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Fractions
Louise Addison
Fraction starter
• One player is dots the other is crosses• Number line from 0 to 6• Roll 2 dice and form a fraction, place this on
number line (use materials if necessary)• Aim is to get 3 marks uninterrupted by your
opponent’s marks on the number line.• If a player chooses a fraction that is equivalent
to a mark that is already there they lose a turn.
Fractions in the new curriculum Level 4
• Number strategies and knowledge• NA4-2 Understand addition and subtraction of fractions, decimals, and
integers.• NA4-3 Find fractions, decimals, and percentages of amounts expressed
as whole numbers, simple fractions, and decimals.• NA4-4 Apply simple linear proportions, including ordering fractions.• NA4-5 Know the equivalent decimal and percentage forms for everyday
fractions.• NA4-5 Know the relative size and place value structure of positive and
negative integers and decimals to three places.
• Probability• S4-4 Use simple fractions and percentages to describe probabilities.
Level 5
• Number strategies and knowledge• NA5-3 Understand operations on fractions, decimals, percentages, and
integers.• NA5-4 Use rates and ratios.• NA5-5 Know commonly used fraction, decimal, and percentage
conversions.
• Patterns and relationships• NA5-8 Generalise the properties of operations with fractional numbers
and integers.
• Probability• S5-4 Calculate probabilities, using fractions, percentages, and ratios.
Level 6
• Number strategies and knowledge• NA6-2 Extend powers to include integers and
fractions.
• Patterns and relationships• NA6-6 Generalise the properties of operations with
rational numbers, including the properties of exponents.
A closer look…
• NA4-2 Understand addition and subtraction of fractions, decimals, and integers.
• NA4-3 Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
• NA5-3 Understand operations on fractions, decimals, percentages, and integers.
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Facilitate shared
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Inquire into the teaching-
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Create a supportive
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E-Learning and
Pedagogy
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QuickTime™ and a decompressor
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QuickTime™ and a decompressor
are needed to see this picture. Encourage reflective
thought and action
QuickTime™ and a decompressor
are needed to see this picture. Make connections to prior
learning and experience
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
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QuickTime™ and a decompressor
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Effective PedagogyTeacher actions promoting
student learning
QuickTime™ and a decompressor
are needed to see this picture.
Enhance the relevance
of new learning
5 views of fractions
€
3
73 over 73 : 7
3 out of 7 3 ÷ 7
3 sevenths
Implications for each…
• 3 pizzas are divided amongst 4 people. How much of a pizza does each person get?
3 out of 7
How would each of the views solve this problem?
of 42
€
3
7
3 over 7
3 : 7
3 out of 7
3 ÷ 7
3 sevenths
3 out of 7 of 42 - ok if factor of 7
0.42857143 of 42
3 over 7 of 42?
1 seventh is 6 so 3 sevenths are 18
3:7 split 12.6 : 29.4
How would each of the views solve this problem?
+
€
3
7
€
3
14
3 over 7
3 : 7
3 out of 7
3 ÷ 7
3 sevenths
0.42857143 + 0.21428571
6 out of 21
3:7 + 3: 14 = 6:21
3 over 7 + 3 over 14
How would each of the views solve this problem?
€
1−3
7
3 over 7
3 : 7
3 out of 7
3 ÷ 7
3 sevenths
1 - 0.42857143
1-3 out of 7 = -2/7 (or 2/7)
1 - 3:7?
1 - 3 over 7 = -2/7 (or 2/7)
1 - 3 sevenths = 4 sevenths
How would each of the views solve this problem?
of
€
3
7
€
1
3
3 over 7
3 : 7
3 out of 7
3 ÷ 7
3 sevenths
0.42857143 of 0.333333333
3 out of 7 of 1 out of 3
3:7 of 1:3
3 over 7 of 1 over 3
3 sevenths of 1 third1 third, split into 7 pieces gives ‘21ths’
So is three 21ths (3/21)
Key ideas of fractions
Fractional vocabulary
One half
One third
One quarter
Don’t know
Implications for teaching
• Use words (pattern and meaning needs to be taught)
• Always refer to the ‘whole’
• Modelling with covered unifix cubes
• This needs to be understood before decimal fractions can be taught
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opportunities to learnQuickTime™ and a decompressor
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Facilitate shared
learning
QuickTime™ and a decompressor
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Inquire into the teaching-
learning relationship
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Create a supportive
learning environment
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E-Learning and
Pedagogy
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QuickTime™ and a decompressorare needed to see this picture.QuickTime™ and a decompressorare needed to see this picture.
