course nm309 fractions, decimals & percentages teaching for understanding from vince

Nope. I t still costs the same for me. Five dollars w orth.

I can’t believe the price of gas these days!

Oh, has it gone up?

You mean you haven’t noticed?

COURSE NM309COURSE NM309

FRACTIONS, DECIMALS & FRACTIONS, DECIMALS & PERCENTAGES PERCENTAGES

TEACHING FOR UNDERSTANDINGTEACHING FOR UNDERSTANDING

From VinceFrom Vince

OBJECTIVESOBJECTIVESDeveloping understandings of fractions Developing understandings of fractions

and decimalsand decimalsDiscuss the difficulties and misperceptionsDiscuss the difficulties and misperceptionsIdentify strategies at different stages Identify strategies at different stages Discuss learning processesDiscuss learning processesDiscuss teaching strategiesDiscuss teaching strategiesExplore activities to use in the classroomExplore activities to use in the classroom

Which family has more girls?

The Jones Family

The King Family

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 problemA fishy problem

Two-thirds of the goldfish are maleTwo-thirds of the goldfish are male

There are 24 male goldfishThere are 24 male goldfish

How many goldfish are there How many goldfish are there altogether?altogether?

Share your strategyShare your strategy

How did you do it?How did you do it?

Discuss your method in groupsDiscuss your method in groups

Who taught you how to do it this Who taught you how to do it this way?way?

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

The Bill Gates questionThe Bill Gates question

Report on Numeracy Project 2003Report on Numeracy Project 2003

The performance of year 7 and 8 students on The performance of year 7 and 8 students on fractions and decimals is well below what would fractions and decimals is well below what would be wished.be wished.

Integration of fractions with proportional Integration of fractions with proportional reasoning would aid understanding of those reasoning would aid understanding of those topics.topics.

Decimals are of particular concernDecimals are of particular concern Decimals need to be taught using the Numeracy Decimals need to be taught using the Numeracy

principles: using materials and imaging before principles: using materials and imaging before number propertiesnumber properties

Why do students have difficulty with Why do students have difficulty with fractions?fractions?

Rational number ideas are sophisticated and Rational number ideas are sophisticated and different from natural number ideasdifferent from natural number ideas

Natural numbers can be represented Natural numbers can be represented individually, rational numbers cannot.individually, rational numbers cannot.

Students’ whole number schemes can interfere Students’ whole number schemes can interfere with their efforts to learn fractionswith their efforts to learn fractions

Students have to learn new ways to represent, Students have to learn new ways to represent, describe and interpret rational numbersdescribe and interpret rational numbers

Rote procedures for manipulating fractions (eg Rote procedures for manipulating fractions (eg making equivalent fractions) may not be enoughmaking equivalent fractions) may not be enough

Initial Fraction Interview: Task 1Initial Fraction Interview: Task 1

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

Why did you choose that many lollies?Why did you choose that many lollies?


22 22 11 22 22 11

55 33 44 88 33 33

Which of these pairs of fractions are Which of these pairs of fractions are equivalent (have the same value)?equivalent (have the same value)?

How did you decide?How did you decide?

5

2


Typical responses:Typical responses: One-quarter is equivalent to two-eighths ‘cos ‘1 One-quarter is equivalent to two-eighths ‘cos ‘1

goes into 4, four times, and 2 goes into 8, four goes into 4, four times, and 2 goes into 8, four times.times.

If you were to simplify it (2/8) it would go down to If you were to simplify it (2/8) it would go down to a quarter. You just halve it.a quarter. You just halve it.

Double one-quarter to get two-eighths.Double one-quarter to get two-eighths.

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


33 == 2121

10 10

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



0.5 0.5 0.25 0.10.25 0.1 0.4 0.4 11 11 11 22

22 44 1010 55

Match each fraction with the equivalent Match each fraction with the equivalent decimal.decimal.



