1 Construction Tasks John Mason Open University & University of Oxford Flötur Selfoss Sept 2008

2 Outline  A suite of task Types for –Engaging learners –Extending & enriching their example spaces

3 Another & Another  Write down a pair of numbers whose difference is 2  and another pair  and another pair that you think no-one else in the room will write down  and another that perhaps no other human being has ever before written down!

4 Another & Another  Write down a pair of numbers whose product is 12  and another pair

5 Another & Another  Write down a pair of numbers whose product is 13  and another pair  and a pair that you think no-one else in the room will write down  and a pair that perhaps no human being has ever written down

6 Example Spaces  The examples that come to mind when you hear a word or see symbols  Dimensions of possible variation  Ranges of permissible change

7 Fractional Difference  Write down two fractions that differ by 3/4  and another pair  and a pair that make it as obscure as possible

8 Constrained Decimal  Write down a decimal number between 2 and 3  and which does not use the digit 5  and which does use the digit 7  and which is as close to 5/2 as possible

9 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 7  and another  Write down one which you think no-one else here will write down.

10 Remainders of the Day (2)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

11 Constrained Quadrilateral  Draw a quadrilateral  which has no right-angles  and which has one pair of equal sides  and which has one pair of parallel sides  and which has three different angles

12 Constrained Quadrilateral  Draw a quadrilateral  with a pair of equal edges  and with a pair of perpendicular edges  and with a pair of parallel edges  How many different ones can you find?

13 Perpendicularity  Draw a quadrilateral which has both pairs of opposite sides perpendicular  Trouble? –Try just one pair of opposite sides perpendicular

14 Sentenced 37 + – 37 = 49 Make up your own like this 3 ÷ 4 = 15 ÷ Make up your own like this What is the ‘like this’ of your example?

15 Distribution  Write down five numbers whose arithmetic mean is 5 –What are the dimensions of possible variation: how much freedom?  and whose median is 6 –how much freedom now?  and whose mode is 7 –how much freedom now?

16 Task Types  Another and Another  One that no-one else will write down  An easy example of … A hard example of … A general example of …  One that will challenge …  Meeting successive constraints All mathematics tasks can be seen as construction tasks

17 For More Details Thinkers (ATM, Derby) Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby) Mathematics as a Constructive Activity (Erlbaum) Structured Variation Grids Studies in Algebraic Thinking Other Publications This and other presentations