More Cheese Means Less Cheese

I wrote last week about the measuring-polygons activity we did to begin our investigation into pi, and hinted that many exciting things happened in class as a result.  Well, here’s one of them – the More Cheese Means Less Cheese Conundrum.

I may be wrong, but I think that actually measuring the shapes (and the What Can You Do With This? work we did several weeks ago) got the kids thinking about measuring ratios in the world around them.  Because in the middle of our work with the polygons, two girls in one of my classes said they had a burning question they’d like to ask everyone.  I love it when kids have burning questions – whether related to math or not – so up they went to the board to share.

Their question was: You know how Swiss cheese has holes?  Well, when you make a block of Swiss cheese bigger, the holes get bigger, right?  So, when you make a block of Swiss cheese bigger, do you actually have less cheese?

The answer to this question was obviously no.  That wouldn’t make any sense, so long as you don’t mean inflate the holes by “make a block of Swiss cheese bigger.”  But, if you’re a sixth grader and your practical understanding of proportion isn’t so strong, this question is baffling.  The holes get bigger – and more holes must mean less cheese, right?

When we put the question to a vote, the class was about 3 to 1 in favor of more cheese means less cheese.  And as the more cheese = less cheese faction shared their arguments (mostly that more cheese means more holes, and more holes means less cheese), the ratio shifted to about 6 to 1.  In a 14-person class, that makes the more cheese means more cheese faction pretty tiny – but they were very vocal for their position.  Debate was heated – but also without much evidence or solid logic either way.

This seemed to me like a very important question to resolve.  I couldn’t have my students going into a grocery store and buying a smaller amount of Swiss cheese thinking that they’d actually be getting more cheese.  But, class was over.  So, instead of having more polygon-measuring homework, I asked them to come up with a compelling argument in favor of their cheese position.  Before I sent them on their way, however, we discussed what could be important to include in a compelling argument.  Evidence, they said.  So, how will you get evidence?  The responses ranged from actually buying and measuring cheese to making an accurate diagram.

By the following class, only one student remained in favor of more cheese = less cheese.  But I couldn’t have asked for a better student to be pro more cheese = less cheese.

This particular student is very good at actively learning.  And by this, I mean he often has misguided ideas, but values solid logic very highly.  When he has an idea, he always has a logical argument for it that he believes to be strong.  And he will stick to his idea and logical argument until you give him a really good reason that the opposite is true.  So, I knew that his classmates would have to really know their stuff to bring him around.

Oddly, our single hold-out’s argument rested on proportion.  The holes must increase as the block of cheese increases, he argued, because the proportion the cheese taken up by the holes must remain constant.  So, the holes increase in volume as the entire cheese increases.  Meaning that the amount of solid cheese must decrease.

The other students practically fought to give an argument that would persuade him otherwise.  Some came to the board and drew models of what would happen with more cheese.  He didn’t buy it – how could you draw a really accurate model on the board?  As he doubted their accuracy, his classmates began to realize that they’d need some numbers to bring him around.  So, one student suggested making a mathematical model.

With numbers at her command, she demonstrated that, given the assumption that the larger piece of cheese is a scaled-up version of the smaller piece of cheese, the holes-to-cheese ratio must stay the same.  If the whole cheese doubles in size, the holes double, but so does the cheese.

The argument was convincing!  Those who preferred picture arguments altered their pictures to fit her mathematical model.  We talked for the rest of the period about whether her mathematical model would work regardless of the holes-to-cheese proportion and how scaling of geometric objects alters and doesn’t alter measurements you make of them.

We’d been working with that last question for almost a week with the polygons and proportions, and we had already talked about how ratios of lengths don’t change with scaling.  So, I could look at this incident – in which they didn’t take that knowledge and immediately apply correctly it to the new context – and be disappointed with my teaching and their learning.  Maybe I could judge myself as more successful if they’d immediately seen the connection between this question and the polygons.  Yes, there are things I can learn about my polygon lesson from the cheese experience – we should spend more time with scaling, and I should probably incorporate scaling into other work we do.

But, in my opinion, at least, learning math isn’t climbing a ladder or building a wall.  Just because something has been “taught” doesn’t mean you’ve topped that rung and you’ll never see it again, or that the brick has been mortared into place and completely dried, forming a foundation for future learning.  Learning math is more like playing a game of Entanglement.  (Play it, if you haven’t.  It’s fun!)  New tiles are laid, connections are made, but that doesn’t mean that you won’t use old tiles in new ways or pass through old connections – intentionally or entirely by accident.  My kids used their knowledge of ratio gained from the polygon problems to learn about cheese, whether intentionally or not.  And now, hopefully, it’s a little more tightly woven into their mesh of mathematical knowledge.

