fraction understanding

Fraction Understanding

This workshop will cover:• Common misconceptions with fractions• Framework levels for fractions linked to the

Mathematics K-6 syllabus• Teaching activities

2CMIT Facilitator training 2009

Starting with a half

Describe halves, encountered in everyday contexts, as two equal parts of an object. (NES1.4)


Starting with half an apple

How do you know that you have equal parts of an object? 6CMIT Facilitator training 2009

Is this halving?

How do you know?


Halving

What is the basis of your decision?


What about a quarter (area or length)?

What is being partitioned?9CMIT Facilitator training 2009

Folding to find halves & quarters


Parallel partitioning

Initial ‘halving’ is often applied without attention to the equality of the parts. Halving is initially used in an algorithmic manner without concern for equality. Vertical parallel lines that work in a rectangular region may also be used in a circular region to produce thirds, fourths or fifths. (Pothier & Sawada, 1983)


Parallel partitioning


Counting parts

Students are expected to demonstrate their understanding by shading in parts of a shape. For example:

710(a) Shade seven-tenths

of the following shape.

New Signpost mathematics 7 p.318

45(b) Shade four-fifths

of the following shape.

Maths Plus Unit 4 Stage 2 p.15

Sometimes it breaks down

Instead of seeing the relationship between the parts and the whole, some students see:

• Parts from parallel partitions• Number of parts (not equal)• Number of equal parts (not a fraction of the

whole)• And we sometimes lose the equal whole


Which is bigger, one-third or one-sixth?

An area model without equal partitioning

(Number of pieces)

Fractions defined by the number of parts


More: Number of partsFractions defined by the number of parts without attention to the equality of parts

16CMIT Facilitator training 2009Year 6 (6221)

Number of parts - equal parts

7458

The bigger the denominator the bigger the fraction!


Number of parts rather than area

6221

Which is the bigger number and how do you know?

Sometimes students attend to the number of parts rather than the equality of the parts. (Vertical and horizontal partitioning)


What about equal wholes?

Which is bigger, two-thirds or five-sixths?

Can 1/4 ever be bigger than 1/2?


The equal-whole

Which is bigger, one-sixth or one-twelfth?

An area model but what happened to the equal wholes?


The importance of the equal whole

• The equal whole is currently missing from our syllabus. It needs to be in our teaching.

• What is ?

6041What could it be for this student?

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Building on what students know

If we wish to build on what students currently know we need to be aware of what that is.

To recognise what students know we need to examine their recordings and explanations.


Problems introducing fraction notation

• When fraction notation is introduced , we introduce it as a way of recording a double count, that is we count the number of parts and then record this first count over the second count as a description of a fraction, eg 2/3

• Developing fraction notation from the double count is an additive interpretation as the whole is ignored.


4/5 + 11/12


The syllabusES1 S1 S2 S3

Describes halves encountered in everyday contexts, as two equal parts of an object

Describes and models halves and quarters, of objects and collections, occurring in everyday situations

Models, compares and represents commonly used fractions & decimals, adds & subtracts decimals to two decimal places, & interprets everyday percentages

(denominators 2, 4 & 8, followed by 5, 10 & 100).

Compares, orders and calculates with decimals, simple fractions and simple percentages

(denominators 2, 3, 4, 5, 6, 8, 10, 12 & 100)


Fraction frameworkHalving Forms halves and quarters by repeated halving in one-direction.

Can use distributive dealing to share.NES1.4NS1.4

Equal partitions

Verifies continuous and discrete linear arrangements have been partitioned into thirds or fifths by iterating one part to form the whole or checking the equality and number of parts forming the whole. Partitioning continuous quantities into specified numbers of equal parts is very difficult for those partitions not based on repeated halving (i.e. other than halves, quarters, eighths, sixteenths, etc). Instead of partitioning to create odd numbers of parts such as fifths, verifying partitions is recommended. Verifying partitioned fractions helps to establish the relationship between the part and the whole and links to Levels 3 and 4 in measurement.

NS2.4

Re-forms the whole

When iterating a fraction part such as one-third beyond the whole, re-forms the whole.

NS3.4

Fractions as numbers

Identifies the need to have equal wholes to compare fractional parts. Uses fractions as numbers, i.e. Creates equivalent fractions using equivalent equal wholes.

NS3.4

Multiplicative partitioning

Coordinates composition of partitioning (i.e. can find one-third of one-half to create one-sixth).Coordinates units at three levels to move between equivalent fraction forms. Uses multiplicative partitioning in two directions.

NS4.3

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Level 1: Halving• Forms halves and quarters by repeated halving

in one-direction• Can use distributive dealing

to share

NES1.4

Describes halves, encountered

in everyday contexts as two equal parts27CMIT Facilitator training 2009

Level 1: Halving

Using halving to

create the 4-partition.

NS1.4

Describes & models

halves & quarters, of

objects and collections 28CMIT Facilitator training 2009

Distributive dealing to share


Level 2: Equal partitions

Verifies continuous and discrete linear arrangements have been partitioned into thirds or fifths by iterating one part to form the whole.NS2.4Models, compares, and represents commonly used fractions and decimals, adds and subtracts decimals to two decimal places, and interprets everyday percentages


Level 2: Equal partitions

An ant crawls around

the outside of this triangle.

If the ant starts at the top,

show me where it will be

when it is ½,1/3 of the

way around?31CMIT Facilitator training 2009

By pouring show me exactly a third of a glass of water


Level 3: Re-forms the whole

When iterating a fraction part such as one-third beyond the whole, reforms the whole unit. fraction.NS3.4Compares, orders and calculates with decimals, simple fractions and simple percentages


Level 3: Re-forms the whole


Level 4: Fractions as numbers

Identifies the need to have equal wholes to compare fractional parts. Uses fractions as numbers ie.

Creates equivalent fractions using

equivalent equal wholes. NS3.4

Compares, orders and calculates with decimals, simple fractions and simple percentages

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Level 4

36

Level 5: Multiplicative partitioning

Coordinates composition of partitioning.

For example the student can find one-third of one-half to create one-sixth

Coordinates units at three levels to move between equivalent fraction forms. Uses multiplicative partitioning in two directions.

NS4.3

Operates with fractions, decimals, percentages, ratios & rates


Level 5: Multiplicative partitioning

If this is ¾ of the strip of paper, where would ½ of the whole piece of paper be?


Multiplicative partitioning


Relational numbers

To address the quantitative misconceptions,students need opportunities to:• see non-examples (particularly to whole number

interpretations)• partition the whole and duplicate the piece to rebuild

the whole• have opportunities to verify the fraction• focus on the attribute (e.g. length) used in the relation• make adjustments• recognise the equal whole (especially Stage 3). 40

Teaching activities• The emphasis should be on verifying the relationship

between one part and the whole.• The transition from fractions as part of a collection or

parts of an object to fractions as numbers is crucial.• To make this step, students need opportunities to

create fractional parts and then increase the number of these parts so that it exceeds the whole.

• The idea of the whole becomes clearer when it is exceeded, so that it is necessary to re-form the whole. 41CMIT Facilitator training 2009

Teaching activities


More teaching activities

• Placing fractions and decimals on the empty number line

• Double number line

• Coloured fractions


fraction understanding

Documents

equality of parts

equalnumber of equal

bigger number

number of partsfractions

equal parts7458the

equal wholes

parallel partitioning6041

following shape