pizza, and I ate two-eighths and my sister ate three-eighths, how much
would be left for our brother?
If my three friends each had a pizza, and each friend gave me two-eighths of
their pizza, would that be fair?
If my sister and brother each had a pie, and my sister gave me a third of hers
and my brother gave me a third of his, would that be fair?
The students can discuss how they add and subtract fractions with the same
denominator.
Students can pose their own problems like the ones above. They might use
fractions with different denominators, so this will naturally lead to discussing and
investigating how they might handle such problems.
138
Adding and subtracting fractions with related
denominators
Content
K&S: Adding and subtracting simple fractions where one denominator is a multiple of another
WM: Pose and solve problems
Materials
Paper and pencils
Following on from students posing their own fraction problems, which might
involve a mix of fractions with different denominators, ask them to discuss which
groups of fractions are easy to add.
Set students in pairs to investigate adding and subtracting a mix of fractions.
Some could investigate a mix of halves, quarters and eighths, others a mix of
fifths and tenths, and others a mix of thirds, sixths and twelfths.
The students should gather together and share their findings about the different
groups of fractions, perhaps using diagrams and models to explain their thinking.
To transfer their thinking beyond the groups of fractions investigated, ask: What
fractions would be easy to add and subtract along with sevenths?
139
Which is closest to a half?
Content
K&S: Comparing and ordering fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Paper and pencils
Set students in pairs to work out which of the following fractions is closest to a
half: five-eighths, four-sixths, seven-twelfths.
They should explain their strategies as they work. They might use diagrams,
models and number lines.
After the students have completed the task, gather them together to share their
findings and the different strategies they used. If they did not use number lines,
suggest to them that using number lines is a strategy worth exploring.
Ask them to investigate how to use number lines to compare fractions and find
equivalent fractions.
You could consider having students help make a class set of number lines, all
the same length but showing different fractions.
140
Which fraction is the largest?
Content
K&S: Comparing and ordering fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Paper and pencils
Write the following fractions on the board and ask students to work out the
pattern: 2 3 4 5
3, 5, 7, 9
(Each denominator is twice the numerator, minus one; or each numerator is half
the denominator, plus a half.)
Ask them to think about which of the fractions is the largest, and set them the
task of investigating in pairs.
Other outcomes covered
PAS3.1a: Records, analyses and describes geometric and number patterns that involve one
operation using tables and words
141
Problems involving unit fractions
Content
K&S: Calculating unit fractions of a collection
Materials
Paper and pencils
Pose problems such as the following.
I made some cakes and iced them in three different colours. One third of the
cakes have white icing. One-sixth have pink icing. 12 cakes have yellow icing.
How many cakes did I make?
How many pairs of footwear do I have if one-third of the pairs are sandals, one-
sixth of the pairs are thongs, one-twelfth of the pairs are slippers, and I have 5
pairs of shoes?
Some students will be able to investigate how to find non-unit fractions of a
collection, such as two-thirds of a collection. They can be set problems such as
the following.
I have a collection of large and small marbles. One-third of my marbles are red.
Five-sixths of my marbles are large. I have three small marbles. How many red
marbles do I have?
142
Multiplying fractions by whole numbers
Content
K&S: Multiplying simple fractions by whole numbers
Materials
Paper and pencils
Pose problems such as the following.
Eight children shared some jelly snakes. If each child got two and a half jelly
snakes, how many did they have to start with?
Six people shared some cakes. They ate one and a third cakes each, so how
many cakes did they have altogether?
Eight children shared some sausages. They were able to have two and a quarter
sausages each, so how many had they started with?
A bathroom floor was covered with large square tiles. There were eight rows of
tiles with nine and a half tiles in each row. How many tiles were used altogether?
143
Multiplying mixed numbers
Content
Materials
Squares of coloured paper; large sheets of paper; glue
Ask students to investigate what happens when they make arrays to work out the
answers to 1½ x 1½, 2½ x 2½, 3½ x 3½, and so on.
Likely student responses are as shown in the photograph below.
144
Adding and subtracting fractions with unrelated
denominators
Content
Materials
Paper and pencils
This is not a topic listed in the syllabus for Stage 3, but is an interesting topic to
extend students, and to observe how students apply their understanding of
fraction concepts.
Pose problems such as the following.
If I ate half a pie and my brother ate one-third of it, how much would be left?
145
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