Algebra

1
Algebra
Promoting Mathematical Thinking
Expressing GeneralityInterpreting Diagrams
John Mason PTIHaringey 2014
2
Conjectures and Assumptions

• Everything said here is a conjecture
• to be tested in your experience!
• Enjoyment (in mathematics) arises from acts
• the exercise of internalised skill
• the use of natural powers
• the development of those powers
• What you get from this talk is most likely what
  you notice happening inside you, in order to be
  sensitised to what learners may be experiencing

3
Matched Array

• In how many different ways can you find how many
  matchsticks would be needed to make an array like
  this with 16 columns and 12 rows?
• Could exactly the same number of matchsticks be
  used to make a different sized array?
• Generalise!
• How many vertices does the array have, and how
  many squares?
• (Check Vertices Matchsticks Squares 1)

4
Matched Array

• I how many different ways can you work out how
  many matchsticks would be needed to make an array
  like this with 16 columns and 12 rows?
• Could exactly the same number of matchsticks be
  used to make a different sized array? Generalise!

Suppose r rows and c columns of squares Then
horizontal matches need r(c 1)
matchsticks Vertical matches need c(r 1)
matchsticks Total r(c 1) c(r 1) 2rc r
c So 2x12x16 12 16 412 matchsticks required

5
Undoing

• Suppose N 2rc r c
• Then 2N 4rc 2r 2c
• 2N 1 (2r 1)(2c 1)
• Since 2x412 1 825 3x5x5x11, selecting two
  factors gives
• 3x11 and 5x5 that is 16 rows and 12 columns
• 3x5 and 5x11 that is 7 rows and 27 columns
• 3 and 5x5x11 that is 1 row and 137 columns
• 5 and 3x5x11 that is 2 rows and 82 columns
• 11 and 3x5x5 that is 5 rows and 37 columns

6
Extensions

• Notice that there are 2rc r c matchsticks
• rc squares
• (r 1)(c 1)
  vertices
• So Vertices Edges Faces 1 (Euler)

Try putting a diagonal matchstick in each
square Try three dimensions Try other arrays
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Sundarams Sieve
Each row and each column is an arithmetic
progression
Claim N will appear in the table iff 2N 1 is
composite
What number will appear in the Rth row and the
Cth column?
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One Sum

• I have written down two different numbers whose
  sum is 1.
• I have squared the larger and added the smaller,
  and
• I have squared the smaller and added the larger.
• Which of my two answers will be the larger?

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One Sum Diagrams
1
1
1
10
One Sum Diagrams
x
y
z
1
c
c
1
b
b
a
a
x
y
z
1
ax (ab)y z a b(yz) cz ax by
cz ay bz az
11
Reading a Diagram
x2 (1-x)2
x3 x(1x) (1-x)3
x2z x(1-x) (1-x)2(1-z)
xz (1-x)(1-z)
xyz (1-x)y (1-x)(1-y)(1-z)
yz (1-x)(1-z)
12
Routes

• How many ways are the for getting from the upper
  left corner to bottom right corner moving only to
  the right or down along an edge?

Denote by r, a move to the right by one edgeand
by d, a move down one edge
Then a journey consist of a sequence of 5 rs and
3 ds in some order Which is the same as the
number of terms involving 5 rs and 3ds in the
expansion of (r d) (r d)(r d)(r d)(r
d) (r d) (r d)(r d) (r d)8
13
(No Transcript)
14
More Routes

• Now how many ways are the for getting from the
  upper left corner to bottom right corner moving
  only to the right or down along an edge?

1
1
1
1
1
1
3
5
7
9
11
1
5
13
25
41
61
1
7
25
63
129
231
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Completing The Square (Babylonian Style)
x2
bx
x
c
x
b
c
16
a
1
a
a
x2
x
x
1
bx
1
c
ax2
x
bx
x
a
1
x
b
c
c
x
a
x
x
a
-
1
c
x
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Completing the Square


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Generalised Completing the Square
a
2b
c



x2
x2
x2
a
2b
c



x2
x2


a



19
Completing the rectangle
a

N
N
a bc
(ax c)(ay b)
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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

25
26 27 28 29 30
31 32 33 34 35
Generalise
Say What You See
Watch What You Do
Justify
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1 2
3
1
4 5 6
7 8
2
13 14 15
9 10 11 12
3
16
17 18 19 20

21 22 23 24
4

25
26 27 28 29 30
5
31 32 33 34 35
Generalise






Justify
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Consecutive Sums
Say What You See
23
Algebra Readings
Say What You See
Say What You See
Expresssymbolically
Expresssymbolically
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Mutual Factors
But there are others!
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Follow Up

• Developing Thinking in Algebra (Sage)
• Thinking Mathematically (Pearson)
• Questions Prompts for Mathematical Thinking
  (ATM)
• J.h.mason_at_open.ac.uk
• Mcs.open.ac.uk/jhm3 (presentations Structured
  Variation Grids)