## [Confab] Circle-Square

The following problem has obsessed me since I first heard about it several months ago from a workshop participant in Boston. I believe it originates from The Stanford Mathematics Problem Book, though I’ve seen it elsewhere in other forms.

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Here’s why I’m obsessed. In the first place, the task involves a lot of important mathematics:

1. making sense of precise mathematical language,
2. connecting the verbal representation to a geometric representation,
3. reasoning quantitatively by estimating a guess at the answer,
4. reasoning abstractly by assigning a variable to a changing quantity in the problem,
5. constructing an algebraic model using that variable and the formulas for the area of a square and a circle,
6. performing operations on that model to find a solution,
7. validating that solution, ensuring that it doesn’t conflict with your estimation from #3.

Great math. But here’s the interesting part. Students won’t do any of it if they can’t get past #1. If the language knocks them down (and we know how often it does) we’ll never know if they could perform the other tasks.

What can you do with this? How can you improve the task?

I’m going to update this post periodically over the next few days with the following:

• two resources I’ve created that may be helpful,
• commentary from some very smart math educators on the original problem and those resources.

Help us out. Come check back in.

Previous Confab

The Desmos team asked you what other Function Carnival rides you’d like to see. You suggested a bunch, and the Desmos team came through.

Man did you guys came to play. Loads of commentary. I’ve read it all and tried to summarize, condense, and respond. Here are your big questions as I’ve read them:

• Is learning to translate mathematical language the goal here? Or can we exclude that goal?
• What role can an animation play here? Do we want students to create an animation?
• What kinds of scaffolds can make this task accessible without making it a mindless walk from step to step? On the other end, how can we extend this task meaningfully?

There was an important disagreement on our mission here, also:

Mr. K takes one side:

It took me about 3-4 minutes to solve – the math isn’t the hard part. The hard part is making it accessible to students.

Gerry Rising takes the other …

If we want students to solve challenging exercises, we should not seek out ways to make the exercises easier; rather, we should seek ways to encourage the students to come up with their own means of addressing them in their pristine form.

… along with Garth:

Put it to the kids to make it interesting.

I’ll point out that making a task “accessible” (Mr. K’s word) is different than making it “easier” (Gerry’s). Indeed, some of the proposed revisions make the task harder and more accessible simultaneously.

I’ll ask Gerry and Garth also to consider that their philosophy of task design gives teachers license to throw any task at students, however lousy, and expect them to find some way to enjoy it. This seems to me like it’s letting teachers take the easy way out.

Lots of you jumped straight to creating a Geogebra / Desmos / Sketchpad / Etoys animation. (Looking at Diana Bonney, John Golden, Dan Anderson, Stephen Thomas, Angelo L., Dave, Max Ray here.) I’ve done the same. But very few of these appleteers have articulated how those interactives should be used in the classroom, though. Do you just give it to your students on computers? To what end? Do you have them create the applet?

Stephen Thomas asks two important questions here:

1. How easy is using [Geogebra, Desmos, Etoys, Scratch] for kids to construct their own models?
2. When would you want (and not want) the kids to construct their own models?

My own Geogebra applet required lots of knowledge of Geogebra that may be useful in general but which certainly wasn’t germane to the solution of the original task. It adds “constructions with a straightedge and compass” to the list of prerequisites also, which doesn’t strike me as an obviously good decision.

Lots of people have changed the wording of the problem, replacing the mathematical abstractions of points and line segments to rope (Eddi, Angelo L) and ribbon (Lisa Lunney Borden) and fencing (Howard Phillips).

This makes the context less abstract, yes, but the student’s work remains largely the same: students assign variables to a changing quantity on the line segment, then construct an algebraic model, and then solve it. The same is true for some suggestions (though not all) of giving the students actual rope or ribbon or wire.

So I’m interested now in suggestions that change the students’ work.

Kenneth Tilton proposes a “stack” of scaffolding questions:

1. If the length of AB is 1, what is the length of AP?
2. What is the ratio of AP to PB?
3. Given Ps, the perimeter of a square, what is the area of the square?
4. Given Pc, the perimeter of a circle, what is the area of a circle?
5. How would you express “the two areas are equal” algebraically?

The trouble with scaffolds arises when a) they do important thinking for students, and b) when they morsel the task to such a degree it becomes tasteless. Tilton may have dodged both of those troubles. I don’t know.

David Taub lets students choose a point to start with. Choosing is new work.

Mr K asks students to start by correcting a wrong answer. Correcting is new work.

It seems to me that a simple model of the problem, (picture of a string) with a failed attempt (string cut into two equal parts) should be enough to pique the kids “I can do better” mode. Providing actual string with only one chance to cut raises the stakes above it being a guessing game.

I think more important would be to start with some “random” points and some concrete numbers and see what happens.

Max Ray builds fluency in mathematical language into the end of the problem:

So I would have my students solve the problem as a rope-cutting problem. Then I would invent or find a mathematical pen-pal and have them try to pose the rope problem to them.

If our mathematical language is as efficient and precise as we like to believe, its appeal should be more evident to the students at the end of the task than if we put it on them at the start of the task.

Gerry Rising offers us an extension question, which we could call “Circle-Triangle.” I’d propose “Circle-Circle,” also, and more generally “Circle-Polygon.” What happens to the ratio on the line as the number of sides of the regular polygon increases? (h/t David Taub.)

On their own blogs:

• Justin Lanier offers a redesign that starts with a general case and then becomes more precise. I’m curious about his rationale for that move.
• Jim Doherty runs the task with his Calculus BC students and reports the results.
• Mike Lawler gives us video of his son working through the problem.

2014 Feb 26. Some of my own resources.

Here’s one way this problem could begin:

1. Show this video. Ask students to tell each other what’s happening. What’s controlling how the square and circle change?
2. Then show this video. Ask students to write down and share their best guess where they are equal.

The problem could then proceed with students calculating whether or not they were right, formulating an algebraic model, solving it, checking their answer against their guess, generalizing their solution, and communicating the original problem in formal mathematical language.