A Recipe for a Recipe for Pi

I’ve been thinking a lot lately about how I plan lessons and units, particularly lessons and units about things I’ve never taught before.  (Partly inspired by this.)  Here’s an anecdote describing how I planned a recent unit (that’s still going on) for my 6th grade classes.  I like how I planned this and how my planning played-out, so I thought I’d share it with you.


Several weeks ago, I thought to myself, “Hmmm, I think it’s about time to do some work with circles in my 6th grade classes.”  But, the crazy thing is, I’d never taught an introduction to pi before.  Sure, I’d taught graphing circles to Algebra 2 students.  And I’d taught quite a variety of things related to circles and pi to trigonometry students.  But, in both of those situations, I’d been able to assume that my students had a working understanding of many important concepts surrounding pi and circles.

This blog may be called, “Recipes for Pi,” but I’d never before taught a group of students what pi is.

Wrapped up in pi are a number of ideas and skills: measuring circles, finding ratios, understanding what ratios of quantities in shapes tell us about the idea of “shape,” understanding irrational numbers, believing in limit processes, understanding the nature of infinity…  This list encompasses some very complex and deep mathematical ideas that the most thoughtful mathematicians and philosophers throughout history have had immense trouble wrapping their heads around.  (Consider Zeno, for instance.)  But, once I’d realized that all of these things were related to pi, and that no understanding of pi would be complete without them, I couldn’t just say to my students, “So, there’s this number, pi.  It’s something like 3.14159.  And here’s how you use it to find the area of a circle.”

How was I going to get my students to really think about these ideas mathematically?

I knew that I wanted my students to somehow derive a value for pi.  I also knew that I wanted them to come to an understanding of the significance of this number, and of ratios of lengths in shapes in general, through hands-on experience.  I had no idea how I wanted to deal with the irrational-number thing, but I knew that it had to come up.  And with that, the idea of an infinite process limiting to a finite quantity.

So, here’s how I planned our attack on pi:  First, I made a list of the large questions that I wanted us to discuss at some point during our investigations.  Here are some of my questions:

–          How do you find the perimeter and area of a circle?

–          What is the importance of pi?

–          What kinds of criteria do we use to distinguish among shapes, and what measurements and ratios of measurements in shapes can be used to distinguish among the criteria?

–          What is an irrational number?

–          How can an infinite process produce something finite?

Then, I picked a place to start – deriving pi and working with quantities that distinguish among shapes – and decided what I wanted my students to do to achieve this.  Meaning, I made a worksheet.

When I plan a class, I always start by making a worksheet.  The plans that I make for lessons or discussions almost always come after I’ve made the worksheet – because I feel like I only really know what I want my students to learn after I’ve thought about problems through which they’ll be doing the learning.

The worksheet I made had four different regular polygons on it – a square, a pentagon, a hexagon, and an octagon.  These polygons all had the same “diameter,” or distance between opposite vertices (for the pentagon, I used the distance between a vertex and the opposite side).  I asked my students to use a ruler to measure each polygon’s diameter and side-length, and then to calculate its perimeter and area.  Finally, I asked them to compute the ratio of the perimeter to the length of the diameter.

I decided to have them measure the side-lengths and diameter with a ruler instead of giving values because I wanted them to produce the data themselves.  I didn’t want them to have the lingering suspicion that I’d rigged the data so that they would all get similar ratios – I wanted this to be a property of the shapes themselves, not the problems I’d created.

For homework, I made them a second sheet of polygons. Some of the polygons had double the “diameter,” others half.  Before the students began measuring, I asked them to predict how the perimeter, area, and ratio of perimeter-to-diameter would change if the diameter was doubled or cut in half.  They then were asked to repeat the measurements we did in class.

After they completed both of those sheets, I planned for us to spend time in class tabulating our data and making observations about patterns and trends.  I hoped that they would find that the ratios were very close across the shapes, while the other quantities they were measuring and calculating were not.

Once I’d made these worksheets and thought about the discussions I wanted us to have about them, I spend a little time listing other activities we could do and discussions we might have.  But, I basically stopped planning.  That’s because I didn’t really know what my students were going to take away from these activities – and I didn’t want to plan anything that didn’t follow their trains of thought and interests.  I try, as much as I can, after the initial raising of a topic, to let our investigations in class be guided by student interest.  I thought about where they might want to take the topic and about how I’d guide them based on where they went, but I basically left it at that and trusted to the curiosity and intelligence of my students.

Briefly, here’s how all of this played out in class.  The decision to make measuring part of the activity totally worked.  Their perimeter and “diameter” measurements were all wildly different – because kids had used different units of measurement – but the ratios were all about the same, across polygons.  This made the ratio-consistency all the more poignant.