Most confidently matched fraction and Most confidently matched fraction and decimal equivalents for one-half and one-decimal equivalents for one-half and one-tenth, were less confident with one-quarter tenth, were less confident with one-quarter and put two-fifths with 0.4 because it was and put two-fifths with 0.4 because it was ‘just the one left’‘just the one left’

Difficulties arose when students were Difficulties arose when students were asked to choose the larger of two asked to choose the larger of two fractions…fractions…

Probing Task 1Probing Task 1

33 22

55 33

Which is larger, three-fifths or two-thirds?Which is larger, three-fifths or two-thirds?



33 55

55 88

Which is larger, three-fifths or five-eighths?Which is larger, three-fifths or five-eighths?



33 33

55 44

Which is larger, three-fifths or five-quarters?Which is larger, three-fifths or five-quarters?



Pick one of the tasks where the student was Pick one of the tasks where the student was incorrect. Hand the student one card and a incorrect. Hand the student one card and a number line marked 0 to 1.number line marked 0 to 1.‘‘Place this fraction on the number line.’Place this fraction on the number line.’‘‘How did you decide?’How did you decide?’

Hand the student the second cardHand the student the second card‘‘Place this fraction on the number line.’Place this fraction on the number line.’‘‘What did you find when you placed your What did you find when you placed your fractions on the number line?’fractions on the number line?’

Misconceptions 1: ‘gap’ thinkingMisconceptions 1: ‘gap’ thinking

Interviewer: ‘Which is larger: Interviewer: ‘Which is larger: 33 oror55 ? ?

55 88

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

Misconceptions 2: Misconceptions 2: ‘comparing to a whole’ thinking‘comparing to a whole’ thinking

Interviewer: ‘Which is larger: Interviewer: ‘Which is larger: 33 oror55 ? ?

55 88

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

Misconceptions 3: Misconceptions 3: ‘larger is bigger’ thinking‘larger is bigger’ thinking

Interviewer: ‘Which is larger 2/3 or 3/5 ?Interviewer: ‘Which is larger 2/3 or 3/5 ?Student 3: Student 3: 22 66 1212 1818

33 99 1818 2727

33 6 6 1212 181855 1010 2020 3030

1818 is larger than is larger than 1818 because 30 > 27 because 30 > 273030 2727

Probing with student AProbing with student A

Chose Chose 33 as larger than as larger than 33 44 5 5

Int: ‘Can you do it another way?’Int: ‘Can you do it another way?’A: ‘I automatically said it.’A: ‘I automatically said it.’

He was given a sheet with empty number He was given a sheet with empty number lineslines

Further probing with student AFurther probing with student A

A placed A placed 33 close to 1. close to 1.

44

On the number line he put On the number line he put 33 twice as far away from 1 as twice as far away from 1 as 33

5 5 44

00 3/43/4 11

00 3/53/5 11

Probing with student BProbing with student B

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

His reason was that three-fifths is His reason was that three-fifths is ‘‘two numberstwo numbers away from being a whole and away from being a whole and

two-thirds is two-thirds is one numberone number away from being away from being a whole’a whole’

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


Int: ‘Think about 2/3 and 3/4.’Int: ‘Think about 2/3 and 3/4.’

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

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


He was given an empty number line.He was given an empty number line.He marked the number line in fifths.He marked the number line in fifths.On the second number line he marked one-On the second number line he marked one-

half, one-quarter and three-quarters by half, one-quarter and three-quarters by eye.eye.

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

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


00 11

1/51/5 3/5 3/5

00 11

¼ ¼ ½ ½ ¾ ¾


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

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

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

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

Probing with student CProbing with student C

To compare 3/5 and 2/3, To compare 3/5 and 2/3,

C said ‘Both go into 15’ and then wrote C said ‘Both go into 15’ and then wrote

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

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


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

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

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

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


00 11

½ ½ ¾ ¾

00 11

½½ 3/5 3/5


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

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

Further probing was required.Further probing was required.

