Transcript Iceland Alg & Geom

Thinking
Algebraically & Geometrically
John Mason
University of Iceland
Reykjavik 2008
1
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.
2
Remainders of the Day (2)
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
…
3
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
…
4
Remainders of the Day (4)
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?
5
Some Sums
1+2= 3
4+5+6= 7+8
= 13 + 14 + 15
9 + 10 + 11 + 12
16 + 17 + 18 + 19 + 20
= 21 + 22 + 23 + 24
Generalise
Say What You See
Justify
Watch What You Do
6
Cubelets
Say What You See
7
Differences
8
1  1 1
1 1 1
7 6 42
2 1 2
1 11
1 1 1  1 1  11
3 2 6
8 7 56 6 24 4 8
Anticipating
1  1 1  1  1
Generalising
4 3 12 2 4
Rehearsing
1 1 1
5 4 20
Checking
1  1  1  1 1 1 1  1  1
Organising
6 5 30 2 3 3 6 4 12
Word Problems
In 26 years I shall be twice as old as I was
19 years ago. How old am I?
40 + 26
?=?
= 2( 40- 19)
?
?
26
19
?
19
9
Mid-Point
Where
can the
midpoint of the
segment joining
two points each
on a separate
circle, get to?
10
Scaling
Q
Imagine a circle C.
Imagine also a point P.
M
P
Now join P to a point Q on C.
Now let M be the mid point of PQ.
What is the locus of M
as Q moves around the circle?
11
Map Drawing Problem
 Two
people both have a copy of the same map of
Iceland.
 One uses Reykjavik as the centre for a scaling by a
factor of 1/3
 One uses Akureyri as the centre for a scaling by a factor
of 1/3
 What is the same, and what is different about the maps
they draw?
12
0
Difference
Divisions
1
2
4–2=4÷2
1
2
–3=4
5
1
3
–4=5
÷1 4
6
1
4
–5=6
1
÷
4
7
1
5
–6=7
÷1 6
4
13
1
÷
2
3
1
–
(-1)
=
-2
1
–0=1
-1
1
oops
1
–2=3
1
1
0
÷-2(-1)
1
÷-1 oops
1
÷
1
2
3
How does this fit in?
3
5
5
Going with the grain
Going across the grain
Four Consecutives
Write
down four consecutive
numbers and add them up
and another
and another
Now be more extreme!
What is the same, and what is
different about your answers?
+1
+2
+3
4
14
+6
One More
What numbers are one more than
the product of four consecutive
integers?

Let a and b be any two numbers, 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.
15
Gasket
16
Leibniz’s Triangle
1
1
2
1
3
1
4
1
5
1
6
17
1
6
1
12
1
20
1
30
1
2
1
3
1
12
1
30
1
60
1
4
1
20
1
60
1
5
1
30
1
6
How Much Information?
How
much information
about lengths do you need in order to work out
–the perimeter?
–the area?
How
few rectangles needed to compose
it?
Design a rectilinear region requiring
18
– 3 lengths to find the perimeter and
– 8 lengths to find the area
More Or Less Perimeter & Area
Draw a rectilinear figure which requires at least
4 rectangles in any decomposition
are
Perimetera
more
19
more
more perim
more area
same
same perim
more area
less
less perim
more area
same
less
more perim more perim
same area less area
same perim
less area
less perim
same area
less perim
less area
Dina Tirosh & Pessia
Tsamir
Two-bit Perimeters
What perimeters are possible
using only 2 bits of
information?
2a+2b
a
20
b
Two-bit Perimeters
What perimeters are possible
using only 2 bits of
information?
4a+2b
a
21
b
Two-bit Perimeters
What perimeters are possible
using only 2 bits of
information?
6a+2b
22
a
b
Two-bit Perimeters
What perimeters are possible
using only 2 bits of
information?
6a+4b
23
a
b
Parallelism
How many
angles do you
need to know
to work out all
the angles?
24
Kites
25
Seven Circles
How many
different angles
can you discern,
using only the red
points?
How do you
know you have
them all?
How many
different
quadrilaterals?
26
Square Count
27
Ratios in Rectangles
28
Some Mathematical Powers
Imagining
& Expressing
Specialising & Generalising
Conjecturing & Convincing
Stressing & Ignoring
Ordering & Characterising
Distinguishing & Connecting
Assenting & Asserting
29
Some Mathematical Themes
Doing
and Undoing
Invariance in the midst of Change
Freedom & Constraint
30
Worlds of Experience
Inner
World of
imagery
World of
Symbol
s
Material
World
enactive
31
iconic
symbolic
Structure of the Psyche
Awareness (cognition)
Imagery
Will
Emotions
(affect)
Body (enaction)
Habits
Practices
32
Structure of a Topic
Language Patterns
& prior Skills
Imagery/Senseof/Awareness; Connections
Root Questions
predispositions
Different Contexts in which
likely to arise;
dispositions
Standard Confusions
& Obstacles
Techniques & Incantations
Emotion
Only Emotion is Harnessable
Only Awareness is Educable
33
Only Behaviour is Trainable
Mathematics & Creativity
Creativity
is a type of energy
It is experienced briefly
It can be used productively or thrown away
Every opportunity to make a significant
choice is an opportunity for creative energy to
flow
It also promotes engagement and interest
For example
– Constructing an object subject to constraints
– Constructing an example on which to look for or
try out a conjecture
– Constructing a counter-example to someone’s
assertion
34