Mr. K has already anticipated my redesign and raised some concerns, all fair. My intent here is more to provoke and less to settle anything.

I’m going to link up this video also without commentary.

2014 Feb 27. Other smart people.

I asked some people to weigh in on this redesign. I showed the following people the original task and the videos I created later.

• Jason Dyer, math teacher and author of the great math education blog Number Warrior.
• Keith Devlin, mathematician at Stanford University.
• Two sharp curriculum designers on the ISDDE mailing list, whose comments I’m reproducing with permission.

Here’s video of a conversation I had with Jason where he processes and redesigns the original version of the task in realtime. It’s long, but worth your time.

I immediately drew a simple sketch – divide the interval, fold a square from one segment, wrap a circle from the other, and then dive straight into the algebraic formulas for the areas to yield the quadratic. I was hoping that the quadratic or its solution (by the formula) would give me a clue about some neat geometric solution, but both looked a mess. No reason to assume there is a neat solution. The square has a rational area, the circle irrational, relative to the break point.

So in the end I just computed. I got an answer but no insight. I guess that reveals something of a mathematician’s meta cognitive arsenal. You can compute without insight, so when you don’t have initial insight, do the computation and see if that leads to any insight.

In the case of the obviously similar golden ratio construction, the analogous initial computation does lead to insight, because the equation is so simple, and you see the wonderful relationship between the roots

So in one case, computation just gives you a number, in the other it yields deep understanding.

Off the ISDDE mailing list, Freudenthal Institute curriculum designer Peter Boon had some useful comments on the use of interactives and videos:

I would like to investigate the possibility of giving students tools that enable them to create those videos or something similar themselves. As a designer of technology-rich materials I often betray myself by keeping the nice math (necessary for constructing these interactive animations) for myself and leaving student with only the play button or sliders. I can imagine logo-like tools that enable students to create something like this and by doing so play with the concept variable as tools (and actually create a need for these tools).

Leslie Dietiker (Boston University) describes how you can make an inaccessible task more accessible by giving students more work to do (more interesting work, that is) rather than less:

If the need for the task is not to generate a quadratic but rather challenge students to analyze a situation, quantify with variables, and apply geometric reasoning with given constraints, then I’m pretty certain that my students would appreciate a problem of cutting and reforming wire for the sake of doing exactly that …

I disagree with people who are saying that this problem as written is inherently bad or artificial. As an undergrad math major, a big part of the learning for me was figuring out that statements worded like this problem were very precise formulations of fundamental insights — insights that often had tangible models or visualizations at their core.

I remember lectures about knots, paper folding, determinants, and crazy algebras that the lecturers took the time to connect to interesting physical situations, or even silly but understandable situations about ants taking random walks on a picnic blanket. For a moment I even entertained the idea of graduate work in mathematics, because I realized that math was actually a pretty neat dance between thinking intuitively and thinking precisely.

Terrence Tao writes about that continuum here.

tl;dr version: Translating this problem from precise to intuitive and intuitive to precise, is part of the real work that research mathematicians (and their college students) do, and not something we should always keep from our students. It’s a skill we should help them hone.

2014 Mar 4. As usual Tim Erickson got here first.

Tagged in:
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### eddi

February 24, 2014 - 8:28 pm -

Dan,

I would think starting off with easier shapes to work with, like rectangles, would help. But then the exact question is too easy (“cut it in half, so you get same size squares”). So modify that a little: You’ve got a rope 12 inches long. Cut it into 2 pieces and make a rectangle out of each piece. Where do you have to cut it so that the bigger rectangle is twice the size of the smaller rectangle.

Once you’ve got that idea down, you can change the shape to square and circle, or equilateral triangle and circle, or what have you.

Great problem, and interesting to think about approaching.

Eddi

2. #### Diana Bonney

February 24, 2014 - 8:36 pm -

It would be cool to use some geometry software like GSP (Geometer’s Sketchpad) to help students play around with the idea and think about it. I made a GSP file… but I’m not sure if it represents the situation accurately. I’ll email it to you so you can take a look.

3. #### Jim Doherty

February 24, 2014 - 8:39 pm -

My first thought is that a visual representation will go a long way toward nudging the students to a place where they can begin to construct a model. I visualize this as soon as I read it. I know many of my students will not.

4. #### Annette Lievaart

February 25, 2014 - 12:00 am -

The size of your problem depends on the goal you want to reach with students doing this exercise. Is it the start of something new? Is it to test if they understood what they’ve learned before? …?

Annette

5. #### David Taub

February 25, 2014 - 12:49 am -

The need to change the language would depend on what a particular class of students is used to – more “everyday” language vs. more “mathematical” language.

Visual aids would be helpful in the beginning, Geogebra after the students worked a little with it first.

I think more important would be to start with some “random” points and some concrete numbers and see what happens.

Something like:

You have a string that is 36 cm long. Choose somewhere to cut the string into two pieces. Make a square out of one piece and a circle out of the other piece. Which has a bigger area? How much bigger is it?

Now switch pieces – make a circle out of the piece you made a square from and a square from the piece you made a circle from. Which has a bigger area? How much bigger is it?

Cut the string at another place and do the same thing again.

Is there somewhere you can cut the string so that the circle and square will have the same area?

6. #### Lisa Lunney Borden

February 25, 2014 - 3:37 am -

Hi Dan,

This is a great problem but there are two things I would do to transform it: Contextualize and Verbify! First I would give it a context (a piece of ribbon) and then I would make it active instead of so static (cutting ribbon to make shapes). So:

I have been given a certain length of ribbon. I want to cut the ribbon into two pieces to make (outline) two shapes, a circle and a square. I want the shapes to have equal areas. Where should I cut the ribbon?

Dan, I envision you creating a little video to show the active nature of this problem – you sitting at your table cutting ribbon and trying to make circles and squares from the pieces.

