Total 7 Posts

## [Confab] Mind Reading & Math

I give out 5-6 sets of three dice. I have the students roll them and then add up all the numbers which cannot be seen (bottom, middles and middles). Once they have the sum, they sit back with the dice still stacked and I “read their minds” to get the sum.

So then I shuffled up the little slips of sequences and started saying, B, your sum is 210. C, your sum is 384. D, your sum is 2440. E, your sum is -24. They were astonished!

These moments seem infinitely preferable to just leaping into an explanation of the sums of arithmetic sequences.

Our friends who are concerned with cognitive load should be happy here because students are only accessing long-term memory when we ask them to roll dice, write down some numbers, and add them. It’s easy.

Our friends who are concerned that much of math seems needless are happy here also. With The Necessity Principle, Harel and his colleagues described five needs that drive much of our learning about mathematics. Kate and Scott are exploiting one of those needs in particular:

The need for causality is the need to explain â€“Â to determine a cause of a phenomenon, to understand what makes a phenomenon the way it is.

[..]

The need for causality does not refer to physical causality in some real-world situation being mathematically modeled, but to logical causality (explanation, mechanism) within the mathematics itself.

Here are three more examples where the teacher appears to be a mind-reader, provoking that need for causality. Then I invite you to submit other examples in the comments so we can create a resource here.

Rotational Symmetry

Here is a problem from Michael Serra’s Discovering Geometry. No need for causality yet:

But at CMC in Palm Springs last weekend, Serra created that need by asking four people to come to the front of the room and hold up enlargements of those playing cards. Then he turned his back and asked someone else to turn one of the cards 180Â°. Then he played the mind-reader and figured out which card had been turned by exploiting the properties of rotational symmetry.

Number Theory

The Flash Mind Reader exploits a numerical relationship to predict which symbol students are thinking about. Prove the relationship.

Here is a little trick I like to call calculator magic. You will need a calculator, a 7-digit phone number and an unwitting bystander. Here goes:

Key in the first three digits of your phone number
Multiply by 80
Multiply by 250
Subtract 250
Divide the number by 2
Surprise! It is your phone number!

A nice trick is this one with dice. A lot of dice. Letâ€™s say 50 or so. You lay them on the ground like a long chain. The upward facing numbers should be completely random. Then you go from the one end to the other following the following rule. Look at the number of the die where youâ€™re at. Take that many steps along the chain, towards the other end. Repeat. If youâ€™re lucky, you already end up exactly at the last die. Youâ€™ll be a magician immediately! But usually, that isnâ€™t the case. What you usually have to do, is take away all those dice which you jumped over during the last step. Tell them that that is â€œthe rule during the first roundâ€. Now the actual magic begins. You tell the audience that they can do whatever they want with the first half of the chain. They may turn around dice. Swap dice. Take dice away. Whatever. As long as they donâ€™t do anything with the second half of the chain. [If you like risks, let them mess up a larger part of the chain.] What youâ€™ll see, is that each and every time, they will end up exactly at the end of the chain!

A few years ago, I found this “trick” on a “maths” site, not sure which, but it was UK. You need 5 index cards. Number them 1, 2, 3, 4, 5 in red ink on the front. On the reverse side, number them 6, 7, 8, 9, 10 in blue ink. Be sure that 1 and 6 are on opposite sides of the same card…same with 2 and 7, etc. Turn your back to the group of students. Have one of the students drop the 5 cards on the floor and tell you how many cards landed with the blue number face up (they don’t tell you the number, just “3 cards are written in blue”). Tell them the total of the numbers showing is 30. The key is that each blue number is 5 more than its respective red number. Red numbers total 15. Each blue number raises the total by 5. So 3 blue numbers make it 15 (the basic sum) + 15 (3 times 5). Let them figure out how you are using the number of blue numbers to find the total of the exposed numbers.

Expressions & Equations

I ran an activity with students I called “number tricks.” (Okay. Settle down. Give me a second.) I’d ask the students to pick a number at random and then perform certain operations on it. The class would wind up with the same result in spite of choosing different initial numbers. Constructing the expression and simplifying it would help us see the math behind the magic. (Handout and slides.)

I do something called calendar magic where I show a calendar of the month weâ€™re in, ask the students to select a day and add it with the day after it, the day directly under it (so a week later), and the day diagonally to the right under it, effectively forming a box. Then I ask them to give me the sum and I tell them their day.

Always a bunch of students figure out the trick, but the hardest part is writing the equation. Every year I have students totally stumped writing x+y+a+b. Itâ€™s really a reframing for them to think about the relationship between the numbers and express that algebraically.

Finally I ask them to write a rule for three consecutive numbers, but I donâ€™t say which number you should find and inevitably someone has a rule for finding the first number and someone has one for finding the middle number. I love that!

Different Bases

Andy Zsiga suggests this card trick involving base 2.

