VIEWS: 15 PAGES: 18 POSTED ON: 7/26/2011
SNP Workshop 5 Fraction Concepts c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Learning intentions 1) To develop teacher understanding of the strategies and knowledge required in the fraction, ratio and proportion domain, in particular, fraction concepts. 2) To enable teachers to plan a lesson sequence for fraction concepts based on existing student knowledge c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Workshop Overview 1) Reflection Time (a) Where are people up to at the moment? [individual report back] (b) What issues/problems do people currently have? (c) What has gone well? Any good activities to share? 2) Learning intentions 3) Starters (a) Uncover one at a time, answer “question” at bottom of page (b) ½ of 56 students may treat halving as an operation (like doubling) introduce ideas of fraction constructs - halving as the operator construct – start list on right hand side of board may wish to integrate this work with / teaching (c) ¼ of 36 halve and halve again relationship between times tables and division and fractional strategies Draw a picture to show what ¼ of 36 looks like. {Ask: What is the whole?] (36 4 = 9 by sharing out counters. Extending this to asking “what is one quarter of 36”) Issue of what happens when a student is eating a block of chocolate with 7 pieces in it. If eat 4 pieces how much is left? – Students saying “3 pieces” are reunitising – changing the unit to the piece rather than maintaining the block as the whole. ie repackaged Getting students to do the problem with sevenths – actually doing 1 – 4/7 (d) 2/3 of 21 built on the idea that 1/3 of 21 is 7 students need to be multiplicative thinkers to be able to answer this 4) (a) What does two thirds mean? Two out of three equal sized pieces So this is not cut into thirds (though some families may share pizzas like this…) c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Does P(X) = 2/3 fit here?… [add this to the list of constructs] (b) draw a picture to show two thirds discuss this as the part-whole construct of fractions (add it to the list). Research suggests that understanding the part-whole construct is essential to understanding fractions. introduce the concept of discrete and continuous fraction models, how they differ and to where they lead [OHT] Which is the bigger fraction? Many miss the comparative nature of fractions – the relationship between the numerator and denominator. (Some research suggests that the failure to understand this is the reason why students have difficulty with fractions) (c) so what do you “understand” when you say you understand what 2/3 means? In 2001 42% of Y7 & 8 students could not name the fractions ½, 1/3 and ¼ Issues of representation – verbal/mental – materials – symbol translations [diagram on board] add in written words. handout 5) Line-ups (a) Fraction activities – fraction line-ups [cards] [get teachers to do these as if they are students] (i) ½ to 1/6 why are you where you are? Example of the need to tie up the material pieces with the symbol and a language based understanding. (Put pieces on OHT to show and hand to those with the cards) (ii) harder ones why are you where you are? Discuss strategies to sort out which is bigger – equivalent, half, bigger or smaller than half, knowing size of piece, etc (iii) what curriculum levels do you think these activities are at? c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc [level 3 – but split into i and ii] What stages do they relate to? [ stage 5 and 8] Wave the maths exemplar with these questions (b) Sorting fractions from Number Sense 4-6 [handout] close to zero, close to one etc building the idea of coordinating the numerator and denominator 6) A progression for fractions (a) Outline of a fraction progression [OHT and handout] 7) More starter questions… [OHT] (a) Discuss the problems in pairs. (b) Choose volunteers to feed back to the group 8) Teaching demonstration using fractions materials Unifix or multilink cubes Chewing gum packs with 5 and 4 pieces Sums to one 4 reds to six yellows as a ratio, as fractions and percentages Fraction circles Naming fractions and size of the pieces Improper fractions and mixed numbers Equivalent fractions Number stick Avalon results – 1/77 naming ¾ and 1¼ correct. Hardest construct for students Measure construct for fractions (add to list) Need to use whole numbers first, and teach conventions of this construct Need to get students to draw a number stick and discuss the key conventions Locating fractions with spaces left, and without marked divisions Starter activity is to use a piece of paper and get kids to fold it in half [progression of student strategies] Fraction dominoes [handout] 9) Some more fraction constructs… (a) quotient construct 2/3 as 2 3 (b) ratio construct with 2/3 as the ratio 2:1 (c) probability construct – how does this utilise the concept of 2/3? (d) handout of fraction constructs 10) Looking higher up this domain… c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc (a) Teaching percentage, ratio, fraction manipulation. [Note that some may decide to do their decimal teaching here too] 11) Identifying resources (a) Rangi’s square pizzas [important resource to support idea that fractions must have equally sized pieces] what curriculum level is the activity at? [2?] What numeracy skills are needed for the activity? [fractions, ] What stage is the activity at? [3-4] (b) Fraction games – set of cards [handout] 12) Group planning for fractions [if time…] [OHT] 13) For next time [OHT] Read articles – fractions Beginning to learn fractions c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Resources needed Beforehand remind people to bring along copies of/materials for one good activity used in the past to teach fractions, along with any equipment needed for the activity and any resources prepared for practice or independent groups relating to the activity. “Number Framework” and Getting Started Booklets. Set of FIO On site Board, pens and duster OHP Sets of “Figure it out” are needed for the planning session To Take Newsprint, blue-tac and vivids Box of numeracy