This is an account of a maths lesson with a class of children aged 8-11. Today we continued work in various bases using Cuisenaire rods. The focus of today's lesson was looking what happens in different bases with regards to carrying or borrowing.

I started by introducing some so-called 'funny-looking calculations' from our work with base-7 last week, demonstrating them on Nrich's Cuisenaire environment:

As we went along, I asked students to write any interesting examples on the board. Here's what was on the board by the end:

Initially, some of the calculations did not specify the colour of the rods, such as the example 6 0 - 0 7 = 5 1. There was some debate as to whether this calculation was true for brown or blue rods, so we had a vote. I asked some of the children to justify their vote. We decided that from now on it would be useful to state the colour of the rods for each calculation.

The class was becoming unsettled for whatever reason (I put it down to being the last day of term, but it might have been the structure of the tasks), so I decided at this point to bring the class together.

I asked them to continue a similar sequence, starting at 4 3 + 2 2 = 6 5, with

*blue*rods. Each person in the class was called upon to call out the next number in sequence. Here is what happened:

The faces on the left were an indicator of the general consensus of opinion. Roughly 50% of the class thought there was a problem with 3 8 + 2 6 = 6 5, and around 60% thought 3 6 + 2 8 = 6 5 was not correct. We went through them using the Cuisenaire environment.

The calculation 3 3 + ___ = 6 5 shown in the photo was originally 3 3 + 2 5 = 6 5. Everyone in the class agreed that this calculation required modification (and all those after it), so I asked them to find the missing number that would make 3 3 + ___ = 6 5 true.

We then talked about number bonds to ten using various rods - see the photo below - and that if we (mankind) had decided to use some other base instead of base-10, then 5 + 5 = 1 0 would be very funny looking.