Thursday, May 14, 2015

Box of Clay

     It's volume time again in grade 5.  Last year Rich and I tackled this important standard with a cereal box project.  This year we added a wonderful item from Illustrative Mathematics called Box of Clay:

  I decided to introduce the task by removing the question and replacing it with a notice and wonder prompt:

This removes the anxiety of having to "figure something out" right off the bat.  The kids have a chance to process the information and enter the task in a comfortable way.

I gave them several minutes to think on their own, then do a turn and talk, then share out with the class.  Most of what they had to say didn't surprise me:
  • There are two boxes.
  • The first box is smaller than the second box.
  • The first box can hold 40 grams of clay.
  • The first box is 2 cm high, 3 cm wide, and 5 cm long.
 But then...
  • There's something wrong.  The first box is supposed to hold 30 grams of clay, not 40.
  • 2 x 3 x 5 is not 40.
  • The volume is measured in grams, not cubic centimeters.
   I was not expecting this.  Many kids were completely thrown off by the two units of measure in the problem: the height, width, and length of the box, measured in centimeters, which they dutifully multiplied to get the volume (30 cubic cm), and the weight of the clay, stated as 40 grams.  
   This issue came up again in their "wonderings":
  • Why 40 grams of clay?  3 x 2 x 5 = 30.  It's supposed to be 30.
  • Why is the unit grams?
  • Is a gram the same size as a centimeter?
  • What's the volume in centimeters and grams?
  • Why are there two different units of measurement?
  • What does 40 grams of clay look like?
  Along with:
  • What's the volume of the second box?
  • What's the volume of both boxes together?
  • How much bigger is the second box?
  • How many grams of clay can the second box hold? (This was the actual question!)
  • How many grams of clay can both boxes hold together?
  • It's a box of clay?  Why is the box made out of clay?  
     Before I go on and explain my next move, I want to reflect on just how powerful this experience was.  Rich and I could have just copied the task and handed it out for the kids to work on.  But providing time and space for the kids to notice and wonder before they got to work revealed some very important understandings, and allowed other very telling misunderstandings and misconceptions to surface.  That, in and of itself, proves the efficacy of the "noticing and wondering" protocol.  But even more, the kids generated the actual question themselves, along with several others they could work on, which turned the experience into a quasi-3-Act.
   I explained to Rich that the kids could begin working on the project the following day, but we would need to start with a whole class demonstration.  That morning I rounded up some supplies:

I felt it was important to clear up the misunderstandings about the units.

I started with a small piece, and continued adding until the two pans balanced.  40 grams of clay.

   We discussed that what was contained in a box could be measured in many different ways, but that a useful measure of the size of a box itself was cubic units.  I'm not sure if the distinction was clear to all, but I felt it was a start, and I had them imagine what a box that could hold 40 grams of clay might look like.  Not very big.
    I was about to get them started working out the problem, when a hand popped up in back.
   "If the clay was flattened out, would it still weigh 40 grams?"
   Again, this was not a question I expected.  And, surprisingly, the class could not agree.  Some kids thought that a flattened piece would weigh less.

Nope.  Still 40 grams.

With that out of the way, it was time to get to work.

What do you know!  The actual question from the Illustrative Mathematics task was one the kids had come up with the previous day during the noticing and wondering activity.

Several kids wanted to build the boxes using centimeter cubes.  I could only drum up a small bag.  They had to improvise.

Ultimately, however, they all drew boxes...


...and there were issues here too.  Is this box really 2 cm high, 3 cm wide, and 5 cm long? 

There were mistakes and false starts.  The erasers got quite a work-out.
But they persevered, and the boxes of clay took shape.
I'm glad we invested the time in having the kids draw the boxes.  Because when it came time to answer the question, we saw quite a bit of this:

Uh oh.
     With Michael Pershan's ShadowCon talk on hint-giving fresh in my mind, I had to do some thinking.  What could I tell these kids to get them on the right track?  I looked at the commentary that Illustrative Mathematics provided to go along with the task:

This picture provided a helpful clue.

