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Welcome to our
Fractions Session
Aims
• To develop our subject knowledge around the
teaching and learning of fractions
– What are fractions?
– Common difficulties and misconceptions with
fractions and where they might arise from
– Ideas for teaching and learning to address these
Key Difficulty Points
How many of your Year 6s do you reckon could answer this
correctly? What would common errors be? Reflect carefully
on how YOU know what the right answers are…
Here are four fractions.
Look at the number line below.
Write each fraction in the correct box.
Key Difficulty Points
Key Difficulty Points
• Find questions like this extremely hard:
1
3
• Which fraction is larger? or
1
?
4
• Can you label the missing fraction?
0
1
4
1
2
1
If the world is the whole, then _____________ is part of the whole.
Equally
Divided
Not
Equally
Divided
The concept and the non-concept
Triangle or not triangle?
Equally divided or not equally divided?
Part whole relationship
“Where there is an explicit division of a whole
into equal parts, children are able to determine
the fraction of the part/parts indicated by
counting the number of parts in the whole and
the number of parts indicated. In figures where
it is more difficult for the children to adopt this
'partitioning' approach, children are required to
analyse the relationship of the particular
part/parts indicated in relation to the entire
whole.
From “Teaching Fractions with Understanding: Part Whole Concept”, nrich
Part whole relationship
A fraction is a multiplicative relationship
between a part and a whole [a ratio concept].
2
“
3
4
”
6
When we say =
we don’t mean that “2 out
of 3 things is the same as 4 out of 6 things” – it
isn’t.
We mean “2 is to 3 as 4 is to 6” [in multiplicative
terms]
Part whole relationship
Approximately what
fraction of the oval is
shaded?
Part whole relationship
How about this one?
Part whole relationship
“Learners begin to see the fraction as a comparison
between the numerator and the denominator (what
Grace calls one-to-many or many-to-many
comparison). This is based on a ratio concept and
Grace argues it also indicates a deeper understanding
of fractions”
Teaching fractions with understanding: part whole concept
(nrich)
Pouring contexts
“See it, feel it”
Ordering Unit Fractions
• When comparing unit fractions, the bigger the
denominator, the smaller the fraction
Ordering Unit Fractions
Showing unit fractions on a number line:
Equivalent fractions through
continuous amounts
• Draw a 0 – 1 number line
• Give your partner a fraction (<1)
• Estimate where the number lies on the
number line
• Can you reflect on how you are able to do
this? How do we develop this skill in children?
Fractions of continuous amounts
Fill this beaker
about 15
18 full
Fill this beaker
about 56 full
Ordering numbers: variation theory
Deepening learning
Prove that 3 < 4 in 3 different ways
4 5
Explain why Sal is wrong…
Teach, Learn, Confuse
Think: Which line is longer?
First:
1
2
Second:
1
3
So far…
Fractions concepts
• Part whole
relationship
• Equally divided
• Same whole
Mastery concepts
• Concept and non
concept
• Many different ways
(the answer is only the
beginning)
• Teach, learn, confuse
3 interpretations of fractions
• Fractions as an operator on a set
1
of the children
4
1
x
3
6
• Fractions as a number in themselves
1
= 0.25
4
• Fractions as division
5
3
(5cm high cake cut into 3 slices = 5 ÷ 3 = =
2
1
3
cm each)
Key Difficulty Point
0
1
2
3
4
Can you mark the number ½
on this number line?
What misconception might arise here?
Why? (Can you relate it to what we have just
discussed?)
5
Embedding fractions as numbers in
themselves
• LOTS of counting!
Year 2: Pupils should count in fractions up to 10, starting from any
1
1
number and using the 2 and 4 equivalence on the number line
(Non Statutory Guidance)
Embedding fractions as numbers in
themselves
Critical phrase in the new curriculum – crops up in non statutory guidance in
Years 2, 3 and 5, but important for ALL!
Numbers in themselves vs operators
on a set
• Work with ‘stuff’ (continuous quantities) moving
between a unit of ‘1’ and a unit which isn’t 1
• “Measuring is the first context in which children
meet rational number” (Anne Watson)
Numbers in themselves vs operators
on a set
• “Here I have two 2 litre bottles. One
3
of them has litre in it [fraction as
4
3
number in itself], and one of them is
4
full [fraction as operator on a set].
Which is which?
• How much squash is in the bottle
3
which is full?
4
• What fraction of the bottle which has
3
litre in it is full?
4
Fractions as a quotient in division
1 share 4 pizzas between 5 people.
How much do they each get?
Deepening learning
Write
17
4
as a mixed number
Calculate 17 ÷ 4
What is the same, what is different?
Time for a pause…
Look back at your original definition of
a fraction. What do you understand it
to be now?
Cuisenaire Rods
• Amazing for learning about fractions!
• Videos of the activity we will be doing are on
NCETM website
• Google ‘Caroline Ainsworth NCETM’
Concepts which that may have given
rise to…
•
•
•
•
•
Converting mixed numbers & improper fractions
Comparing fractions
Equivalences
Addition of fractions
Multiplication of fractions
Have a go
• Role of the teacher is to:
– Decide what is ‘one’
– Plan the sequence of rods
• Work through the ‘quarters example’
• Have a go at planning an example which would lead
to reasoning with sixths
Adding and subtracting fractions
Adding and subtracting fractions
8
10
+
8
10
8 ‘of those things called tenths’
2
+ 10
0
8
10
6
+ 10
1
6
10
1
2
Starting early!
• Link to addition and subtraction units lower down
the school
Adding and subtracting fractions
1
3
+
1
2
Need to understand common multiple. Actually pretty
hard to find a really good model for this.
+
=
Adding and subtracting fractions
a
b
L4
Tier 4-6
L5
L6
0.05
0.04
0.15
0.03
0.45
0.17
L5
Tier 5-7
L6
L7
0.23
0.06
0.49
0.17
0.80
0.50
L6
Tier 6-8
L7
L8
0.58
0.24
0.91
0.59
0.99
0.86
Multiplying fractions by whole
numbers
Fractions of quantities
(Fraction x whole number)
3
4
x 12 OR
12 x
3
4
12
Fraction ÷ Whole Number
1
3
÷2
Fraction x Fraction
2
3
x
2
5
• Children need an understanding of multiplication
that is about more than repeated addition.
• The area model is a PERFECT tool for this.