It’s time to look at another typical geometry problem to make over. This time Dan Meyer (@ddmeyer) presented this problem for revision.

Dan decided to go in this direction for the revision, which, for the record I really like. I would encourage you to check it out.

I took a try at it, too. I’ll let you decide which you like better.

I like this problem’s basic core idea. Looking at the volume of a sphere (the meatball) and the volume of the cylinder (the cooking pot), in general, this is a pretty tasty set-up (pun intended). Especially considering that I am always a fan of problems that make use of food.

But…

For this problem, food and cooking were actually more of a problem that a support.

First, the cooking pot is sitting on a hot burner and I’ll be the first to tell you, a cooking pot doesn’t have to be full to spill over. So, the question of whether or not sauce will spill over is a bit more complicated that it might seem at first.

Second, meatballs aren’t spheres. They are irregular and rarely are two of them congruent.

So, my first thought was to choose spherical objects that are all congruent: for example, baseballs. Coaches regularly carry baseballs around in 5-gallon buckets, so there is our cylindrical container.

And I figured I’d deliver the task in a video simply because videos tend to improve engagement on their own.

Now, once I made the video (and some meaningful conversation was had among those who are better at this than I am) I found that my task had one glaring drawback. When you put baseballs in a bucket, they don’t pack tightly. There is air between them. A lot of it, in fact.

So, now it seems like if we are to use this video for instruction, we would need to change the question in to multiple parts.

1. How many baseballs can we fit into the bucket? (This would likely end up being a demo or a lab where we collect data. Tricky to calculate this.)

But then we supplement the question above by…

2. How much volume is wasted by packing that many baseballs in the 5-gallon bucket.

This would get back to the original content. Likely the cylindrical volume would need to a unit conversion, and then some analysis of the collective volume of the collection of baseballs.

Now, if we could ind a way to check it. The first thought I had was to fill the bucket with water. Put the baseballs in to displace the water out of the bucket. Take the soggy baseballs back out of the bucket. Find the volume of the water that’s left.

Problems with this idea: 1. Baseballs float which is going to effect the manner in which the water is displaced. 2. Baseballs absorb water. This means that some of the none displaced water would get removed with the baseballs and not counted.

Hmm… I thought of filling the bucket with baseballs and then topping the bucket off with sand. Which would solve both of the above problems, it would also give me an opportunity to make a beach trip.

Any other ideas out there?

“Packing” problems are fairly advanced, but the following no-frills problem is meant for the 6th year:

http://fivetriangles.blogspot.com/2013/06/78-cylinder-and-ball.html

How do you feel like that problem compares to the original meatball problem at the top of the article? Do you like it better? worse? What are your thoughts?

Ok, a bit about our teaching philosophy.

The most important measure of where a problem should be placed in a teaching sequence is the most difficult mathematical skill that is involved. This problem and its variants are all distilled into two formulae, volume of a cylinder and volume of a sphere, and plugging numbers into those formulae. Basically, you’re given all the numbers to plug in, save one. The missing number is the increase in depth of the liquid. No matter how you add fluff to this problem, e.g, video, meatballs, etc., it’s the same problem, and those formulae are found in Common Core, Grade 8.

Because there is π, but no square roots, negative numbers, etc., these problems really could be moved into grade 6 or earlier. That’s where our cylinder-and-ball problem originally appeared.

You can add measuring the meatballs, but that’s a grade 2 skill, perhaps. Measuring the pot: the same. What’s to be gained by adding that? Adding peripheral details to the problem does not increase its complexity.

You can put a lip on the pot, but its effect can only be estimated, not calculated.

Here’s a problem where a change in horizontal area can be used to increase the problem’s complexity. That, for us, makes it far more sophisticated:

http://fivetriangles.blogspot.com/2013/03/51-hot-bath.html

We don’t advocate “guessing”, ever. Guessing is not for a mathematics class and the word “guess” shouldn’t be in the lexicon. Estimating can belong, perhaps, but that doesn’t add to this problem’s complexity, either. Put a pea in the pot, it won’t overflow, put in a softball, it will. What does it matter if you correctly “guess” that 4 meatballs didn’t make it overflow, but 5 did? It should suffice to know that there’s going to be a particular number and calculating that number is the essential skill.

Everyone already knows the more you add to the pot, the deeper it will get.

If you want to impart visualization skills, create a visual problem:

http://fivetriangles.blogspot.com/2012/04/basic-geometric-shapes.html

We actually think the meatball problem and its variants are fairly low level from beginning to end, because there’s no problem solving. It doesn’t need a makeover; it just needs to be recognized for what it will always be: it never rises past a simple plug-in-the-numbers calculation problem.

That’s why our ball-and-cylinder problem is a 5 minute task. That’s all the class time that such low level calculations warrant.

We understand these attempts at “real life” problems pervade American mathematics education, but dragging out a 5 minute task into an entire 45 minute class or beyond is really doing a disservice to students.

I’m understanding your philosophy. Question: what type of activity would be worth an entire 45-minute class period?

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We like concise problems, more than activities, that lend themselves to varied approaches. Here’s a problem that relies on one theorem: the sum of the angles of a triangle = 180°:

http://fivetriangles.blogspot.com/2013/07/81-fireworks.html

(It may or may not help to know the polygon corollary: sum of angles = 180(n-2), where n is the number of sides.)

This is not a one- or two-step simple calculation problem.

A few comments/solutions have shown up in cyberspace since we posted this problem. One called it a “chasing angles” problem, something we’ve never heard of, but that might be one approach. Over on tumblr, a few people posted solutions, and they were all different. One began by looking at the septagon in the middle of the diagram; another looked at the whole diagram as a “sharper 7 pointed star”, as opposed to a blunter star; a third looked only at various overlapping triangles. Another approach might be started by adding lines: by connecting all of the star’s points and making a septagon around the outside.

Each approach can be tried individually or in groups, and the class can reach consensus or not as to the relative merits of each approach. Proper class preparation means the teacher should be familiar with many approaches beforehand and can prompt or give hints as necessary.

Although our idea of a good problem may differ from others, one thing about choosing a problem well is that throughout the solution process, the grade level difficulty should be maintained. This missing angle problem never slacks off.

One well-conceived, complex problem can easily consume a 45 minute class.

Hope this answer wasn’t too pedantic.