Saturday, 30 January 2016

The L Shaped Room

So, Year 4 can multiply a teens number by a one-digit number, using Cuisenaire rods, and the grid (box) method. At least when we do that as a specific lesson.
Most of the students can multiply two small two-digit numbers this way:

And when we give a simple real-life situation, like we did in this 3-Act task last week, they could use it.

In the next few weeks we should abstract away from area questions and see whether they reach for and can use this tool for non-area situations. But J, my partner teacher in the other Year 4 class would like to do some more area situations, but more complicated. L-shapes, that kind of thing.

Well, Graham Fletcher has provided a nice one that we'll do on Tuesday, one of his great 3-Act tasks. It's Paper Cut. I'm really interested to see how students respond to this, without telling them too much.

So, for Wednesday... an L shape...

Our staff room is L shaped, so before I went home yesterday I had a look. I think it could work...
 It's carpeted with our half-metre long squares. From the back of the fridge on the left across to the door on the right is the length of eight carpet squares.
Rounding a little, and adding a grid, I've got the dimensions, measured in carpet-tile lengths:
I think I'll suggest we start by giving them this shape, and asking what they notice and wonder. This has gone well before. And then, what? 

Perhaps off individually to write in their journal what they noticed and what other people noticed?

Then, if it hasn't come up, introduce the idea of area, by showing the staff room picture and plan, with the question 'How many carpet tiles would you need for this room?'

How shall we go then? Perhaps individual reflection on how to do it first, then sharing in pairs, then having a go at it in pairs. 

What would you do?

Notes in response to John's comment:

The 3rd act could perhaps be in image form:
or as a video?

This room is a better start, if we round some of those dimensions
This would do for choosing an L shape:

Read how it went in the next post.


  1. I like this and want to try it. The flow from notice/wonder to individual to partners and group makes sense to me. I wonder if making a paper version that students could cut and measure might support understanding around whole/part relationships of different polygons. Of course they will build with rods, right? It wold be interesting to provide some related room shapes - a "t" or an "x" etc. - and try to create strategies or rules for figuring area and perimeter based on partially labelled measurements.

    1. Rods - I planned to have them on hand, but not to too obviously point to them. I'm interested in whether anyone reaches for them. When one does, others are likely too.

      I was thinking to give a paper copy of my last diagram, not on a cm scale though, so this time there would not be direct measurement with rods. (Maybe I should have a cm scale ready, now I think about it, in case anyone gets stuck at that point.)

      I agree, more rooms would be good. Perhaps another L first (I'm sitting in one at home now, give or take some funny bits, I could use that.) They could look at some more partially labelled rooms...

  2. What if you started with the photo of the staff room and did a notice and wonder with that? For that matter, can you just bring them to the room with their notebooks and have them do the notice and wonder right there? It's possible that students might notice something about the room's shape, also possible they might notice something about the floor covering. Lots to notice in that room actually. If they're good at mathematizing with their notice and wonders it's possible that a student might come close to the question of how much carpet you need to cover the floor. You could measure it together and then scale down the dimensions to get that picture you would have shown originally.

    1. Yes, someone (thank you, Mary Pardoe) tweeted: "Suggest challenge them to 'guess' (estimate) in m-squares the area (record estimates) before they start to try to 'work it out'" And it made me realise that, yes, the noticing and wondering, and of course the estimating should go right back.
      I should come clean and say that I've lied about those tiles, or maybe just not told the whole truth. They don't cover the room; there's an end to the right of the photo which is the kitchen which has lino on the floor.
      I hadn't thought about taking them in, although I did think about making a quick video. So, if I said, see what mathematical things you notice about the staff room... I wonder how that would go...
      We could definitely do that 8 tile measurement across the room...
      I think your instinct is right, Joe, to give a sensory and direct experience of the thing we're trying to mathematise. I'll ponder the logistics...

    2. I really like the idea of having students visually see the room OR removing the task away from the traditional paper copy. The question then becomes "How to introduce the task?"
      Here's a thought...what if you started numbering the tiles in the top corner of the room and videotaped it. Keep numbering the tiles with post-its and stop after you've done about 12.

      Act 1 Questions: how many post-it/tiles will there be? How big is the room?
      Act 2: you give them the dimensions of the room

      Joe's suggestions are bang on, so this just my 2 pent piece.