equipment Multilink or unifix Fraction circles Number stick Fractions dominoes Handouts Issues of representation Sorting fractions (from Number Sense 4-6) Progression for fractions Fraction constructs Set of fraction cards and games to use them c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Starters 1) Half of 56 2) Quarter of 36 3) Two thirds of 21 4) Half an apple At which stage do you think it would be reasonable to pose each of these problems? What underlying knowledge is needed? c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Discrete and continuous fraction models Continuous fraction model Fractions are met as cutting up an entire object This single object soon extends to multiple objects to be shared For example:sharing two apples between 5 people Leads to the concept that fractions are divisions Discrete fraction model Fractions are met through sharing a collection of objects Problems initially lead to discrete, whole number answers Later problems merge with the continuous as “remainder” objects are cut c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc A teaching progression for fractions Discrete Continuous Stage 2 Stage 2 Grouping [and sharing] Understanding terms like half an apple, quarter of an orange Understand ordinal numbers (second, third, fourth…) Stage 3 Recognise the words for ordinal numbers Stage 3/4 Stage 3/4 Skip counting [and skip Idea that fractions are sharing and repeated of “equal size and shape” subtraction] Recognise symbols for Doubling and Tens ordinal numbers [Halving and Fives] Symbols for halves, Solves sharing problems thirds, quarters, fifths … involving whole numbers by dealing out the Identifying what fraction amount is shaded c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Stage 4/5 Stage 4/5 Quarters and thirds. Symbols for all fractions with denom 2, 3, 4, 5, 10 Fraction of a number by [at least] – inc improper addition fractions Solves sharing problems Ordering fractions with by dealing out the the same denominator amount, dividing up left- overs by dividing and Order unit fractions sharing halves then qaurters… Stage 5/6 Stage 5/6 Fraction of a number by Placing fractions on and multiplication [simple reading fractions from fractions] number lines Solves sharing problems Converting improper by dealing and fractions and mixed appropriate division (& numbers sharing) of remainders Stage 6/7 Stage 6/7 Fraction of a number by Equivalent fractions (& multiplication [more simplifying fractions) c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc complex fractions] [simple denominators] What fraction? Items Fraction – decimal – with same units) percentage conversions [simple cases] Fractions as divisions Stage 7 +/- (same denominator) Stage 7/8 Stage 7/8 What fraction? Ordering fractions with (involving unit different denominators conversion) Given fraction conversions – eg eighths to decimals and percentages Reciprocals Simple +/- (different denominators) c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc NOTE: For a more detailed explanation of what some of these statements mean, refer to the fraction questions in the diagnostic interview and examples in the Number Framework c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc A teaching progression for fractions Discrete Continuous Stage 2 Stage 2 Grouping [and sharing] Understanding terms like half an apple, quarter of an orange Understand ordinal numbers (second, third, fourth…) Stage 3 Recongnise the words for the ordinal numbers Stage 3/4 Stage 3/4 Skip counting [and skip sharing and Idea that fractions are of “equal size and repeated subtraction] shape” Recognise the symbols for th ordinal Doubling and Tens [Halving and Fives] numbers Symbols for halves, thirds, quarters, Solves sharing problems involving whole fifths … numbers by dealing out the amount Identifying what fraction is shaded Stage 4/5 Stage 4/5 Quarters and thirds. Symbols for all fractions with denom 2, 3, 4, 5, 10 [at least] – inc improper fractions Fraction of a number by addition Ordering fractions with the same Solves sharing problems by dealing out denominator the amount, dividing up left-overs by dividing and sharing halves then Order unit fractions qaurters… Stage 5/6 Stage 5/6 Fraction of a number by multiplication Placing fractions on and reading fractions [simple fractions] from number lines Solves sharing problems by dealing and Converting improper fractions and mixed appropriate division (& sharing) of numbers reaminders c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Stage 6/7 Stage 6/7 Fraction of a number by multiplication Equivalent fractions (& simplifying [more complex fractions] fractions) [simple denominators] What fraction? Items with same units) Fraction – decimal – percentage Fractions as divisions conversions [simple cases] Stage 7 +/- (same denominator) Stage 7/8 Stage 7/8 What fraction? (involving unit Ordering fractions with different conversion) denominators Given fraction conversions – eg eighths to decimals and percentages Reciprocals Simple +/- (different denominators) c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc More starter questions… 2 1) Find 4 fractions equivalent to 7 2) Simplify the following 36 (a) 72 28 (b) 35 3) In groups of 2 or 3, create a brief outline for a lesson that gets students to learn how to 1 4 1 to convert 3 3 (and vice versa) without being taught an algorithm c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc Group planning for fractions In pairs or threes 1) Pick one stage of the progression 2) Think of as many relevant fraction based activities as possible for that stage Consult the Getting Started book Consult the activity books Consult Figure It Out Think of good activities that were used in the past – and still fit Think them up Record your activities on newsprint 3) Reference everything – so someone doing the later planning can find them c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc For next time… 1) Read through the fraction booklet, and the readings provided 1) Continue planning the fraction lesson series c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc