Scott Hills:

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.

Kate Nowak:

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 — 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 180°. 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.

Jehu Peters:

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
Add 1
Multiply by 250
Add the last 4 digits of your phone number
Add the last 4 digits of your phone number again
Subtract 250
Divide the number by 2
Surprise! It is your phone number!

Sander Claassen:

A nice trick is this one with dice. A lot of dice. Let’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 you’re at. Take that many steps along the chain, towards the other end. Repeat. If you’re lucky, you already end up exactly at the last die. You’ll be a magician immediately! But usually, that isn’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 “the rule during the first round”. 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 don’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 you’ll see, is that each and every time, they will end up exactly at the end of the chain!

Isabel Wiggins:

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.)

Kate Nerdypoo:

I do something called calendar magic where I show a calendar of the month we’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. It’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 don’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.

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 buyer, asking “Is the Money Duck worth its price?”
• 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.

Featured Task Designers

Featured task designer Fawn Nguyen:

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.

Possible task:

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?

Featured task designer Robert Kaplinsky:

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.

Featured Comments

Bowen Kerins:

My favorite question here would be to set a price for the duck and ask the seller’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 “Should I buy it” question, and the real learning comes from shifting them around to the sellers side of the problem.

Dan Anderson:

I’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 “ducks” 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. It’d be up to them if they’d like to buy that groups ducks.

Jennifer Potier:

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.

Kevin:

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

Questions that students raised included: “How likely would it be for the package to have 10, 20 or 50 dollars?”, “Is the money planted in the middle of the soap? If so, would someone break the soap to get at the money?” [in which case the soap didn’t matter at all in the pricing] and “What is the quality of the soap?” [If it was a good quality soap many were willing to pay more]. Some students started taking the seller’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.

Katy Engle:

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

Jason Dyer:

From what I’ve heard, nearly all of the ducks are $1 ducks. It’s like buying a lottery ticket – you expect to lose, but it can be fun for some people anyway.

Bit from an Amazon review of a different money soap:

“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 that’s likely your odds.

Task Proposal

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:

1. What is a probability distribution and how do we represent it?
2. What does an impossible probability distribution look like? Why is it impossible?
3. If you’re a seller, what kind of probability distribution is bad for business and why?
4. Rank these distributions in order of “I’d definitely buy that for $5!” to “I definitely wouldn’t buy that for $5!”
5. What would a fair price be for each of these distributions so that over time you wouldn’t lose or win any money?”

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?

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

• The Cannon Man’s graph is piecewise quadratic and linear.
• The Bumper Car’s graph is piecewise linear, which has thrown a bunch of students.
• The Ferris Wheel’s graph is sinusoidal.

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.