Reasoning about the number of matches needed to build squares that
share their sides.
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
These alphabet bricks are painted in a special way. A is on one
brick, B on two bricks, and so on. How many bricks will be painted
by the time they have got to other letters of the alphabet?
I don't know about you but I find myself counting the tiles in
particular ways. Maybe even your much younger brother or sister
tends to count things that are around them. Keep your eyes open for
patterns in tiles and you could discover a lot!
I was in an old church in Cambridge, where I live, and I looked
at the floor.
This is what I saw:-
What a design!!
There were these things to count:-
. . . the tiles making up the three squares
. . . the dark green tiles that made a border for the squares
. . . the pale yellow tiles that filled in the gaps into the long
When I got home I drew this design out (I had made some notes in
the church, which is a good idea as it's sometimes hard to remember
exactly how it was).
I then drew a 2 square version:-
and a 1 square!!
Well, it's really up to you to count and compare the numbers of
particular tiles in one or all of these three designs.
See what number patterns you can come up with. I guess there are
Now I wonder what would happen if we had four squares? I suppose
we could put two lots of 2 together:-
I was not altogether happy with this, but maybe you are, and
I decided to remove that centre line and come up with:-
Mind you, you could just continue with things in a straight
line. What do you think?
Try this out for yourself and maybe extend the pattern for even
greater numbers of squares.
Someone might be asking "What about smaller sized squares?" You
then might have something like :-
You could try some more like these.
What rules do you decide on, so that the pattern works out in the
Perhaps something rather different appears with different sized
squares and you could follow that line!!
And for those who really are fed up with squares why not look at
the same idea with triangles!!
Yes, here we are :-
for the small one and my next size up is like this :-
But what happens when we want more of the central triangles?