Enrichment in Mathematics – Algebra
Algebraic Magic Square
Teachers Notes
This activity is an extension of the Year 7 ‘Magic square’
It revises simplifying expressions and practises problem solving skills.
It is a good practical activity and so it lends itself to a paired card sort.
Discussions about “Where did you start?”, “Why did you try that?”, “What rules
did you find that helped you?” will help formative assessment of problem
solving skills and understanding of the concept of simplifying expressions.
Pupils may need to start the exercise by looking at a numerical magic square,
to try to find rules or characteristics which can be applied to algebraic magic
squares.
Solution (A number focus version is also included in Y7 Number 1)
Numerical magic square
2
7
6
9
5
1
4
3
8
Other possibilities are the above but rotated 90°
To help pupils get started, they might want to consider what the numbers all
add up to. (45)
From that, they know that each row, column or diagonal add up to 15, as it
represents one third of the square.
They might then want to arrange the numbers in order of size and find the
middle number, 5. Does this need to go in the centre? (Yes)
Then take combinations from each end of the numbers i.e. 1 goes with 9, 2
with 8 and so on. Keep these pairs of numbers together when placing in the
square, so that they go on either side of the middle number in a straight line.
1
2
3
4
5
6
7
8
9
Pupils may then use trial and error/improvement to get the correct
arrangement. They might see that all the numbers in the corners are even.
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Algebra
Algebraic magic square
Now apply the same rules to the algebraic magic square.
Add up the expressions to get a total of 36a + 45b
Divide by 3 to get the total for a row/column or diagonal 12a + 15b
They may notice that the cards contain all the ‘b’s from b up to 9b, so does a
similar arrangement as the numerical magic square work?
Yes!
3a + 8b
8a + 3b
a + 4b
2a + b
4a + 5b
6a + 9b
7a + 6b
7b
5a + 2b
Other possible arrangements are 90° rotations of this square.
Other ways are to find the middle square by putting all the cards in order of
size (by a or b) and finding the middle square.
Now match up cards from each end of the row, so 7b has to go with 8a + 3b
and so on.
b
2b
3b
4b
5b
6b
7b
8b
9b
Now trial and improvement will get you there in the end!
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Algebra
Algebraic Magic Square
a + 4b
2a + b
8a + 3b
5a + 2b
7a + 6b
7b
6a + 9b
4a + 5b
3a + 8b
Can you rearrange the numbers so every row,
column and diagonal add to the same total?
Worcestershire Numeracy Team
Enrichment Activities