Today we started by reviewing last lesson, and then I invited the children to draw a (number) line of length 30cm on a piece of A3 paper, marking the ends 0 and 1. I asked them if they could find as many fraction names as possible for each centimetre mark, using Cuisenaire rods or otherwise. I demonstrated this using the orange rod (length 10cm), which fits into 30cm three times, allowing me to mark 1/3, 2/3 and 3/3 on the number line.

Following this, I invited them to reflect on (a)

__how__they worked out the fractions, and (b) what they found out. Here are their responses, along with comments based on conversations I had with each child.

Charlie found it difficult to get started, but after a second demonstration he was able to use the rods to find multiple fraction names for each mark on the line. He wasn't sure what he had found out!

Annie often appears to be one of the least confident children in the class, but today she used the rods very successfully to find many fraction names for each mark on the number line. I was surprised that she found it easier to work alone.

Elsie developed her own ingenious method: she divided the length of each rod into 30, to work out the fraction name of each

*rod,*which she*then*used to write down fraction names for each mark!Caitlin initially used the method of counting along using the rods, before switching to Elsie's method (above). It seemed that using

*two*methods allowed her to think more deeply about what was happening.Emily developed a similar method to Elsie (dividing rod lengths into 30), finding some

*very*interesting fraction names, for example 2/3.75 (two brown rods)! She used the black rod to write fraction names such as 1/4.2857142 = 7/30.Today, each child decided how and who they were going to work with. Violet worked beautifully with Emily, getting a lot of joy out of the unusual fraction names they were discovering. Violet is dyslexic; I really like the detail in her descriptions here.

Callum is one of the youngest children in the class, and sometimes finds the work difficult - there are children with two years more school experience in this room. He initially thought that the light green rod was 1/3 because it had length 3cm. Placing ten green rods on the number line, he realised that it was 1/10. From this point forwards, he could access the task, and was engaged in some deep mathematical thinking. For example, he was trying to work out

*why*the blue rod didn't "work" (fit into the whole), conjecturing that it was because it was "odd".Jack thought he had found "all" of the possible fraction names, so I pointed out that he didn't have halves and quarters. Here he describes his ingenious method to create marks for quarters!

Aiden quickly decided that he didn't need the rods to complete the number line. When he had done this, he then decided to find fraction names for each rod. He conjectured that the pink rod was 1/9; I suggested it wasn't which he confirmed by laying 9 pinks rods in a line, and crossed it out. He was working on this problem when the lesson ended.

Alexis is another of the younger children, and also finds the work we do challenging at times. Sometimes she doesn't ask for help, but today she did, which was really encouraging. I sat with her for some time explaining how she could use the light green rods (length 3cm) to mark tenths on the line. She then worked on finding other fractions for herself. It was a joyful moment when she told me that the yellows (length 5cm) represented sixths!

Even though I talked with every child during the lesson at least once, there is often someone who 'slips through the net'. This happens more than I would like with this child, whose name I've decided not to give here. This is something I must become more aware of in the moment. I spoke with her early in the lesson and she seemed to be on the right track, but it seems that she must have gradually, quietly lost her way.

Charlotte enjoys working on her own. She is another student who I find more difficult to approach, but I had a couple of conversations with her during the lesson. She said that she had made a few mis-takes early on, which had confused her, but then managed to correct what she had done. I really like the way she used an example with diagrams to describe how she worked the fraction names out.

Connor decided to sit with his friends today, which resulted in them not attending to the task as well as they might have. This at least provided an opportunity to talk about self-regulation. Eventually, he managed to get on track. He thought that because 1/5 and 1/6 were next to each other, that the next marks would be 1/7, 1/8, 1/9, ... and had labelled the fifteenths as ninths. (see image above). When we put the red rods on the line, he realised this was not correct.

Harry was "stuck" at one point, but a little more modelling allowed him to access the task. There are some lovely descriptions here. He was hindered slightly by the odd mis-take, such as counting 11 light-greens (length 3cm), rather than 10!

After writing all of the thirtieths on the line (quite a common approach), Milos decided to work systematically to find the rest of the fractions names in order, starting with the smallest denominator (thirds), up to the largest (fifteenths).

Sophie's descriptions are wonderfully detailed. She realised by trial and error that there wasn't a rod with the name "one quarter". She also got into a dispute with Milos about the name of the red rod. Milos had laid out fifteen reds and said they don't quite fit into a whole, but Sophie suggested that this was just a measurement error: "There must be fifteen reds in a whole, because 30 divided by 15 is 2. So a red rod

*must*be 1/15."Ellie is another child who I find it difficult to approach; I get the sense she does not want to be approached. When I came to talk to her, she had just written "I don't understand!" in the descriptions. I felt she was a bit upset; I talked to her as gently as I could. It appeared that she had found out more than she had written down, saying for example that the pink rod "didn't fit". We worked on the red rod together, and she worked on some other rods. I asked her at the end of the lesson if she felt better about what she had done in the lesson, and she said she did.

Sarah produces some wonderful pieces of mathematics, and today was no exception. Her work speaks for itself.

Minnie often seems to resist what I offer. Her response to the question "what did you find out?" is to the point! At the end of the lesson I asked her what she might do in the future if she finds herself in the position of "knowing everything already". She was not sure how to answer; I asked her to think about it. I was reminded of something Fritz Perls once said about being a therapist, that applies equally well to being a teacher:

If the therapist disapproves of resistances, he might as well give up.

At the end of the lesson I asked: What is a fraction name for the blue rod? A few children offered 9/30, and then Sarah called out 1/3.33333... and then Caitlin offered 3/10.

I'll start the next lesson with a couple of questions like "Which marks had three fraction names?" and then draw attention to the relationship between the numerators and denominators.

I'll start the next lesson with a couple of questions like "Which marks had three fraction names?" and then draw attention to the relationship between the numerators and denominators.