with negative numbers

geomagic squares - 3D

more of Lee Sallows's work

geomagic squares

Lee Sallows's ever so impressive work on making geometric magic squares was described in the Guardian recently and many examples can be found on Lee's website

the shapes combine to make the same overall rectangle (or other shape)

one way to use these with students is to present them with the magic square shapes and ask them what the totals are (as shown below), possibly with some of the shapes missing

these solutions show how, remarkably, the shapes combine to create the same rectangle (or other shape) - with the same properties as a number magic square

multiplication

multiply across the rows:
2 x 7 x 6 etc
then add these

now multiply down the columns:
2 x 9 x 4 etc
then add these

does this work for other magic squares?










using a calculator, square the rows as 3-digit numbers and add them:
276^2 + 951^2 + 438^2

then do the same thing with the columns, reading downwards:
294^2 + 753^2 + 618^2

does this work for other magic squares?

decimal addition and subtraction practice

fraction practice

rotations and reflections

find all the ways that the 1 to 9 magic square can be rearranged (and still be a magic square) - using rotations and reflections

finding patterns - swaps

create a magic square using the first nine odd numbers: 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15  and 17

then create a magic square using the first nine even numbers ( 2 up to 18 inclusive)

how is this easy after making one with the odd numbers?

magic square using the first nine odd numbers

















using the first nine even numbers

what happens when you half these?

in these magic squares, 2 of the numbers have been swapped over

without bothering about the middle number, students try to 'unswap' these magic squares

then they look at their squares, sort out what the central number will be and find patterns (there are many)

then they can try to create a different magic square (not a refection or rotation of these) with 10 in the middle:

they can do this by adjusting from the middle number moving outwards along both of the diagonals

what happens if a = 2b?

how can you avoid involving negative numbers?

bigger numbers

what patterns do students notice in this magic square?

this unusual magic square, discovered by Lee Sallows, is such that if you write out the numbers as words and then count the letters in each word - this results in another magic square!

using addition and subtraction, with maybe some appreciation of patterns, students can attempt to complete some or all of these

simple magic squares