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### OBJECTIVES

### Models for fractions

### FRACTION NUMBER SENSE

### FRACTION NUMBER SENSE

### FRACTION NUMBER SENSE

### FRACTION NUMBER SENSE

### FRACTION NUMBER SENSE

### FRACTION NUMBER SENSE

### Operations with fractions

### EXTENDING UNDERSTANDING

### Course NUM309

Developing understandings of fractions and decimals

Discuss the difficulties and misperceptions

Identify strategies at different stages

Discuss learning processes

Discuss teaching strategies

Explore activities to use in the classroom

Before, tree A was 8m tall and tree B was 10m tall. Now, tree A is 14m tall and tree B is 16m tall.

Which tree grew more?

A B A B

Before Now

A fishy problem

- Two-thirds of the goldfish are male
- There are 24 male goldfish
- How many goldfish are there altogether?

Share your strategy

- How did you do it?
- Discuss your method in groups
- Who taught you how to do it this way?

You have a fish tank containing 200 fish and 99% of them are guppies. You will remove guppies until 98% of the remaining fish are guppies. How many will you remove?

The Bill Gates question

Report on Numeracy Project 2003

- The performance of year 7 and 8 students on fractions and decimals is well below what would be wished.
- Integration of fractions with proportional reasoning would aid understanding of those topics.
- Decimals are of particular concern
- Decimals need to be taught using the Numeracy principles: using materials and imaging before number properties

Why do students have difficulty with fractions?

- Rational number ideas are sophisticated and different from natural number ideas
- Natural numbers can be represented individually, rational numbers cannot.
- Students’ whole number schemes can interfere with their efforts to learn fractions
- Students have to learn new ways to represent, describe and interpret rational numbers
- Rote procedures for manipulating fractions (eg making equivalent fractions) may not be enough

Initial Fraction Interview: Task 1

This is three-quarters of the lollies I started with. How many lollies did I start with?

Why did you choose that many lollies?

Initial Fraction Interview: Task 2

221221

5 3 4 8 3 3

- Which of these pairs of fractions are equivalent (have the same value)?
- How did you decide?

Initial Fraction Interview: Task 2

Typical responses:

- One-quarter is equivalent to two-eighths ‘cos ‘1 goes into 4, four times, and 2 goes into 8, four times.
- If you were to simplify it (2/8) it would go down to a quarter. You just halve it.
- Double one-quarter to get two-eighths.

All were successful except one student who said that one third and two thirds were equivalent because ‘the bottom is the same’

Initial Fraction Interview: Task 3

3 = 21

10

What number do you need to write in the box so that the fractions are equivalent?

How did you decide?

Initial Fraction Interview: Task 4

0.5 0.25 0.1 0.4 1112

2 4 10 5

Match each fraction with the equivalent decimal.

How did you decide?

Initial Fraction Interview: Task 4

- Most confidently matched fraction and decimal equivalents for one-half and one-tenth, were less confident with one-quarter and put two-fifths with 0.4 because it was ‘just the one left’
- Difficulties arose when students were asked to choose the larger of two fractions…

Probing Task 4

- Pick one of the tasks where the student was incorrect. Hand the student one card and a number line marked 0 to 1.

‘Place this fraction on the number line.’

‘How did you decide?’

- Hand the student the second card

‘Place this fraction on the number line.’

‘What did you find when you placed your fractions on the number line?’

Misconceptions 1: ‘gap’ thinking

Interviewer: ‘Which is larger: 3 or 5 ?

5 8

Student 1: ‘Three-fifths is larger because there is less of a gap between the three and the five than the five and the eight’.

Misconceptions 2: ‘comparing to a whole’ thinking

Interviewer: ‘Which is larger: 3 or 5 ?

5 8

Student 2: ‘Three-fifths is larger because it is two numbers away from being a whole and five-eighths is three away from being a whole’.

Misconceptions 3: ‘larger is bigger’ thinking

Interviewer: ‘Which is larger 2/3 or 3/5 ?

Student 3: 261218

3 9 18 27

3 61218

5 10 20 30

18 is larger than 18 because 30 > 27

30 27

Probing with student A

Chose 3 as larger than 3

4 5

Int: ‘Can you do it another way?’

A: ‘I automatically said it.’

He was given a sheet with empty number lines

Further probing with student A

A placed 3 close to 1.