QuickTime™ and a decompressor
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QuickTime™ and a decompressor
are needed to see this picture. Encourage reflective
thought and action
QuickTime™ and a decompressor
are needed to see this picture. Make connections to prior
learning and experience
QuickTime™ and a decompressor
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QuickTime™ and a decompressor
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QuickTime™ and a decompressor
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Effective PedagogyTeacher actions promoting
student learning
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Enhance the relevance
of new learning
• NA5-5 Know commonly used fraction, decimal, and percentage conversions.
• 3 4
• 3 8
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QuickTime™ and a decompressor
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Provide sufficient
opportunities to learnQuickTime™ and a decompressor
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Facilitate shared
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QuickTime™ and a decompressor
are needed to see this picture.
Inquire into the teaching-
learning relationship
QuickTime™ and a decompressor
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Create a supportive
learning environment
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are needed to see this picture.
E-Learning and
Pedagogy
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QuickTime™ and a decompressor
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QuickTime™ and a decompressor
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thought and action
QuickTime™ and a decompressor
are needed to see this picture. Make connections to prior
learning and experience
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
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QuickTime™ and a decompressor
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Effective PedagogyTeacher actions promoting
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Enhance the relevance
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Use fraction strips to work out:
€
21
2÷
3
4
WHOLE
1
WHOLE
1
HALF
21
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
QUARTER
41
Two more questions…
€
4 ÷1
3
WHOLE
1 THIRD
31
THIRD
31
THIRD
31€
1
3÷ 4
THIRD
31
Also…
€
4 ÷1
3=
€
1
3÷ 4 =
€
5
6÷
1
3
SIXTH
61
SIXTH
61
SIXTH
61
SIXTH
61
SIXTH
61
THIRD
31
THIRD
31
THIRD
31€
1
3÷
5
6
THIRD
31
SIXTH
61
SIXTH
61
SIXTH
61
SIXTH
61
SIXTH
61
WHOLE
1
HALF
21
QUARTER
41
THIRD
31
HUNDREDTHS
FIFTH
€
15
SIXTH
61
TENTH
101
Algebraic generalisation
• NA4-8 Generalise properties of multiplication and division with whole numbers.
• NA5-8 Generalise the properties of operations with fractional numbers and integers.
• NA6-6 Generalise the properties of operations with rational numbers, including the properties of exponents.
The EGG technique
Explain the strategy
Give other examples
Generalise using Algebra
Example:
Task 1
9 + 9 + 9 + 5 + 5 + 5 = 3 × 9 + 3 × 5
6 + 4 + 6 + 4 + 6 + 4 = 3 × (6 + 4) 9 + 9 + 9 – 5 – 5 – 5 = 3 × 9 – 3 × 5 6 – 4 + 6 – 4 + 6 – 4 = 3 × (6 – 4)
Lightbulb Moments
• How useful is the algebra this generates?• How could you use this in your classroom?• Who could you use this with?• What connections between ideas can you
make?• What thinking is involved?• What issues could arise...
Task 2
15 + 16 = 15 + 15 +1 = 2 15 + 1
19 + 20 = 20 + 20 – 1 = 2 20 – 1
9 + 10 + 11 = 9 + (9+1) + (9+2) = 3 9 + 3
9 + 10 + 11 = (10–1) + 10 + (10+1) = 3 10
9 + 10 + 11 = (11–2) + (11–1) + 11 = 3 11 – 3
Task 3
12 × 13 = 12 × 12 + 12 × 1 =12 2 + 12
13 × 12 = 13 × 13 – 13 × 1 =13 2 – 13
Task 4
7 × 32 = 7 × 30 + 7 × 2 7 × 39 = 7 × 40 – 7 × 1
Task 5
32 42 = 30 40 + 2 40 + 30 2 + 2 2
32 48 = 30 50 + 2 50 + 30 -2 + 2 -2
39 42 = 40 40 + -1 40 + 40 2 + -1 2
39 49 = 40 50 + -1 50 + 40 -1 + -1 -1
Task 6
9 9 9 9 9 9 9 = 97
92 95 = (9 9) (9 9 9 9 9) = 97
97 = 9 9 9 9 9 9 995 9 9 9 9 9 = 92
(94)3 = 94 94 94
= 912
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