Patterns, sequences and fractions

When my daughter and I play with the Cusineire rods we always seem to run out of reds. So I ordered a second packet the other day, and when it arrived it looked like this:

I can’t get over how weird it looks with the colours being different, but it’s here now so we are going to have a play with it anyway.

Yesterday we got then out and she started making shapes:

These looked like some things I’d seen in the ATM book (Ollerton et al. 2017) on using rods. In the book they called them boats and this made sense to me as far as their shape went. I asked her which the odd one out was and she identified the one with a purple base. I asked why and she said that it was because all the others got one smaller each time as we went up. So I asked her to try and change it so it fit the pattern.

First she did this. Which was interesting. She misinterpreted what I was asking (I was trying to get her to alter the boat that didn’t fit) and made a whole new boat that would in fact have been the next in the pattern. I was quite impressed and we discussed this for a while and then I asked what other ones we could make:

Again she’d found many boats that fit the pattern, but the pesky purple base still didn’t, so we discussed it and corrected it.

We than talked about how many each boat would be worth if the grey cubes were worth one.

She calculated a few and noticed they were going up in 3s, so we predicted some bigger ones and counted to check.

I then asked her why they went up in 3s, and she didnt know so I got her to try and add some rods to one to make it be the same shape as a bigger one:

She could then see we were adding one to each level and there were 3 levels.

We then looked at other patterns we could make:

We started with these L shapes and I asked her if she could find them all:

Again we talked about their value if the small cubes were one. I asked her to predict what the values would go up by. She initially said one. I asked her why and aa she was explaining she changed her mind to do. We worked some out to check.

Then she decided to make squares:

This led to similar discussions a d investigations into how much they were increasing by.

Around this point I asked what would happen if the white cube was 2 or 3 and we discussed that. I then asked what would happen if a larger number was 1. She didn’t understand at first so we made some partitions:

I said, if the brown one is “one”, how much is the red one worth. She reasoned that it must be worth a half as there were 2 that fit a brown. We then discussed the grey and blue values.

And then had a similar discussion around this:

Which was interesting and gave her a better understanding of fractions, she had never met 1/6 before.

She then started making trains the same lengths:

We talked about why these were the same, ie the commutative property although we didn’t use the word today. I then asked her to see which other rods she could make the same length with:

We discussed why these worked and others didn’t. Linked it to times tables, discussing the terms multiple and factor. We then wrote the number sentences that they would represent if the grey cube was 1:

After this she started to see what other shapes she could make. She made some triangles:

We talked about these triangle and what they were called. She hadn’t heard the terms equilateral or isoceles before. She knew what a right angle was but hadn’t heard the term “right angled triangle”. I asked her if she could make another right angled triangle. She noticed the lengths were 3,4 and 5 so she tried 6,7 and 8:

We discussed why it wasn’t a right angled and she then suggested doubling each side:

We discussed this a little and finished for the day.

A fun and enjoyable time for both of us, and the playing we are doing is both helping her mathematically, and helping me understand the rods and see where they might be useful in my teaching going forward. So it’s a win win.

This has been the 5th post in a series on use of manipulatives in maths, you can find the others here:

Manipulatives – the start of a journey

Fun with Cusineire

Meaning making with manipulatives

Playing with Cusineire

Reference

Ollerton, M. Gregg, S. Williams, H. 2017. Cusineire – from early years to adult. Derby: Association of Teachers of Maths