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Being Rational about RatioBeing Rational about Ratioandand

Multiplicative ReasoningMultiplicative Reasoning

John MasonJohn Mason

Northampton AcademyNorthampton Academy

Dec 2009Dec 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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Doing & Undoing: xDoing & Undoing: x

What action undoes the action ‘multiply by 3’?What action undoes the action ‘multiply by 3’? What action undoes the action ‘divide by 5’?What action undoes the action ‘divide by 5’? What single action has the same effect asWhat single action has the same effect as

‘multiply by 3’ then ‘divide by 5’? ‘multiply by 3’ then ‘divide by 5’? What actions undo the previous compound What actions undo the previous compound

action?action?

Generalise!

What action undoes ‘multiply by What action undoes ‘multiply by ‘? ‘?

multiply by 5 then divide by 3

divide by 3/5

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Doing & Undoing: ÷ intoDoing & Undoing: ÷ into

What action undoes ‘divide into 24’?What action undoes ‘divide into 24’? What action undoes the compound actionWhat action undoes the compound action

‘divide by 6’ then ‘divide into 24’?‘divide by 6’ then ‘divide into 24’? What action undoes the compound actionWhat action undoes the compound action

‘divide into 6’ then ‘divide into 24’?‘divide into 6’ then ‘divide into 24’?

Generalise!

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Doing & Undoing: exponentsDoing & Undoing: exponents

What action undoes ‘square’? What action undoes ‘square’? What action undoes ‘take the cube-What action undoes ‘take the cube-

root’?root’? What compound action undoes What compound action undoes

‘square’ then ‘take the cube-root’?‘square’ then ‘take the cube-root’?Generalise!

What action undoes ‘raise 10 to the What action undoes ‘raise 10 to the power of’?power of’?

What action undoes ’take the power to What action undoes ’take the power to which 10 must be raised to give’?which 10 must be raised to give’?

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PopulationPopulation

What is the approximate density of people What is the approximate density of people in this room?in this room?

What is the population density of England?What is the population density of England?– population 61 x 10population 61 x 1066; area about 1.3 x 10; area about 1.3 x 106 6 kmkm22??

What would be the population density if What would be the population density if everyone in England arrived in Northants area everyone in England arrived in Northants area about (2.3 x 10about (2.3 x 1033 km km2)2)

What if everyone in the world arrived What if everyone in the world arrived (population about 6.7 x 10(population about 6.7 x 1099)?)?

Suitably informat

ive units?

What do you

need to know?

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Densities & Other Named Densities & Other Named RatiosRatios

What other densities can you come up with?What other densities can you come up with?– pens per person; coins per person; names per person; pens per person; coins per person; names per person;

siblings per person;siblings per person;– gm per cmgm per cm33; objects per cm; objects per cm22; shops per km; cm water ; shops per km; cm water

per hourper hour Density changesDensity changes

– People per kmPeople per km22 per year in a region per year in a region What other ‘named’ or ‘per’ ratios are commonly What other ‘named’ or ‘per’ ratios are commonly

used? used? – miles-per-gallon; litres-per-mile; miles-per-gallon; litres-per-mile; – miles-per-hour; minutes-per-mile;miles-per-hour; minutes-per-mile;– teaspoons-per-poundteaspoons-per-pound– calls-per-minute; minutes-per-call;calls-per-minute; minutes-per-call;– £-per-hour; £-per-minute£-per-hour; £-per-minute

In what context?Which way up?

How do you think about ‘per’?How is it related to Arithmetic Mean?

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Density ChartsDensity Charts

Estimate the density of blue in each of Estimate the density of blue in each of the following, THEN calculatethe following, THEN calculate

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Elastic Mathematics Elastic Mathematics Stretching an elastic band, what do you Stretching an elastic band, what do you

notice?notice? Hold one hand still and stretch the Hold one hand still and stretch the

elastic. What is the same and what is elastic. What is the same and what is different?different?

What is the same, and what is What is the same, and what is proportionateproportionate??

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Berry PickingBerry Picking

Me You Pot

If you have 330, how many do I have and how many in the pot?If there were 1899 berries altogether, how many does each get?

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ScalingScaling

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Fraction OperatorsFraction Operators

Raise your hand when you can see something that isthree – sevenths of something elsefive – sevenths of something elseseven – fifths of something else

What fraction of the whole is the yellow?What fraction of the whole is the cyan?What fraction of the yellow is the cyan?

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Same & ProportionateSame & Proportionate

What is the same and what proportionate What is the same and what proportionate about the relative sides of the squares?about the relative sides of the squares?

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Area RatiosArea Ratios What is the ratio of the shaded area to the whole?What is the ratio of the shaded area to the whole?

(Idea due to Doug French)

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Reasoning Without NumbersReasoning Without Numbers

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Revealing ShapesRevealing Shapes

The coloured shapes on the right are somewhere in the white cells on the left.By clicking on a shape and then on a cell, you can test your conjectures as to which shape is where.

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Mill-Dienes-Bloor AnimationsMill-Dienes-Bloor Animations

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Wine-Water TransfersWine-Water Transfers

In the basement there are two casks CIn the basement there are two casks CAA and C and CBB of possibly different sizes and known mixtures of possibly different sizes and known mixtures WWAA : w : wAA and W and WBB : w : wBB of wine to water. of wine to water.

In the middle of the night Person A transfers In the middle of the night Person A transfers a bucketful from Ca bucketful from CAA to C to CBB, mixes thoroughly, , mixes thoroughly, then transfers a bucketful from Cthen transfers a bucketful from CBB back to C back to CAA..

In the middle of the night Person B transfers a In the middle of the night Person B transfers a pitcher-full from Cpitcher-full from CBB to C to CA,A, mixes thoroughly, mixes thoroughly, and then transfers a pitcher-full from Cand then transfers a pitcher-full from CAA back back to Cto CBB..

By comparing the new mixtures in the By comparing the new mixtures in the morning, can you determine who transferred morning, can you determine who transferred first?first?

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Magic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

7

– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?

2

2

Any colour-Any colour-symmetric symmetric

arrangement?arrangement?

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More Magic Square ReasoningMore Magic Square Reasoning

– = 0Sum( ) Sum( )