```Fractions, Decimal Fractions
and Percentages a closer look.
Jill Smythe
Numeracy Facilitator
(Based on work by Peter Hughes)
Fractions:
Pirate Problem
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•
•
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(While you are waiting - there is a prize for whoever
gets the answer first!)
Three pirates have some treasure to share. They
decide to sleep and share it equally in the morning.
One pirate got up at at 1.00am and took 1/3 of the
treasure.
The second pirate woke at 3.00am and took 1/3 of
the treasure.
The last pirate got up at 7.00am and took the rest of
the treasure.
Do they each get an equal share of the treasure?
If not, how much do they each get?
The Rope Activity:
Objectives:
• Identify the progressive strategy stages of
fractions, decimal fractions, percentages and
ratios.
• Further develop teacher’s confidence and content
knowledge of fractions, decimal fractions,
percentages and ratios.
• Explore key ideas, equipment and activities used
to teach knowledge and strategy of the above.
Developing Proportional
Thinking:
• Order the scenarios.
• How well did you do? Check by using The
Number Framework: Book 1 (Pg 15).
• Highlight all the fractional knowledge
across the stages (Pg18-22).
Fraction Knowledge Test:
•
•
Draw 2 pictures: (a) one half (b) one eighth
Mark 5 halves on a number line
•
12 is three fifths of what number?
•
What is 3 ÷ 5?
•
Draw a picture of 7 thirds
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Write one half as a ratio.
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The ratio of kidney beans to green is 3:4. What fraction of the beans are green?
Order these fractions:
2/4, 3/4, 2/5, 7/16, 2/3, 6/49
Now include these % and decimals into your order
30%, 75%, 0.38, 0.5
Group discussion
Fraction questions
Why do we have fractions?
Shape on board - divided in 3. Are they thirds?
Views of Fractions:
What does this fraction mean?
3
7
Views of Fractions:
Does this fraction mean?
3÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
The Problem with Language:
Use words first before using the symbols:
e.g. one half not 1 out or 2
How do you explain the top and bottom numbers?
1
2
The number of parts chosen
The number of parts the whole has been
divided into
Models of Fractions:
• Show 3 quarters using any of the materials
Continuous Model:
• Models where the object can be divided in any way
that is chosen.
e.g. ¾ of this line and this square are blue.
0
1
Discrete Model:
• Discrete: Made up of individual objects.
e.g. ¾ of this set is blue
Whole to Part:
• Most fraction problems are about giving students
the whole and asking them to find parts.
• Show me one quarter of
this circle?
Part to Whole:
• We also need to give them part to whole
problems, like:
• 5 is a quarter of this
number.
What is the number?
Teaching Fractions:
• What do you see as some of the
confusions associated with the
teaching and understanding of
fractions?
Misconceptions with Fractions:
• Charlotte believes that one eighth is bigger than
one half.
1/2  1/3  1/4  1/8
• Why do you think Charlotte has this
misunderstanding?
• How would you address this misconception?
• What equipment would you use?
Misconceptions with Fractions:
• Jenna says the following:
¼ + ¼ + ¼ = 3/12
• Why do you think Jenna has this
misunderstanding?
• How would you address this misconception?
• What equipment would you use?
Misconceptions with Fractions:
• A group of students are investigating the books they
have in their homes.
1
• Steve notices that 2 of the books in his1 house are
fiction books, while Andrew finds that 5 of the books
his family owns are fiction.
• Steve states that his family has more fiction books than

Andrew’s.
Consider….
Is Steve necessarily correct?
Why/Why not?
What action, if any, do you take?
Key Idea:
The size of the fraction depends on the size of
the whole.
• Steve is not necessarily correct because the
amount of books that each fraction represents is
dependent on the number of books each family
owns.
1
1
• For example: 2 of 30 is less than 5 of 100.

Misconceptions with Fractions:
• Anna says
7
is not possible as a fraction.
3
Consider…..
7
• Is 3 possible as a fraction?
• Why does Anna say this?
• What action, if any, do you take?
Key Idea:
A fraction can represent more than one whole.
Can be illustrated through the use of materials and
diagrams.
Question students to develop understanding:
• Show me 2 thirds, 3, thirds, 4 thirds…
• How many thirds in one whole? two wholes?
• How many wholes can we make with 7 thirds?
Misconceptions with Fractions:
• You observe the following equation in Bill’s work:
Consider…..
• Is Bill correct?
• What is the possible reasoning behind his answer?
• What, if any, is the key understanding he needs to
develop in order to solve this problem?
Key Idea:
To divide the number A by the number B is to find out
how many lots of B are in A. When dividing by some
unit fractions the answer gets bigger!
