Transcript Slide 1

Fractions, decimals, and
percentages.
What’s the point?
Outcomes
By the end of this workshop you should be
able to:
• Explain how students’ strategies for
addition, subtraction, multiplication and
division affect their learning of fractions.
• Teach lessons that address the main
confusions students have about fractions,
decimals and percentages.
Models of fractions
Make up a chart showing how this fraction
can be represented:
⅔
Think about the ways in which your models
might be sorted into groups.
Discrete and Continuous
The research literature on fractions reports
two main fraction models:
Discrete: Made up of individual objects.
e.g. ¾ of this set is blue
Continuous Models
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
Part to Whole
Most fractions problem are about giving
students the whole and asking them to find
parts.
We also need to give them part to whole
problems, like:
¼ of a number is 5.
What is the number?
Why teach both models?
Students understanding of fractions,
decimals and percentages will depend on
understanding both models.
Consider this problem:
What decimal does the arrow on this
number line point to?
0
3
Continuous becomes Discrete
0
0
0
1
2
3
1
2
3
1
2
3
Note that the decimal system is a constrained
division as it always involves expressing
numbers by subdividing into ten parts.
Geometry or Number?
Some fraction tasks are about Geometry,
and not much about number.
For example, fold your circle into thirds.
Multiplication and Division are the
Key!
In NumPA students were
asked to find ⅓ of 12.
The context was sharing
out jellybeans onto a
birthday cake.
What strategies did they use?
How did this relate to their strategy stage for
multiplication and division?
Kieran’s Constructs for Rational
Number
Rational Numbers
PartWhole
Comparisons
Measures
Operators
Quotients
Ratios and
Rates
Part-Whole Constructs
Use the circle pieces.
Find as many ways that you can to make
one (whole) circle.
Record your ways using fraction symbols:
½+⅓+
1/6
1/
6
½
⅓
More Part-Whole Constructs
This is ⅓ of one whole strip.
⅓
If it is cut into quarters, four equivalent
pieces, what will each new piece be called?
1
12
Holding Onto One
To operate effectively with fractions
students need to be able to subdivide one.
This constitutes a substantial mind shift for
students whose number world has been
based on counting ones (discrete items).
1
12
Where do fractions live?
Fractions are numbers that are parts of
one, like ¼ and ½.
So fractions are less than one.
True of false?
Use fraction strips you have been given to
develop a lesson about the relationship
between fractions and whole numbers.
Fraction Scales
1
1
0
1
1
0
0
⅔
1
1
2
1
2
Students need to connect each “new” set of numbers they
encounter with the whole numbers that they know about.
This applies equally to fractions, decimals, and integers.
Fractions as Measures
Make 2½ or 5/2 with your fraction strips.
1
1
½
How many lots of ¾ fit into 2½?
How could you write this in equation
form?
1
1
½
Harder Divisions (AP+)
Make ⅔ with your fraction strips.
Compare it to ¾.
How many ¾’s fit into ⅔?
How could you record this?
⅓
¼
⅓
¼
¼
Rethinking Multiplication and
Division
What is ⅓ of ¾?
Why might a student think that this
should be written as ⅓ ÷ ¾ instead of
⅓ × ¾?
Fractions as Operators
Solve this problem using your fraction strips:
⅝+¾=?
⅛
⅛
¼
⅛
⅛
⅛
¼
¼
¼
1
¼
⅛
⅛
¼
⅛
⅛
⅛
Harder Additions and Subtractions
Use the fraction strips to solve
these problems:
¾+⅔=?
5/ – 4/ = ?
3
5
Why was multiplicative thinking critical
in solving these problems even
though they involved addition and
subtraction?
Fractions as Operators Continued
Scaffolding problems can help students
work through to difficult number properties:
⅓ of 1/5 = ?
So what is ⅔ of 1/5 = ?
1/15
1/5
1/15
1/5
1/15
Generalisation
⅔ of 1/5 = 2/15
1/15
1/151/5
So what is…
⅔ of 4/5 = ?
1/15
1/15
1/15
1/15
1/15
1/15
1/15
1/15
Fractions as Quotients
Quotients are the answers to division
problems.
For divisions with whole number answers,
like 12 ÷ 4, fractions are not needed.
But…what about 8 ÷ 3 = ?
Use paper circles to show how students at
different stages might solve this problem.
Decimats for Decimals
Decimals are special cases of equivalent
fractions. The decimal system says that
fractions must be expressed as tenths,
hundredths, etc.
Use your decimats to solve this problem:
7÷4=?
How does your answer compare with what
the calculator gives as the answer?
More Decimat Divisions
Solve these problems with decimats:
3÷5=?
7÷8=?
5÷3=?
Why do some decimals recur?
What other decimals do you know that recur?
Decimal Multiplication
0.4 x 0.5 = 0.2
Ratios and Rates
What is the difference between a ratio and a
rate?
Both are multiplicative relationships.
A ratio is a relationship between two things
that are measured by the same unit,
e.g. 4 shovels of sand to 1 shovel of cement.
A rate involves different measurement units,
e.g. 60 kilometres in 1 hour (60 km/hr)
Ratios
Make up this paint recipe using unifix cubes.
24 yellow cubes with 18 blue cubes (24:18).
If this recipe makes a big pot of paint, what
might go into smaller pots with the same
colour?
What is the key multiplication idea here?
Equivalent Ratios
Equal subdivisions
Double Number Lines
This could be shown on a double number
line:
0
12
24
0
0
4
9
12
18
24
0
3
9
18
Paint Mixtures
Make up these paint recipes.
For each recipe show how it might be shown
on the circular colour chart.
18 yellow
6 blue
14 yellow
21 blue
18:24 Again
What is this recipe if shown as percentages?
What makes this ratio difficult?
Calculating Percentages
Percentage strips help students to see that
calculating percentages is like mapping a
fraction onto a base of 100.
Leonne got 18 out of her 24 shots in.
What was her shooting percentage?
0
10
20
24
Leonne’s Percentage
0%
0
10%
20%
30%
40%
10
50%
60%
70%
80%
90%
20
So Leonne’s 18 out of 24 maps onto 75 out
of 100 (75%).
How does this relate to how you would
calculate 18/24 as a percentage?
100
%
24
Using Percentage Strips
Use your percentage strips to work out
these percentage problems:
Keith got 16 out of 25 for his spelling test.
What percentage was that?
Renee paid $360 000 for her house.
It has increased in value to $480 000.
What percentage of the original price is it
worth now?
Rates
These are examples of rate problems:
Ian can write 3 reports every hour.
How long will it take him to write 21 reports?
On the high country sheep station there are
20 sheep per square kilometre.
The station has an area of 400 square
kilometres.
How many sheep are there?
Review
Do you understand all of these constructs?
Rational Numbers
PartWhole
Comparisons
Measures
Operators
Quotients
Ratios and
Rates
Extra for Experts
At Kiwi School 40% of the children are girls.
At Tui School 70% of the children are girls.
If the schools meet for sports day, what
percentage of all the children are girls?
Kiwi school has twice as many children as Tui
School.
More Extra
On of a tank of petrol the car can make
one full trip.
How much of the trip can it make on of a
tank?
A fishy problem
• A fish tank holds 200
fish and 99% of them
are blue.
• How many blue fish
do you take out to
make this 98%?