4

On the number line he put 3 twice as far away from 1 as 3

5 4

0 3/4 1

0 3/5 1

Probing with student B

Student B said correctly that 2/3 was larger than 3/5.

His reason was that three-fifths is

‘two numbers away from being a whole and two-thirds is one number away from being a whole’

He applied the same reasoning to 3/5 and 5/8 arguing that ‘three-fifths must therefore be bigger’

Probing with student B

Int: ‘Think about 2/3 and 3/4.’

B: ‘I think they are equal. Not just because they are one away form being a whole. This (3/4) is 75% and 2/3 is about 75%.’

He didn’t have any idea of how he could check how close 2/3 was to 75%

Probing with student B

He was given an empty number line.

He marked the number line in fifths.

On the second number line he marked one-half, one-quarter and three-quarters by eye.

From his diagram he concluded that ¾ was bigger than 3/5

He reiterated that ¾ was 75% and used a calculator to show that 3/5 was 60%

Probing with student B

To compare 3/5 and 5/8, B subdivided the second number line from quarters into eighths by eye

He then said ‘5/8 is bigger- it is a bit ahead of 3/5. My old method doesn’t work.’

Int: ‘Consider one-half and four-eighths’

B: ‘ my old method would say that ½ is bigger but they are the same’

Probing with student C

To compare 3/5 and 2/3,

C said ‘Both go into 15’ and then wrote

2/3 as 10/15, and 3/5 as 9/15.

To compare 3/5 and 5/8, C first said that ‘3/5 is bigger by one’. He then converted both fractions to the same denominator (24/40 and 25/40) and said that 5/8 was bigger.

Probing with student C

He converted 3/5 and ¾ to 12/20 and 15/20 and correctly concluded that ¾ is bigger.

Using number lines to compare ¾ and 3/5, he divided the first number line by eye into quarters and marked one-half and three-quarters.

He placed one-half on the number line below in a corresponding position.

He said that ‘three-fifths is smaller than three-quarters and marked three-fifths to the right of one-half and the left of three-quarters on the number line

Probing with student C

He placed 3/5 and ¾ approximately where we would expect.On a pencil and paper test his response would be OK…

However it was not clear to the interviewer why student C had placed the fractions where he did.

Further probing was required.

Further Probing with student C

Int: ‘Can you place 3/5 on the number line?

Int: ‘Where would 1/5 be?’

C: ‘one-fifth is more than one-half (I think)’

He then placed one-fifth to the right of one-half.

Int: ‘where would one-third and one-quarter be on the number line?’

He placed these two fractions in between one-half and one-quarter.

Findings

Procedural competencecan disguise whole number thinking about fractions

- eg scaling up to equivalent fractions is a rote technique and students may relate new numerators and denominators as discrete whole numbers

Whole number thinking

- Treats numerators and denominators as discrete whole numbers (gap thinking and larger is bigger)
- Treats the ‘gap’ as a whole number not a fraction

Conclusions

To overcome whole number thinking students need to:

- Make multiple representations of fractions using discrete and continuous quantities
- Use a number line to represent and compare fractions
- Check results and estimate answers
- Deal explicitly with whole number thinking

Discrete models

Sets for counting

counters, blocks, beans

Continuous models

Area for dividing and shading

circles, triangles, rectangles

Number lines

rope and paper strips for folding

double number lines

Developing an understanding of Fraction includes:

Representing the fraction as an expression of a relationship between a part and a whole and relationships among parts and wholes.

Regardless of the representation used for a fraction and regardless of the size, shape, colour, arrangement , orientation, and the number of equivalent parts, the student can focus on the relative amount

Recognising that in the symbolic representation of a fraction the denominator indicates how many parts the whole has been divided into, and the numerator indicates how many parts of the whole have been chosen

FRACTION NUMBER SENSE

Developing an understanding of Fraction Number Sense includes five different but interconnected subconstructs:

(Kieran 1976,1980)

1

- Part-Whole,
- e.g. ‘3 parts out of every 4’

CLASSROOM EXAMPLES

Fold a strip of paper into four equal parts (quarters). What are three of these called?

Fold each quarter into three equal parts.

What are the new parts called?

What are three of these new parts called?

Can you see 3/5 of something?

Can you see 5/3 of something?

Can you see 2/3 of 3/5?

Can you see 1 divided by 3/5?

Can you see 3/5 divided by 2?

Half the candy bar is how many sticks?

2 sticks is what part of the bar?

If you have half and I have 1/3, who has more?