    While looking back at the pictures of the boxes they drew, I could ask something like, "How many of the first size boxes could fit inside the second box?"  And if that didn't work, I could try something more explicit: "The first box is 30 cubic cm and can hold 40 grams of clay.  What if there were two of those boxes?  What would be their volume?  How many grams of clay could two boxes hold?"
I tried it out the following day:

As fortune would have it, our next unit is on rates.  Think this will be useful?
     And as the kids finished working on the main question, they started in on some of the others they had brainstormed during the notice and wonder phase:

Yes, I made a mistake wording question B.  The kids caught it right away.

     Though they are not all created equal, there are many other rich tasks on the Illustrative Mathematics site, and I have encouraged our teachers to look there for ideas and supplemental lessons.  Reflecting on the experience with Box of Clay, I would offer these suggestions:

  • Try introducing the task with the question removed and replaced with a notice and wonder prompt.  What you find may surprise you, and will help inform the direction of the task moving forward.
  • Take a careful look at the commentary that accompanies the task.  
  • Take Michael Pershan's ShadowCon advice.  Try to anticipate potential wrong turns, and plan out in advance what "hints" will put students back on track.
     And there's something else about Illustrative Mathematics that has rocketed it towards the top of my "go-to" list: the project's CEO is Bill McCallum, and Phil Daro and Jason Zimba are listed as Senior Advisors.  Who are they?  Only the leaders of the team that wrote the Common Core State Standards for Mathematics.  Who better to create and select questions, problems, and activities that accurately reflect the standards.  I've got only one question for them:
    Why is the box made out of clay?


  1. Wonderful, how the noticing and wondering opens this up, and moves the class from being passive to active! And how questions like, what happens if you flatten it? can come up in this climate. A lovely lesson!

    A few thoughts:

    Questions like this could be made easier, if needed, by having the box full of water (let's hope there are no holes in the clay box!) and then I gram = 1 cm³ and our plastic cubes are the same density too. (Though I think the difference with clay added an interesting challenge!)

    It must be harder working with g and cm³ in a country that uses pounds and ounces lots too!

    We've ordered lots of those plastic interconnecting centimetre cubes - so at least one person can make the solid model for us. Also, you know how I like triangle grid paper - that makes drawing these cuboids more satisfying, and maybe gives a good idea of how many cubes are in there. Not that drawing them on blank paper isn't good too!

    1. Thanks Simon! I have tons of those cm cubes, but they were back in my room, not in that particular classroom. I do think that having a solid model would have helped. I'm going to try it myself on triangle grid paper.
      I've often thought that we're at a disadvantage because our kids need to work in both measurement systems. They often wind up learning neither very well. Using the metric system as a matter of course also promotes decimal place value understanding, so you may have a built-in advantage there too.

    2. Re-reading my point about the water, I hope it didn't sound patronisingly obvious. I often miss, or need reminding about the obvious things myself. And I didn't know how au fait you are with metric - as you say, I have the luck to be in the home of metric!

    3. No worries Simon! Keep reminding. And I'm curious: do kids in Europe learn anything at all about the US Customary system? Is it part of a curriculum?

    4. In Britain, despite decades of metric in schools, it's still miles and mph on the roads, still a pint of beer. So yes, conversion charts and so on. As for French state schools, I suspect not. I'll check.

  2. The task is already rich, and I wonder what happens if/when they move on to computing the volume of cylinders. It might be interesting to use an essentially similar task, but double just the radius, double just the height, and double both the radius and the height. Students could explore the effect on total volume (and, specifically, observe that the radius change has quadratic effects).

    1. Great idea for vertical planning. The folks over at Illustrative Mathematics might like a task like that, especially because it build on previous learning. I think that volume of cylinders is a grade 8 standard.

  3. Joe, the box isn't made of clay. The prompt says it can hold 40 grams of clay.

    1. It was an attempt at a little humor, responding to one of "wonders" from a student. Not so successful I guess!