The wildly different measurements also sparked the question that I’d most of all wanted them to ask: “Which measurement is a property of regularity of shape, not size or necessarily the exact shape – the perimeter, or the ratio between the perimeter and the ‘diameter’?”  We had a very focused, deep discussion – generated by the students – about size, shape, and regularity, and the importance of proportion to each quality of a geometric object.

When they saw the ratio, in the polygons and in the circles we measured next, students who already knew something about pi said things like, “Hey – that’s pi!” and then rattled off a ton of decimal places.  So we didn’t really spend time deriving a good approximation for pi.  But, they did come to understand the importance of pi for circles and the relationship between ratios of measurements and the nature of shape.

And now questions about ratios and infinite-finite quantities and types of numbers have been cropping up all over the place… More about that to come!

MArTH Madness!

A note for Sam Shah:  This post is inspired by a conversation we had today.  I’m going to wax education-philosophical for a while, and I apologize to you ahead of time because I know you don’t like to read that kind of stuff.  But I promise – PROMISE – that I’m going to give some practical lesson advice at the end of the post.  So please don’t drop off reading in the middle, ok?


If you read the note for Sam, you’ll know that this post is inspired by a conversation I had with my dear friend and education-world colleague Sam Shah today.  Sam – being a totally awesome person – dropped by our first annual Mathematical Art Festival today to see what all the hoopla was about.

Want more pics?  Check out our facebook page!

Yep – we had an all-day Mathematical Art (or MArTH) Festival today at Saint Ann’s.  And it was a BLAST!  We called it MArTH Madness.  (Paul Salomon’s brilliant idea.)  This didn’t just happen out of the blue – Paul Salomon, Justin Lanier, Max Neesemann, and I are leading a Mathematical Art seminar for high school students this semester, so MArTH has been happening around the school for a little while now.

This was the first time we did it with the whole school, and it was great fun!  We had a bunch of different stations, led by teachers and students in our seminar – Escher tessellations, math doodling, 3D constructions, unit origami, 3D printing, computer-aided design… Kids and teachers made so much beautiful and fascinating art, and everyone had tons of fun.

As Sam and I were observing the creative chaos of the 3D constructions station (in the cafeteria, with ZOME tools), I asked him what he thought of it all.  I won’t pretend to quote him directly, because I don’t remember exactly what he said – and, Sam, I apologize if I don’t get it quite right.  But he said something along the lines of: I wish we could do this at my school, but we can’t – because your students already really like math, and mine don’t.

In the moment, I disagreed with him that he couldn’t do this at his school.  But I agreed with him that our kids already like math.  This wasn’t quite true.

I think I can say that a good number of my students like my classes.  And I think I can say that a good number of students at Saint Ann’s in general like their math classes.  But do most of our students like math?  I don’t know about that.

But I don’t think their love of math was what was driving our students to tackle MArTH with so much enthusiasm today.  I think it was simply that we gave them things to do that they were into.

Here’s something I think is true:  People, and kids specifically, really like to make things.  They also really like to test boundaries.  MArTH is a perfect combination of these two things.  You get to make really cool-looking things, and the way you make them is by following and pushing at the boundaries of mathematics.  So, regardless of how much they like math, I think most kids would be really excited by mathematical art.

When students ask for “practical applications” of mathematics, I don’t think they’re necessarily asking for more problems having to do with “practical” things, like interest rates, grocery shopping, or draining water.  I think they want something to do with the math they’re learning – a way to make something with it.  MArTH is precisely that.


This looks like great fun, but how do I bring MArTH into my class, you may ask?  Say I’m an Algebra 2 teacher.  I’m totally not going to take a class period to make sonobe cubes or Appolonian gaskets.  That would not be relevant to my curriculum.  Alrighty.  I can see why you wouldn’t want to do that.

But you’ll be teaching transformations of functions at some point, right?  Maybe you’re doing it right now.  Here’s a MArTH activity for you – in which your students will see function transformations in action AND made a beautiful work of mathematical art of their very own.

  1. Get your students computers with some good graphing software.  Graphing calculators just won’t do it – the graphics are abominable.  And you might want to be able to print at some point.
  2. Have them pick a basic function, or pick one for them.  Something like f(x)=x^2 will do nicely.
  3. Tell them to choose a single type of transformation to make to the function.  Shifts, stretches, flips.  Then…
  4. Tell them to make small transformations of that type to the function.  Graph each new function on the same screen.
  5. Huzzah!  Art!
  6. Here’s the clincher, where the art really becomes their own: You now have to let them take it where they will.  They may make the craziest of transformations.  But, so long as they make small changes of their transformation type and graph them, they are making discoveries about the visual effects of algebraic transformations of functions.  And they are making their own art.

I did this not too long ago with my algebra students, and wrote about it here.

If you do this, please let me know how it goes!  If you want more MArTH activities, you know who to ask!  And if you have any great MArTH activities of your own, you know who to share them with!