How much more?

How much is half and 1/3 together?

What part remains for someone else?

How much of the candy bar is half of a third?

How many times will 1/3 fit into ½?

- Operator, i.e.
- ‘3/4 of something’

- MEANING

3/4 gives a rule that tells how to operate on a unit (or the result of a previous operation), that is find 3/4 of something.

CLASSROOM EXAMPLES

A photo measures 26cm x 15cm. You want a copy made which has each side three quarters of its original length. How big will the copy be?

You have a collection of bubble gum cards. You divide the collection into 4 equal piles and give your friend three of the piles. How much of the whole collection do you give them?

- Ratios and Rates, i.e.
- ‘3 parts of one thing to 4 parts of another’

- MEANING

3:4 means 3 parts of A to 4 parts of B, where A and B are of like measure (ratio) or of different measure (rate)

CLASSROOM EXAMPLES

Sally mixes 12 tins of yellow paint with 9 tins of red paint.

Tane mixes 8 tins of yellow paint with 6 tins of red paint.

Each tin holds the same amount. Whose paint is the darkest shade of orange? How do you know?

- Quotient, i.e.
- ‘3 divided by 4’

MEANING

3/4 is the amount when each party gets when 3 units are shared equally among four parties.

CLASSROOM EXAMPLES

There are three chocolate bars to share equally among four people. How much chocolate bar will each person get?

Three pizzas for five children:

- How much pizza does each child eat?
- How much of the pizza does each child eat?

- Measure, i.e.
- ‘3 measures of 1/4’

- MEANING

3/4 means the putting together of three 1/4 units

CLASSROOM EXAMPLES

There are 7 otters and 5 sea-lions in Marineland. At feed time each otter gets one-quarter of an eel, and each sea-lion gets one third of an eel. Which hgroup of animals takes the most eels to feed?

- COUNTING FROM ONE / IMAGING TO ADVANCED COUNTING
- Common language
- Initially focus on unit fractions with 1 as numerator, but it is also important to introduce non unit fractions like ¾
- Use continuous models and discrete
- Use whole number strategies to anticipate result of equal sharing
- PAGES 23 & 25 BOOK 3, PAGE 2 BOOK 7

ADVANCED COUNTING TO EARLY ADDITIVE

Students must realize that the symbols for fractions tell how many parts the whole has been divided into (denominator), and how many of those parts have been chosen (numerator). The terminology is not as important as the understanding.

Students need to appreciate that fractions are both numbers and operators. It is vital to develop an understanding of the home of fractions among the whole numbers

PAGE 27 BOOK 3, PAGE 5 BOOK 7

EARLY ADDITIVE TO ADVANCED ADDITIVE

Early additive students are progressing towards multiplicative thinking. Fractions involve a significant mental jump for students because units of one, which are the basis for whole number counting, need to be split up (partitioned), and repackaged (re-unitised).

PAGE 30 BOOK 3, PAGE 14 BOOK 7

ADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVE

- Use a diverse range of strategies involving multiplication and division with whole numbers including:

Compensation from tidy numbers 28 x 7 as 30 x 7 - 2 x 7

Place value 64 x 8 as 60 x 8 + 4 x 8

Reversibility and commutativity e.g., 84 ÷ 7 as 7 x = 84 or

2.37 x 6 as 6 x 2.37

Proportional adjustment e.g., doubling and halving

Changing numbers e.g., 201 ÷ 3 as (99 ÷ 3) + (99 ÷ 3) + (3 ÷ 3)

ADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVE

- Solve division problems with remainders and express answers in fractional, decimal and whole number form
- Use written working forms or calculators where the numbers are difficult and / or untidy
- BOOK 3 Page 33 / BOOK 7 Pages 21 & 22

ADVANCED MULTIPLICATIVE TO ADVANCED PROPORTIONAL

To become strong proportional thinkers, the students need to be able to find multiplicative relationships in a variety of situations involving fractions, decimals, ratios and proportions.

It is also important that they see the relationships between the three views of fractional numbers: proportions, ratios and fractional/decimal operators. (Examples page 31, Book 7)

BOOK 3 Page 35 / BOOK 7 Pages 30 & 31

Experience with operations on fractions using manipulatives in contextual problem solving settings

Select appropriate visual models, methods and tools for computing with fractions, decimals and percents

Explain methods for solving problems by developing and analyzing algorithms for computing with fractions, decimals and percentages

Understand the meaning and effects of arithmetic operations with fractions, decimals and percents

Division with fractions

- People seem to have different approaches to solving problems involving division with fractions.
- How do you solve a problem like this one?