• No he is not correct. The correct equation is
•
Possible reasoning behind his answer:
1/2 of 2 1/2 is 1 1/4.
– He is dividing by 2.
– He is multiplying by 1/2.
– He reasons that “division makes smaller”
therefore the answer must be smaller
than 2 1/2.
Fractional Knowledge:
Having secure fractional knowledge is the key to
operating successfully with fractions…
Stage 4: Identifies and writes common fractions.
Stage 5: Orders unit fractions.
Stage 6: Identifies any fraction including improper
fractions.
Stage 7: Knows equivalent fractions for halves, thirds,
quarters, fifths and tenths.
Stage 8: Orders fractions, decimals and percentages.
(Taken from IKAN assessment)
Exploring Book 7 & FIO:
Folding Fractions Figure it Out Number Lev.2 Pg.18
Stage 5-6: Birthday Cakes Page 26
Focus on the following :
• What key knowledge is required before beginning this stage.
• Highlight the important key ideas at this stage.
• The learning intention of the activity.
• Work through the teaching model (materials, imaging,
number properties).
• Possible follow up practice activities.
• Link to the planning units and Figure It Out.
Fractions as Operators
Sione has 35 marbles.
At the end of play he only has 3 sevenths of his
marbles left.
At the end of lunch he has only two fifths of the
marbles he started lunchtime with.
How many marbles does he have left?
Can you draw a grid to show your thinking?
Fractions as Operators
So we are finding out a fraction of a fraction.
2/5 of 3/7
Fractions as Operators
So we are finding out a fraction of a fraction.
2/5 of 3/7
We start with the 3/7 of 35
Which is 15.
Fractions as Operators
So we are finding out a fraction of a fraction.
2/5 of 3/7
We start with the 3/7 of 35
Which is 15.
Now we need to work out
2/5 of 15
Which is 6.
Multiplying with Fractions
• When you multiply by some fractions
the answer gets smaller
• This is ⅓ of one whole strip.
• If it is cut into quarters, four
equivalent pieces, what will each new
piece be called?
1/12
1/3
Summary of key ideas:
• Fractional language - emphasise the “ths” code.
• Fraction symbols - use symbols with caution. Combine
language, symbols and a visual representation to
consolidate understanding.
• Continuous and discrete models - use both.
• Go from Part-to-Whole as well as Whole-to-Part.
• Fractions are always relative to the whole.
• Begin to teach fraction when children are at stage 2!!
• Use the teaching model to develop conceptual
understanding.
Introducing Decimal Fractions:
Present the class with unifix cubes that have been
wrapped in tens. Each represents one bar of
chocolate with 10 pieces. How many pieces/tenths
would each person get?
2÷5=?
Discuss:
• Why should we introduce decimal fractions
by division of whole numbers rather than
telling students that that the places after
the decimal point represent tenths,
hundredths, thousandths and so on?
Reason:
• The whole must be broken into ten parts –
canon of place value (ten for one)
• Teaching division of whole numbers leads
E.g 3 ÷ 7 = 3 sevenths of one whole
• This is an essential precursor to decimal
fractions
Solve these using Materials:
•
•
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•
7
8
6
5
2
÷
÷
÷
÷
÷
2
5
4
2
4
=
=
=
=
=
?
?
?
?
?
Wholes
Wholes
Wholes
Wholes
Wholes
and
and
and
and
and
?
?
?
?
?
Tenths
Tenths
Tenths
Tenths
Tenths
Why have we used words?
• Words emphasise what fractions really are.
• Stops students thinking of fractions as division,
out of, over or ratios.
• Develops a deeper understanding of what fractions
represent.
• The number 3.5 should be read as 3 and 5 tenths
rather than 3 point 5.
Solve this:
• 4.5 ÷ 3 =
• 4 wholes ÷ 3 = 1 whole for each share
with 1 whole left over. This 1 whole is split
into ten tenths – so there are now fifteen
tenths altogether. 15 tenths ÷ 3 = 5
tenths for each share. Each share is 1
whole and 5 tenths (1.5)
I Whole take one large piece of paper
• Make 1 tenth
• Make 1 hundredth
Becoming familiar with the
equipment:
• Decimal Fraction Mats (book 4, page 8 and
book 7 pg 45).
Prove that 1.7 is bigger than 1.68 using one
of the above materials.
Misconceptions with Decimal
Fractions:
• Jill believes that 0.45 is less than 0
• Why do you think Jill has this misunderstanding?
• How would you address this misconception?
• What equipment would you use?
Misconceptions with Decimal
Fractions:
• John says the following:
“one half is the same as 0.2”
• Why do you think John has this misunderstanding?
