Today I had only P5 and P6 children (no P7s), and decided to spend a bit of time exploring equivalent fractions.
Here is my initial plan, which I simplified. There is a lot here, enough for a few lessons. This is OK; I often over-plan (with regards to quantity), and then discard things in response to what the children are doing.
The set of questions I had created in the above plan is, I feel, a different type of over-planning. I am aware that an important part of what I need to do in this classroom is to create opportunities for choice and creativity. There is always the struggle between freedom and constraint.
In the event, we started with this, from Cuisenaire - Early Years to Adult:
Amidst the many conversations, I recall a few comments, such as: "Black doesn't have anything," and, "A quarter of orange would be 2.5."
After a while, we stopped for a discussion. The children had found many (unit) fractions, and suggested that we could write them in various ways (e.g. 1/2, 2/4, 3/6, ...). One of the P6s said: "This is simplifying", and explained what she understood by that. All of the P6s said they recognised simplifying, but the P5s didn't. At this point, I felt I could have rearranged the groups into P5/P6 pairs, but I didn't act on that impulse.
In hindsight, it would have been useful to explore non-unit fractions here, but we didn't. I feel now that I moved on too quickly from this to the next task.
The next task was a simplification of this from Practising Mathematics: Developing the mathematician as well as the mathematics:
I wrote five questions on the board:
There are a couple of problems with the design of this activity. One is that most (if not all) of the children found one answer for each, not as many as they could. I think this is a natural thing to do, getting the 'right' answer, particularly if you are used to working in workbooks. I don't think it is easy to switch from 'getting the answer', to getting a range of answers by substituting any value you want into one of the boxes. They may have worked on each example more fully had I only presented one example, and then another...
However, a more serious flaw in the design was that questions 2 and 3 are very difficult to solve without already knowing about equivalent fractions! There was a moment in which I realised: "Ah, this isn't going to work!" I may not have imagined this part of the lesson as fully as I might, or perhaps my familiarity with equivalent fractions blinded me to some of the problems the children would face (as described by Brent Davis in his recent interview in MT261).
Since the lesson, I have also wondered if Cuisenaire rods were the most useful medium with which to work here. At the least, we would have needed to consider non-unit fractions, and perhaps build fraction walls with the rods. Something for next time.
I found myself explaining, and explaining again. Some of the children were becoming unresponsive, not listening. They were no longer working together in pairs, so some children were not getting the support they required to access these problems. The ones who could do the problems found it difficult to explain to the others what they were doing. Of course they did!
I sensed frustration, initially with the children (!), but as soon as I noticed my frustration, I realised it was due to the problem in the design of the activity. I then realised that I must simplify what I was asking them to do. I replaced the questions on the board with just two: ?/6 = 1/? and ?/8 = 1/?. The children started working together again, and all of them began to find solutions.
I then asked children to make up their own examples 'like this'. There was a wide range of examples, including some that "didn't work" such as ?/5 = 1/?. After some time doing this, I decided to present one for everyone to work on, ?/12 = 1/? Bringing the children together in this way brought some energy; they all worked together to find as many solutions as they could. I offered alternatives to those children who had found 'all 6', such as ?/12 = 3/?.
We discussed solutions and how we got them. I then asked each child to describe a moment or example for which they understood something they hadn't until then. Here are some of their responses:
Elsie (P5) said she understood what to do when she created and solved her own example, ?/10 = 1/?.
Harry (P5) knew he understood what was happening when he found all 6 solutions to the last question.
Ellie (P6) started to understand when she wrote "2/4 = 1/2?", found it was correct, and then found lots of other examples.
It "all clicked" for Milos (P6) when he saw the pattern in the answers to ?/12 = 1/? in the discussion at the end: "If you have four over twelve, then there are three fours in twelve, so it is the same as one third."
Minnie (P6) was pleased she had found four answers to the last question.
Caleb (P5) said he "sort of understood equivalent fractions", but couldn't explain how to find them.
Jack (P6) said: "I felt like I had accomplished something when I worked out 1/12 = 3/36 in a logical way."
Alexis (P5) was pleased that she had found the solution 2/8 = 1/4 without any help.
I then asked the children to prepare themselves to tell the P7s next week about what they had done.