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PROGRESSION IN NUMBER AND
ALGEBRA
Objectives
The objectives vary depending on which attainment target(s) are relative to the learning objectives of your lesson
Addition ArithmagonsAddition Arithmagons
3
8
4
12 11
7
8 + 4 = ?8 + 4 = ?
8 + 3 = ?8 + 3 = ?
4 + 3 = ?4 + 3 = ?
The number in the square The number in the square is equal to the sum of the is equal to the sum of the
circles connected to itcircles connected to it
Addition ArithmagonsAddition Arithmagons
3
8
4
12 11
7
Sum of the circles?Sum of the circles?
Sum of the squares?Sum of the squares?
ΣΣ = 2 x = 2 x Σ Σ
15
30
Addition ArithmagonsAddition Arithmagons
3
8
4
12 11
7
Can you make Can you make 1515
another way?another way?
ΣΣ = 15 = 15
ΣΣ = = ++
Addition ArithmagonsAddition Arithmagons
3
8
4
12 11
7
ΣΣ = = ++
ΣΣ = 2 x = 2 x Σ Σ
ΣΣ = = Magic Magic
NumberNumber+ = + = Magic Magic
NumberNumber
Addition ArithmagonsAddition Arithmagons
10
4
7
11 14
17
Sum of the squares?Sum of the squares?
Magic number?Magic number?
42
21
??
??
??
12
Addition ArithmagonsAddition Arithmagons
15
-1
11 14
27
Sum of the squares?Sum of the squares?
Magic number?Magic number?
52
26
??
??
??
12
Addition ArithmagonsAddition Arithmagons
15
-1
11 14
27
So, if the numbers So, if the numbers in the squares do in the squares do
not satisfy the not satisfy the triangle inequality triangle inequality
then you get zero or then you get zero or negative numbers?negative numbers?
6.5 9.5
4.5
Addition ArithmagonsAddition Arithmagons
11 14
16
Sum of the squares?Sum of the squares?
Magic number?Magic number?
41
20.5
??
??
??
4½ 7½
6½
Addition ArithmagonsAddition Arithmagons
11 14
12
So, if the sum of the So, if the sum of the squares is odd then squares is odd then
you get fractions you get fractions and/or decimals and/or decimals
numbers!numbers!
6a a
3a
Addition ArithmagonsAddition Arithmagons
9a 4a
7a
Sum of the squares?Sum of the squares?
Magic number?Magic number?
20a
10a
??
??
??
2b + 2 b
b + 6
Addition ArithmagonsAddition Arithmagons
3b+8 2b+6
3b+2
Sum of the squares?Sum of the squares?
Magic number?Magic number?
8b + 16
4b + 8
??
??
??
2b + 2 b
b + 6
Arithmagons and equationsArithmagons and equations
3b+8 2b+6
3b+2
Sum of the squares?Sum of the squares?8b + 16
So 4b + 8 = 28
and b = 5
We are told that the We are told that the sum of the circles = sum of the circles =
2828
b = 5
2b + 2 b
b + 6
Addition ArithmagonsAddition Arithmagons
3b+8 2b+6
3b+2 12 5
11
23 16
17
=
Arithmagon: Arithmagon: GeneralisationGeneralisation
Sum of the squares =
a + b + b + c + a +c
= 2a + 2b + 2c= 2(a + b + c)
So, sum of squares = 2 x sum of circles
What do we notice?
a c
b
a +b b + c
a + c
Arithmagon: Arithmagon: GeneralisationGeneralisation
That the sum of a circle and a square opposite each other
= a + b + c
What do we notice?
a c
b
a +b b + c
a + cThe The EndEnd
