Download + 1

A Natural
Way To
48
27
49
29
52
30
14
55
Summerschool
2015
17
8
18
41
21
20
19
60
40
38
65
64
39
63
62
37
36
59
42
22
7
35
58
23
1
Algebra
43
9
6
34
57
10
0
16
44
24
2
5
33
56
25
3
15
45
11
4
32
54
26
13
31
53
46
12
28
50
51
47
61
Martin Kindt
Freudenthal Institute
Algebra
the art of ‘thingification’
And now is a good moment to open up Pandora’s box
and explain one of the most powerful general weapons
in the mathematician’s armory, which we might call the
‘thingification of processes’.
Ian Stewart in Nature’s Numbers
Algebra at school
RESTRICTIONS
equations
inequalities
linear
programming
PROCESSES
(CHANGE)
operations
functions
graphs
PATTERNS
&
FIGURES
sequences
figured
numbers
ICME Berkeley 1980
1
1
2
3
5 8
13
Guess the rule !
? ?
?
?
2
3
5 8
13
2
3
5 8
89
13 21
5
34 5
1
1
1
1
f0 = 1
f10 = 89
f20 = 10946
f1 = 1
f11 = 144
f21 = 17711
f2 = 2
f12 = 233
f22 = 28657
f3 = 3
f13 = 377
f23 = 46368
f4 = 5
f14 = 610
f24 = 75025
f5 = 8
f15 = 987
f25 = 121393
f6 = 13
f16 = 1597
f26 = 196418
f7 = 21
f17 = 2584
f27 = 317811
f8 = 34
f18 = 4181
f28 = 514229
f9 = 55
f19 = 6765
f29 = 832040
Another pattern ?
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
Even or Odd? Explain your strategy!
A
E
C
D
B
F
From ‘PATTERNS and SYMBOLS’ (MIC)
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
ODD + ODD = EVEN
ODD + EVEN = ODD
choose five consecutive Fibonacci-numbers ....
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
+
24 = 3  8
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
+
39 = 3  13
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
+
699 = 3  233
representation of any 5 consecutive Fibonacci-numbers
first
in the style of
second
third
fourth
fifth
Euclid
PROOF!
+
a
a
b
b
+
a+b
b+a+b
a+b+b+a+b
a
b
b
a
a
b
b
b
a
a + (a + b + b + a + b) = (a + b) + (a + b) + (a + b)
b
+
a
a +  2a + 3b 
b
=
a+b
3a + 3b
=
3  (a + b)
a + 2b
2a + 3b
Progressive symbolization
more ‘Fibonacci-exercises’
* Take any subsequence of nine consecutive numbers.
The sum of the first and the ninth number equals ....
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
* Compare the sum of any six consecutive numbers with the 5th in the row ....
Which relation do you observe? Prove that it’s valid for all groups of six.
* Design your own Fibonacci-exercise
PATTERNS
&
FIGURES
Ian Stewart
A natural number is an idea that has
long ago been thingified so thoroughly
that everybody thinks of it as a thing.
figured numbers
Which pattern has the biggest number of dots?
Same question ....
Productive exercises
* Design a dice-pattern with 625 dots
* Design another dice- pattern with a number of dots
between 100 and 1000
Nikomachos of Gerasa
(ca. 100 AD)
sum of ‘odds’ = square number
sum of ‘evens’ = oblong number
Introduction to
Arithmetic
sum of consecutive numbers
=
triangular number
square number
n2
oblong number
oblong number
n  (n + 1)
or
n2 + n
triangular number
1
--- n
2
 (n + 1)
Air-show
a squadron flied in a ‘W-formation’
W-numbers
0
1
2
3
pattern-number
0
1
2
3
4
5
6
number of dots
1
5
...
...
...
...
...
Formula?
0
1
2
W=4n+1
3
0
1
2
W = [2n] + [2n + 1] = 4n + 1
3
W = double V  1
=
+
W = (1 + 2n) + (1+ 2n)  1 = 1 + 4n
-
W-squadrons sometimes can transform in square(dron)s
* Which ones? Why?
Sequence of W-numbers
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57
0
2
6
12
Sequence of W-numbers
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57
Is every
odd square
a W-number ?
2n + 1
2n + 1

