This post is about a maths lesson for a group of 8 - 11 year old children. This lesson was the fifth in a sequence of lessons in which the children are using Cuisenaire rods to explore place value.

In this lesson I wanted to introduce

*operations*with the rods. I chose to use

**black**rods (i.e. base-7), as I thought it would create some interesting calculations, and it fitted into an idea I had of using rods of increasing length through the sequence of lessons (ending with orange).

I deliberated over the number the calculations should be equal to, and chose 4 4 because I thought it might allow

**'strange-looking'**calculations (i.e. calculations that are not the same in base-10) such as 2 5 + 1 6 = 4 4.

Perhaps a different 'target' number might have been better for this, as not many children came up with strange-looking calculations. However, it seems as though it wouldn't have mattered which number I chose, as the children generally did not come up with such examples throughout the lesson (see below).

Here are some of the childrens' responses:

The children quickly started using other operations besides addition. The excitement this task generated was further evidence of the enjoyment they get from creating questions (see lesson 3).

The question

*'think of a calculation that no one else in the class will have'*was designed to encourage the children to come up with unusual or interesting examples, which I then invited them to write on the board:The children spent some time working on deciding whether they agreed with these calculations. The bottom right example raised some interesting questions about placement of equals signs.

The example 1 0 4 - 3 0 was perhaps most interesting, as it is one of those strange-looking calculations. Some children suggested that the answer for this one should be 7 4, but one of the children spontaneously wanted to explain to the whole class why it was indeed 4 4, using the rods.

Here's are the children's responses to the second task (with captions):

I love the calculation

*'5 6 - 5 6 + 5 6 - 5 6 + 5 6 etc.'.*Finally, I gave the children a choice of three tasks:

There was a reasonably even spread between the three tasks, and some interesting examples, including this one:

There were some lovely examples throughout the lesson, but I was hoping that more of those strange-looking calculations would appear, to highlight the difference between working in base-7 and base-10.

As a plenary, I wrote the example 1 6 5 + 2 4 6 (in blacks) on the board, and invited the class to make conjectures. They were reticent at first, but as I started writing conjectures on the board without evaluation, many of the children offered one or more solutions, as can be seen here on the right hand side of the board:

As a plenary, I wrote the example 1 6 5 + 2 4 6 (in blacks) on the board, and invited the class to make conjectures. They were reticent at first, but as I started writing conjectures on the board without evaluation, many of the children offered one or more solutions, as can be seen here on the right hand side of the board:

Next lesson we will start by working on a few more addition and subtraction examples like this, perhaps individually, as it might be useful to see what the children have picked up.

I think it is also time for some reflection. I might ask them to write about what they have most enjoyed about our work so far, and what they would like to do more/less of.

We finished by chanting the two times table in base-7 as a class, and I wrote it on the board as they went (as can also be seen in the photo above). It was a lovely end to the lesson.

The next thing we will work on is times tables. There are some surprising patterns in the times tables in base 7, and it will also be interesting to look at patterns in times tables in base 6.

The next thing we will work on is times tables. There are some surprising patterns in the times tables in base 7, and it will also be interesting to look at patterns in times tables in base 6.