Further Probing with student CFurther Probing with student C

Int: ‘Can you place 3/5 on the number line?Int: ‘Can you place 3/5 on the number line?Int: ‘Where would 1/5 be?’Int: ‘Where would 1/5 be?’C: ‘one-fifth is more than one-half (I think)’C: ‘one-fifth is more than one-half (I think)’He then placed one-fifth to the right of one-He then placed one-fifth to the right of one-

half.half.Int: ‘where would one-third and one-quarter Int: ‘where would one-third and one-quarter

be on the number line?’be on the number line?’He placed these two fractions in between He placed these two fractions in between

one-half and one-quarter.one-half and one-quarter.

Further Probing with student CFurther Probing with student C

00 11

½ ½ 1/3 1/4 1/51/3 1/4 1/5

FindingsFindingsProcedural competenceProcedural competence can disguise whole number can disguise whole number

thinking about fractionsthinking about fractions eg scaling up to equivalent fractions is a rote eg scaling up to equivalent fractions is a rote

technique and students may relate new numerators technique and students may relate new numerators and denominators as discrete whole numbersand denominators as discrete whole numbers

Whole number thinkingWhole number thinking Treats numerators and denominators as discrete Treats numerators and denominators as discrete

whole numbers (gap thinking and larger is bigger)whole numbers (gap thinking and larger is bigger) Treats the ‘gap’ as a whole number not a fractionTreats the ‘gap’ as a whole number not a fraction

ConclusionsConclusions

To overcome whole number thinking To overcome whole number thinking students need to:students need to:

Make multiple representations of fractions Make multiple representations of fractions using using discretediscrete and and continuouscontinuous quantities quantities

Use a number line to represent and Use a number line to represent and compare fractionscompare fractions

Check results and estimate answersCheck results and estimate answers Deal explicitly with whole number thinkingDeal explicitly with whole number thinking

Models for fractionsModels for fractions

Discrete modelsDiscrete modelsSets for countingSets for counting

•counters, blocks, beans counters, blocks, beans

Continuous modelsContinuous modelsArea for dividing and shadingArea for dividing and shading

•circles, triangles, rectangles circles, triangles, rectangles Number lines Number lines

•rope and paper strips for folding rope and paper strips for folding •double number linesdouble number lines

FRACTION NUMBER SENSEFRACTION NUMBER SENSE

Developing an understanding of Fraction includes:Developing an understanding of Fraction includes:

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

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

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


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

(Kieran 1976,1980)(Kieran 1976,1980)


Part-Whole, Part-Whole,

e.g. ‘3 parts out of every 4’e.g. ‘3 parts out of every 4’ 11CLASSROOM EXAMPLESCLASSROOM EXAMPLES

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

Fold each quarter into three equal parts. Fold each quarter into three equal parts.

What are the new parts called?What are the new parts called?

What are three of these new parts called?What are three of these new parts called?

WHAT CAN YOU SEE ?WHAT CAN YOU SEE ?

Can You See It?Can You See It?

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?

Big Stix Chocolate BarBig Stix Chocolate Bar

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. Operator, i.e.

‘‘3/4 3/4 of of something’something’

MEANINGMEANING3/4 gives a rule that tells how to operate on a unit (or the result of 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.a previous operation), that is find 3/4 of something.

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

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

22

Thinking Up And DownThinking Up And DownThinking Up And DownThinking Up And Down


Ratios and Rates, i.e. Ratios and Rates, i.e.

‘‘3 parts of one thing to 4 parts of another’3 parts of one thing to 4 parts of another’

MEANINGMEANING3:4 means 3 parts of A to 4 parts of B, where A 3:4 means 3 parts of A to 4 parts of B, where A and B are of like measure (ratio) or of different and B are of like measure (ratio) or of different measure (rate)measure (rate)

CLASSROOM EXAMPLESCLASSROOM EXAMPLESSally mixes 12 tins of yellow paint with 9 tins of red paint.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.Tane mixes 8 tins of yellow paint with 6 tins of red paint.

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

33


Quotient, i.e. Quotient, i.e.