• How would you address this misconception?
• What equipment would you use?
Key Idea:
Decimals are special cases of equivalent
fractions in that they always involve tenths,
hundredths, or thousandths, etc.
• John is not correct. ½ is 0.5 ( 5 tenths) and 0.2 is
two tenths
1
1
2
5
• Practise in needed
in order to convert
fractions to
equivalent decimals

Misconceptions with Decimal
Fractions:
• Mary has to order the following fractions from smallest
to largest:0.67
0.8
0.532
• She orders them as follows:0.8
0.67 0.532
Consider….
Is Mary correct?
Why/Why not?
What are the misconceptions?
What action, if any, do you take?
Misconceptions with Decimal
Fractions:
• Jacob added the following decimals together:3.4 + 1.8 = 4.12
• Why do you think Jacob has this
misunderstanding?
• How would you address this misconception?
• What equipment would you use?
The Connection between Fractions and
Percentages:
What does % mean?
• In mathematics, a percentage is a way of expressing a number as
a fraction of 100 (per cent meaning "per hundred"). It is often
denoted using the percent sign, % For example, 45% (read as
"forty-five percent") is equal to 45 hundredths or 0.45.
• What do we need to do to fractions so that it can be read as a
percentage?
• What key mathematical knowledge do children need
to be able to do this?
Hot Shots
Book 7 P 47 - 49
Extending Hot Shots P 56 - 60
% Problems
• 20% of 150 is
• 20% of
•
is 30
% of 150 is 30
Question (in context)
The local dairy farmer is selling
20% of his herd of 150 cows. How
many is he selling?
20% of 150 is 

0%
20%
Rewrite in maths
language
150
100%
How do we use the lines to get the answer?

0%
20%
150 divided by 5 = 30
20 x 5 = 100
Find 10% :
0%
150
100%
150 divided by 10

150
20%
100%
So 10% = 15
So 20% =30
15 x 2

0%
20%
10 x 2
15 x 10
150
100%
10 x 10
There are 30 students in Room 16.
40% are girls.
How many girls are there in the class?
What is the maths?
(Mathematize it)
40% of 30 is 
30

__________________________________________
0%
40%
100%
How do we use the lines to get the answer?
30 divided by 5 = 6

0%
20%
30
40%
100%
20 x 5 = 100
20% = 6
So 40% = 12
Find 20% :
0%
20%
30 divided by 5

30
40%
100%
20% = 6
so 40% =12
3x4
0

0
40%
10 x 4
3 x 10
30
100%
10 x 10
Sarah went shopping for a new bike which cost \$350.
When she got to town there was a sale and she got
20% off the price. What did she pay?
Did she pay more or less?
How much less?
So instead of paying 100% she only paid?
Show all this on the number lines
\$350
0%
80%
100%
Additional problems using the
Double Number line!!!
• Emily’s team won the basket ball game 120-117. Emily
shot 60% of the goals. How many goals did Emily get?
• John scored 104 runs in a one day cricket match, that
was 40% of the teams total. How many runs did his
team score altogether?
• In a bike race, 30% of cyclists drop out. 42 riders
finish the race. How many cyclists started the race?
Ratios:
• In the rectangle below, what is the ratio of
green to blue cubes?
• What is the fraction of blue and green cubes?
• Can you make another structure with the same
ratio? What would it look like?
• What confusions may children have?
More on Ratios….
• Divide a rectangle up so that the ratio of its blue
to green parts is 7:3.
• Think of other ways that you can do it.
• What is the fraction of each colour?
• If I had 60 cubes how many of them will be of
each colour?
A Ratio Problem to Solve:
• There are 27 pieces of fruit. The ratio of fruit that I get
to the fruit that you get is 2:7. How many pieces do I
get?
• How many pieces would there have to be for me to get
8 pieces of fruit?
• What key mathematical knowledge is required?
Which recipe will make the
darkest green?
• A: One part of blue with three parts yellow
(1:3)
• B: Four parts of blue with eight parts
yellow (4:8)
• C: Three parts of blue with five parts of
yellow (3:5)
A diagnostic question
• A recipe needs 6 eggs, 4 kg of flour and 8
teaspoons of flavouring.
• What would the recipe be if 6 kg of flour is
used?
• What would a successful answer look like?
Thought for the day:
• Smart people believe only half of what they
hear.
• Smarter people know which half to believe.
Teaching progression
Start by:
Using materials, diagrams to
illustrate and solve the problem
Progress to:
Developing mental images to help
solve the problem
Extend to:
Working abstractly with the
number property
Materials
Images
Knowledge
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