Transcript SMC 2007 Algebra

Learning
to Think and to Reason
Algebraically
and the
Structure of Attention
John Mason
SMC
2007
1
Outline
Some
assumptions
Some tasks
Some reflections
2
Some assumptions
A
lesson without the opportunity for
learners to generalise
is not a mathematics lesson
Learners
come to lessons
with natural powers to make sense
Our
job is to direct their attention
appropriately and effectively
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Grid Sums
To move to the right
one cell you add 3.
To move up one cell
you add 2.
??
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In how many different ways can you work out a value for
the square with a ‘?’ only using addition?
Using exactly two subtractions?
4
Grid Movement
((7+3)x2)+3
is a path from 7 to ‘?’.
What expression
represents the reverse of
this path?
What values can ‘?’ have:
x2
÷2
?
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- if only + and x are used
- if exactly one - and one ÷
are used, with as many
+ & x as necessary
What about other cells?
Does any cell have 0? -7?
Does any other cell have 7?
-3
5
+3
Characterise ALL the possible
values that can appear in a cell
Varying & Generalising
What
are the dimensions of possible
variation?
What is the range of permissible
change within each dimension of
variation?
You only understand more if you
extend the example space or the
scope of generality
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Number Line Movements
Imagine a number line
Rotate
it about the point 5 through 180°
– where does 3 end up?
–…
Now
rotate it again bit about the point -2.
Now where does the original 3 end up?
Generalise!
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Number Spirals
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2
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5
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3
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2
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CopperPlate
Multiplication
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796
7964455
64789
64789
30
2420
361635
54242840
4236423245
28634836
497254
5681
63
5160119905
Four Odd Sums
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Tunja Sequences
-1 x -1 – 1 = -2 x 0
0 x 0 – 1 = -1 x 1
1x1–1=
2x2–1=
3x3–1=
4x4–1=
12
With
the
Grain
0x2
1x3
2 x 4 Across the Grain
3x5
Tunja Display (1)
…
…
(-1)x2 - (-1) - 2 = (-2)x3 - 1
0x2 - 0 - 2 = (-1)x3 - 1
1x2 - 1 - 2 = 0x3 - 1
2x2 - 2 - 2 = 1x1 - 1
3x2 - 3 - 2 = 2x1 - 1
4x2 - 4 - 2 = 3x1 - 1
5x2 - 5 - 2 = 4x1 - 1
…
1x3 - 1 - 3 = 0x2 - 1
2x3 - 2 - 3 = 1x2 - 1
3x3 - 3 - 3 = 2x2 - 1
4x3 - 4 - 3 = 3x2 - 1
5x3 - 5 - 3 = 4x2 - 1
…
…
…
…
…
…
…
…
…
(-1)x3 - (-1) - 3 = (-2)x2 - 1
0x3 - 0 - 3 = (-1)x2 - 1
Run Backwards
Generalise!
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Tunja Display (2)
…
4x(-1)x2 - 2x(-1) - 4x2 = (-4)x3 - 2
4x0x2 - 2x0 - 4x2 = (-2)x3 - 2
4x1x2 - 2x1 - 4x2 = 0x3 - 2
4x2x2 - 2x2 - 4x2 = 2x3 - 2
4x3x2 - 2x3 - 4x2 = 4x3 - 2
4x4x2 - 2x4 - 4x2 = 6x3 - 2
4x5x2 - 2x5 - 4x2 = 8x3 - 2
4x6x2 - 2x6 - 4x2 = 10x3 - 2
…
4x3x3 - 2x3 - 4x3 = 4x5 - 2
4x4x3 - 2x4 - 4x3 = 6x5 - 2
4x5x3 - 2x5 - 4x3 = 8x5 - 2
4x6x3 - 2x6 - 4x3 = 10x5 - 2
…
Run Backwards
Generalise!
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Structured Variation Grids
Generalisations in two dimensions
Available free at
http://mcs.open.ac.uk/jhm3
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One More
What numbers are one more than the
product of four consecutive
integers?
Let a and b be any two numbers, at least one of
them even.
Then ab/2 more than the product of:
any
number,
a
more
than
it,
b
more
than
it
and
a+b more than it,
is a perfect square,
of the number squared plus a+b times the
number plus ab/2 squared.
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Remainders of the Day (1)
Write
down a number which when
you subtract 1 is divisible by 5
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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Remainders of the Day (3)
Write down a number which is 1 more
than a multiple of 2
 and which is 2 more than a multiple
of 3
 and which is 3 more than a multiple
of 4
…

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Remainders of the Day (4)
Write down a number which is 1
more than a multiple of 2
 and 1 more than a multiple of 3
 and 1 more than a multiple of 4
…

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Magic Square Reasoning
2
2
6
7
2
1
5
9
8
3
4
Sum( ) – Sum(
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What other
configurations
like this
give one sum
equal to another?
Try to describe
them in words
) =0
More Magic Square Reasoning
Sum(
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) – Sum(
) =0
Perforations
How many holes
for a sheet of
r rows and c columns
of stamps?
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If someone claimed
there were 228 perforations
in a sheet,
how could you check?
Gasket Sequences
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Toughy
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7
6
5
4
3
2
1
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Powers
Specialising
& Generalising
Conjecturing
Imagining
Ordering
& Convincing
& Expressing
& Classifying
Distinguishing
Assenting
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& Connecting
& Asserting
Themes
Doing
& Undoing
Invariance
Freedom
& Constraint
Extending
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Amidst Change
& Restricting Meaning
Some Reflections
Notice
the geometrical term:
– It requires movement out of the current
space into a space of one higher
dimension in order to achieve it
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Attention
Gazing
at wholes
Discerning details
Recognising relationships
Perceiving properties
Reasoning on the basis of
properties
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John
Mason
J.h.mason @ open.ac.uk
http://mcs.open.ac.uk/jhm3
Developing Thinking in Algebra (Sage
2005)
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