‘‘3 divided by 4’3 divided by 4’

MEANINGMEANING

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

CLASSROOM EXAMPLESCLASSROOM EXAMPLES

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

44

Three pizzas for five children:Three pizzas for five children:

How much pizza does each How much pizza does each child eat?child eat?

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


Measure, i.e. Measure, i.e.

‘‘3 measures of 1/4’3 measures of 1/4’

MEANINGMEANING3/4 means the putting together of three 1/4 units3/4 means the putting together of three 1/4 units

CLASSROOM EXAMPLESCLASSROOM EXAMPLES

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

55

EARLY STAGESEARLY STAGES

COUNTING FROM ONE / IMAGING TO ADVANCED COUNTINGCOUNTING FROM ONE / IMAGING TO ADVANCED COUNTING

- Common languageCommon language

- Initially focus on unit fractions with 1 as numerator, but it is Initially focus on unit fractions with 1 as numerator, but it is also important to introduce non unit fractions like ¾also important to introduce non unit fractions like ¾

-Use continuous models and discreteUse continuous models and discrete

-Use whole number strategies to anticipate result of equal Use whole number strategies to anticipate result of equal sharingsharing

PAGES 23 & 25 BOOK 3, PAGE 2 BOOK 7PAGES 23 & 25 BOOK 3, PAGE 2 BOOK 7

STAGES 4 TO 5STAGES 4 TO 5

ADVANCED COUNTING TO EARLY ADDITIVEADVANCED COUNTING TO EARLY ADDITIVE

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

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

PAGE 27 BOOK 3, PAGE 5 BOOK 7PAGE 27 BOOK 3, PAGE 5 BOOK 7


EARLY ADDITIVE TO ADVANCED ADDITIVEEARLY ADDITIVE TO ADVANCED ADDITIVE

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

PAGE 30 BOOK 3, PAGE 14 BOOK 7PAGE 30 BOOK 3, PAGE 14 BOOK 7

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

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

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

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

2.37 x 6 as 6 x 2.372.37 x 6 as 6 x 2.37

Proportional adjustment e.g., doubling and halvingProportional adjustment e.g., doubling and halving

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

ADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVEADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVE


•Solve division problems with remainders and express answers Solve division problems with remainders and express answers in fractional, decimal and whole number formin fractional, decimal and whole number form

•Use written working forms or calculators where the numbers Use written working forms or calculators where the numbers are difficult and / or untidyare difficult and / or untidy

BOOK 3 Page 33 / BOOK 7 Pages 21 & 22BOOK 3 Page 33 / BOOK 7 Pages 21 & 22

ADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVEADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVE

STAGES 6 TO 7 (cont)STAGES 6 TO 7 (cont)

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

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

BOOK 3 Page 35 / BOOK 7 Pages 30 & 31BOOK 3 Page 35 / BOOK 7 Pages 30 & 31

ADVANCED MULTIPLICATIVE TO ADVANCED PROPORTIONALADVANCED MULTIPLICATIVE TO ADVANCED PROPORTIONAL


Operations with fractions Operations with fractions

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

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

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

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

Division with fractionsDivision with fractions

People seem to have different approaches to solving People seem to have different approaches to solving problems involving division with fractions. problems involving division with fractions.

How do you solve a problem like this one? How do you solve a problem like this one?

=÷2

1

4

31

Division with fractionsDivision with fractions

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

What would you say would be a good story of model for What would you say would be a good story of model for

??

=÷2

1

4

31

=÷2

1

4

31

DECIMALSDECIMALS

Develop sound understanding of fractions Develop sound understanding of fractions first.first.

Develop understanding of decimals as special Develop understanding of decimals as special fractionsfractions

Develop understanding of percentage once Develop understanding of percentage once decimal understanding is sounddecimal understanding is sound

Highest Number: 0._ _ _Highest Number: 0._ _ _

Play in pairsPlay in pairs

Aim is to make the largest 3 digit numberAim is to make the largest 3 digit number

1.1. Take turns to roll the dieTake turns to roll the die

2.2. Place card in column of choice on your Place card in column of choice on your sheetsheet

3.3. If you get a repeat, ignore and roll againIf you get a repeat, ignore and roll again

What is a good strategy?What is a good strategy?

