Composite Figures in Context: The Wedding Cake Problem

What you see below is the original post from 2013. Since that time, this problem has taken on some alternate forms. One alternative was suggested by Jeff in Michigan and appeared in a post in March of 2014. The other alteration I made myself in Desmos in May of 2015. Feel free to read on and consider checking out the many different ways there are to approach this interesting (and delicious) mathematical situation.

We’ve started our 3-D unit.

Once we get into the volume and surface area measures for 3-D figures, the textbook leads us to shapes called “composite figures” that look like this.

source: Geometry, copyright Holt, Rhinehart, Winston, 2007, Pg 684 #8

This can be a tricky image for student to try to work with, mostly because they’ve never seen anything that looks like that before. But they’ve seen composite figures. They are everywhere. But, removing the context can be enough to take this very applicable, contextual concept and make it abstract enough to be confusing.

In reality, composite figures are wonderfully applicable. (I say again, they are everywhere!) So, here’s my question: Why do we insist on giving them an abstract picture to start with? Why not start them with one of the many composite figures that will draw the students into a real context.

I present exhibit A: The Wedding Cake.

Photo Credit: Flickr user “kimberlykv” – used under Creative Commons

Your basic wedding cake, like the one shown at the top is three cylinders of differing sizes stacked on top of each other. What I like about the Wedding Cake is that the measures of volume and surface area matter in real time and without too much more background than a bit of story-telling (which I love to do).

Now, let’s toss an additional cylinder into the mix.

Now, you’ve got yourself a math problem!

Question to the students: How much can the baker of the above cake expect to spend on the lemon frosting that is on the exterior of that cake?

… and see where they go with it.

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Cereal and Peanut Butter: The Unexpected Lesson Plan

The unplanned econmonics lesson helped it all make sense…

We’ve spent the last few days picking apart The Ritz Cracker Problem, Episode I. I designed this problem about two years ago and this is the first time I have unleashed it onto a group of students. I wasn’t sure what to expect. I set my learning goals and after some individual deliberations, we started big group conversation with the question that you see below.

The learning goals had to do with volume and surface area

Translation: If you stack 16 crackers up and then split them into two stacks of eight, can we simply avoid using the volume and surface area formulas by simply dividing the values for the 16-cracker stack in half?

As the discussion continued on this point, it became clear that you could divide the volume in two, but the same wouldn’t be accurate for surface area. The explanation for this became a bit of a sticking point for some.

Then peanut butter and cereal come to the rescue. (I teach in Battle Creek, MI. Cereal is involved in everything we do, after all.) I never even thought of this image. It never crossed my mind.

I asked them why a 16 oz. container of peanut butter or cereal could be, perhaps, $2 but double the amount would almost certainly be less than $4. How is that manageable for the company selling the product?

Now, that is fairly complex answer in reality, but for our purposes in class, the students were able to explain and understand that bigger packages allow the company to push way more product for a minimal increase in packaging. Bigger cereal boxes allow Kellogg’s to sell more of what they make (cereal) while not having to spend time and money messing around with what they don’t make (boxes).

Translation: Combining smaller packages allows can allow for big changes in volume without a correspondingly big change in surface area. The peanut butter and cereal did it!

… and I never saw it coming.

I Hope You Struggle Well

How prepared are our students to struggle well?

One year while I was saying my final goodbyes toward the end of a final class period of the school year, I scandalized my students quite effectively. It had been one of the most enjoyable classes that I’d ever worked with and I told them so. That part didn’t scandalize them. What did was my justification for why I enjoyed them so much.

” All year long, you all struggled well. You struggled together.”

They looked surprised. Some looked indignant, as though I was being sarcastic with my original statement. But that was sincere praise and it is some of the highest praise that I will give to a math student. What I meant was that they had been a resilient group who understood the learning process with its ups and down. They accepted that it wasn’t going to always be easy and worked through it. Sometimes they struggled and they struggled well.

We need to encourage the ability to struggle well.

