So I took a page out of Bryan Meyer’s blog and turned it into this three-act task.

Two release notes here:

**This task isn’t worth much if you don’t start with intuition.** You should point to this image and ask your students to intuit the location of a fair horizontal cut. At the moment, I think my best option is to print out that frame and pass it out to students so they can each draw their own lines. What I *need*, though, is a digital system where students can adjust that line precisely to their liking and then tap submit.

After that, the students see a composite of their classmates’ guesses.

This does two things. One, it ratchets up engagement. We want to know what the answer is and who guessed closest. Two, the mathematical model gets a lot of credibility when its solution falls right in the middle of our field of guesses.

**This task isn’t worth much if you don’t end with generalization.** The initial task sets the hook, but it resolves quickly into computation. Where this task (and others like it) light up the board is when we say, “Okay, now tell me where to make the cut for any size wedge of cheese. Any angle. Any radius.”

The ideal outcome on a digital device is that the student comes up with an abstract function with respect to theta and r, enters it into the device, and then that abstraction gets *concretized right on the original image*. The student sees the *result* of her model on a dynamic cheese wedge. She adjusts the theta slider and sees both the wedge and the cut adjust dynamically according to her function.

That’s the ladder of abstraction right there –Â from intuition to generalization.

**Featured Commentary**

There’s an interesting back-and-forth in the comments with one side claiming that the obviousness of the vertical cut makes the horizontal cut kind of contrived and another side saying it doesn’t matter.

## 24 Comments

## Andrew Stadel

June 29, 2012 - 2:30 pm -Bryan is on FIRE!

He had a piece in my recent Elmo’s Microwave Travel.

Dan, I love the theta slider (I’d call it feta slider, ha!). Seriously, I really enjoy the thought of using a touch tablet to make estimates and collect them for use in the lesson. I know you’d have me climbing the ladder.

Plus, I love cheese!

## Steven Peters

June 29, 2012 - 2:55 pm -It feels a little contrived because you could just make a vertical cut along the axis of symmetry. What if you had an ice cream cone that was filled with ice cream and one person starts and the other gets to finish it (ignoring the public health aspects)? How far from the top does the first person get to eat before switching over? That’s a more natural scenario for splitting that type of shape along that horizontal axis. Plus it’s in 3d if you’re into that sort of thing.

## Marshall Thompson

June 29, 2012 - 3:58 pm -Three observations:

This might be a situation where old tech trumps new tech. Transparencies overlayed would show the distribution of guesses very nicely.

I need to find out where I can get a monogrammed cutting board.

@SP: YUCK. I am not eating the second half of your ice cream cone!

## Bryan Meyer

June 29, 2012 - 4:48 pm -I love the section about the potential for extension and generalization with this task. The group of students I worked with on this task didn’t take it in that direction (although I pursued it with a colleague and found the results interesting and surprising). In “attending to precision” (*Common Core alert*) with solving the original task, some trigonometry is required in pinpointing an exact location for the cut. My students understanding of angle relationships didn’t allow them to find this. As a result, we took the opportunity to investigate angle relationships and create a generalization that allowed them to produce a more precise solution than before. I think this speaks to two things:

1. we should select tasks that offer an opportunity for students to engage in/DO mathematics as opposed to ones that allow them to implement/problem-solve with concepts they already “know.”

2. “rich” tasks are both flexible and extendable in response to the individual community of students. We should let the students (not the task or OUR intentions) dictate the direction that it should go.

Contrived? For students, maybe. For me, no…because it was a question that arose out of a situation in my daily life. I think there is a lot to think about here in terms of school as a place to pursue the curiosities of STUDENTS (something I am still working on and that I know Dan does a lot of). Regardless, students found the question perplexing and, as a result, were happy to go the distance with it. I don’t see context as a way to help students see applicability. I see it as a natural starting point because it offers them an opportunity to create their own mathematics, one that is based in their ways of living and experiencing their world.

## Dan Anderson

June 29, 2012 - 6:31 pm -I’m with

Stevenon the contrived point. I like the idea for sure, but my brain doesn’t get over the “just cut it vertically!” statement. Not sure about a students take on it though.What about splitting a slice of pizza or pie where someone doesn’t want any of the crust?

## Chris Sears

June 29, 2012 - 6:43 pm -I was going to comment on the cutting the cheese vertically, but people beat me to it. Especially if you are about to spread it on a bagel.

Love the sound effects.

## Kelly Berg

June 29, 2012 - 8:50 pm -Hmm. I am thinking the newest TI-Nspire might allow for students to submit answers via the Nav hats. I’m going to play around with the image and the calculator to see if it’s possible. Heck, throw the photo on a Cartesian plane and have them write horizontal line equations. You can gather the central tendencies, etc. Thanks for next week’s project! I’ll let you know what I find.

## Nathan Kraft

June 30, 2012 - 4:48 am -This desperately needs a third act. Cut it and weigh the two pieces.

