Tuesday, 6 May 2014

Magic Squares

I had a parent send me a link to a site that has free e-books about maths on it.

I'm always keen on free things (it's the Scottish heritage) so I went and had a look. There was a lot of books that weren't going to be that useful in Year 5 as they were aimed at university level maths. However, I did find several books that really intrigued me. I am interested in old maths text books and there were several from over 100 years ago.

One that caught my eye was called "Magic Squares and Cubes" by William Symes Andrews, published in 1908.

Well, this looked promising. Magic squares are always fun and they can't have changed much in 100 years, since they've been around since Aristotle.

So, looking through, I found some really great examples, some instructions on how to construct magic squares of different sizes, and then this diagram:

Without telling them what they were looking at, I asked my class to find patterns.

Here's what they found:

Lots of numbers increasing by 10 arranged vertically!

If you look carefully you can see that the number under each multiple of 9 is the next consecutive number. For example, under 36 is 37.

And yes we found that every row and column adds up to 369!

And the sum of all columns and rows is 6642, or 18 x 369.

It took a bit of leading to see that this grid had been constructed by using an elongated "knight's move" - four squares to the right and 1 up. 

Alternatively, you could go five squares to the left and one up because there are 9 squares in each row (5+4 = 9). 

Which made the kids think of the globe - you can travel either East or West and you will meet up at the same destination. 

Good thinking!

I wonder what other treasures I am going to find in these old books?

Friday, 2 May 2014

Area and Perimeter Exploration

So we were looking at area a while ago. This led us into a conversation about the properties of 2D shapes. And then we spent some time with perimeter.

Now had come the moment I dread - when area and perimeter collide in the minds of the students and they come away dazed and confused. (How many times have you seen that? Hopefully we had clarified each of the concepts clearly enough.)

Anyway, the time had come to pose a favourite question of mine:

In fact, I wrote a post about this almost 2 years ago.

Here's what we came up with this time:

This group chose to use a hexagon for their example. The perimeter of the yellow hexagon was 15cm. Good stuff.

They found that 2 trapeziums (trapezia?) had the same area as the hexagon but a different perimeter. Well done.

Another group chose to use the Base-10 blocks. They made a 10x10 square (a=100; p=40) and then reorganised the blocks to make a 5x20 rectangle (a=100; p=50).

So just to be annoying, I asked them what would happen if they put all the blocks end-to-end? Same area but what would be the new perimeter? And then if you cut that rectangle in half long ways....

So, everybody happy. Yes - you can have the same area but with different perimeters.

Next question:

Previously I have done this as two separate activities but today we had time and we were on a roll so we kept going.

And here is what the kids came up with:

Happy with this one - nicely set out and very clearly explained.

An interesting solution - not using regular shapes. 
Solves the problem but not a lot of explanation.

Another way to do this.

The post I did 2 years ago that showed this activity was with Year 4 students. This time I am working with Year 5. Not quite as bright and colourful as the younger kids, not quite as diverse in their solutions and somewhat more structured in their presentation of solutions.

Wonder whose expectations have changed - theirs as students or mine as a teacher?