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Fractions, decimals, and percentages.

Whats 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 - ?
- 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.

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?

Continuous becomes Discrete

- 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 ? 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?

Kierans Constructs for Rational Number

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 ? 1/6

More Part-Whole Constructs

- This is ? of one whole strip.
- If it is cut into quarters, four equivalent
- pieces, what will each new piece be called?

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).

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

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.
- How many lots of ¾ fit into 2½?
- How could you write this in equation
- form?

Harder Divisions (AP)

- Make ? with your fraction strips.
- Compare it to ¾.
- How many ¾s fit into ??
- How could you record this?

Rethinking Multiplication and Division

What is ? of ¾? Why might a student think that

this should be written as ? ¾ instead of ? ¾ ?

Fractions as Operators

- Solve this problem using your fraction strips
- ? ¾ ?

Harder Additions and Subtractions

- Use the fraction strips to solve
- these problems
- ¾ ? ?
- 5/3 4/5 ?
- 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
- ? of 1/5 ?
- So what is ? of 1/5 ?

Generalisation

- ? of 1/5 2/15
- So what is
- ? of 4/5 ?

1/5

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. - Butwhat 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 (2418).
- 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

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

1824 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?

Leonnes Percentage

- So Leonnes 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?

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?

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?

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