Call for Submissions

Where else have you seen mind-reading lead to math-learning? Are there certain areas of math where this technique cannot apply?

2014 Oct 30. Megan Schmidt points us to all the NRich tasks that are labeled “Card Trick.”

2014 Oct 30. Michael Paul Goldenberg links up the book Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks.

## These Horrible Coin Problems (And What We Can Do About Them)

From Pearson’s Common Core Algebra 2 text (and everyone else’s Algebra 2 text for that matter):

Mark has 42 coins consisting of dimes and quarters. The total value of his coins is \$6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem.

The only math students who like these problems are the ones who grow up to be math teachers.

One fix here is to locate a context that is more relevant to students than this contrivance about coins, which is a flimsy hangar for the skill of “solving systems of equations” if I ever saw one. The other fix recognizes that the work is fake also, that “solving a system of equations” is dull, formal, and procedural where “setting up a system of equations” is more interesting, informal, and relational.

Here is that fix. Show this brief clip:

Ask students to write down their best estimates of a) what kinds of coins there are, b) how many total coins there are, c) what the coins are worth.

The work in the original problem is pitched at such a formal level you’ll have students raising their hands around the room asking you how to start. In our revision, which of your students will struggle to participate?

Now tell them the coins are worth \$62.00. Find out who guessed closest. Now ask them to find out what could be the answer â€“ a number of quarters and pennies that adds up to \$62.00. Write all the possibilities on the board. Do we all have the same pair? No? Then we need to know more information.

Now tell them there are 1,400 coins. Find out who guessed closest. Ask them if they think there are more quarters or pennies and how they know. Ask them now to find out what could be the answer â€“ the coins still have to add up to \$62.00 and now we know there are 1,400 of them.

This will be more challenging, but the challenge will motivate your instruction. As students guess and check and guess and check, they may experience the “need for computation“. So step in then and help them develop their ability to compute the solution of a system of equations. And once students locate an answer (200 quarters and 1200 pennies) don’t be quick to confirm it’s the only possible answer. Play coy. Sow doubt. Start a fight. “Find another possibility,” you can free to tell your fast finishers, knowing full well they’ve found the only possibility. “Okay, fine,” you can say when they call you on your ruse. “Prove that’s the only possible solution. How do you know?”

Again, I’m asking us to look at the work and not just the world. When students are bored with these coin problems, the answer isn’t to change the story from coins to mobile phones. The answer isn’t just that, anyway. The answer is to look first at what students are doing with the coins â€“Â just solving a system of equations â€“Â and add more interesting work â€“Â estimating, arguing about, and formulating a system of equations first, and then solving it.

This is a series about “developing the question” in math class.

Featured Tweets

I asked for help making the original problem better on Twitter. Here is a selection of helpful responses:

2014 Oct 20. Michael Gier used this approach in class.

One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

## [Confab] Money Duck

Confab time! Let’s make some magic here. This is a Money Duck. It’s soap.

My sense is this is an object with a lot of potential for a math teacher. I’d like to know how you’d harness that potential.

A particular question I’m wrestling with is whether or not to put the student in the position of:

• the seller, asking “How should I price the Money Duck?”

Over the next few days I’ll update this post with comments I’ve solicited in advance from some of my favorite curriculum designers. I’d love to add your thoughts to the pile.

I’d be more interested in having students be sellers rather than buyers. Buyers’ incentives seem more qualitative: soap quality, residue, allergy, shape, smell.

Being sellers, students could do a lot with cost analysis given production constraints.

1. Show the picture. It’d be cool to have time lapsed video of soap breaking down under a shower stream or, to not waste water, one could actually document soap usage over time.
2. Ask students how much they’d be willing to buy it for.
3. Decide on a central measure of their answers to use (mode or median?) or go with a small range of values. (I actually surveyed my algebra class: out of 36 kids, 13 said they’d be willing to pay between \$4 and \$6, while 15 were willing to pay \$7 to \$10 for a bar.)
4. Students are now sellers of the soap. How much should they price it knowing what people are willing to pay?
5. Give constraints: cost of raw material, time to make, number of workers, etc.
6. If 5,000 bars were made, what is the distribution of the bills in them? What about 106 bars?

Thinking about it, I would position the student as the seller of the money duck for two main reasons. Ultimately, knowledge of the quantities of each denomination would be needed by both the seller and buyer. Unfortunately, the buyer would most likely never have access to that information (since I assume that it is not like a lottery where they divulge the odds of winning) and would have to guess whereas the seller could reasonably have that information.

In terms of determining the selling price:

• The seller thinks: profit + cost of “real money” + production cost.
• The buyer thinks: amount buyer would regularly pay for soap + amount of money buyer could win + amount buyer would pay for the novelty of having a money duck soap – minus whatever margin they are hoping to profit.