## Chris Friberg

June 30, 2012 - 5:23 am -Kelly, I was thinking the same thing about the TI-Nspires. It’s definitely a good image to send out to the handhelds and let the kids draw their lines and share. I have found many of the photos that Dan and followers have posted would make great TI-Nspire problems because it’s so easy to let students choose the graphs or measuring tools they want to place right on top of the images. His basketball shot was a great one to do the same when exploring quadratics. I need to find time to make some Nspire versions of these great ideas. I’ll share when I do.

## mr bombastic

June 30, 2012 - 8:15 am -I am wondering how the measurements provided in Act II affect the difficulty and influence the student’s approach to the problem. I definitely think the thickness should be provided.

When I did something similar to this, I just taped paper over the object, passed it around, and had them mark their guesses. Worked great and my students were very curious about the answer.

I don’t think the contrived aspect of the problem would be an issue at all for my 9th grade algebra students. The use of video actually seems a little contrived for this lesson — I would just bring in a cheese wedge.

I disagree that this task isn’t worth much if you don’t end with a generalization. There are all kinds of relationships between the angle, radius, arc, and secant that can be investigated, even for students that don’t know any trigonometry. Looking at a wedge with an angle a little less than 180 deg changes things quite a bit as well.

## Kathy Sierra

June 30, 2012 - 8:24 am -Add me to the list of people stuck on, “vertical cut”. But I couldn’t immediately come up with an oddly-shaped thing that didn’t also feel contrived. I love the idea of it, though. But doesn’t it need to be something of a non-symetrical shape where there IS no alternate obvious way to split it in half?

Though I am also loving the idea of the classroom where one student finally stands up and says, “HELLO… Vertical, people, vertical!”. :)

## Michael Paul Goldenberg

June 30, 2012 - 9:59 am -On the inability to suspend disbelief (i.e., to ignore the vertical cut option): first, there’s already been a great suggestion vis-a-vis pizza slices and crust-avoidance. Many soft cheeses have rinds that some folks would wish to avoid or minimize. There are issues where cutting with or against the grain come into play (though probably not involving cheese, particularly soft cheeses). I suspect there are other ways to have this task make sense on realistic grounds.

Second, there are lots of situations where we accept counter-intuitive rules/restrictions without batting an eye. Why in the world would anyone agree that a good game for humans is one where, except on throw-ins, none of the players save the goalie use his/her hands (you know, the ones with the opposable thumbs), the body’s most sensitive, flexible, ingenious, creative extensions of the mind?

Yet, soccer/football remains the most popular team sport on the planet for some odd reason and other than William R. Robinson, my mentor in graduate school when I studied literature at the University of Florida in the ’70s, I’ve never heard anyone say, “only Europeans could come up with such an artificial and stupid idea for a sport.” (Bill played football under Bear Bryant at the University of Kentucky in the early ’50s, was a semi-pro baseball player, and an avid handball enthusiast (note: a sport that virtually requires that one become equally adept with both hands and, hence, the antithesis in many ways of soccer).

Nostalgia aside, my point is that we routinely accept restrictions that make a task somehow more challenging and/or intellectually and even aesthetically pleasing. It’s quite plausible that students can see the value of accepting “illogical” rules in order to make a particular mathematical issue more concrete. Certainly, I’ve been able to do this with inner-city Detroit kids, who sometimes get an enormous kick out of the most implausible “frame tales.” I did Towers of Hanoi with three classes on Thursday and while a few wanted to know why the monks couldn’t move more than one disc at a time, no one asked why it was necessary not to put a larger disc on top of a smaller one. There’s no “logical” reason not to do that, and it would cut the minimum number of moves dramatically. So everyone bought into one constraint, and most bought into both, and I even had one or two students who were a bit concerned about the impending date for the end of the world (but we didn’t have time to explore that).

I’m not advocating that we simply throw any dumb-seeming restriction at kids in the classic tradition of pseudo-context, but I’m not opposed to puzzles with rules that might be inconvenient or counter-intuitive, placed in the context of situations like slicing cheese horizontally. There’s a huge difference between the question being asked in this problem and the one where a football punt leads to asking about the vertical line of symmetry, something that simply has no connection to the actual situation. That’s a clear-cut case of phonying up a situation to ask a math question that simply isn’t relevant. Here, the math question and the task are the same. That there might be a different task and question that emerge from the same basic situation minus the restriction of horizontal cuts doesn’t matter, or at least not to me.

## Dan Meyer

June 30, 2012 - 11:43 am -Okay, okay, everyone gets full credit for coming up with the vertical cut. But the task appealed to me (and maybe also to Bryan and Bryan’s students)

becauseof its quarter-turn past normal, not in spite of it.Moreover, not every task that exists outside the walls of the classroom is an explicit statement that “this is how people use math outside the classroom.” I mean, look at Will It Hit The Hoop? again. People want to know if the ball goes in. It’s perplexing. But no basketball player has ever used math to answer that question.

That difference interests me.

Bryan:So we can assign the initial question (the cheese wedge) but not any follow-ups? Seems like a distinction without a difference.

