Square of Area 10
Today, I going to be videoed teaching a mixed ability class of year 7 students (11 or 12 yr old) and I’ve been given the Squares of Area 10 investigation to teach. You should be able to see the lesson guide (don’t think it quite counts as a lesson plan). And the SMART notebook file I used.
I’m quite intrigued by the opportunity to teach mixed ability and am very interested to see how it goes. I’ll update with my reflections after the lesson…
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Ok, so the lesson is over and here are some of my thoughts. You probably need to look through the word document above first.
The investigation was certainly interesting and all the students seemed to understand that they were trying to make squares with different areas. Perhaps even areas they’d never thought were possible.
I loved the fact that we got a page of conjectures:
The red tick was because we confirmed that conjecture was true. We couldn’t confirm or reject the others at this stage. I picked up on one person’s conjecture and channelled the class into trying to find the area of 10.
I was struck by how the weaker students were often drawing rectangles rather than squares even after being reminded by myself or another student about the equal sides property of squares. Other weak students were very resistant to even trying to draw a square that was ’tilted’. I didn’t know the students well enough to know if this was due to lack of ability, lack of confidence, laziness or some other reason. The other teacher trying this lesson at the same time had a work sheet that may have helped at this point.
Only a couple of students seemed willing to try splitting the square into triangles to find the area convincingly. The majority of them wanted to just count squares and/or put together parts of squares. This was a bit of an issue as I wanted one of the outcomes of the lesson to be about finding areas of triangles. Still, they were finding the areas correctly so it was difficult to complain about it or insist on an unecessary method.
By the time I got to using the notation, I could tell that some students were getting left behind because they hadn’t drawn any tilted squares so couldn’t add to the common table of results. We pretty much ran out of time by then but I think the students who couldn’t help create the table could still have got involved in finding rules/patterns from the table:
We didn’t have time to look for patterns but I liked the fact and would have drawn out some thinking about Tommy’s comment (bottom right) that “Surely the 1,3 square is the same as the 3,1 square?”
Lots of things to think about and I’d be interested to hear your thoughts…
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The following is from Mandeep who taught the same lesson at the same time as me:
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I taught the lesson – ‘Area of 10’ – to a mixed ability group which was videoed at the same time as Dave’s lesson. I started with asking volunteers to draw simple squares on the board and count their area. They all seem to be comfortable with this, although a few of them were not very sure whether it was outside (boundary) of the shape or the space inside, which was clarified. I then asked them to draw a different kind of square but volunteers kept drawing simple straight squares but with different areas. I therefore decided to introduce a tilted square (of area 2) and some of the students were able to draw these squares easily.
I asked them what they think about squares of areas between 1 and 20. i.e. can they draw them all?’ I did realise at this point I had asked the question too early. I should have let them explore and think about different conjectures. I decided to introduce [vector notation: 1,2], and showed them how to calculate the area of this kind of square and asked them to find for themselves whether it was possible to make the squares with areas between 1 and 20. After giving them some time to experiment on their own, I realised they were drawing different kind of squares and not following the vector notation I introduced earlier. Perhaps they needed clear instructions written on the board as to what they needed to do. I had to therefore direct them to just one kind of square and introduced another square [vector notation: 1,3] of area 10.
In order to lead them towards the generalised formula I had to instruct them to focus on just these kinds of squares. The worksheets that I prepared became very handy at this point as I was able to direct them towards special squares, their notations and area of these squares much more easily. Some of the students were able to draw some more squares e.g. [vector notation: 1,4], squares with area of 17. The less able students were able to calculate the area of squares on the worksheet.
In the end I was able to get their attention to the pattern in the table drawn on the board. They could see that areas were going up in odd numbers and in order to relate the area with the notation they came up with:
Notation

