Earlier this month
Graham Fletcher and
Joshua Greene got together for a tag team takedown of a textbook lesson on finding the volume of rectangular prisms:

It never really had a chance.

Their analysis and make over suggestions are spoton. And I hate to pile on, but what makes this especially egregious is that exploring volume of rectangular prisms, a major grade 5 content standard, can be, well, fun! Rich and I have had success with activities described
here and
here, and I'll offer up another one that we tried out with the fifth graders last year.
Andrew Stadel has written about the potential that
starting arguments in math class has to reinforce both
practice standards (especially numbers 1 and 3) and engage kids in learning, and I agree. For this project, I collected five boxes and asked the kids to guess their order from smallest to largest as measured by volume. I didn't want the boxes to be too similar, just similar enough that it wouldn't be too obvious.

I chose these five...


...and gave the kids time to get their hands on them. 

Argument started! 
With the intellectual need now built, it was time for the kids to resolve the dispute. They first had to estimate the volume of each box...

...then find the necessary measurements. We decided on nearest tenth of a centimeter to reinforce working with decimals. 

We let them use a calculator because who wants to do all those calculations by hand?


The data was recorded.

And, due to faulty measuring, improper rounding, and incorrect number crunching, the argument
continued to rage! In fact it took several days for the class to come to agreement on the volume of each box and the correct order. But they were days filled with engagement, collaboration, discussion, and multiple content and practice standards.
The activity isn't all that imaginative, and it's not that hard to prep for. All you need are some boxes, rulers, and calculators. And while it's not as easy as asking the kids to take out their books and do this...
. ...I guarantee you'll have more fun.
I wonder if this could be an evil packager story. (Maybe as a follow up?) Your company has reduced how much ___ in their product, and want you to design a box that seems like it has more...
ReplyDeleteLove this lesson.
Thanks John! I like your follow up. The concept of playing around with reducing or enlarging the measurements, and the effect that would have on the box, would be interesting to explore.
DeleteDelightful. There isn't a container large enough to contain
ReplyDeletemy gratitude!
Thanks Turtle!
ReplyDeleteNice activity! I love anything to do with measuring and real data.
ReplyDeleteThinking about data, I wonder about helping students with the shift between mental modes: real world vs idealized world. A lot of early years math is discrete with small numbers, so there isn't much of a difference (3 buttons is exactly 3 buttons). However, measurements and decimals can easily be confusing. When is 10.1 cm x 12.3 cm = 124.23 sq cm exactly and when is it the same as 120 sq cm (b/c the error is larger than 4 sq cm)?
While looking at the pictures, I was also wondering about filling the boxes with something. Could be interesting discussions sparked filling the boxes with large objects vs small, regular shapes vs irregular.
Just the other day, while perusing the CCSSM document it struck me that the approach to volume was very narrow and had only a small relation to the actual world. How is a volume measure presented "out there" ? As pints, gallons, liters, milliliters, NOT as cubic inches. Filling boxes with tiny objects, beans, sand, whatever, and pouring from one box to another tells a lot. We have rulers for measuring length. What do we use for measuring (not calculating) volume? Make one! Draw attention to the "volume = base area X height" for boxes, and use a transparent material for the (open one end ! ) box.
DeleteThis can then be used for measuring the volumes of glasses, cups, odd shaped containers, pill bottles ....
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Thanks both Howard and Joshua for your comments and observations. Regarding the decimals, this did provoke a discussion about what they all meant. It gave rounding some kind of meaning, rather than it being an exercise to be done for its own sake. A box with a volume of 614.125 cubic cm begged the question: what is 0.125 cubic cm?
DeleteWe build the volume concept with a series of estimation180type tasks where the kids have to estimate how many wooden cubes will fit in boxes of various sizes, with the reveal being me actually filling them up. But in my experience exploring volume by filling containers with beans, sand, and other types of objects is something done maybe in preschool or kindergarten and then never again. What a shame!
A 3d shape lesson done in 3d. Imagine that! I love how this shows how easy it can be to make math more realistic and hands on. 2 minutes of grabbing things if my shelf and I can turn a worksheet into a hands on lesson. V
ReplyDeleteThanks so much! If you try it out let me know how it goes.
DeleteLove the way you introduced the need to solve this through an argument Joe! Wondering what would happen if we started the unit on volume this way. Students would use nonstandard items to measure at first. How many __________ can you fit in the box. Just wondering if would give the need to be more precise through standard measurement.
ReplyDeleteInteresting thought...with no previous experience with volume, what would they come up with? How would they go about determining the sizes? Would they combine linear measures in some way? Would they hit on filling each up with some type of standard unit? I thought of this activity more as practice after introducing volume, not as the introduction itself, but maybe next year we'll try it this way. As always, thanks for making me think!
DeleteThank you for sharing! Love the competitive element of this lesson and taking it out of the textbook and into the box! I can see a Minecraft extension here:)
ReplyDeleteThanks Catherine! If you have you kids work on something similar in Minecraft let me know how it goes.
DeleteJoe, I'm thinking your cuboid argument would have set us up really well for our series of lessons on making a fruit juice carton last year.
ReplyDelete(You remember the one, http://y4ist.blogspot.fr/search/label/carton )
Something to think about for this year... I think the contention really gets everyone's focus on the question!
Yes! And I loved the fruit juice cartons. Could mash that up with John's "evil packager" idea above.
DeleteI really like how this is lesson begins with estimation and takes off on a journey of "need to know" discovery with a lot of bumps in the road that engage students further. I do like Graham's suggestion of beginning a unit on volume with this task. This might even be adaptable for surface area in 6th grade. Hmmmm...
ReplyDeleteThanks Mike! I think it could adapt to surface area...kids could order the boxes based on how much wrapping paper you would need to wrap each one up as a gift. In fact I think Dan had a go at that here:
Deletehttp://blog.mrmeyer.com/2014/purposefulpracticedandycandies/
Excellent. I've seen this before. Thanks for the mental nudge.
DeleteJust worked through this lesson with teachers, then cotaught it with a 5th grade class! Though we certainly didn't finish it today, there was a deep discussion about dimensions and volume.
DeleteWe began the lesson by showing the boxes and asking what they notice and wonder. They immediately started wondering which had the largest volume and narrowed it down to 2, though a couple were skeptical that they all had the same volume. The best part about this was that the elusive 3dimensions all came out during this brief discussion. This was the beginning of their unit on volume.
Icing on the cake: a couple of students came up to us afterward and said, "I really liked our math lesson today. It was cool." Thanks for sharing!
That's great! Thanks for letting me know. The NJGA connection strikes again!
DeleteThank you, Joe! This is what I love about your blog  you take what you've done, break it down, and make it applicable for others to use with any curriculum. I may not be in the same situation as the teachers you're working with, but I can still use your creative thinking. In that way I can use my time to find different sized boxes (or whatever it is) and prepare to teach MY students without being overwhelmed by thinking about reenvisioning my entire curriculum or even the next unit; which is incredibly overwhelming as an elementary school teacher. (Add that to Dan Meyer's list of reasons why the #MTBOS is so powerful!) Thank you for doing what you do! ~Julie Howard
ReplyDeleteThanks so much Julie! I'm so glad you find what I'm doing here to be useful. It's what keeps me motivated to continue experimenting, exploring, and sharing.
ReplyDelete