Download Flotur Construction Tasks

Construction Tasks
John Mason
Open University & University of Oxford
Flötur
Selfoss
Sept 2008
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Outline
A
suite of task Types for
– Engaging learners
– Extending & enriching their example spaces
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Another & Another
Write
down a pair of numbers whose
difference is 2
and another pair
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!
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Another & Another
Write
down a pair of numbers whose
product is 12
and another pair
and another pair
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Another & Another
Write
down a pair of numbers whose product
is 13
and another pair
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
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Example Spaces
The
examples that come to mind
when you hear a word or see
symbols
Dimensions of possible variation
Ranges of permissible change
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Fractional Difference
Write
down two fractions that differ
by 3/4
and another pair
and another pair
and a pair that make it as obscure as
possible
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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
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Remainders of the Day (1)
Write
down a number which when
you subtract 1 is divisible by 7
and another
and another
Write down one which you think
no-one else here will write down.
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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?
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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
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Perpendicularity
Draw
a quadrilateral which has both
pairs of opposite sides
perpendicular
Trouble?
– Try just one pair of opposite sides
perpendicular
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Sentenced
37 +
– 37 = 49
3 ÷ 4 = 15 ÷
Make up your
own like this
Make up your
own like this
What is the ‘like this’
of your example?
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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?
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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
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All mathematics tasks
can be seen as
construction tasks
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
http://mcs.open.ac.uk/jhm3
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