The problem you present is a classic example of how we can make math problem linguistically challenging by an over reliance on static or nominalized language. When we make it active, it opens it up to more students. I have actually written about the need to “Verbify” mathematics in a “For the Learning of Mathematics” article which you can download here: http://showmeyourmath.ca/sites/default/files/02-Borden.pdf. Feel free to share.

7. #### Howard Phillips

February 25, 2014 - 4:54 am -

A rancher needs to build two new corrals for his cows. Each is to contain 100 cows. He gives the job to his two sons, Joe and Wayne. Joe makes a square corral and Wayne makes a circular one, and between them they use 400 yards of fencing. How much fencing did Joe use?
At least this version is more practical (slightly!).
The difficulty with the mathematical description of the problem is its static nature. The ‘line segment AB’ bit is designed to muddy the problem. Once one sees that a piece of string is what is meant the difficulties vanish.
How about putting a square on the line at one end and a circle of the same area at the other end, rolling them towards each other by one full turn and seeing what you get (a gap which can be thrown away or a change in the area, so that the objects meet, at a sensible point of course).
Also, the problem as given is as artificial as some of the worst ‘Word problems’. Clearly designed for students who are already ‘into’ math.

8. #### Gerry Rising

February 25, 2014 - 5:59 am -

It seems to me that many of the responses miss the point of the exercise. If we want students to solve challenging exercises, we should not seek out ways to make the exercises easier; rather, we should seek ways to encourage the students to come up with their own means of addressing them in their pristine form. We want them, not us, to generate the interesting ways cited by responders of exploring the problem.

9. #### Stephen Thomas

February 25, 2014 - 6:52 am -

So I played around for an hour and created a simple visualization for this problem, that given the proper guidance, kids could easily construct themselves using Etoys. Which in addition to the “important mathematics” listed in 1-6, would add item 7 Computational Thinking (using computers to think and model with).

Here is a post to a video of the project: http://youtu.be/8Nl-XytAtMc

FYI Tried posting this before but didn’t see it, then got a “duplicate comment detected”

Cheers,
Mr. Steve

10. #### Garth

February 25, 2014 - 7:34 am -

My first question was “Why would a kid want to do this in the first place?” As a math geek (most math teacher seem to be math geeks) I think the question looks fun. Most kids are not math geeks. Gerry Rising has the point on a problem like this. Put it to the kids to make it interesting.

11. #### Kenneth Tilton

February 25, 2014 - 8:03 am -

Let’s make it more “real world”: “Two trains leave NYC and Boston. When they meet….”

Seriously, it is a great teaching problem. I am imagining a Socratic “stack” of questions leading them to a solution providing the minimum of help needed by each learner, with the idea that, if they cannot answer a question, they push it onto the unanswered stack and go on to the next slightly easier question, popping the stack of unanswered questions after answering any successfully to see if they can now handle it:

0. If the length of AB is 1, what is the length of AP?
1. What is the ratio of AP to PB?
2. Given Ps, the perimeter of a square, what is the area of the square?
3. Given Pc, the perimeter of a circle, what is the area of a circle.
4. How would you express “the two areas are equal” algebraically?

That’s just off the top of my head. I see a gap already been 0 and 1.

12. #### Dan Allen

February 25, 2014 - 8:10 am -

I like the idea of the Geogebra applet but it’s also important for students to look at general cases in higher level courses. The geogrbea applet could be misunderstood as “the answer” by students who do not know the meaning of “arbitrary” and the possibility of multiple answers.

13. #### Angelo L.

February 25, 2014 - 10:28 am -

Dan,

I also jumped straight to the technology to make a simulation (on GSP). If I were to do this with my class, I think that, as Eddi said, I would use rope.

Give each pair of students a piece of rope or string and a pair of scissors. Give them the problem and challenge them to find the maximum area for a square and a circle. Go from there to the technology (GSP, DESMOS etc.) and the more formal algebra.

14. #### Stephen Thomas

February 25, 2014 - 10:42 am -

@Dan Anderson
Nice work with Desmos. Couple of thoughts:
1) Have a version where the Square and Circle overlap
2) Make it easier to change line length in one step
2) show/hide gridlines option

15. #### Joe Mako

February 25, 2014 - 11:06 am -

My first question was: What is the ratio of AP to PB? That way we could know where P is for any length of AB.

Here was my process:
PB = 2 *pi*r
area = pi*r^2
AP = 4*x
area = x^2
x^2 = pi*r^2
x = sqrt(pi)*r
AP = 4*sqrt(pi)*r
AP/PB = (4*sqrt(pi)*r)/(2 *pi*r)
simplified to:
AP/PB = 2/sqrt(pi)
(where r>0)

so for any positive length of AB,
length of AP = (2*AB)/(2+sqrt(pi))

16. #### EC

February 25, 2014 - 11:15 am -

I have a comment re: use of the word “form” here. Substituting “is equal to” makes the meaning more clear for me- is this reducing the required complexity/ use of mathematical language?

17. #### Howard Phillips

February 25, 2014 - 11:27 am -

Another point – Why does the problem formulation have names for the points. This gets in the way.
Without mentioning the words rope or string it would still be the same problem if it said

Here is a line ______________________

Pick a point on the line.
Make a square from the piece on the left.
Make a circle from the piece on the right.
Will these have the same area?
…..

The question as originally put should now scream at you!!!

18. #### Stephen Thomas

February 25, 2014 - 11:31 am -

@Dan Anderson
Nice! and quick ;)

@John Golden
Your Geogebra is also very good.

Now a few more comments/questions for all solutions:
1) How easy is using <pick your tool, Geogebra, Desmos, Etoys, Scratch, etc) is it for kids to construct their own models
2) When would you want (and NOT want) the kids to construct their own models.
3) Challenge: Can you make a version where they find the answer via a series of successive approximations? Say where they pick the middle of the segment then can decided Left or Right and go half way each time until they get "close enough".

