1 Being Rational about Ratio and Multiplicative Reasoning John Mason Northampton Academy Dec 2009 The Open University Maths Dept University of Oxford Dept of Education

2 Doing & Undoing: +  What action undoes the action of ‘adding 3’?  What action undoes the action of ‘subtracting 5’?  What single action has the same effect as ‘adding 3’ followed by ‘subtracting 5’?  What actions have the same effect as undoing the previous compound action? Generalise! adding 5 then subtracting 3 subtracting (3 – 5)

3 Doing & Undoing: – from  What action undoes the action of ‘subtracting from 12’?  What single action has the same effect as ‘subtracting five’ then ‘subtracting from 12’? Generalise!

4 Doing & Undoing: x  What action undoes the action ‘multiply by 3’?  What action undoes the action ‘divide by 5’?  What single action has the same effect as ‘multiply by 3’ then ‘divide by 5’?  What actions undo the previous compound action? Generalise!  What action undoes ‘multiply by ‘? multiply by 5 then divide by 3 divide by 3/5

5 Doing & Undoing: ÷ into  What action undoes ‘divide into 24’?  What action undoes the compound action ‘divide by 6’ then ‘divide into 24’?  What action undoes the compound action ‘divide into 6’ then ‘divide into 24’? Generalise!

6 Doing & Undoing: exponents  What action undoes ‘square’?  What action undoes ‘take the cube-root’?  What compound action undoes ‘square’ then ‘take the cube-root’? Generalise!  What action undoes ‘raise 10 to the power of’?  What action undoes ’take the power to which 10 must be raised to give’?

7 Population  What is the approximate density of people in this room?  What is the population density of England? –population 61 x 10 6 ; area about 1.3 x 10 6 km 2 ?  What would be the population density if everyone in England arrived in Northants area about (2.3 x 10 3 km 2)  What if everyone in the world arrived (population about 6.7 x 10 9 )? Suitably informativ e units? What do you need to know?

8 Densities & Other Named Ratios  What other densities can you come up with? –pens per person; coins per person; names per person; siblings per person; –gm per cm 3 ; objects per cm 2 ; shops per km; cm water per hour  Density changes –People per km 2 per year in a region  What other ‘named’ or ‘per’ ratios are commonly used? –miles-per-gallon; litres-per-mile; –miles-per-hour; minutes-per-mile; –teaspoons-per-pound –calls-per-minute; minutes-per-call; –£-per-hour; £-per-minute In what context? Which way up? How do you think about ‘per’? How is it related to Arithmetic Mean?

9 Density Charts  Estimate the density of blue in each of the following, THEN calculate

10 Elastic Mathematics  Stretching an elastic band, what do you notice?  Hold one hand still and stretch the elastic. What is the same and what is different?  What is the same, and what is proportionate?

11 Berry Picking MeYouPot 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?

12 Scaling

13 Fraction Operators Raise your hand when you can see something that is three – sevenths of something else five – sevenths of something else seven – 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?

14 Same & Proportionate  What is the same and what proportionate about the relative sides of the squares?

15 Area Ratios  What is the ratio of the shaded area to the whole? (Idea due to Doug French)

16 Reasoning Without Numbers

17 Revealing 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.

18 Mill-Dienes-Bloor Animations

19 Wine-Water Transfers  In the basement there are two casks C A and C B of possibly different sizes and known mixtures W A : w A and W B : w B of wine to water.  In the middle of the night Person A transfers a bucketful from C A to C B, mixes thoroughly, then transfers a bucketful from C B back to C A.  In the middle of the night Person B transfers a pitcher-full from C B to C A, mixes thoroughly, and then transfers a pitcher-full from C A back to C B.  By comparing the new mixtures in the morning, can you determine who transferred first?

20 Magic Square Reasoning –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2 Any colour-symmetric arrangement?

21 More Magic Square Reasoning –= 0Sum( )Sum( )