You can download my slides here. Hopefully they make sense without the chat to go alongside it!
At La Salle Education's PTConf19, I presented on the curricular development of the topic of Indices beyond the six laws we normally teach in Scotland.
The corresponding resource booklet is available here should you wish to use.
Sharing the Area Model part of my presentation from La Salle Education's PTConf on Friday in the hope that it's of use to people. Thanks to everyone who came and spoke to me or contacted me afterwards.
Examples in this presentation go from from multiplication in Primary/early Secondary Education through to Advanced Higher.
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.
In what's turned out to be a hectic week I've decided to write an account of my S1 lesson yesterday quite simply because I enjoyed it and want to remind myself of it in the future.
I was ready to begin examining Factors with my S1 class. In the past, I would have referred to this as 'doing Factors', 'teaching Factors' or something to that ilk, and would have relied heavily on direct instruction followed by what may be perceived by pupils as a boring exercise which no clear aim. However, I decided we were going to 'examine' Factors using various discussion points together.
I started off by asking the class who knew what a factor was. It should be noted at this point that this is a group of pupils who are moving at a fast pace and their pre-requisite knowledge of the basics of number appear strong. Various definitions were offered, all of which told me they knew what a factor was. I checked this using statements like 7 is a factor of 35, 10 is a factor of 96, 3 is a factor of 81. For each, they told me on their mini whiteboards if the statement was true or false, and on asking them how they knew, I was met with a variety of answers (I didn't have the chance to take photos as the lesson was moving fast):
Q: 10 is a factor of 96
A: 96 isn't an answer in the 10 times table
A: Numbers end in a zero if 10 is a factor
A: 0 in units column
A: 96/10 = 9 r 6 so it can't be a factor
During this, the girls in the class had to leave for an assembly but I decided to continue with the lesson. We then went on to list the factors of a numbers. Examples were 6, 24, 11 and 21. On reflection during the lesson, I hadn't thought these numbers through as well as I could have, although I could see what I had been thinking: 6 - even, 24 - larger even, 11 - odd and prime, 21 - odd.
We found the factors of each and I asked the pupils if they agreed with the following statement: the larger the number the more factors there are. As they thought, I reordered the numbers on the board and they told me I was wrong but one pupil said 'Well maybe you're not completely wrong. Maybe the larger the even number, the more factors it has.'. So before I could even tell them we would explore this conjecture, another pupil asked if we could try a larger number to see if Xpupil was right. We had listed the factors of 6 and 24 and added 36 and 48 to our list. At this point, the conjecture was holding true, so I asked them to list the factors of 50 and when they realised the conjecture wasn't true some were surprised. It was a nice moment as they wanted to find out and prove it was true but then realised it wasn't.
We then went through all of the numbers we had used and I asked them if they noticed anything about the number of factors. Someone said most of them had an even number of factors. Someone else said they all had an even number of factors apart from 36. I nodded just before someone shouted out that 36 is a square number followed by various reactions of agreement. So we found the factors of other square numbers and found that all square numbers have an odd number of factors.
We then examined the factors of prime numbers and 1.
This may be the sequence which most peoples' Factors lessons may take, but the important thing for me was that I was changing how I approached a lesson like this: less jotter work - I didn't feel like finding the factors of various different numbers in an exercise would have been productive time spent for these pupils; more discussion - I know from pupil reactions that they were getting something from this.
The most significant moment for me was the conjecture which one child made and the 'noticing' which took place by all...being mathematical at its best?
Thursday 27th September
During the last half hour of today's time with my S4 class, I delivered an introduction to simultaneous equations. Given time constraints at the moment, I know I'm not going to teach simultaneous equations as in depth as I would love to so I will only be focussing on solving by elimination over the next few lessons.
I told the class we were doing a new topic to do with equations. They have previously studied equations in great depth so I displayed the first task on the board and gave them the instruction to organise the equations into groups.
A few pupils asked what kind of groups I wanted and I told them I was interested in what they noticed about the equations and they had to decide on their own groups.