Cheers,
Mr. Steve

19. #### David Taub

February 25, 2014 - 11:41 am -

I got curious about extending the problem to cutting the string to make an arbitrary polygon and a circle with the same area (just for fun). This is what I came up with for where to put the point (given as the fraction of the polygon section). I apologize for the cumbersome “in text” mathematical notation:

1/(1+sqrt((pi/n)cot(pi/n)))

20. #### Jon Tyndall

February 25, 2014 - 11:56 am -

I’d say Howard Phillips has nailed it perfectly when it comes to lowering the barrier to entry and by getting the kids to pose the question you want them to answer. At this stage no tech is necessary, just a board and a marker. I think I’d only use tech in act3 to show that their answer looks right.

21. #### Dave

February 25, 2014 - 12:24 pm -

Read the question this afternoon and did a little GeoGebra doodling as a means of getting students to understand the question without giving the answer.
A great part of learning math is learning how to read, and visuals can only help this process.

22. #### Dave

February 25, 2014 - 12:25 pm -

Read the question this afternoon and did a little GeoGebra doodling as a means of getting students to understand the question without giving the answer.
A great part of learning math is learning how to read, and visuals can only help this process.

Here is my doodle.
http://ggbtu.be/m90339

23. #### Jered

February 25, 2014 - 2:42 pm -

I played with it for about 15 minutes before finding the ratio of AP to PB. I’m sure there would be many varied approaches, but they are related as follows:

2:√(pi)

24. #### Max Ray (@maxmathforum)

February 25, 2014 - 3:21 pm -

I too went quickly to a visual model, using Geometer’s Sketchpad. Because Sketchpad, like Geogebra and Desmos and ETools(?) is a geometric construction tool, I ended up with a square and a circle based on the measurements of AP and PB, both of which kind of hang out at the end of the line. The whole time my brain is going “can’t the line kind of wrap up to make the square and the circle?” but because I couldn’t model it that way with my tool, I couldn’t see the interpretation of using part of the line (or rope, or ribbon, or whatever) to physically form the circle or the square.

Gerry wonders if our goal is to get the kids to do the problem or to get them to translate the precise mathematical language… and if it is to get them to translate the language, what we would do then.

If the goal is to get kids to do the math, I really like the simplicity of having some rope and asking kids to make guesses about where to cut the rope so that a circle formed from one and a square from another have equal area. It lets them get into the problem, make some guesses, test possibilities, and will require them to set up the exact same algebraic expressions to eventually solve the problem. This also lowers the cost of entry by a LOT. Any kid can point to a spot on a rope. Very few kids can start working on expressions for area in terms of PA/AB (or whatever it is y’all did to solve this).

But… if we do all the interpreting for them, then we will have kids who can solve real problems that they define for themselves, and any problem interestingly posed, but who will look at dry old mathematician talk and turn up their noses. And I DON’T think that the reason kids should learn to read problems like this and get to work is because they’ll have to on tests. That would suggest that there’s no other reason that precise language might be important in math. I think there are good reasons for precise language and students should encounter those reasons.

I also think we learn to read by learning to write, often.

So I would have my students solve the problem as a rope-cutting problem. Then I would invent or find a mathematical pen-pal and have them try to pose the rope problem to them. Think of the Writing Instructions for a Peanut Butter & Jelly Sandwich task so beloved by math & science teachers. This pen pal would come back at them with questions and non-answers until they had mathematized the problem as selecting an arbitrary point P on line AB of arbitrary length, forming a square whose perimeter was equal to PA and a circle whose circumference was equal to PB, such that the areas enclosed by the circle and square were equal. Such nitpicking might include wondering what it means to “form a circle from the rope” or why you couldn’t just measure the length of the rope or how a rope could have area, or whether you could cut the rope along its long axis, thus making two long, skinny pieces of rope, etc. The point of modeling with math is to be able to abstract away those pesky real-world details that get in the way of solving a reasonable version of the question… and I’d want my students to experience that kind of writing, so they could begin to make sense of that kind of reading. I’ve never tried this so I don’t know how it would go, but I could imagine some version of it leading to frustration but also insight.

Last, of course, we’d look at the precise math version Dan presents and compare it to our informal version and think about what benefits and costs each has. We’d practice translating some other informal math presentations of questions into fancy ones, and vice versa (e.g. translating “pick two numbers that add up to 25. Now multiply them. How big an answer can you get?” into “x and y are real numbers such that x + y = 25. Find the maximum value of xy, or explain why there is no upper bound on xy.”

25. #### Sue Popelka

February 25, 2014 - 6:35 pm -

This problem is rich in Algebra and Geometry–it’s got perimeters/areas of squares and circles, the quadratic formula, simplifying radicals, factoring, compound fractions, and simplifying rational expressions. I have not seen it before, but I really enjoyed it. I worked through it, got the correct answer and then later checked your list of seven mathematics ideas involved; I had used all of them. I found the ratio of the length of the “square” piece to the length of the “circle” piece to be
(1-sqrt(pi/4))/(sqrt(pi/4)-pi/4)=1.129. I teach AP Calculus BC and find that what students struggle most with in Calc is the Algebra. This problem really stretches those skills. I am going to use this problem in my Calc classes next week. I plan on presenting it just as it was here and then have the students cut various lengths of strips of paper to try to get the areas to be equal before they attempt the algebra.

26. #### Yvette

February 25, 2014 - 8:15 pm -

If I read the post, it seems to me that the question is “How do we teach/ensure that the language is understood? At some point, a child will not have someone who will simplify the language.
They can only model the problem on their own IF they can understand the question.
I’m just a parent, but it seems to me that the first thing would be to ascertain comprehension.
Get a highlighter, underline key words. Encourage them to look up definitions.
The using those words, illustrate or model the problem. Draw a diagram.
If you can’t get past that point, then finding the right formula is going to be a challenge.
My son is younger, but we’ve been trying to use math language in daily language and find ways for him to retain the meaning. For example: Perimeter …”Secure the perimeter!!!” As in the distance around the object. Which as a typical young boy, he relates to.
Personally, I always found working through math questions it helped to read it once all the way through. Then read it a 2nd or even 3rd time. Draw the diagram one phrase at a time. TMI to retain all that in short term memory. I feel, at least with my kid, that they feel they should get it the first time through. Not sure math is like that. It’s okay to re-read, ponder and draw.