  3. I like this and Graham's activity raises neat ideas for college algebra, too.(

    What I wrote before was that it would be interesting if pairs could pick up to 4 dimensions for which to ask. Then you might see some really different but related methods to solve, which would have lots of connections.

    Will they be able to go count squares for act 3?

    Also wondering how many more measurements they'd need to find the areas of 8048 and 8049.

    1. These comments are really helping me think about this. Sometimes it's just the obvious. Yes, B048 is all carpet, and smaller. There's a couple of students that could get boggled by the big numbers and lose the main idea, that we can see and calculate the L shape in a number of ways.

      So I really like the idea of asking them to choose their own L shape. They should do that after seeing an L shape room, but before calculating one. It's looking like a 2-day lesson. Am I trying to put too much in?

      More generally, there's a problem with the real world. The carpet squares never line up neatly, there's always pesky fractions. Or lino. This is why we need each other to find the neat situations, because they don't come up that often!

      (I've added some illustrations for this reply up in the post.)

  4. Thanks, Simon, for posting this work and your proposed lesson. Your post has generated some Area “big question” thoughts in my mind! I’m wondering what your goals are for these two lessons (Paper Cut and L-shape) and do you see one lesson building on the other? I am assuming you are looking to see if the students will apply their understanding of the area model of multiplication to an L-shape area. With the 3-act I am assuming you are using this to help them see that area comes in different shapes than rectangles and give them an opportunity to compare area without iterating area-units. What have I missed? From a teacher perspective, how do you see the scope and sequence of area, both in general and with these two problems?

    Regarding Area scope and sequence, I was fortunate to discover the works of Dr. Michael Battista on students levels of understanding the major math content areas, including Geometric Measurement which includes Area. ( Battista's work on Levels of Understanding of Geometric Measurement illuminates two types of geometric reasoning: Non-Measurement (NM) and Measurement (M). NM does not use numbers. For area an example would be that students would compare the area of two shapes by cutting one shape into pieces and putting those pieces on top of the other shape. Fletcher's Paper Cut 3-Act is an example of NM. The Measurement reasoning of Area, "M", "...involves using [iterating] numbers to indicate how many units are contained in an object. This is the work you've been doing with the Cuisenaire Rods and Place Value blocks. Of course NM and M are connected but they are different ways of looking at Area and I am wondering how you will see children making connections between the 3-Act and the L-shape lesson to deepen their understanding of area?

    Mark Pettyjohn and I have discussed how we believe students do not have the deep understanding of area that we often think they do. In our work with 4th-9th grade students many understand the idea using the area model for multiplication but it is not built on a strong understanding of what area really means, like your teacher’s lounge scenario. I am wondering how you think they would they see the connection between what you’ve been doing in class to the idea of the area of a room, L-shaped or otherwise?

    So, what to do about building that foundation? I think this is where Battista’s work is critical for me. I have a goal of creating lessons around the idea of NM Area, both paper and pencil lessons, and contextual. Then I should pursue how we use that understanding to connect to finding the Measurement aspect of area and iterating units with rectangles, triangles and composite shapes. I believe our work in Room 118 has been too light on NM to date. THANKS to your blog post and your lesson questions you have given me the opportunity to reflect on my own practice and solidify the scope and sequence of Area. THIS is the REAL value of social media, in my mind. Thanks a million, Simon! I cannot wait to see what your lessons reveal!

    1. Thanks Nina - really interesting, and it's great to be given something to go away and look at. I've just started reading the preview of Battista's book on amazon. I like his division between M and NM. And I like the way he's dividing the learning in to lots of small steps, into levels of sophistication. I'd like to read more, and look carefully at the sequences he sets out. I know I'm sometimes guilty of lurching forward a bit too abruptly in my sequence of lessons, so I appreciate this kind of analysis.
      It looks like from Battista's point of view, I'm looking for this level of sophistication (p18):
      M3 Student correctly operates on composites of visible area-units
      I'm wanting to see if this can be done flexibly, when the area isn't the familiar rectangle, to see if students can use additive or subtractive reasoning about shapes to arrive at answers.
      I'm also trying to build a real-world connection, (although the real world won't always keep itself neat). I want students to connect their experience of area outside of diagrams to what they see in a diagram.

      I'll be out of the country immediately after these lessons, but I'll try to blog about what we do!

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