Today, I was lucky enough to be asked to teach a third grade math class because the teacher was going to be out. Since I have never taught the Arranging Chairs activity in third grade, I was excited when two of the other third grade teachers, Jen and Devon, wanted to plan with me yesterday. Before meeting I read up in ** Children’s Mathematics: CGI**,

*Carpenter, Fennema, Franke, Levi, Empson*to think more about this idea of equal groups, meets arrays, meets area model builds. Here is one piece I found that connects them in a very nice way.

We decided to change the Ten-Minute activity from a time activity to a dot image number talk. We thought since the students have been doing so many dot images involving equal groups, that it would be interesting to see how they thought about one image with a missing piece. We were curious if students would use any structure of an array to think about how many dots were in the picture. The board ended like this…

For the most part, students either added rows (so they were seeing the array structure) or looked at the symmetry of the picture. They had so many more strategies they wanted to share, but for times sake, I did a quick turn and talk so they could share their ideas with someone before they left the carpet. Because I heard a student talk about filling in the middle, I asked him to describe to the group what he and his partner talked about. He said, “You could fill in the missing dots and then do 4,8,12,16 minus 2.” I heard the word array thrown around so I asked them to tell what they knew about arrays. A few students built upon one another’s definition ending with something with rows and columns.

Next, we introduced the activity on the carpet right after the Number Talk. *You have 12 chairs to arrange in straight rows for an audience to watch a class play. You want to arrange the chairs so that there will be the same number in every row with no chairs left over. How many arrangements can you make?* They talked to a neighbor and I took one example, a 6×2, and constructed it on the board. We talked about what that would look like on the grid paper. The grid here felt like a very natural way to move students between arrays as equal groups to rectangular arrays. They went back to their table, with cubes, and worked on making as many arrangements as they could. We shared them as a group.

They talked about the commutativity in the rotation of the arrays. We discussed the fact that since we were talking about seating arrangements in this activity, we would consider them two different ways to arrange the seats. This is where I saw the arrays as such a beautiful way of visualizing commutativity in a much different way than they previously had discussed in rearranging number or groups and group sizes.

Next, each group was given a number to create as many arrays as they could, cut and paste them on a piece of construction paper. Choosing the numbers for each group was something we spent a lot of time in during our planning. We wanted to be sure that noticings around sets of numbers such as primes, composites, evens, odds, and squares would surface, as well as relationships between different sets of numbers, we tried to be really thoughtful around this. We came up with a first set of numbers and then decided on a second number to give that same group if they finished early. So, this list is first number/second number (although we knew not all would get to the second one).

11 / 27 – Prime number and then an odd that wasn’t prime

25 / 5 – Odd square number and then relationship to a multiple they did of that number.

16 / 8 – Even square number and then halving on dimension

9 / 18 – Odd composite and square and then double a dimension

24 / 12 – Even number and then half a dimension (we didn’t think they would get to this one because 24 has quite a few to cut out:)

18 / 36 – Even number to compare with another group and then double a factor (36 could also relate to other groups numbers in various ways)

15 / 30 – Odd composite and then double a factor. We didn’t think they would finish 30.

13 / 14 – Prime number and then how adding one more chair changes what you can make.

Extras for groups done both: 64, 72, 128. (No one got there)

Thanks to a lovely fire drill in the middle of class, some groups did not get to a second number or if they did, did not get to finish. This is the point where you realize how amazing it is to have more than 1 teachers in the room! Everyone could walk around and listen to their conversations while they worked. We heard everything from frustration/wonderings about prime numbers because they thought there had to be more than one (and the rotation) to excitement when they finally got a second number with more. Here a few of the (close to) final products:

On Monday they will hang them up and walk around to do a notice/wonder about all of the different numbers around the room, but we really wanted them to think about their work today before jumping into comparing others. I also really wanted to capture what they were frustrated by, liked about their number, were thinking about in the moment and were left wondering. So, I asked them to write about what they noticed and wondered about their work today. I expanded on the prompt a bit to avoid, “I notice I could make 4 arrays,” and I said, “You could tell me why you liked your number or didn’t, what you think made your number easy or hard, or what you realized as you were making them.”

There were some beautiful responses that I cannot wait for Andrea (their teacher) to hear on Monday because they were so excited to share!

A nice noticing that could lead to largest perimeter with the same area:

An informative noticing and wonder about commutativity to keep in mind when planning…

Wonderful comparison of why they feel evens are easier than odds, but also great wonderings about “Is that really all you can do?” with prime numbers and why?

I talked to this student and he was using the 12’s for 24 but had trouble articulating it in his journal.

Loved this one wanting a number in the hundreds because it would be more challenging and don’t miss the bottom piece about subtraction!

