Transcript 3 over 7 - aucksecmaths

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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Effective Pedagogy
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Provide sufficientTeacher actions promoting
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E-Learning and
Pedagogy
5 views of fractions
3÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
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?
3
7
3 out of 7
of 42
3 out of 7 of 42 - ok if factor of 7
3÷7
0.42857143 of 42
3 over 7
3 over 7 of 42?
3:7
3:7 split 12.6 : 29.4

3 sevenths
1 seventh is 6 so
3 sevenths are 18
How would each of the views solve
this problem?
3
7
+
3
14
3 out of 7
6 out of 21
3÷7
0.42857143 + 0.21428571
 7
3 over
3:7
3 sevenths
3 over 7 + 3 over 14
3:7 + 3: 14 = 6:21
How would each of the views solve
this problem?
3
1
7
3 out of 7
3÷7
3 over
7
3:7
3 sevenths
1-3 out of 7 = -2/7 (or 2/7)
1 - 0.42857143
1 - 3 over 7 = -2/7 (or 2/7)
1 - 3:7?
1 - 3 sevenths = 4 sevenths
How would each of the views solve
this problem?
3
7
3 out of 7
3÷7
3 out of 7 of 1 out of 3
0.42857143 of 0.333333333
3 over 7

of
1
3

3:7
3 sevenths
3 over 7 of 1 over 3
3:7 of 1:3
3 sevenths of 1 third
1 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
QuickTime™ and ato learn
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me™ and a
Enhance the
decompressor
are needed to see t his pict ure.
of newQuickTime™
learning
and a
decompressor
are needed to see this picture.
decompressor
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QuickTime™ and a
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are needed to see this picture.
QuickTime™ and a
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Effective Pedagogy
QuickTime™ and a
decompressor
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decompressor
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Provide sufficientTeacher actions promoting
student learning
Create a supportive
learning environment
Facilitate shared
learning
Inquire into the teachinglearning relationship
E-Learning and
Pedagogy
• NA5-5 Know commonly used fraction, decimal, and
percentage conversions.
• 34
• 38
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reflective
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thought and action
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Make connections to
prior
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learning and experience
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decompressor
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decompressor
are needed to see this picture.
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QuickTi me™ and a
decompressor
are needed to see t his pict ure.
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decompressor
are needed to see t his pict ure.
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTi me™ and a
decompressor
are needed to see t his pict ure.
opportunities
QuickTime™ and ato learn
QuickTi relevance
me™ and a
Enhance the
decompressor
are needed to see t his pict ure.
of newQuickTime™
learning
and a
decompressor
are needed to see this picture.
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
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are nee ded to s ee this picture.
Effective Pedagogy
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
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Provide sufficientTeacher actions promoting
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Create a supportive
learning environment
Facilitate shared
learning
Inquire into the teachinglearning relationship
E-Learning and
Pedagogy
Use fraction strips to work out:
1 3
2 
2 4
WHOLE
WHOLE
1
1
QUARTER
1
4

QUARTER
1
4
QUARTER
1
4
QUARTER
1
4
QUARTER
1
2
HALF
1
4
QUARTER
1
4
QUARTER
1
4
QUARTER
1
4
QUARTER
1
4
QUARTER
1
4
Two more questions…
1
4
3
1
4
3
WHOLE
1
THIRD
1
3
THIRD
1
3

THIRD
THIRD
11
33
THIRD
1
3
THIRD
THIRD
11
33
THIRD
1
3
Also…
1
4 
3
1
4
3
5 1

6 3
SIXTH
1 5

3 6
1
6
SIXTH
THIRD
1
3
THIRD
1
3
SIXTH
1
6
1
6

SIXTH
1
6
SIXTH
1
6
SIXTH
THIRD
THIRD
11
33
THIRD
1
3
SIXTH
1
6
1
6
SIXTH
1
6
11
33
THIRD
THIRD
SIXTH
1
6
SIXTH
1
6
THIRD
1
3
HUND REDTHS
HUND REDTHS
HUNDREDTHS
HUND REDTHS
HUND REDTHS
HUND REDTHS
HUND REDTHS
HUND REDTHS
TENTH
1
10
1
6
SIXTH
1
5
FIFTH
QUARTER

1
3
THIRD
HALF
WHOLE
1
1
2
1
4
FIFTH
1
5

THIRD
1
3
HUND REDTHS
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:
6+6+6+6=4 6
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
=122 + 12
13  12 = 13  13 – 13  1
=132 – 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
95
=9999999
99999
= 92
(94)3 = 94  94  94
= 912
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