This is from Discovering Geometry.
What I Did
Basically, I three-acted the heck out of it. Which means:
- Reduce the literacy demand. Let’s encode as much of the text as we can in a visual.
- Add perplexity. That visual will attempt to leave students hanging with the question, “What’s going to happen next?”
- Lower the floor on the task. The problem as written jumps straight to the task of calculation. We can scaffold our way to the calculation with some interesting concrete tasks.
- Add intuition. Guessing is one of those lower-floor tasks and this problem is ready for it.
- Add modeling. We’ll ask students “what information would be useful here?” before we give them that information. That’s because the first job of modeling (as it’s defined by the CCSS) is “identifying variables in the situation and selecting those that represent essential features.” The task as written does that job for students.
- Create a better answer key. Once we’ve committed to a visual representation of the task, it’ll satisfy nobody to read the answer in the back of the book. They’ll want to watch the answer.
Here’s the three-act page. Leave a response to see the entire lesson.
Show this video to students.
Ask them to write down a guess: will the sauce overflow? Ask them to guess how many meatballs it’ll take. Guess guess guess. It’s the cheapest, easiest thing I can do to get students interested in an answer and also bring them into the world of the task.
Ask them what information would be useful to know and how they would get it. Have them chat in groups about what’s important.
If they come back at you telling you they want the radius of the pot and the radius of the meatballs, push on that. Ask them how they’d get the radius. That’s tough. Is there an easier dimension to get?
Someone here may ask if the lip of the pot matters. It isn’t a perfect cylinder. Give that kid a lot of status for checking those kinds of assumptions. Tell her, “It may matter. It isn’t a perfect cylinder but modeling means asking, ‘Is it good enough?'”
Give them the information you have.
Let them struggle with it enough to realize what kind of help they’ll need. Then help them with the formula for cylinder and sphere volume. Do some worked examples.
Once they have their mathematical answer, have them recontextualize it. What are the units? If that lip matters, how many meatballs will it matter? Should you adjust your answer up or down?
Then show them the answer.
Surprisingly close. The student who decided to add a couple of meatballs to her total on account of that lip is now looking really sharp.
Let’s not assume students are now fluent with these volume operations. Give them a pile of practice tasks next. Your textbook probably has a large set of them already written.
Help I Need
- Raise the ceiling on the task. My usual strategy of swapping the knowns and unknowns to create an extension task is failing me here. Watch what that looks like: “The chef adds 50 meatballs to a different pot and it overflows. Tell me about that pot and its sauce level.” I’m not proud of myself. Can you find me a better extension? I’ll give highest marks to extensions that build on the context we’ve already worked to set up (ie. don’t go running off to bowling balls and swimming pools) and that further develop the concept of volume of spheres and cylinders (ie. don’t go running off to cubes or frustums).
What You Did
Over on the blogs:
- Andrew Shauver also turns to video but (inadvertently, I think) makes the problem much harder.
- Beth Ferguson switches to a live demonstration and posts a great list of scaffolding questions.
Over on the Twitter:
- Max Ray, Michelle Parker, and Terry Johanson are all inside my head.
- Ignacio Mancera poses a similar situation but suggests doing it live in the classroom. I don’t accept the premise that “real” always beats “digital” – there are costs and benefits to consider – but I think Ignacio and Beth have the right plan here. If you have the materials, do it their way.
- Scott McDaniel suggests changing the context from meatballs in sauce to ice cubes in an iced mocha because kids drink iced coffee but don’t make spaghetti. This introduces a pile of complications (like the non-spherical shape of the ice cubes and the non-cylindrical shape of the cup and the fact that the ice will float at the top of the cup) for unclear benefits. Time and again in this series I’ve tried to convince you that changing the context of a task does very little compared to the changes we can make to the task’s DNA. Does someone (Scott?) want to make the case that the following task is a significant improvement over the original?
Call for Submissions
You should play along. I post Monday’s task on Twitter the previous Thursday and collect your thoughts. (Follow me on Twitter.)
If you have a textbook task you’d like us to consider, you can feel free to e-mail it. Include the name of the textbook it came from. Or, if you have a blog, post your own makeover and send me a link. I’ll feature it in my own weekly installment. I’m at email@example.com
2013 Jul 16. A makeover from Chris Hunter in the comments. (I had forgotten how weird Orbeez look in water.)
We just talked about this problem and your makeover at Math for America. One idea was to up the stakes: I’m putting this jar of water on top of a student’s phone. How many balls can I put it before it spills over? If you are sure you are right, put your phone under the jar…