She was not as much of a fan of the square as I was when I walked up, she said it is, “just the same when we turn it” and I said, “That is an awesome thing!” (I meant her noticing, but I think she thought it was about the square:)

I will leave you with this one that struck me as “We always have more to learn.” I cannot wait to see her working with fractional dimensions in 5th grade!

I cannot wait for the gallery walk and noticings and wonderings from the entire group of numbers. I am also really excited to see this work move into rectangular arrays and seeing students’ strategies around multiplication evolve and how they take this work and form relationships between multiplication and division.

Great day in 3rd grade and I have to say, I think Jen, Devon and I planned really well for this one!

-Kristin

Joe SchwartzGreat lesson! I love how thoughtful you were in assigning the numbers. It’s really a model of how to plan. I’d like to think about this: The original problem is about seating arrangements for a play. So where’s the stage? And what’s an optimal arrangement? 1 row of 12 would give everyone a front row seat, but what about the kids on the ends? 12 rows of 1 would make no sense. Maybe 2 rows of 6? What would adding more chairs do to the optimal arrangement? I think it would be a good way to get the kids thinking about the meaning of the arrays and a way to start some interesting arguments. Kids may even think about variables such as the size of the stage or the heights of the kids in the rows.

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mathmindsblogPost authorI love those questions Joe! We did talk the stage position when doing the 12, but it was not revisited during the construction. Is this when decontextualizing happens? But then to bring it back would be really fantastic… Have to think more on this too! Thanks!!

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Simon GreggI thought about contextualising it too. I wondered about saying that one of the 17 children in my class was going to perform and we’d like to have a rectangular array of audience chairs for the other 16. Then we could re-enact the arrays after they had all been discovered with our actual chairs. (Best here if they had something to actually perform! – a haiku maybe…) Hopefully the humour of 16 chairs one behind the other, wouo

I love that you plan so carefully and collaboratively. I’m not sure I’d do this with the choice of numbers. Like, I did with our Cuisenaire factors, I’d probably just say, choose a number between x and y from the hundred square, take it with you, and then get exploring. I imagine we’d get examples of the different kinds of numbers… But, I’d like to hear what I’m missing.

The thing which makes me think “I want some of that!” is how your students write. Although I’ve got maths “journals” this year, I’m still not really getting them writing reflectively like this.

OK – this week… I’m thinking that I might send them off after, or just shortly before when I’d normally end, a dot image time, and ask them to write how they saw the image, and how someone else did, and perhaps what’s in common and what’s different about the two ways. Perhaps they could say what the they thought the most obvious and least obvious ways of seeing the dots were… Anyway, however I approach it, I’ve got to get going on getting more reflection!

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mathmindsblogPost authorSimon, thank you so much for your thoughtful comment, you and Joe have me thinking about all of the possible contextual opportunities! Reflecting on it, the choice of numbers was probably more in the interest of time in seeing the sets of numbers, properties and relationships. I love the way you do this so masterfully though and I am wondering if another third grade class (of one of the teachers who planned/taught with me) who hasnt taught this yet could try out free choice of numbers. Would you then pose numbers if, let’s say, no prime numbers were chosen? I really love this idea, I just would like to see certain things come out in the share.

As for the journals, I think your students do it within their work all of the time, so are you thinking just more reflective writing? I thinking designing the prompt is so interesting based on what is happening during the lesson!

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howardat58Hello again.

Did you see my post from last year ? Here it is:

https://howardat58.wordpress.com/2015/02/03/commutative-distributive-illustrative-ly/

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mathmindsblogPost authorHi Howard! I had not seen that post and that is such a wonderful visual of what is happening with the properties. Thank you so much for sharing!

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howardat58Hi Kristin. I am glad you liked them. There is an element of overkill in a lot of the treatments of “the properties”, and also in my view some real horrors. Here’s one for you, it’s about the distributive law:

https://howardat58.wordpress.com/2015/01/21/more-bad-language-in-math/

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maryleenewmanI read your post and we did a very similar lesson this week in my third grade class. I did pick numbers for the groups to use and it was interesting to see where the hang-ups were. The groups with 27 found it very difficult to come up with any arrays other than 1 x 27 and 27 x 1.

Most students are comfortable with skip counting or using circles and stars for multiplication. They figure out the groups or arrays and can determine the total number of objects, but they struggled with dividing the chairs into equal rows because they didn’t know in advance if the rows would be even, and they were reluctant to have “leftovers.” Division is not as neatly packaged for them.

Students were more willing to experiment with the unifix cubes than on paper, and when they did figure out that there could be rows of 3 or 9 when they had 27 chairs, they were thrilled. It was fun for them to solve a hard problem.

I can tell we need to follow up with more activities using the unifix cubes so students get more comfortable with division as division and not just the inverse operation to multiplication.

Thank you for sharing your thoughts and lessons!

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