If you rolled a 5 where would you put it?If you rolled a 5 where would you put it?

Why?Why?

How does your strategy change if How does your strategy change if the winner is the closest to 0.5?the winner is the closest to 0.5? You use a 10 sided die with numbers 0 to 9? You use a 10 sided die with numbers 0 to 9? You play with 4 place value positions?You play with 4 place value positions?

Highest NumberHighest Number

Why do students have difficulties with Why do students have difficulties with decimals?decimals?

•The relative size of decimals. The relative size of decimals. Ordering 0.4, 0.23, 0.164.Ordering 0.4, 0.23, 0.164.

•Decimal place value. Decimal place value.

•Multiplication and division ideas extended from Multiplication and division ideas extended from whole numbers. whole numbers.

‘‘When you multiply the numbers get bigger When you multiply the numbers get bigger When you divide the numbers get smaller’ When you divide the numbers get smaller’

1. Whole number thinking1. Whole number thinking

0.25 is larger than 0.60.25 is larger than 0.6 … …because 25 is larger than 6because 25 is larger than 6So 1.5 is smaller than1.45 So 1.5 is smaller than1.45 but 1.50 is bigger than1.45but 1.50 is bigger than1.45

The decimal point is seen as a separator of two The decimal point is seen as a separator of two number systems. No understanding of place number systems. No understanding of place value to the right of the decimal point.value to the right of the decimal point.

Work with money tends to reinforce this Work with money tends to reinforce this misperception as do ‘tricks’ like adding zeros misperception as do ‘tricks’ like adding zeros

2. Denominator focused thinking2. Denominator focused thinking

0.4 is larger than 0.51 0.4 is larger than 0.51 … … because 4 is for tenths and the 1 is for because 4 is for tenths and the 1 is for hundredths and tenths are bigger than hundredths and tenths are bigger than hundredthshundredths

The longer the decimal number, the smaller it isThe longer the decimal number, the smaller it is

3. Reciprocal thinking3. Reciprocal thinking

0.2 is bigger than 0.3 0.2 is bigger than 0.3 … … because you have one of two pieces (halves) because you have one of two pieces (halves) versus one of three pieces (thirds)versus one of three pieces (thirds)

Numbers after the decimal point indicate how Numbers after the decimal point indicate how many pieces there are, of which you have one.many pieces there are, of which you have one.Fraction / decimal equivalents confusion so 1/5 Fraction / decimal equivalents confusion so 1/5 the same as 0.5 the same as 0.5

4. Negative number thinking4. Negative number thinking

5.3 is bigger than 5.45.3 is bigger than 5.4… … because 5.3 is 3 units away from 5 and 5.4 is because 5.3 is 3 units away from 5 and 5.4 is 4 units away. 4 units away.

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

5. Money thinking5. Money thinking

2.45 is bigger than 2.452 2.45 is bigger than 2.452 … … because decimals beyond 2 decimal places because decimals beyond 2 decimal places are not really real – they are either ignored or are not really real – they are either ignored or they are regarded as diminishing the earlier they are regarded as diminishing the earlier number. number.

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

6. Place value thinking6. Place value thinking

4.08 is bigger than 4.7 4.08 is bigger than 4.7

8.0527 is smaller than 8.548.0527 is smaller than 8.54

… … you ignore the zero because zero is nothing.you ignore the zero because zero is nothing.

Confused about concept of zero as a Confused about concept of zero as a placeholder.placeholder.