I think that the seller’s situation is much more manageable. We can have Act Two information to determine the profit and production cost. The math will come from determining the cost of the “real money.” As for the buyer, there are many assumptions that will not fall as conveniently into Act 2. There may be disagreement as to what a buyer would pay for soap, the novelty cost, and profit margins. Accordingly, I think it is much easier to do the seller situation.

My favorite question here would be to set a price for the duck and ask the sellerâ€™s question, what distribution should I use for the bills? But only after the kids have determined that an equal distribution is profitable for the buyer.

Mr K:

My thought is that the kids will gravitate to the â€œShould I buy itâ€ question, and the real learning comes from shifting them around to the sellers side of the problem.

Iâ€™d set up many sets of two types of groups, producers (sellers) and consumers (buyers). The producers would determine what distribution of the bills (ha) go in the ducks and then set a price based on that info. They make up 20 â€œducksâ€ with their distribution. The consumers would go up to a producer and be given the odds of each type of bill and the price of the duck. Itâ€™d be up to them if theyâ€™d like to buy that groups ducks.

how about using the money duck to construct a survey. Survey students as to what odds (probability) of winning any prize would encourage them to muy multiple money ducks.

I presented the Money Duck to my grade 4/5 class this morning with the question: â€œWhat is the most that you would pay for a Money Duck bar of soap?â€ Group conversations were animated as was the class discussion.

Questions that students raised included: â€œHow likely would it be for the package to have 10, 20 or 50 dollars?â€, â€œIs the money planted in the middle of the soap? If so, would someone break the soap to get at the money?â€ [in which case the soap didn’t matter at all in the pricing] and â€œWhat is the quality of the soap?â€ [If it was a good quality soap many were willing to pay more]. Some students started taking the sellerâ€™s point of view and gave ideas how they could increase profits.

The discussion lasted nearly half an hour. Incidentally, the average price the students were willing to pay was \$5. (Figuring that the soap itself was worth 3 or 4 dollars, and then factoring in the minimum prize of 1 dollar).

Clearly this idea has potential at multiple grade levels.

How many money ducks would I have to buy to be guaranteed to score at least one duck with a \$50 bill inside?

From what Iâ€™ve heard, nearly all of the ducks are \$1 ducks. Itâ€™s like buying a lottery ticket â€” you expect to lose, but it can be fun for some people anyway.

“I used to work at a warehouse for online gag gifts until it went out of business I had four boxes of these prob 200 bars or so and never got more than a five.”

I think legally they only need to have one duck in the entire country that has a \$50 in it to claim there could be a \$50, so thatâ€™s likely your odds.

First, show this video.

Ask students to tell their neighbor how much they’d be willing to pay for the Money Duck. Find the high and low in the class.

Now there are a series of questions I’d like students to confront, including:

I’m trying to progressively formalize a) this new, strange representation of probability and b) the calculation of expected value.

The first two questions assist (a), basically asking “What is this thing?” and “What isn’t this thing?”

The next three questions assist (b), applying progressively more nuance to the concept of expected value. First, the concept is either/or. (“Who gets screwed?”) Then it becomes ordinal. (“Rank ’em.”) Then numerical. (“Put a price on ’em.”)

Kids will struggle at different moments in this sequence, but that’s okay because the purpose of the sequence isn’t that they discover the concept of expected value. The purpose of the sequence is to make make my eventual formal explanation of expected value much more comprehensible. (See Schwartz on contrasting cases.)

So explain how mathematicians calculate the expected value of a distribution. Now let’s go back and calculate the exact expected value of the distributions in #5.

Here are those questions and screens packaged as a Keynote slideshow (also PowerPoint, if you must) and as a handout.

Now let’s show them the answer, what the Virginia Candle Company actually charges for the Money Duck.

Now every student should create a distribution that results in some profit at this price over time.

Fin.

Follow on with practice and assessment.

What would you change, add, or subtract from this sequence?

## [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 â€“ the math isnâ€™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 â€œthe two areas are equalâ€ 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 â€œI can do betterâ€ 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 â€œrandomâ€ 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 â€“ 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 mathematicianâ€™s meta cognitive arsenal. You can compute without insight, so when you donâ€™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 â€” 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. Itâ€™s a skill we should help them hone.

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

## [Confab] Design A New Function Carnival Ride

The early feedback on Function Carnival has been quite kind. To recap, a student’s job is to graph the motion on three rides:

But we found ourselves wondering if there were other rides and other graphs and other great ideas we had missed. So we’re kicking this out to you in this week’s Curriculum Confab:

What would be a worthwhile ride to include in Function Carnival? What would you graph? Why is it important?

I’ll post some great responses shortly.

Previously:

In the last confab, we looked at a math problem inspired by Waukee Community School District’s decision to let their buses idle all night. Molly showed us how to make a good problem out of it, and a lousy problem also. Great confabbing, people.