## Bryan Meyer

June 30, 2012 - 3:09 pm -Dan:

“So we can assign the initial question (the cheese wedge) but not any follow-ups? Seems like a distinction without a difference.”

That was not my intention, but I can see how it might be received that way. For the students I worked with, the generalization that we ended with was one about angle relationships (in particular, tangent); one that they created by by investigating multiple scenarios, looking for patterns, and generalizing/justifying in a way that was useful for them.

My hunch is that it can be tempting to teach/tell students this relationship in the service of having them ‘solve’ the problem. I supposed my only concern is that interesting tasks like these serve as springboards to students creating math and that what they create is inspired by their current ways of knowing (as opposed to other outside influences).

I think the extension you suggest is a logical and interesting next step. One that could yield powerful results for students provided they are willing to investigate.

## Garrett Gray

June 30, 2012 - 8:46 pm -Very cool problem! I’ve enjoyed playing around with this, and I think my students will have a good experience with this. I’ve been teaching Math for 6 1/2 years in an English Program in Bangkok, Thailand, but I’ve just recently discovered this math teacher blog world.

Mr Bombastic brings up an interesting point on how things change if we use larger angles. I like how this leads to the question of how we find the cutoff angle and how we have to use a different method if the angle is bigger than x. Finding x is also interesting as it reduces to an innocent looking x = sin(x/2) and the possibility of using graphical intersections to solve an equation.

Thanks for providing so many interesting ideas in this blog, Dan!

## Garrett Gray

June 30, 2012 - 8:53 pm -Oh yeah, one more thing. A nice extension to send them home thinking about that might interest the still curious students is the following:

Insist that the cut starts at the point where the radius intersects the arc and extends in a straight line to the opposite side.

## Simon Clough

July 1, 2012 - 8:16 pm -I like the task. I think that the second act could possibly be improved by simply showing the circular container with the remaining 7 wedges with its diameter (10cm). Hopefully the students could deduce that 8 even wedges would have an angle of 45 deg and a radius of 5 cm.

I think that some kind of interactive app/software that could be overlayed to analyse these images would be awesome. A ruler with an adjustable scale for a start, and why not a protractor. The collaborative aspect of comparing results certainly is a great idea.

Keep up the great work Dan, this is my first time post after lurking around for a while. Interesting to see where all this leads…

Cheers

Simon

Mullumbimby, Australia

## Steve Phelps

July 2, 2012 - 8:14 am -I have an Nspire file that I used last year for exactly this kind of activity. Kids needed to graph the equation of a line that they thought would cut a given region in half. I will do my best to find it…

Now, I wonder how a GeoGebra file might look?

## Fawn Nguyen

July 3, 2012 - 12:53 pm -Then there are times when we WANT to learn more (and teach less) by saying, “Okay, a vertical cut is obvious, but obvious is easy. Let’s try to think about the horizontal cut…”

It would be contrived if a textbook made me do it, but if it’s put out as an invitation in the tone of “What do you think?” then I want to play the game.

## darren white

July 4, 2012 - 1:27 pm -This task has reminded me of an activity I set for a class of 15/16 year olds. I took in a round of Camembert Cheese. The task was to make 2 parallel cuts to slice the cheese into 3 equal parts.

They quickly rejected the Horizontal Cuts as trivial. They spent 100 minutes collaborating on the task then went home to finalise their solutions. the next lesson they presented their solutions and then took a vote on which was most convincing. Finally out came the chees knife. we weighed the results on scientific scales to see how accurate the solution was before eating the cheese. The solution was not very accurate for lots of reasons but the students were highly engaged, did an amazing amount of cognitive accelerating maths, had fun and ate cheese so all were happy

## darren white

July 4, 2012 - 1:37 pm -By Horizontal cut I meant, cutting it in the same way as the video cuts the beigal

## Susan Russo

July 4, 2012 - 4:20 pm -I’ve Geogebra-ed the issue:

http://www.geogebratube.org/student/m13424

I wish I were a student again……………….

## Steven Peters

July 5, 2012 - 6:28 am -After reading a few of these comments, I realized a scenario in which the horizontal cut makes a lot more sense than the vertical cut from a practical perspective: when the slice is very thin (think cheesecake rather than pizza). In that case the length of the cut in the horizontal direction is much, much shorter than the length of a vertical cut. From my own experience, it can be tricky to make long straight cuts, so I would prefer the horizontal cut if it was much shorter than the vertical cut.

Regarding the whole matter of whether contrivance is a problem or not: I think you can still get mileage out of the problem, but I was distracted by the vertical cut issue, and I figured others might have been as well. It felt like it was almost there, so I was trying to help brainstorm a more natural scenario for the horizontal cut. In my opinion, I think it’s more practical when the slice is very thin.

## Jared Cosulich

July 9, 2012 - 1:29 pm -Dan,

If you want to collaborate around building this type of application I would be happy to help. I’m working on a number of puzzle based educational apps that work similarly to what you are looking for:

http://puzzleschool.com

Anyways let me know if you’re interested.

Best,

Jared