7. Zero on number line7. Zero on number line

0.03 is bigger than 0.00 0.03 is bigger than 0.00

……But 0.03 is smaller than 0.0 But 0.03 is smaller than 0.0

… … and 0.0 is not the same as 0.00 and 0.0 is not the same as 0.00

Unsure of relationship between zero, one and Unsure of relationship between zero, one and decimal numbersdecimal numbers

8. Task Experts8. Task Experts

Get most tasks correct but make a few Get most tasks correct but make a few errors with no patternerrors with no pattern

No clear way of dealing with decimals but No clear way of dealing with decimals but know that whole number thinking is not know that whole number thinking is not correct.correct.

9. Not consistent9. Not consistent

UNDERSTANDING DECIMALSUNDERSTANDING DECIMALS

•Decimal place valueDecimal place value

•Ordering decimalsOrdering decimals

•Relationship between fractions and decimalsRelationship between fractions and decimals

•Placing decimals on a number linePlacing decimals on a number line

•Operating with decimals: Operating with decimals: Addition and subtraction as extensions of whole Addition and subtraction as extensions of whole number operations number operations Multiplication and division as extension of Multiplication and division as extension of operations with fractionsoperations with fractions

EXTENDING UNDERSTANDING EXTENDING UNDERSTANDING

Write down a number that is:Write down a number that is:

Bigger than 3.9 and smaller than 4Bigger than 3.9 and smaller than 4

Bigger than 6 and smaller than 6.1Bigger than 6 and smaller than 6.1

Bigger than 0.52 and smaller than 0.53Bigger than 0.52 and smaller than 0.53

Bigger than 8.9 and smaller than 8.15Bigger than 8.9 and smaller than 8.15

Zero insertion taskZero insertion task

Which is bigger? Which is bigger?

2.35 or 2.350?2.35 or 2.350?

2.305 or 2.35?2.305 or 2.35?

2.035 or 2.35?2.035 or 2.35?

QUESTIONSQUESTIONS

•How important is context?How important is context?

•Is money an appropriate context for development of Is money an appropriate context for development of decimal understanding?decimal understanding?

•What language should we use for decimals?What language should we use for decimals?

•Should we consider the decimal point as a marker for Should we consider the decimal point as a marker for the ones place?the ones place?

•Is Multiplicative thinking needed for understanding of Is Multiplicative thinking needed for understanding of fractions and decimals?fractions and decimals?

SUGGESTIONSSUGGESTIONS

•Promote Promote connections connections between decimals, fractions and other between decimals, fractions and other mathematical contexts such as metric measures and mathematical contexts such as metric measures and percentagespercentages

•Decimal place value Decimal place value languagelanguage 0.1 is ‘1 tenth’, 0.01 is said as ‘1 0.1 is ‘1 tenth’, 0.01 is said as ‘1 hundredth’, 3.14 as ‘3 and fourteen hundredths’ or ‘3 and 1 tenth hundredth’, 3.14 as ‘3 and fourteen hundredths’ or ‘3 and 1 tenth and 4 hundredths’and 4 hundredths’

•MultiplicativeMultiplicative thinking is essential for understanding of thinking is essential for understanding of equivalent fractions and ratios. The equivalent fractions concept equivalent fractions and ratios. The equivalent fractions concept is essential for understanding decimals and percentagesis essential for understanding decimals and percentages

•Consider the decimal point as a marker for the ones place Consider the decimal point as a marker for the ones place rather than a barrier that separates the wholes from the fractionsrather than a barrier that separates the wholes from the fractions

Course NUM309Course NUM309

Developing understandings of Developing understandings of fractions and decimalsfractions and decimals

Discuss difficulties and misperceptionsDiscuss difficulties and misperceptionsIdentify strategies at different stages Identify strategies at different stages Discuss learning processesDiscuss learning processesDiscuss teaching strategiesDiscuss teaching strategiesExplore activities to use in the classroomExplore activities to use in the classroom

Oh, one more thing. Oh, one more thing. Cut that pizza into six slices. Cut that pizza into six slices.

I can’t eat eight.I can’t eat eight.

Thanks for your contribution todayThanks for your contribution today

course nm309 fractions, decimals & percentages teaching for understanding from vince

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