The Grill Fries Problem

That's two medium russets.

That’s two medium russets.

So, let’s suppose that I cut up two medium russet potatoes like so and put them on that grill you see up there.

How many medium russets will the grill hold and (perhaps more importantly) how many people can I feed using those fries as a side dish?

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Double Stuf Oreos: Are they really double the Stuf?

Double Stuf: A factual statement or clever marketing trick?

Double Stuf: A factual statement or clever marketing trick?

I was inspired by this post by Nathan Kraft (@nathankraft1) in which he engages his staff in a question about Oreo cookies. (Mr. Kraft was quick to inform me that Christopher Danielson (@Trianglemancsd) was the inspiration for his post. I do want to give credit where it’s due.)

I decided to see what my third hour students would do with it. So, last week Friday I showed them the picture and we started discussing the a variety of aspects of Oreos (some of which were more useful than others). Then, it happened. One students asked:

“Is the stuff of a Double Stuf really double of the stuff in a single stuff?”

The beauty of this activity is that the students were able to become involved in the formation of the solution process. They practically all had a prediction. First idea, would double the stuff be twice as tall?

It didn't appear to be double by height.

It didn’t appear to be double by height.

The above image represents what multiple students observed. It was an awesome opportunity to discuss conclusions. What conclusion can we draw from the observation we just made?

Either it was double the stuff and it wasn’t manifesting itself in its height, or it wasn’t double the stuf. (Often, the student’s original predictions colored their conclusion to these observation.)

Double by what measure? Mass?

Double by what measure? Mass?

Next idea was mass. Gave in impromptu call to Mr. Corcoran, the chemistry teacher, who loaned us some scales. But what do we measure? The whole cookie? That opened up another important question? Is the same wafer used for both the standard and the double-stuff?

After some quick diameter and mass measurements, it seemed like there was no meaningful difference between the two. But, just to be safe, each student scraped the Stuf from a standard and a Double Stuf and set to the scale to get a mass measurement.

What math class looked like today.

What math class looked like today.

Then we compiled the results.

The mass of the stuff scraped off a sample of standard and Double Stuf Oreos.

The mass of the Stuf scraped off a sample of standard and Double Stuf Oreos.

Each group took a moment to deliberate and concluded that, for the most part, it seems that the Double Stuf is appropriately named. Some groups seemed to think that, if anything, the Double Stuf contained more than double the Stuf.

This activity contained so much of what makes contextual, collaborative learning valuable. Authenticity, source of error, conclusions that were not clear, but needed to be discussed. Students needed to listen, speak and rephrase when others didn’t understand.

It also had the beautiful feature of me not knowing the answer and they knew it. So, there wasn’t the temptation to treat me like the math authority, as though all math learning begins and ends with the Teacher’s Edition.

And for an added bonus, the AP Stats class meets next door at the same time and so, we were able to strike a deal to rerun the trial with the guidance of the stats class for a broader, students-teaching-students experience.

I’ll report back with our findings.

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.

taken from Geometry, copyright Holt, Rhinehart, Winston 2007, Pg 684 #8

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

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.

This blog does not endorse Betty Crocker, General Mills, or any of their products.

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.

Cereal and Peanut Butter: The Unexpected Lesson Plan

The unplanned econmonics lesson helped it all make sense...

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

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.

The Snickers Problem: The Aftermath

What math class looked like this week

What math class looked like early this week

There is nothing quite like watching the students get there hands dirty with the mathematics. This is especially true when the students are literally getting their hands dirty. In this case, it was with caramel, chocolate and nougat.

Here is some of the aftermath of the students exploring The Snickers Problem. (Check it out for a description.)

Snickers #2

To me, the value of this assignment lies in its ability to draw the students in. This problem featured 100% engagement. It featured the type of proportional reasoning that shows up in endless amounts in our unit on Similarity and Dilations. The students were analyzing the size comparison of a fun-sized Snickers and predicting the number of peanuts.

Others preferred the more antiquated utensils...

Others preferred the more antiquated utensils…

This was a fantastic activator. Formal proportional reasoning is difficult for many students, but informal proportional reasoning is intuitive. Many of the students found that their predictions were pretty close. (Although, we did find some Snickers that had remarkably low numbers of peanuts.)

Knife and fork worked for some

Knife and fork worked for some

And the fantastic questions that come up when students are thinking mathematically.

Is this prediction close enough?

What do we do about half-peanuts and peanut pieces?

Their prediction was right, but ours was wrong… but we predicted the same thing! Our snickers had different number of peanuts!

These are all evidence of students having to have an experience with the mathematics. That’s something I’d like to see more of.

The Citrus Problem

20121214-210043.jpg

If we are going to convince our students that math is usual in everyday experiences, then we need to show them everyday experiences and have them do math.

And a box of oranges like that is an everyday experience for me. In fact, that’s my kitchen in the picture. So, how many oranges will fit in that crate?

And for a real challenge, how many will fit into the dark brown basket beside it?

Remember, the power is in the explanation.