I am teaching maths once a week to a class of 8-11 year olds. This is the ninth (and last) in a series of lessons on arithmetic in different bases using Cuisenaire rods.

Today started with children giving feedback to each other about the clarity of the explanations in these videos they made last lesson, which they did with sensitivity and insight. I considered spending this lesson making second versions of the videos, but decided not to, mainly because I have been asked to teach the children about 'time' next week, and wanted to spend the last lesson on this sequence exploring orange rods (base ten). The feedback will be useful for the next time we make videos.

After we discussed the videos, the children worked on the task below in pairs, comparing subtraction calculations in blue (base nine) and in orange (base ten):

Today started with children giving feedback to each other about the clarity of the explanations in these videos they made last lesson, which they did with sensitivity and insight. I considered spending this lesson making second versions of the videos, but decided not to, mainly because I have been asked to teach the children about 'time' next week, and wanted to spend the last lesson on this sequence exploring orange rods (base ten). The feedback will be useful for the next time we make videos.

After we discussed the videos, the children worked on the task below in pairs, comparing subtraction calculations in blue (base nine) and in orange (base ten):

A key difference in today's lesson was that I did not give the children rods, to see if they could perform the calculations mentally. Here are some examples of the children's work:

There are some lovely calculations here, and a beautiful explanation at the bottom of one of them:

*"If you don't exchange the blues or the oranges for whites you get the same answer."*

I had planned to give the children around 40 minutes to do this task (and a similar one involving multiplication), but I sensed a loss of energy in the room after around 25 minutes. This has not always been the case. It was not clear to me whether it was the task, a shared mood among the children (it was the 1 December?), or whether it was over-familiarity with a particular way of working

I decided to bring the class together, to guide exploration on the subtraction task, using the board as a focal point, in order to alter the structure and pace of the lesson. I decided in those few moments to invite each pair to offer an example of a calculation that was

*different*in blues and oranges, which I wrote on the board. I considered also inviting them to offer a calculation that was the

*same*, but felt at the time that this might add to complexity and loss of energy.

Here are the examples the children offered:

I asked if anyone had anything to say about what had been written. One of the children then made the conjecture

*There were a few qualifications to this conjecture: (1)*

**"Oranges are always one more than the blues".***"If these [calculations] are all correct"*, and (2)

*"It's not always true, if you don't exchange,"*as can be seen at the bottom of the board in the following image:

**

There followed much uncertainty about the calculations for 1 0 0 - 8 8, so I gave each pair a few minutes to work on it. I wrote all of the answers to their calculations on the board.

The centre of gravity for the lesson was unusually at the board, in order to generate a shared focus/energy. It was as though I was a minute-taker for the children's ideas, writing down conjectures without evaluation. The board was the medium through which ideas were reflected back to the children:

Around half of the pairs had both calculations correct. I invited someone to explain how to do it using the rods, and was pleased when one of the less confident children offered what turned out to be a lovely clear explanation. The rest of the class gathered round, and affirmed what she had done.

All of this work brought the following amendments to the conjecture:

**as can be seen below:**

*"At least one of the numbers in the orange calculations is one more than the blues unless you don't exchange!"*

The (green) writing on the board is a reflection of the evolution of the children's ideas.

It was noticed at this stage that the calculations for 1 1 3 4 8 - 8 2 4 were not 1 apart, but rather 1 0 0 apart, another counter-example to the earlier conjecture. There were a few minutes left, and my decision to centre the lesson on the board had resulted in mounting energy; it seemed to me as though all of the children were now 'active'.

This building of excitement led me to propose a final challenge:

**There was a real buzz as all the pairs worked to find an example. Here is what they came up with:**

*Create a calculation in which all of the digits of the answer change when calculated in blues and in oranges.*

Some lovely examples here, and for me some evidence that they understood what we had been doing with the rods.

We rounded off this sequence of lessons with a discussion about calculations using orange rods. A number of children had noticed that calculations in oranges were just like 'normal' calculations, and some of them realised they could use their calculators again to check their answers!

Next week, time.