Division with fractions

- Imagine you are teaching division with fractions. To make this meaningful for kids, something that many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story problems to show the application of some particular piece of content.
- What would you say would be a good story of model for

?

DECIMALS

- Develop sound understanding of fractions first.
- Develop understanding of decimals as special fractions
- Develop understanding of percentage once decimal understanding is sound

Highest Number: 0._ _ _

Play in pairs

Aim is to make the largest 3 digit number

- Take turns to roll the die
- Place card in column of choice on your sheet
- If you get a repeat, ignore and roll again

Highest Number

What is a good strategy?

If you rolled a 5 where would you put it?

Why?

How does your strategy change if

- the winner is the closest to 0.5?
- You use a 10 sided die with numbers 0 to 9?
- You play with 4 place value positions?

Why do students have difficulties with decimals?

- The relative size of decimals.
- Ordering 0.4, 0.23, 0.164.
- Decimal place value.
- Multiplication and division ideas extended from whole numbers.
- ‘When you multiply the numbers get bigger
- When you divide the numbers get smaller’

- 0.25 is larger than 0.6
- …because 25 is larger than 6
- So 1.5 is smaller than1.45
- but 1.50 is bigger than1.45
- The decimal point is seen as a separator of two number systems. No understanding of place value to the right of the decimal point.
- Work with money tends to reinforce this misperception as do ‘tricks’ like adding zeros

2. Denominator focused thinking

0.4 is larger than 0.51

… because 4 is for tenths and the 1 is for hundredths and tenths are bigger than hundredths

The longer the decimal number, the smaller it is

0.2 is bigger than 0.3

… because you have one of two pieces (halves) versus one of three pieces (thirds)

Numbers after the decimal point indicate how many pieces there are, of which you have one.

Fraction / decimal equivalents confusion so 1/5 the same as 0.5

5.3 is bigger than 5.4

… because 5.3 is 3 units away from 5 and 5.4 is 4 units away.

Thinks of decimals as measuring a distance from the whole number. The distance is seen as diminishing the whole number.

2.45 is bigger than 2.452

… because decimals beyond 2 decimal places are not really real – they are either ignored or they are regarded as diminishing the earlier number.

These students may not be noticed in class as they are successful dealing with decimals up to hundredths.

4.08 is bigger than 4.7

8.0527 is smaller than 8.54

… you ignore the zero because zero is nothing.

Confused about concept of zero as a placeholder.

0.03 is bigger than 0.00

…But 0.03 is smaller than 0.0

… and 0.0 is not the same as 0.00

Unsure of relationship between zero, one and decimal numbers

8. Task Experts

- Get most tasks correct but make a few errors with no pattern
- No clear way of dealing with decimals but know that whole number thinking is not correct.

9. Not consistent

- Decimal place value
- Ordering decimals
- Relationship between fractions and decimals
- Placing decimals on a number line
- Operating with decimals:
- Addition and subtraction as extensions of whole number operations
- Multiplication and division as extension of operations with fractions

Write down a number that is:

Bigger than 3.9 and smaller than 4

Bigger than 6 and smaller than 6.1

Bigger than 0.52 and smaller than 0.53

Bigger than 8.9 and smaller than 8.15

- How important is context?
- Is money an appropriate context for development of decimal understanding?
- What language should we use for decimals?
- Should we consider the decimal point as a marker for the ones place?
- Is Multiplicative thinking needed for understanding of fractions and decimals?

- Promote connections between decimals, fractions and other mathematical contexts such as metric measures and percentages
- Decimal place value language 0.1 is ‘1 tenth’, 0.01 is said as ‘1 hundredth’, 3.14 as ‘3 and fourteen hundredths’ or ‘3 and 1 tenth and 4 hundredths’
- Multiplicative thinking is essential for understanding of equivalent fractions and ratios. The equivalent fractions concept is essential for understanding decimals and percentages
- Consider the decimal point as a marker for the ones place rather than a barrier that separates the wholes from the fractions

Developing understandings of fractions and decimals

Discuss difficulties and misperceptions

Identify strategies at different stages

Discuss learning processes

Discuss teaching strategies

Explore activities to use in the classroom

Oh, one more thing. Cut that pizza into six slices. I can’t eat eight.

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