Transcript Fraction-workshop

Something to Think About

What are good activities for helping
students to learn their basic facts?
Secondary Numeracy Project
Fraction,
Decimal,
Percents
Fraction Constructs
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Fractions are needed when whole units
are inadequate to get a job done.
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Part-Whole: continuous & discrete
Continuous fraction model
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Fractions are met as cutting up an
entire object
– e.g. one quarter of an apple
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This single object soon extends to
multiple objects to be shared
– e.g. sharing two apples between 5 people
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Leads to the concept that fractions are
divisions
2
– e.g.
25
5
Discrete fraction model
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Fractions are met through sharing a
collection of objects
– e.g. 4 kids share 12 cookies
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Problems initially lead to discrete, whole
number answers
Later problems merge with the
continuous as “remainder” objects are cut
– e.g. 4 kids share 10 cookies
Handout: A Teaching
Progression for Fractions
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Stages – Continous & Discrete
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Any surprises?
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Look at Knowledge & Strategy Tests
Knowledge Test: Fractions
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Stage 4 (3 sec)
9: Write one half as a fraction.
10: Write one sixth as a fraction
11: Write one third as a fraction
12: Write one quarter as a fraction
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Stage 5 (5 sec)
13: Which of these fractions is the smallest?
14: Which of these fractions is the largest?
1
3
1
5
1
4
1
2
1
6
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Stage 6 (5 sec)
15: Which of these numbers is the same as
eight-sixths?
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Stage 7 (5 sec)
6
2
2
1
1
8
6
14
16: Which of these fractions is the same as
two-thirds?
17: Which of these fractions is the same as
three-quarters?
2
6
6 8
8 12
9
12
3
9
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Stage 8
18: Which of these fractions is the smallest?
(10 sec)
2
5
3 5
10 16
19: Which of these fractions is the largest?
(10 sec)
2
3
5 7
9 10
Strategy Test: Fractions
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Stage 2-4 (materials)/Stage 5 (facts):
This cake has been cut into thirds. Here are
twelve jellybeans to spread out evenly on
top of the cake. You eat one third of the
cake. How many jellybeans do you eat?
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Stage 6:
– What is
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3
4
of 28?
Stage 7:
– 12 is 2 of a number. What is the number?
3
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Stage 8:
– There are 21 boys and 14 girls in Ana’s
class. What percentage of Ana’s class are
boys?
– It takes 10 balls of wool to make 15
beanies. How many balls of wool does it
take to make 6 beanies?
SNP Teaching Model
Use a concrete representation (Materials/Diagrams)
Encourage imaging and visualisation (Imaging)
Push to the inherent property and generalisation (Abstraction)
How can we use this with fractions?
Fraction Strips
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Work in groups of 8
To start each person has a different colour
paper
Fold paper in eighths – make sure everyone
folds the same direction so that all eighths are
the same size
Cut 1/8 strips (along fold lines)
Share out your strips so that everyone ends
up with 8 strips of eight different colours
Cut strips as directed to match magnets
Uses of fraction strips
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Order unit fractions and fractions with
the same denominator and explain why
they are larger or smaller
– Which is bigger? Why?
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Order fractions visually using materials,
including improper fractions and explain
what the numerator and denominator
mean.
– Make pairs of fractions. Which is bigger?
Which is the bigger fraction?
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Many students miss the comparative
nature of fractions – the relationship
between the numerator and
denominator. (Some research suggests
that the failure to understand this is a
reason why students have difficulty with
fractions)
Complexifying the constructs
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What fraction is shaded pink?
What fraction is shaded pink?
3
4
48
64
48 = 3 x 16
64 = 4 x 16
Operational Construct
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¼ of 36
– halve and halve again
– relationship between times/division and
fractional strategies
– ¾ of 36
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0.25 of 36
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25% of 36
Book 7
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Changing up the way the question is
asked.
A Teaching Progression for
Percentages
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Requires high levels of thinking
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Decimals (decimal fractions)
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Other handouts
Maths is......explaining...
Remember the long term goal –
improving their algebraic thinking:
One thing you can do to help this every
day is to ask questions in class:
What’s the answer?
How did you get the answer?
Why does it work
?
always
Fraction Teaching Resources
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Numeracy Folder – Number – Frac-Ratio
– Key Progressions for Fractions, Decimals,
Percentages, Rates and Proportions
– A teaching progression for fractions
– Intro to Fractions Worksheets
– HIBS material
– New Numeracy Books
– No Nonsense Number Fractions, stages 5-6
– NZMaths (DecimalFractionsOfASet,
AddingFractions)
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Rime
10 Ticks (Pack 4 Level 4 and Pack 5 Level 2)
Activity: In groups of 2 or 3
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Create a brief outline for a lesson that
gets students to learn without being
taught an algorithm
How to convert 1 1 to 4 (and vice
3
3
versa)
2
How to find 4 fractions equivalent to
7
How to simplify (a) 36 (b) 28
72
35
Activity 2
1)Pick one stage of the progression
2)Think of as many relevant fraction based
activities as possible for that stage
 Consult the Getting Started book
 Consult the activity books
 Consult Figure It Out
 Think of good activities that were used in the
past – and still fit
 Think them up