27. #### Jered

February 25, 2014 - 9:13 pm -

I should clarify that I started with a circle and square with congruent areas. I used 36. So the perimeter was 24. I solved for the radius which was 6/(sq rt pi). Circumference then is 12*(sq rt pi). Compare the two lengths [24 : 12*(sq rt pi)] and this simplifies to the ratio I listed above: 2 : (sq rt pi). I ran this by other math teachers and at least one of them used quadratics to get there. I thought this was fascinating, but he was fascinated that I started with the shapes. He called that “working backwards” but it was just the way I naturally came into the problem.

28. #### Avery

February 25, 2014 - 10:15 pm -

I’m going to be Downer Debbie here. This problem doesn’t do it for me. I looked at it using technology and came up with an answer, but didn’t feel that this answer helped me understand the “story” behind the problem any more deeply. I then solved the problem algebraically, but was once again left with a feeling of “ok then” versus “whoa.” I wasn’t surprised by the answer. Using technology, I didn’t find a “simple” answer that just screamed “something else must be going on here” or “there must be a simpler solution.” There aren’t any variations that I am yearning to solve. I’m not sure how a student might use this to explore a new concept or idea. Every visual representation I thought up missed the mark in relating the length of the lines and the “perimeters” of the shapes.

So I wasn’t going to post this comment. I didn’t want to come across as rude or mean, the person yucking other people’s yum. But then I realized this was silly. The fact that this problem didn’t inspire me doesn’t mean that Dan’s an idiot simpleton because it DID inspire him. On the contrary, it’s a great reminder that there isn’t a quick answer to what makes for a good problem, and this answer isn’t the same for everyone.

29. #### Mr K

February 26, 2014 - 7:23 am -

I put this off for a bit, due to grading (Argh!).

It took me about 3-4 minutes to solve – the math isn’t the hard part. The hard part is making it accessible to students.

If I make an interactive visual model (my first instinct) I am depriving them of the opportunity to create their own models. If I give them concrete numbers (The line is one meter long) I eliminate a lot of the value of the problem.

It seems to me that a simple model of the problem, (picture of a string) with a failed attempt (string cut into two equal parts) should be enough to pique the kids “I can do better” mode. Providing actual string with only one chance to cut raises the stakes above it being a guessing game.

Following that, it’s a matter of scaffolding their modeling (CCSMP #4!) encouraging (CCSMP #1!) them to climb the ladder of abstraction (CCSMP #2!).

I’ve got a dead day at the end of this week. Maybe I’ll try this then..

30. #### Max Ray (@maxmathforum)

February 26, 2014 - 7:28 am -

Mr. K, the idea of only one piece of string and only one chance to cut is nice because it makes the concept of “an arbitrary length” meaningful. You won’t tell them the string length in advance, they need to be prepared for any length. Cool!

31. #### Mr K

February 26, 2014 - 7:46 am -

On further thought, I really like the idea of the introduction being:

“I cut this string in half – do the two shapes I made out of it have the same area?”

It asks a simpler question, without doing the work for them, and in answering it they’ll be developing some of the components of the model they’ll need to use later.

32. #### Andy Brown

February 26, 2014 - 8:40 am -

Vary interesting comments. As Don Small probably battled his whole career teaching cadets at West Point, mathematics is only about modeling, even the arithmetic we put on the paper lest we forget what makes us happy to do.

33. #### Ignacio Mancera

February 26, 2014 - 12:00 pm -

I’ve been reading some of the comments and I really don’t think that one should add a real/fake context such as a ribbon, a corral or any of the kind. I do think, on the contrary, that purely mathematical questions should be presented to the students once in a while, so that they can appreciate them “for the sake of it”. I find the problem appealing and I resent the idea that “most” kids won’t find it so.

That being said, the difficulty of the language on which the problem is stated is of course an obstacle. My first idea was also to go to Geogebra and make a visualization, but I still thinks would struggle with the idea of “make a circle out of it”. I imagine that most of my students would misunderstand its meaning with “make a circle using that as its radius/diameter”. The same applies with the square and the perimeter: my first thought was that AP was the side of it, not its perimeter.

1. Draw a segment AB and a point P on it.
2. Draw a square using AP and a circle using PB.
3. Is it possible for the the square and the circle to have the same area? If so, where should we put P?

With this formulation, kids would still have different ideas: they might draw the circle using PB as its radius or as its diameter. Perhaps some kid (let us name him Dan) would draw the square using AP not as the side but as the diagonal of the square. We should take that chance to make them believe that THEY have created a new problem: “Ok class, new situation: what about Dan’s approach? How would that change the problem?”

After that, some leading might carry them towards new possibilities: “What if AP is…?” Or maybe: “What other thing might AP represent in a square?”. From that, we might get our problem as a more elaborated version of the simpler and more intuitive first one (but, actually, it will be much easier to solve it after having solved the rest!)

34. #### Howard Phillips

February 26, 2014 - 2:05 pm -

Me again!
Referring to comment no. 32 Max Ray (@maxmathforum), I like the approach, but he doesn’t go quite far enough. So:
———————–
Kick it around.
Find out where to use a bit of math, and what it entails (representation, solving equations etc)
Get an answer and check it for reasonableness.

Finally, maybe as a separate activity, figure out how to present the problem in mathematical fashion, ie abstraction, use of technical terms etc.
———————-
This way the students will have a much better idea about how to deal with problems already formulated in abstract terms.

Oh, and mathematicians (and engineers) do not happen upon precisely formulated abstract problems and then solve them, they do more or less exactly what I have described above. Also they don’t spend hours trying to prove theorems unless they are convinced that the theorems are true.

The problem as presented is really only suitable for math specialists, and as such quite a nice one.