2n + 1
4n2 + 2n
+
4n2 + 4n + 1 = 4(n2 + n) + 1
oblong number
4  oblong number + 1 = square number
4  n  n + 1  + 1 =  2n + 1 
2
number-strip
0
1
2
3
4 5
6
7
8
9
Which colour has 2015 ?
10
0
1
2
3
4 5
6
7
8
9
10
Add a white number to a blue one, what do you get?
Does the colour of the answer depend on the chosen
white and blue number?
+
2
1
3
6
9
12
4
6
9
12
7
9
12
10
5
8
11
14
17
20
23
15
18
21
24
15
18
21
24
27
15
18
21
24
27
30
12 15
18
21
24
27
29
31
13
15 18
21
24
27
30
33
36
16
18 21
24
27
30
33
36
39
19
21 24
27
30
33
36
39
42
22
24 27
30
33
36
39
42
45
using dot patterns
White + Blue = Red
+
=
=
+
R
W
B
R
R
W
B
W
W
B
R
B
B
R
W
0
1
2
3
4 5
6
7
8
9
10
What is the colour-pattern of : 0, 10, 20, 30, 40, ... ?
And what of : 0, 100, 200, 300, 400, ...?
And of : 0, 1000, 2000, 3000, 4000, ... ?
What is the colour of 2015 ?
2000 + 10 + 5
has same colour as
2 + 1+ 5 =8
The colour of a number =
the colour of the sum of its digits
Formulas
0
1
2
3
4
5
R = 3n
6
7
8
W = 1 + 3n
9
10
11
B = 2 + 3n
12
13
14
15
16
17
3n
1 + 3n
2 + 3n
n = 0,1,2,3, ...
start number
2
+5
7
+5
arithmetic
sequence
12
+5
17
+5
22
+5
27
2 + 5n
expression
n = number of steps
equal
steps
Adding
number-strips
(‘thingification’)
4
1
9
4
14
7
19
10
24
+
13
29
16
34
19
4 + 5n
+
1 + 3n
=
=
1
4
+5
4
9
14
19
24
+5
7
5
+8
+3
12
+3
19
10
+
13
=
26
33
29
16
40
34
19
47
4 + 5n
+
1 + 3n
=
5 + 8n
+8
2
5
8
5

11
14
=
17
20
5

=
2
10
+3
5
25
+15
+3
5
5


8
40
11
55
14
=
70
17
85
20
100
2 + 3n
+15
=
10 + 15n
0
0
0
1
1
2
1
7
4
6
3
19
9
12
6
37
16
20
10
61
0
0
0
1
1
2
1
7
4
6
3
19
9
12
6
37
16
20
10
61
25
30
15
91
36
42
21
127
n2
n2 + n
nn + 1
---------------------2
???
0
0
0
1
1
2
1
7
4
6
3
19
9
12
6
37
16
20
10
61
25
30
15
91
36
42
21
n2
n2 + n
nn + 1
---------------------2
?
127
???
1, 7, 19, 37, 61, ...
+6
1
+12
+18
+24
+30
7
19
37
Hexagonal numbers
1, 7, 19, 37, 61, ...
+6
+12
+18
+24
+30
‘honey-comb-numbers’
7
19
37
32+1=7
3  6 + 1 = 19
3  12 + 1 = 37
7
19
37
32+1=7
3  6 + 1 = 19
3n(n + 1) + 1
3  12 + 1 = 37
7
19
37
61+1=7
6  3 + 1 = 19
6  1--- n  n + 1  + 1 = 3n  n + 1  + 1
2
6  6 + 1 = 37
7
19
37
1 + 6 + 6  2 + 6  3 + ... + 6  n =
1 + 6 (1 + 2 + 3 + ... + n) =
1 + 6  1--- n  n + 1  = 3n  n + 1  + 1
2
48
27
49
29
52
30
14
54
26
13
55
34
57
18
35
58
8
7
6
41
21
20
19
60
40
38
63
62
61
65
64
39
37
36
59
42
22
9
1
17
43
23
10
0
16
44
24
2
5
33
56
11
4
32
45
25
3
15
31
53
46
12
28
50
51
47
Three propositions about algebra education
1
During the first two years of secondary education,
teachers should pay a great deal of attention to the
arithmetic side of algebra, i.e. algebra in relation
to (mental) arithmetic, number patterns, number theory
and combinatorial counting problems.
2
It is advisable to use historical contexts.
Babylonian, Egyptian, Greek, Indo and Arabic
mathematics have a great deal to offer in the
area of concrete (and discrete) algebra
3
Once students attain reasonable mastery over
enough tools or techniques, algebra can and should - be used to make proofs , for example
of special properties of natural numbers.
Homework (1)
* Make two numberstrips, one with the oblong numbers and
the other with the hexagonal numbers.
Adding these strips pairwise, you will find a strip of ...
square numbers!
How to prove this?
Homework (2)
* Check and prove the three given formulas to the diagonals.
Give the three missing formulas.
48
3(n + 1)2
27
49
29
52
30
14
54
34
23
8
7
6
18
35
42
22
9
1
17
43
24
2
5
44
10
0
16
33
25
3
15
45
11
4
32
55
26
13
31
53
46
12
28
50
51
47
21
20
19
36
41
3n2 + 4n + 1
39
38
37
40
3n2 + 3n
Welcome to Holland
3
0
4
1
5
2
9
12 15
18
7 10 13 16
19
8 11 14 17 20
6