There is a logical conflict that exists in education. We want kids to succeed (meaning get good grades… meaning get a lot of answers correct… meaning be able to do what a teacher asks at least 85% of the time.) We also want rigor. (Meaning things that are difficult… meaning tasks that students are NOT able to do… at least, not at first.)

Between the initial rigorous task and the final grade-gathering answer there should be a fairly substantial amount of time. Now, what exists in that time between the introduction of the rigorous task and the submission of the final answer depends largely on the nature of the instructor, the nature of the task, and the nature of the student.

What should happen is that during that time the students are learning how to struggle well.

Not all attempts end well. That’s okay.

But, depending on the nature of the instructor, the task, and the student, struggling well is not an inevitability.

I know the anxiety of watching the students struggle. The students struggle because they don’t know what to do. It’s the instructor’s job to make sure they know what to do, right? If they don’t know what to do, there is a temptation to believe that I am not doing my job well.

Additionally, some tasks are more conducive to struggling well than others. The task needs to be engaging. The task needs to inspire collaboration. The task needs to have several steps. It helps for the task needs to have an answer with enough uncertainty that testing and reasoning about the answer is necessary.

Many students don’t want to struggle. They want to know answers… now. They want their work to be easy and are willing to object when it isn’t. Beyond that, though, most students have to learn how to struggle well (which is a little bit of a bizarre statement to some). They need to learn how to be patient, how to ask questions, how to test their own thought-processes, how to explain their thoughts to others and listen to other explanations, how to be creative and how to participate in creative processes with others. These aren’t innate qualities in many young people, but if young people can learn these skills, the possibility for intense, meaningful and lasting math understanding is real.

So, because of the incredible upside, I hope our students receive rigorous tasks. I hope they need to be patient. I hope they need to collaborate. I hope they need to reason about their answers with others. I hope they learn to be comfortable being a little uncomfortable. I hope they recognize learning as a process.

I hope they are learning to struggle well.

Risks and Rewards of Partial Credit

photo credit: Flickr user “tehbieber” – used under Creative Commons

Let me tell you what’s been on my mind lately. Partial credit. By partial credit, I mean assigning a point value to a student response that is less than the highest possible point value for that problem because the student didn’t get the final answer correct, but the solution process wasn’t completely incorrect. I have traditionally used this as a grading technique for for multi-step tasks.

In some ways it seems natural. You are giving a student points for evidence of learning. If a student is attempting a 3-point task, fumbles up on one single step and so ends with a wrong answer, shouldn’t the student get to salvage a bit of “credit” for performing portions of the problem correct?

In some other ways, though, it would seem natural that if a student gets incorrect final answers for most or all of the problems on a math assignment, the score wouldn’t be very good. However, through the use of partial credit grading, it is possible for a student to earning a decent grade on an assignment having not actually gotten any of the questions correct.

This problem has blown up on the last unit test for my Algebra II students. This is a class where I am one of three people involved in teaching and planning and I have been out-voted to create exclusively multiple-choice tests, which, becomes a serious problem if students aren’t getting final answers correct. The grades that went into the book were stress-inducing for many of my students.

So, therein lies my conflict. I hope that blogosphere will be willing to chime in to help me.

It could be that I’m too liberal with my use of partial credit. (It has occurred to me to award no more than half-points to a student response that has an incorrect final answer, not sure how I feel about it.)

It could also be that I’m worrying for nothing. (It has occurred to me that if a student can earn 80% of the points on an assignment, perhaps they, in fact, know 80% of the math and so it is an appropriate grade. The 20% they are lacking simply consists of parts that mess up the final answer.)

It could be that this is a problem only because of the difference in assessment styles. I am not a big multiple-choice fan and so I often give constructed response questions.

A quick editorial statement: This all seems to be an issue because our system seems addicted to assigning grades to everything. My favorite solution to this problem would be to eliminate the assigning of point values and grades, but that seems like a long shot at this point…

Thanks for your help everyone.