35. #### M Ruppel

February 26, 2014 - 6:35 pm -

Dan et al,

I think this is a great problem that is unfortunately tied up in some confusing language:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

The biggest thing that stands out to me is what does it mean to “find P” – P isn’t a point with coordinates, it isn’t a fixed distance from A, etc. What we are really after is a ratio between 2 lengths, and the problem doesn’t make that clear.

If I were posing the problem to kids, I would make the question itself much simpler and concrete: Here’s a piece of string. You are going to cut it into 2 pieces so that you create a circle and a square that have the same area.

Setting it up, I would want to establish with the students that chopping it in half makes the circle bigger than the square, and ask why. Then I would ask them to take a guess, and then begin exploring. Each group would only get one piece of string, which would require them to compute the area in terms of the perimeter, rather than experimenting with cutting.

I think a lot of commenters want to use Geogebra or something here, but I’m not sure that gives a satisfying answer, because it’s necessarily approximate, and not based on manipulating the formulas. I would ask kids to do this totally by hand. Some might make successive approximations (what if I do a 60:40 split, 65:35, etc) until they have it, but the challenge of having to do it by hand may motivate them to think about it w/ algebra.

Great problem, not great set up.

36. #### Dan Meyer

February 26, 2014 - 7:36 pm -

You folks have brought a lot of depth here. I’ve done two things now:

One, I’ve summarized, condensed, and responded to a lot of your commentary at this link.

Two, I’ve added several resources I’ve created which I hope will serve less to dampen debate and more to toss a few dry logs onto the fire here.

37. #### Max Ray (@maxmathforum)

February 26, 2014 - 8:05 pm -

Dan, you’re a visual wizard. I was making my Sketchpad sketch in the hopes of making a wordless presentation of the video* but I couldn’t figure out a way to suggest how the location of P was building the circle and square. With your video, I would present the task wordlessly first, using videos 1 and 2 (or 1 and 3? Or 3, 1, and 2?) and then introduce the precise language once we had an intuition and played the “stop me when they’re equal…” game.

But… I’m also not convinced that the video adds much that the rope doesn’t… depending on what language game we’re playing. If my students understand the rope as a physical stand-in for a math line, just like the pixels on the screen are a physical stand-in for the math line, then I don’t see any difference. If they take the rope too literally, then we’ve changed the task in subtle (maybe important, maybe not?) ways.

*Having students make their own sketch felt like one of many problem-solving methods students might use, and not one I’d want to enforce or say is “the” way we’re going to make sense of the problem. It’s part of the solving, not part of the sense-making.

38. #### Mr K

February 26, 2014 - 11:57 pm -

I can’t tell – do you think new work a good thing or a bad thing?

To me it feels like it’s a byproduct of trying to make the problem more accessible without actually making it easier. In that sense, I’m okay with it.

39. #### David Taub

February 27, 2014 - 2:06 am -

It wouldn’t be too hard to throw together a javscript app that presents a line on screen and the students can click (drag?) on a point anywhere on the line.

The line would then break and form a circle and square with the two pieces.

A few questions on whether it is worthwhile and if so some of the details.

1. Difference between Dan’s video is the students get to play with where the points go, so the intro is more interactive. Is this helpful or not?

2. Requires students have access to individual/group devices (advantage of javascript is it will work in any browser)

3. How much information should it give? This could in theory be controlled by the teacher in some way.

4. Could just give relative sizes of the shapes. For example, “the area of the square is 5% bigger than the area of the circle”.

5. Or it could give some kind of actual value based on an arbitrary line length.

6. Options to show a grid in the background for estimating relative sizes.

7. Options to add numbers to the line – or, possible to have an optional box where students can type in a fraction instead of clicking for where the dot will go when they feel ready to try and be more precise.

8. Teacher controlled options to show the calculations being done to find the two areas.

9. Options to change to other polynomials (makes the programming a bit more complicated, but maybe not overly so).

10. If allowing for a range of polygons, perhaps a table listing the percents based on number of sides and watching them get closer and closer to 50% as the number of sides increases (I did this for fun in excel, and was personally surprised at how fast it started to converge).

Oh, and Dan, a small personal recommendation for your video – when you have the lines being “sucked up” into the circle and square, you leave behind a doted line to show where it was, but that is no longer there. I would really recommend changing the color of this dotted line so it is no longer the same color as the original line it is replacing. Possibly the same relative shade but much darker so the difference it more clear. It was kind of hard at first to see what was happening. I think a color change would make that stand out a lot more. Just my 2 cents.

40. #### Justin Lanier

February 27, 2014 - 4:14 am -

Hi Dan,
When you say you’re curious about my rationale for my redesign, do you mean you have questions about it, or just that you thought there was something interesting about it?

41. #### Kenneth Tilton

February 27, 2014 - 4:33 am -

Left unsaid (my fault) was that only the first question is offered up front, and that is for the gold medal (or whatever). The next question/hint must be requested, and that drops them to silver.

I remember watching kids using a Carmen San Diego game in school. They just punched clue-clue-clue until it pretty much gave them the answer. The good news? Kids aren’t dumb. :)

btw, I do not understand the objections to the GeoGebra implementation. It is a great intermediate step in a progression, one in which the kids could play with the value of P and see the area trade-off and then perhaps appreciate, “Oh, so this is why we have Algebra (and Geometry): Instead of wiggling P back and forth we can just calculate the answer directly from the givens.” In their own words, of course. :)

42. #### Mike Lawler

February 27, 2014 - 5:07 am -

My quick reaction to the questions you asked:

(1) Is learning to translate mathematical language the goal here? Or can we exclude that goal?

This problem works well for both cases, I think. If your goal with the students is to learn to translate mathematical language, then I think that both the videos you’ve create, Dan, and the other interactive elements created by commentators are great starting points. They help students see what’s going on in the problem and provide a nice starting point.

However, this problem is also a great question for more advanced settings where the students are already comfortable translating into mathematical language. In these settings I would suggest a modification to the videos / visuals and to the problem to allow the students to explore some interesting connections between the geometry and algebra in this problem:

Rather than taking a point on the line, take a point in the plane. Show the same geometry in the video – how one line segment folds into a square and the other curls up into the circle – without giving any hint of the location of the point on the line the problem is asking about. Now there are three follow up questions from the video:

(a) Find the point on the line segment that causes the square and the circle to have the same area.

