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Construction TasksConstruction Tasks

John MasonJohn MasonOpen University & University of OxfordOpen University & University of Oxford

FlFlööturturSelfossSelfoss

Sept 2008Sept 2008

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OutlineOutline

A suite of task A suite of task Types Types forfor– Engaging learnersEngaging learners– Extending & enriching their example Extending & enriching their example

spacesspaces

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Another & AnotherAnother & Another

Write down a pair of numbers whose Write down a pair of numbers whose difference is 2difference is 2

and another pairand another pair and another pairand another pair and another pair that you think no-and another pair that you think no-

one else in the room will write downone else in the room will write down and another that perhaps no other and another that perhaps no other

human being has ever before written human being has ever before written down!down!

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Another & AnotherAnother & Another

Write down a pair of numbers Write down a pair of numbers whose product is 12whose product is 12

and another pairand another pair and another pairand another pair

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Another & AnotherAnother & Another

Write down a pair of numbers whose Write down a pair of numbers whose product is 13product is 13

and another pairand another pair and another pairand another pair and a pair that you think no-one else in and a pair that you think no-one else in

the room will write downthe room will write down and a pair that perhaps no human and a pair that perhaps no human

being has ever written downbeing has ever written down

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Example SpacesExample Spaces

The examples that come to The examples that come to mind when you hear a word or mind when you hear a word or see symbolssee symbols

Dimensions of possible Dimensions of possible variationvariation

Ranges of permissible changeRanges of permissible change

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Fractional DifferenceFractional Difference

Write down two fractions that Write down two fractions that differ by 3/4differ by 3/4

and another pairand another pair and another pairand another pair and a pair that make it as and a pair that make it as

obscure as possibleobscure as possible

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Constrained DecimalConstrained Decimal

Write down a decimal number Write down a decimal number between 2 and 3between 2 and 3

and which does not use the digit and which does not use the digit 55

and which does use the digit 7and which does use the digit 7 and which is as close to 5/2 as and which is as close to 5/2 as

possiblepossible

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Remainders of the Day (1)Remainders of the Day (1)

Write down a number which Write down a number which when you subtract 1 is divisible when you subtract 1 is divisible by 7by 7

and anotherand another and anotherand another Write down one which you think Write down one which you think

no-one else here will write down.no-one else here will write down.

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Remainders of the Day (2)Remainders of the Day (2)

Write down a number which when Write down a number which when you subtract 1 is divisible by 2you subtract 1 is divisible by 2

and when you subtract 1 from the and when you subtract 1 from the quotient, the result is divisible by quotient, the result is divisible by 33

and when you subtract 1 from that and when you subtract 1 from that quotient the result is divisible by 4quotient the result is divisible by 4

Why must any such number be Why must any such number be divisible by 3? divisible by 3?

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Constrained QuadrilateralConstrained Quadrilateral

Draw a quadrilateralDraw a quadrilateral which has no right-angleswhich has no right-angles and which has one pair of equal and which has one pair of equal

sidessides and which has one pair of and which has one pair of

parallel sidesparallel sides and which has three different and which has three different

anglesangles

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PerpendicularityPerpendicularity

Draw a quadrilateral which has Draw a quadrilateral which has both pairs of opposite sides both pairs of opposite sides perpendicularperpendicular

Trouble?Trouble?– Try just one pair of opposite Try just one pair of opposite

sides perpendicularsides perpendicular

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SentencedSentenced

37 + – 37 = 49Make up

your own like

this

3 ÷ 4 = 15 ÷ Make up

your own like

thisWhat is the ‘like this’

of your example?

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DistributionDistribution

Write down five numbers Write down five numbers whose arithmetic mean is 5whose arithmetic mean is 5– What are the dimensions of What are the dimensions of

possible variation: how much possible variation: how much freedom?freedom?

and whose median is 6and whose median is 6– how much freedom now?how much freedom now?

and whose mode is 7and whose mode is 7– how much freedom now?how much freedom now?

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Task TypesTask Types

Another and AnotherAnother and Another One that no-one else will write One that no-one else will write

downdown An easy example of …An easy example of …

A hard example of …A hard example of …A general example of …A general example of …

One that will challenge …One that will challenge … Meeting successive constraintsMeeting successive constraints

All mathematics taskscan be seen as

construction tasks

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For More DetailsFor More Details

Thinkers (ATM, Derby)Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby)Mathematics as a Constructive Activity (Erlbaum)

http://mcs.open.ac.uk/jhm3

Structured Variation GridsStudies in Algebraic ThinkingOther PublicationsThis and other presentations