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

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Starting with a half Describe halves, encountered in everyday contexts, as two equal parts of an object. (NES1.4) 5CMIT Facilitator training 2009

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? 7CMIT Facilitator training 2009

Halving What is the basis of your decision? 8CMIT Facilitator training 2009

What about a quarter (area or length)? What is being partitioned? 9CMIT Facilitator training 2009

Folding to find halves & quarters 10CMIT Facilitator training 2009

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) 11CMIT Facilitator training 2009

Parallel partitioning CMIT Facilitator training 2009

Counting parts Students are expected to demonstrate their understanding by shading in parts of a shape. For example : 7 10 (a) Shade seven-tenths of the following shape. New Signpost mathematics 7 p (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 14CMIT Facilitator training 2009

Which is bigger, one-third or one-sixth? An area model without equal partitioning (Number of pieces) Fractions defined by the number of parts 15CMIT Facilitator training 2009

More: Number of parts Fractions defined by the number of parts without attention to the equality of parts 16CMIT Facilitator training 2009 Year 6 (6221)

Number of parts - equal parts 7458 The bigger the denominator the bigger the fraction! 17CMIT Facilitator training 2009

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) 18CMIT Facilitator training 2009

What about equal wholes? Which is bigger, two-thirds or five- sixths? Can 1/4 ever be bigger than 1/2? 19CMIT Facilitator training 2009

The equal-whole Which is bigger, one-sixth or one- twelfth? An area model but what happened to the equal wholes? 20CMIT Facilitator training 2009

The importance of the equal whole The equal whole is currently missing from our syllabus. It needs to be in our teaching. What is ? 6041 What could it be for this student? 1 3  CMIT Facilitator training 2009

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. 22CMIT Facilitator training 2009

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. 23CMIT Facilitator training 2009

4/5 + 11/12 24CMIT Facilitator training 2009

The syllabus ES1S1S2S3 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) 25CMIT Facilitator training 2009

Fraction framework HalvingForms halves and quarters by repeated halving in one-direction. Can use distributive dealing to share. NES1.4 NS1.4 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

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 parts 27CMIT 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 29CMIT Facilitator training 2009

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. NS2.4 Models, compares, and represents commonly used fractions and decimals, adds and subtracts decimals to two decimal places, and interprets everyday percentages 30CMIT Facilitator training 2009

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 32CMIT Facilitator training 2009

Level 3: Re-forms the whole When iterating a fraction part such as one-third beyond the whole, reforms the whole unit. fraction. NS3.4 Compares, orders and calculates with decimals, simple fractions and simple percentages 33CMIT Facilitator training 2009

Level 3: Re-forms the whole 34CMIT Facilitator training 2009

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 35 CMIT Facilitator training 2009

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 37 CMIT Facilitator training 2009

Level 5: Multiplicative partitioning If this is ¾ of the strip of paper, where would ½ of the whole piece of paper be? 38CMIT Facilitator training 2009

Multiplicative partitioning 39CMIT Facilitator training 2009

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 42CMIT Facilitator training 2009

More teaching activities Placing fractions and decimals on the empty number line Double number line Coloured fractions 43CMIT Facilitator training 2009