(b) If we extend the line segment in both directions, are there any other points on the line segment where the square and circle will have the same area? This question asks for some geometric insight and can be answered without calculation, though finding the exact point obviously requires calculation.

(c) Now find all of the points in the plane where the square and the circle have the same area. The shape of this set of points will likely be a surprise to the students.

(2) What role can an animation play here? Do we want students to create an animation?

I do not believe that the animation is necessary, but I don’t see any down side in the animations. Non-animated pictures drawn by the students would also be helpful, and the suggestions of trying out the original problem with string might also be fun for students. There are many different ways to approach the problem, which is part of what makes it so interesting.

(3) What kinds of scaffolds can make this task accessible without making it a mindless walk from step to step? On the other end, how can we extend this task meaningfully?

My suggestion on extending the problem in the answer to question (1) probably covers most of my thoughts on this one. For me personally, when I read the problem on Tuesday morning, walking through that extension with my son was something I was really excited about.

43. #### Mr K

February 27, 2014 - 9:34 am -

Here’s what I’m trying today. It’s working surprisingly well with my Geometry students. The Algebra students may require some formulas, we’ll see.

The green fields are built in in two separate steps. But aside from that, there’s the whole problem.

Also, stock photo. I’ll take that strike.

44. #### Dan Meyer

February 27, 2014 - 2:13 pm -

Mr. K:

I can’t tell – do you think new work a good thing or a bad thing?

New, more interesting work is a good thing. In its current form the problem asks students to translate, formulate, and solve, in that order. I’d rather move translate to the back end of the task and add new, more interesting work to the front end.

Justin Lanier:

When you say you’re curious about my rationale for my redesign, do you mean you have questions about it, or just that you thought there was something interesting about it?

Both. I’m interested in your rationale for moving from the general to the specific. That seems to make the problem less accessible rather than more, though maybe there’s something to be gained I’m not seeing.

Max Ray:

But… I’m also not convinced that the video adds much that the rope doesn’t… depending on what language game we’re playing. If my students understand the rope as a physical stand-in for a math line, just like the pixels on the screen are a physical stand-in for the math line, then I don’t see any difference. If they take the rope too literally, then we’ve changed the task in subtle (maybe important, maybe not?) ways.

The video and the rope share a lot of features and each has features the other lacks.

I like the rope (the physical one, not a picture or the word) because it places a lot of weight on making the right cut the first time. It discourages guess and check in a great way.

The video, meanwhile, allows students to make a cost-free guess and compare it to the rest of the class. You can guess with the rope but if we all have ropes of different lengths, it’s going to make the share-out less meaningful. (This is to say nothing of the fact that the thought of passing out scissors to everyone in the class gives me a certain queasy feeling the video doesn’t.)

The video also allows me to ask the sense-making question, “What do you see happening here? Put it into words.” It’s hard to ask the same question of a rope that’s sitting on your desk.

45. #### Dan Meyer

February 27, 2014 - 2:47 pm -

Another update:

I interviewed some really smart folk – Jason Dyer, Keith Devlin, and the crew from a curriculum design mailing list I frequent – and asked for their opinions on this task and my redesign.

I recorded video of my interview Jason and pasted all of that in the new update. I found it extremely enlightening and I hope you do too.

46. #### Mr K

February 27, 2014 - 4:23 pm -

BTW – I think Jim’s extension of asking whether that solution to the problem is also the minimum possible area of the two shapes is pretty genius. Definitely calculus level, but it totally justifies calculus.

47. #### Keith Devlin

February 28, 2014 - 6:37 am -

Mike Lawler’s extension is neat. Super question to raise with student: “What about the other root?”

48. #### David Taub

February 28, 2014 - 6:47 am -

In case anyone is interested, here is a quick first draft of a javascript version that allows the student (or teacher) to click on a point on the line and then it will show the two pieces forming a square and a circle.

For now, no additional information is given. This is similar to Dan’s video, but allows the students to choose their own points.

http://hem.bredband.net/taub/circlesquareline.html

49. #### Stephen Thomas

February 28, 2014 - 7:47 am -

Peter Boon’s comment caught my attention, in particular “imagine logo-like tools that enable students to create” well Etoys is a logo like tool that students can easily use to model this problem. I created a a blog post showing how with 3 Objects and 9 lines of tile scripting kids could easily model this problem.

Etoys was designed as an educational tool for teaching children powerful ideas in compelling ways and a media-rich authoring environment and visual programming system for kids to create. So not only can it model this problem but many more Fractions, Multiplication etc. Etoys Illinois has some great resources.

Cheers,
Mr. Steve

50. #### Robert Woodley

February 28, 2014 - 8:08 am -

Not sure if anyone is still reading the comments, but my 2 cents:

The first sentence of the problem makes me feel so sad for the students that have to deal with it.

There are 3 problems here:
Problem 1: What is the radius of a circle with an area equal to a square of side X?
Problem 2: What is the ratio of the perimeters of the 2 objects in Problem 1?
Problem 3: If P1 + P2 = 1, what is P1 and P2?

Problem 1 is the most interesting one. It is a general concept that can find many applications and will appeal to the geeks in the class. (A fun side-line is to extend it to N dimensions – for high schoolers of course). It should be solved using algebra.

But the problem starts off using the language of geometry. It hides the mystery and excitement of solving a more general principle. It hides the beauty of the math behind a nuts-and-bolts problem that is poorly articulated and a problem one would never encounter in real life.

As stated it is neither general nor useful!!!! Neither of theoretical nor applicable use!

Crikey.

51. #### Max Ray (@maxmathforum)

February 28, 2014 - 2:02 pm -

I disagree with people who are saying that this problem as written is inherently bad or artificial. As an undergrad math major, a big part of the learning for me was figuring out that statements worded like this problem were very precise formulations of fundamental insights — insights that often had tangible models or visualizations at their core.

