progression in number and algebra. objectives the objectives vary depending on which attainment...

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