What's in a name?
Problem
What's in a Name? printable sheet
Have a look at these two squares of numbers:
What do you see?
What is the same about the two squares? What is different?
'Magic' squares are square grids with a special arrangement of numbers in them. The arrangement is special because the numbers in each row, column and diagonal add up to the same total. (We could call this the 'magic total'.)
Looking at the left hand square above, if we add the top row of numbers, 1 + 5 + 7, we get a total of 13:
If we add the left hand column of numbers, 1 + 3 + 4, we get a total of 8.
So, we know already that this square is not a magic square as these two totals are different.
Use the numbers 1 to 9 to create a magic square.
Can you find another way of doing it?
And another? And another? ...
What do you notice about your magic squares?
Getting Started
What is the total of all nine numbers? How might this help you to work out the 'magic total'?
Student Solutions
Shivangi from the Garden Suburb Junior SchoolӬ had an interesting idea of choosing what the total for six additions (three across and three down) should be. That did not work so Shrivanga tried splitting her total into groups of three numbers to see if that would produce an answer. Here is what she said:
To solve the problem, I first chose to choose a number for my magic square. Then I decided to break down my number so the numbers would add up to the beginning number. I did not succeed with that solution. Another is that I tried to divide my number equally so they add up to my number. This solution might not work for everyone but it worked for me because my number is 15.
Aleksandra and Hannah from The Glasgow AcademyӬ sent in:
735
249
681
A teacher from Blue Gate Fields Junior SchoolӬ sent in the following, explaining that the students wrote these comments during the lesson next to their workings out. They haven't found an answer yet but are continuing to try to solve the problem:
"I am trying to put a big number and then a small number in each row and column to try to get the same total because if I put all the bigger numbers together then the total will be bigger than other rows or columns."
"1+2+3+4+5+6+7+8+9 = 45 and there need to be three digits in each row so I think the totals have to make 15 so I am adding numbers together to make 15 before writing them in the squares."
"I am placing the large numbers at the sides and the small numbers in the middle so they are mixed up."
"I am mixing up the small numbers and the big numbers because if you put all the big numbers together or the small numbers together, you would get different totals."
Thank you so much for sending us your thinking. It is really helpful to know how you are going about the problem.
Lily and Harry from Harrison Primary School wrote:
Ӭ
So when we started we tried to find what number each row needed to add up to so to do that we needed to find the mean and that was 15 so we put 5 in the centre because it was 3 x 5 = 15 and each row had 3. So then we started to put the other numbers around the edge until we got
816
357
492
Then after we got one we just started to move the numbers around to get many different possibilities:
438
951
276
And
294
753
618
And so on we also flipped the numbers around so they where backwards.
Thank you for those solutions. Well done!
Teachers' Resources
What's in a Name?
What's in a Name? printable sheet
Have a look at these two squares of numbers:
What do you see?
What is the same about the two squares? What is different?
'Magic' squares are square grids with a special arrangement of numbers in them. The arrangement is special because the numbers in each row, column and diagonal add up to the same total. (We could call this the 'magic total'.)
Looking at the left hand square above, if we add the top row of numbers, 1 + 5 + 7, we get a total of 13:
If we add the left hand column of numbers, 1 + 3 + 4, we get a total of 8.
So, we know already that this square is not a magic square as these two totals are different.
Use the numbers 1 to 9 to create a magic square.
Can you find another way of doing it?
And another? And another? ...
What do you notice about your magic squares?
Why do this problem?
Magic squares offer an opportunity for learners to practise mental arithmetic in a meaningful way. They will be performing many, many calculations but with a particular goal in mind, as opposed to isolated, repetitive practice. The idea of the square being 'magic' can stimulate children's curiosity and provide the motivation for exploring the mathematics.
Possible approach
Display the two squares and ask children what they see. Allow them to look in silence first, then give them chance to talk to a partner. Rather than taking any feedback immediately, invite the class to look again and consider what is the same about the two squares, and what is different. Again, give everyone some silent time before asking them to talk to each other.
Facilitate a whole group discussion which will allow pairs to share their 'noticings'. Try to encourage other learners to comment and respond, rather than all the conversation going through you. Draw out the common features of the two squares and lead into an introduction about magic squares. In order to clarify what makes a square magic, you could involve everyone in finding out the totals of
each row, column and diagonal in one or both of the squares given. You can then emphasise that both of these examples are not magic because neither of them has a 'magic total'.
Set children off to create their own magic square/s. Having digit cards available will be useful so that they can try out different arrangements of the numbers easily. You may also wish to print off copies of this recording sheet, which contains lots of blank three by three squares.
You could give out copies of larger individual three by three grids and ask particular pairs to record one of their solutions on to it. Pin up all the solutions on the board/wall for everyone to see and, at a suitable point, bring everyone together again to look at them. What does the class notice about all the solutions they
have found? This is a great opportunity to talk about same and different. Although the solutions do look different at first glance, closer inspection will reveal that the same three numbers always appear together on a row/column/diagonal.
Key questions
How will you find out what the 'magic total' is?
How will you remember the magic squares that you have found?
How are you checking that your square is magic?
How will you make sure you have found all the magic squares?
Possible support
Having digit cards available will mean that everyone can access this task through trial and improvement.
Possible extension
Encourage learners to explain why there is only one three by three magic square. This could, for example, involve working systematically to find all the ways of using 1 with two other numbers, all the ways of using 2 etc. and using this to decide where each number has to be placed.
Some children might enjoy exploring four by four magic squares. There are considerably more solutions this time so rather than asking them to find them all, you could suggest that they found some examples.