Building and summing sequences with cubes!

This has come up a couple of times recently and seeing a twitter discussion on 'brilliant activities using manipulatives' I decided to write this down quickly. This is a beautiful and simple activity to teach arithmetic sequences using manipulatives. Of course this could be done in lots of ways and so this blog is really only giving the outline. What is nice about this activity is the possible age ranges it could be used with!

1st - ask students to make a visual representation of the sequence 1, 5, 9, 13 .... and so on depending on howmany cubes you have. you can get all sorts of different shapes. I have been doing this activity for years and always see something new! The picture on the left is one possible answer, the one below another.

2nd - ask how many cubes it will take to build the next, 10th, 100th pattern etc which is really asking students to think about the structure of the sequences and work towards a formula that is built on intuition.

3rd - ask how many terms of the sequence you can build with the cubes you have - you get some wild answers! Test it by building them

4th - Once they are built arrange them like the picture below...

5th - Carefully get students to demonstrate how this can be made in to a rectangle like the the picture below...

and that the area of this rectangle is the sum of the sequence and that the formula for the sum is easily derived from the way this was put together.

As I said, this is just a quick blog to sow the seeds of an idea. There is so much going on here, so many alternate paths that it has become for me an essential activity!