I remember lectures about knots, paper folding, determinants, and crazy algebras that the lecturers took the time to connect to interesting physical situations, or even silly but understandable situations about ants taking random walks on a picnic blanket. For a moment I even entertained the idea of graduate work in mathematics, because I realized that math was actually a pretty neat dance between thinking intuitively and thinking precisely.

tl;dr version: Translating this problem from precise to intuitive and intuitive to precise, is part of the real work that research mathematicians (and their college students) do, and not something we should always keep from our students. It’s a skill we should help them hone.

52. #### Dan Meyer

February 28, 2014 - 2:07 pm -

Max Ray:

Translating this problem from precise to intuitive and intuitive to precise, is part of the real work that research mathematicians (and their college students) do, and not something we should always keep from our students. It’s a skill we should help them hone.

Good word.

53. #### Mike Pac

March 1, 2014 - 8:15 am -

To me, this problem doesn’t beg for a need to be precise in finding P, and therefore doesn’t inspire a need to pursue/develop a generalized answer. Once I can get to the point of visualizing a square in relation to a circle (like in the animations), my intuitive sense of “about equal” in area is satisfying enough. I think the important part of this problem is getting to the visualization itself, and not actually finding the precise answer. In other words, this problem doesn’t create the same need as some of your other problems presented do – I’m thinking about the water tank problem, where no one wants to sit for multiple minutes to see how long the tank is going to take to fill, therefore creating a need to develop a model.

54. #### Jered

March 1, 2014 - 8:31 am -

@Mike Pac – OK, but to be fair that’s just you. I mean, if approximate is OK, couldn’t just splitting the rope in half be “good enough?” After all, the area of a square and of a circle can’t be that far apart.

Or can they?

Let’s say the rope is 48 inches long. The square will end up being 12 x 12, or 144 sq in, and the circle will end up having a circumference of 48, so a radius of about 7.64. The area of this circle is then approximately 183.4 square inches. That’s 27% bigger! But by shifting the cut just under an inch and a half (3% of the rope’s length), the areas become equal.

That’s amazing to me. How does 3% = 27%? Why are the areas of a square and a circle with congruent perimeter/circumference measurements so far apart? And why is it only 3% of the rope that makes the difference? And why do so many fencelines intersect at 90º instead of being rounded?

Maybe this problem in and of itself isn’t interesting, but I bet there are other approaches that could be taken to make it interesting – either to you or to your students.

Sometimes giving a problem to solve just feels like same-ol same-ol. Perhaps you could give them an answer and having them find out how to get it.

I just think there is *something* you can do with just about every problem to make it accessible and interesting.

55. #### Mike Pac

March 1, 2014 - 9:08 am -

@Jered….Your enthusiasm for this problem is great, and I’m sure it’s infectious to your students. “Interesting” is a subjective experience. I am giving my initial impressions of the problem, and also how I think many of my students would view it. One can view “by shifting the cut just under an inch and a half (3% of the rope’s length), the areas become equal” as exciting as you do, or “oh wow, my intuition was only an inch and a half off.” I’m not saying that this problem can’t be interesting. To me, it didn’t strike the need to really dive in and explore any further. Mentioning fencing, to me and many students, also makes the problem less interesting.

56. #### Chris Hill

March 3, 2014 - 1:40 pm -

I’m really surprised no one has mentioned this yet (or I missed it in my skimming comments).

What does that second solution mean? Pose that extension question to your Pre-calculus and above students. I have some initial thoughts, but I haven’t checked up on them with any thorough reasoning.

57. #### Tim Erickson

March 4, 2014 - 9:33 am -

Sorry to be so late to this confab, but so much has been great there’s not much for me to add. Two things: Dan has kindly referenced a page on my blog that talks about ways to attack the max/min version of this problem using data.
http://bestcase.wordpress.com/2012/09/13/reflection-on-modeling/
if that is of any use.

But I also want to give a shout-out to Mr K (27 Feb, #54, above), and here is the link:
http://mathpl.us/posters/circlesquare.001.jpg

What I love is the precise level of vagueness of the question. If that makes sense. You cut the shoelace, make the circle and square, and the question is, “What are the two areas?” It may be a commonplace here (sorry if I’ve missed it), but I think this type of question is important, if only because a really great answer is, “it depends.”

Which motivates the payoff question: “Depends on what?” (Followed by, HOW does it depend…”). I made a bunch of physics labs where the vague question was an essential part of the intro — enough so that I eventually got field-test students to recognize the ploy and shout, “It depends” in unison whenever they sensed one of these.

This is no replacement for a really good first act. But if you don’t have one, a vague question may be a suitable substitute.

58. #### Dan Meyer

March 4, 2014 - 10:44 am -

Tim Erickson:

This is no replacement for a really good first act. But if you don’t have one, a vague question may be a suitable substitute.

I’ve been going around a couple different places and claiming that “asking questions about the question” is an essential modeling act. But this requires a certain level of comfort from teachers with questions that aren’t fully specified. Not just questions where the information exists but has been withheld for a minute, but questions where the question requires follow-up questions. Like, “What do you mean by that? What form is the answer going to take? What does that word mean in this context?”

I find teachers evenly split in their perception of those ill-defined questions as advantages or disadvantages in class, as features or bugs.

59. #### Brian Miller

March 11, 2014 - 6:57 pm -

I used the original question and handed it to students after they turned in a test – and just waited to see what they would do. A lot of them initially punted on it. They didn’t know how to do it and there was no help so they flipped it over and started drawing on it.

After everyone finished the test I decided to try and be a teacher and scaffold a bit. I put the point right in the middle of AB and asked them if the perimeters were the same here. They said “yes”. Then I asked them if the areas were the same – and this when they made me proud and told me “no” because if the perimeters of the square and circumference were the same, then the circle would have a larger area because the circle has the largest possible area for a fixed perimeter. Thus point P had to be closer to B then A. I think that realization made them more proud that it would have if I had already provided them the visual evidence.