Area

Rule

1 1 
2

1 + 1

1 2 
5

1 + 2 + 2

1 3 
10

1 + 3 + 3 + 3

1 4 
17





As they started to come up with the idea of 1 + 1, they followed it by having 1 + 2 + 2 and then 1 + 3 + 3 + 3. I had to then insist on how to express 2 + 2 in a different way and someone said 2^2 followed by 3^2 for 3 + 3 + 3.
So they got to the pattern of 1 + 2^2, 1 + 3^2 and 1 + 4^2. It was followed by someone generalising the result and said 1 + n^2. I wasn’t left with any time to evaluate roughly how many students were able to understand the last part.
When preparing for this lesson I did think about the grid on the board interfering with the grid on the notebook. Next time when I teach this lesson I must make sure this doesn’t become a barrier at the start of the lesson. The next thing I will do different will be to let them come up with some conjectures themselves as Dave did in his lesson. The worksheets that I prepared were very helpful as it helped less able students to work on some skills e.g. to understand the notation, to calculate the area of special squares, to practice drawing these squares and to look at the table and recognise the pattern. Towards the end of the lesson I found myself rushing through few things which needed more time to sink in e.g. patterns in the table, etc. Next time hopefully I will gain time from start of the lesson and utilise that time at the end of the lesson focusing on patterns and algebraic rule.
There were lots of special skills involved to get a good result in the end. Firstly the ability to come up with conjectures, secondly draw the special squares, thirdly to calculate the area of these squares followed by learning vector notations and then to recognise the patterns and to generalise rules.
I agree with Dave when he said that part of the lesson went towards how to draw these special squares. I wanted the focus of the lesson to find the pattern and perhaps more time could be invested beforehand building up this skill so that there was one less barrier to cross whilst teaching this topic.
I found being in mixed ability group the less able students were motivated by the more able students. Quite a few times during the lesson I observed less able students looking around the class and observing what other students were doing to draw the squares, couple of them even went up to another table to learn how to draw squares and how to count the area of the squares. There was a sense of ‘I am stuck, so what can I do to get over the difficulty’. Some raised hands to get help from me, some used worksheets and some helped each other.
I taught mixed ability six years ago where I was involved in teaching a different curriculum (teaching all subjects to year7s in a project based curriculum) and remember feeling then unsatisfied at the end of a few lessons as every student didn’t end up at the same level. I had the similar feeling at the end of this lesson too. As Dave said most times we are more into settled teaching and are therefore used to a different kind of outcome. I will try this lesson again with a different group using the experience gained and hopefully will get a better response.
Love this lesson! If I didn’t know you better I’d say you nicked my lesson!!http://chrismaths.blogspot.com/2011/02/squareofarea10.htmlFab Fab lesson.
Thanks Chris. I’ve seen it on your site now but I promise I didn’t steal it! (Well, not from you anyway.)You should make sure you ask students to come up with conjectures. Brilliant thing to do in itself but sometimes also feels like they’ve created the investigation themselves too!
From what I read here it feels like there was a lot of mathematics going on in the room during this lesson. Something certainly resonates with me in your last comment when you said that it feels as if they created the investigation themselves through collecting conjectures as this often feels the case for me – so even if you know in your mind where you want the investigation to go – the students own the problem for themselves – which of course is more engaging as a learner. The addition on writing students names against each conjecture is also a way of giving them ownership.I think when you work with students on a problem like this – you need to have in mind what you want the focus to be. So for those less able students – if you wanted them to practice drawing squares then it seems to me that there is no problem with them spending time doing this. If, however, you wanted the focus to be on finding areas of squares then they may have needed some support (perhaps in the form of a work sheet – but not necessarily) or if you wanted the focus to be finding patterns then you may need to provide something different. A task is still open if you provide some resources that support the weaker students ??? they can still think mathematically.The issue of working the area out in a convincing way (not estimating) is something that comes with time ??? something about whenever you are working on mathematics ??? you need to be convincing and I think this is one of those classroom cultures that needs to be established from day 1 with a new group. Therefore it is difficult to do this in a one off lesson. The other problem with one off lessons is that you can???t draw out the thinking as you would have liked to have done at the end with Tommy???s comment. In a series of lessons this could be next lesson???s starting point.I suggest you do the same task with you year 8s???
Thanks for the comments Tracy. <br/> <br/>I agree that the feeling of student driving the lesson forward (even if they’re not) is a very strong one. It was something that was commented on in my AST assessment lessons. <br/> <br/>I think it’s going to take a bit of getting used to the idea that students will not make the same progress as this is what we’re used to in setted teaching. Of course, this rarely actually happened so it isn’t that different but it is a change of mindset to go in to a lesson fully expecting and encouraging different progressions. <br/> <br/>No doubt with more lessons, we could have developed Tommy’s ideas and made further progress with the drawing of squares. I’d like to have another try with pushing a more collaborative gp work approach (maybe complex instruction) and allow more opportunity for use of ICT. <br/> <br/>Thanks again as it’s really useful to be nudged into thinking about my teaching and learning. <br/>Dave
Great comments from Mandeep (see main post).I think it’s great that having both taught the lesson, there are key points we can pick up from each other. This is something that hardly ever happens with the way we normally teach.I think the paragraph I highlighted in blue is a good sign that mixed ability teaching has benefits.Don’t forget that this investigation is not meant to be completed in one lesson!
Late to the game reading this one, but my first thought was that this is a grand exploration for students. Very openended for inviting conjectures with the real challenge being encouraging students to prove the validity of their hypotheses. As I read this, I also began to wonder if any squares could be represented more than one way. How many can recognize that [1,3] and [3,1] squares are absolutely identical. Further, which squares can be represented more than one way? Are there any that could be represented both "straight" and "tilted"? We know this last statement is true, but that is part of what is so special about integral solutions to the Pythagorean Theorem. Very pretty. I hadn’t conceived of walking into the Pythagorean Theorem this way (not that the PT was your point at all here), but this could be a glorious place to plant a seed for later discovery or continued investigation. Thanks.@chrish_1974: Nice insights on your approach, too.Final question to which I don’t have an answer. Is there a largest area of a square that can NOT be determined using this grid?
Thanks for your comments @Chris_HarrowInteresting questions and it’s inspired his blog post here: http://casmusings.wordpress.com/2011/07/29/area10squares/#commentsI've also commented on that post with some more of my thoughts.Dave
In regards to the final paragraph from Mandeep: I don’t think the point of mixed ability grouping is for every student to be at the same point at the end of the lesson. The point is that they’ve stretched their thinking and learning beyond where they were at the beginning of the lesson. Has their understanding of areas,or triangles or problem solving strategies deepened because of this lesson? If yes, then the lesson was a success. Thanks for this post, it was really enlightening to hear how two teachers presented the same lesson.
<html><head></head><body bgcolor="#FFFFFF"><div>Hi Mary and thanks for your comment. </div><div>I agree with you and having now spent a year teaching a mixed ability year 7 class I have a much better idea of what progress in a mixed ability lesson looks and feels like. </div><div>It’s definitely not sensible to expect every student to be at the same place at the end of th lesson and I wonder if this sensible even in wetted classes too?<br><br></div></body></html>