Another interesting lesson with my S1 class.The class had studied multiplication of integers the previous day and had understood this at the initial point of instruction so when they arrived in class they were met with the following task. This task proved to be excellent for pupils. It allowed them to revise what they had learned the previous day but also provided them with the opportunity to think more about what they had learned and apply this as they weren't just answering more multiplication questions. It was evident from their scribblings on their mini whiteboards that they had a good understanding of the work from the previous day. Around the room, I noticed various things happening. Some pupils were finding it difficult to come up with more than a few of their own questions, so I encouraged them to think of positive integers which would have a product of 8 so they could adapt accordingly. Some pupils observed that they were coming up with the 'same thing' over and over again e.g. -2 x 4, 2 x -4, 4 x -2, -4 x 2 so we chatted about what is the same and what is different about these and how if you have the 2 and the 4 either of these could be negative to give -8 as an answer. One pupil had written 2^2 x 2 and kept putting a negative in front of the 2^2 and rubbing it out and then in front of the 2 and rubbing it out. I asked if she could rewrite her expression twice and put the negatives back in the two places so we could discuss each new expression. She was struggling with the -2^2, something which I had planned on discussing with the class later on in the lesson. I had a chat with her about this and how her expressions would both give -8 and asked her to write what she had come up with on the board. By this point, I had asked several pupils to do the same: As a class, we discussed some of the questions which had been generated. We discussed the one near the top (-2 x -4 x -1) which is different from the others as it involves three integers. The pupil who came up with this question seemed surprised that she was the only one who had written a question like it - she asked her friends why they hadn't come up with something similar. We have recently studied Decimals and one child came up with a pattern of questions like the following: ±0.8 x ∓10 ±0.08 x ∓100 ±0.008 x ∓1000 ±0.0008 x ∓10000 She had kept this pattern going for a while. It showed me her understanding of multiplying decimals by powers of 10 was good. We have also explored Order of Operations a few times now so the question -(2 x 4) generated some discussion and allowed pupils to revise this. Later in the lesson, I noticed a pupil had written the question below and we chatted about whether the brackets were essential or not. I then continued with the lesson where we were going to explore negative integers raised to a positive power. We recapped on what we already knew i.e. positive integers raised to a positive power: I asked the pupils what would happen if the 2s were negative 2s. One of the pupils immediately said that the answers would be negative. I said to the class "Peter has made a conjecture. His conjecture is that powers of negative numbers will be negative". I wrote a version of this on the board and repeated his conjecture again. I felt in the moment it was important to introduce the word 'conjecture' to the pupils. I want a culture of noticing and conjecturing in my classroom so I feel this word should be known to them. One pupil then asked what a conjecture was. Before I got a chance to answer, the child who had conjectured originally said it was like a prediction which I thought was a nice definition at that point in time for the pupils. We then started exploring... ...and Peter shouted out that his conjecture was only sometimes true. We continued exploring... ...and another pupil shouted out that negatives raised to odd numbered powers would be negative. There were a few gasps around the classroom, some nods of approval and agreement and someone shouting 'oh yeah, so they are'. The last section of this slide which I had wanted to explore with the class had already been done during the first task of the lesson (-2^2 x 2). One pupil mentioned at this point that Peter's conjecture would be correct in these instances which was a nice link from what had just happened in the lesson. The lesson was exciting as the pupils found out about powers of negatives by exploring and noticing and I believe this will help them remember more in the long term. It was a significant learning episode for me because it once again encouraged a culture of noticing and conjecturing instigated by pupils' questions, my questions and discussions.
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AuthorS McKenna Archives
March 2021
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