The discussion around the room was fascinating. One pupil kept trying to convince her partner that since a few of the equations equated to 10 that these had to be in a group of their own. Others were arguing that the equations with fractional variables had to be in a group by themselves. I found myself really intrigued by what was going on, especially from pupils who sometimes aren't the most motivated about Maths. I found myself not even wanting to go and speak to the pupils about the task. I just wanted to listen and to let them be in the moment without me interrupting. A pupil asked again at this stage if I was looking for a specific answer. I found myself telling her there was no right or wrong answer and that I was just interested in what they noticed. This is strange as I'd set the task hoping that they'd see some of the equations had one variable and some had two variables but in the moment I decided it didn't matter if they saw what I wanted them to see as the dialogue around the room was so interesting.
I eventually wandered around the room and asked a few questions of certain pupils.
This whiteboard was interesting as the pair have some equations in two of their groups. They were still in the middle of a heated discussion about where certain equations should be so I left to it so I could see more of what was going on around the room.
This pupil told me that he'd noticed he could rearrange some of the equations to make other on board. He was trying to group them to see which ones were the same as each other when rearranged.
Another pupil at one stage told me that if you rearranged then some of the equations would look like y = mx + c. I'll be honest, this was something I hadn't even thought of and certainly didn't expect them to notice.
As I continued wandering around the room I noticed this whiteboard i.e. the answer I'd wanted originally when I set the task. The pupils in this pair told me well they've all got x, they've all got y and they've all got x and y. They had hit the nail on the head but I was still more intrigued by some of the other groups' chat.
We then had a discussion about already knowing how to solve equations with one variable and could we solve one equation with two variables. I then told them that we would find out how to do this soon but we needed to be able to do a few other things first. They completed a substitution task where I gave them values of two variables and asked them to ensure these values satisfied two equations involving these variables.
If you've not already read Kris Boulton's (@Kris_Boulton) Simultaneous Equations blog post then I recommend you do so.
I used his task and the class were adding equations in no time. The minimally different questions confused them at first as they thought the first two questions were the same at first glance but then they noticed what was changing each time. Some pupils decided not to rub out their answers but just change the part which required amendment.
After adding several equations, I displayed 10x + 5y = 12 and 2y + 3x = 8 on the board. This was met with some ohs, oohs and ahs...something which has never happened in this class before. Every one of them got it correct and someone even noticed it had been the same as the very first question. Then the bell rang. I'll continue tomorrow with this task which moves onto subtracting equations.
I loved this half hour of my lesson as the reasoning the pupils were giving in the first task was great. Some might not have noticed straight away what I wanted them to originally but in a way this made it an even better lesson.
Reflections on a thought provoking and pedagogical changing day.
I have just taught my S4 National 5 class about Isosceles Triangles in Circles. On Saturday, John and Anne used the question ‘how do you know?’ and mentioned that they didn’t like the question ‘why?’. Right now, I can’t think of a lesson more appropriate to constantly ask ‘how do you know?’ than today’s lesson. Once the class were working, I went straight over to a pupil who normally finds becoming familiar with new concepts difficult. She said she was getting on fine today and knew what she was doing but I was interested in her thinking and the processes she was going through in her head. She said she would find the shaded angle by adding 77o and 77o and taking this away from 180o.
Me: Why are you adding another 77o? You only have one 77o.
A: This other angle is 77o.
Me: How do you know this?
A: The triangle is isosceles.
Me: How do you know this?
A: Because there are two radius’ and they make an isosceles triangle.
She said she was fine. She told me how she was going to find the first answer. However, did I really know she was fine until I questioned her further?
I left her certain that she knew how she would start tackling similar questions.
I approached a few other pupils in the class and had a similar conversation but on approaching two pupils who work well together and normally help each other. One said he didn’t really understand what was going on. He knew what he was looking for but couldn’t work out how he was meant to do that. He mentioned that he could see the triangle but didn’t know which sides were the same to make it isosceles. The pupil next to him said ‘yeah, but you just see it’. We talked through the properties of the circle again and drew more diagrams and he then saw why the triangle was isosceles which I was happy with. He then managed more of the questions comfortably, but what concerned me more was the other pupil who said she just saw it. Was she quickly becoming an expert or was she guessing? I was aware I hadn’t been around the class to see quite a number of the other pupils yet so moved but I think now I should have waited and questioned her further. Did she have a good understanding already or was it guessing? I’ll find out tomorrow as she’ll be one of the first pupils I speak to tomorrow asking ‘how do you know?’ until I am sure of her understanding.