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Fraction Understanding. This workshop will cover:. Common misconceptions with fractions Framework levels for fractions linked to the Mathematics K-6 syllabus Teaching activities. Starting with a half.
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This workshop will cover: • Common misconceptions with fractions • Framework levels for fractions linked to the Mathematics K-6 syllabus • Teaching activities CMIT Facilitator training 2009
Starting with a half Describe halves, encountered in everyday contexts, as two equal parts of an object. (NES1.4) CMIT Facilitator training 2009
Starting with half an apple How do you know that you have equal parts of an object? CMIT Facilitator training 2009
Is this halving? How do you know? CMIT Facilitator training 2009
Halving What is the basis of your decision? CMIT Facilitator training 2009
What about a quarter (area or length)? What is being partitioned? CMIT Facilitator training 2009
Folding to find halves & quarters CMIT 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) CMIT Facilitator training 2009
Parallel partitioning 6041 CMIT Facilitator training 2009
4 5 7 10 Counting parts Students are expected to demonstrate their understanding by shading in parts of a shape. For example: (b) Shade four-fifths of the following shape. (a)Shade seven-tenths of the following shape. New Signpost mathematics 7 p.318 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 CMIT Facilitator training 2009
Fractions defined by the number of parts Which is bigger, one-third or one-sixth? An area model without equal partitioning (Number of pieces) CMIT Facilitator training 2009
More: Number of parts Fractions defined by the number of parts without attention to the equality of parts Year 6 (6221) CMIT Facilitator training 2009
Number of parts - equal parts The bigger the denominator the bigger the fraction! 7458 CMIT Facilitator training 2009
Number of parts rather than area 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) 6221 CMIT Facilitator training 2009
What about equal wholes? Which is bigger, two-thirds or five-sixths? Can 1/4 ever be bigger than 1/2? CMIT Facilitator training 2009
The equal-whole Which is bigger, one-sixth or one-twelfth? An area model but what happened to the equal wholes? CMIT Facilitator training 2009
1 1 + 6 3 6041 The importance of the equal whole • The equal whole is currently missing from our syllabus. It needs to be in our teaching. • What is ? What could it be for this student? 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. CMIT 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. CMIT Facilitator training 2009
4/5 + 11/12 CMIT Facilitator training 2009
The syllabus CMIT Facilitator training 2009
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 CMIT Facilitator training 2009
Level 1: Halving Using halving to create the 4-partition. NS1.4 Describes & models halves & quarters, of objects and collections CMIT Facilitator training 2009
Distributive dealing to share CMIT 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 CMIT 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? CMIT Facilitator training 2009
By pouring show me exactly a third of a glass of water CMIT 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 CMIT Facilitator training 2009
Level 3: Re-forms the whole CMIT 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 CMIT Facilitator training 2009
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 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? CMIT Facilitator training 2009
Multiplicative partitioning CMIT 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).
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. CMIT Facilitator training 2009
Teaching activities CMIT Facilitator training 2009
More teaching activities • Placing fractions and decimals on the empty number line • Double number line • Coloured fractions CMIT Facilitator training 2009
