Here, a monster is made up of three cards: head, torso, and legs. In Bears vs Babies, a monster can be just a head or a head with one to four body parts. I’ve simplified the task to get at the fundamental counting principle.

Act 2

What information would be helpful to have here?

How many head, torso, and legs cards are there?

Students may want to act this out. Give them these cards. Encourage them to find a systematic way of counting the possibilities. How can the number of monsters be determined from the number of head, torso, and legs cards? Start with heads and torsos, if need be.

Act 3

The reveal…

Introduce tree diagrams. Connect this representation to students’ strategies. These might help:

If you do know the four digits, how many combinations¹ could there be?

Act 2

Students may ask to see the four digits.

Remember to ask later if this information matters. That the digits are 1, 3, 4, 5 doesn’t; that there are four different digits — no repetition — does.

Act 3

My hope is that this resolution feels sort of anticlimactic — that Raymond Reddington’s “Now there’s only twenty-four combinations” on the screen doesn’t measure up to students’ shared strategies in the classroom.

Elizabeth Keen’s “Could be thousands of combinations” prior to Red’s sand trick could be an extension. At first viewing, it seems far-fetched that the character — an FBI profiler — doesn’t understand that there are exactly ten thousand four-digit possibilities (0000, 0001, 0002, …, 9999). But has Liz assumed that the digits cannot repeat? If so, how many combinations could there be? Students can no longer answer this question by systematically listing and counting each possibility.

I imagine this task as an introduction to, not an application of, permutuations. It provides a context for students to develop — not practice! — methods of counting without counting. Don’t bother if you’re anticipating a lot of knee-jerk 4!s from your students.

The first question that came to my mind was “How many different phrases are there?” To determine the number of possibilities for a task, we multiply the number of choices for each stage of the task. Phrases are generated by stringing together a verb, adjective, and noun. There are 53 choices of verbs, 65 choices of adjectives, and 66 choices of nouns. So, there are 53 × 65 × 66 = 227 330 “finely crafted phrases of educational nonsense.”

The teacher who posted the education jargon generator writes “I would be remiss if I did not thank my district’s professional development staff for introducing me to many of these gems.” Hah!

Wait! I resemble this remark. I’ve been prepping three all-day workshops that my team and I are facilitating giving this week. While I do use many of these words individually, I hope that I get called on it if I ever string three of them together, be it in person or on this blog.

A recent “Shark Tank” episode featured two entrepeneurs pitching MiX Bikini, the world’s first interchangeable swimsuit. Here’s a sneak peek:

Two things piqued my interest.

Thing One: The Product

“It’s no secret women love to stand out, but there is nothing worse for [a] woman than being at the beach and seeing another girl in the same bikini,” one partner says.

Nothing? Really?

Here’s how it works:

First, assuming [a] woman is not offended by the claim above, she selects a style of bikini top (halter or triangle). Next, she chooses one of 40 colours/patterns for the bikini top. She does this twice (right and left). She then selects a style of bikini bottom (classic or ‘scrunchie’) and picks out one of 33 colours/patterns. (In the “Shark Tank” video, the second model switches out the back bottom. On the Mix Bikini website, the front & back of the bikini bottoms always match.) The bikini tops must be connected. Customers must choose between rings or strings. Rings are available in 10 colours, strings in 9. Of course, bikini tops also need neck strings (right and left). Double neck strings come in 9 colours, rings & strings in 10.

This begs the question… How many Frankenkinis (sp?) are possible?

The website advertises it is possible to create thousands of bikinis.

Thousands? Try millions.

What number do you get? What assumptions do you make? Is fuschia & leopard print different than leopard print & fuschia? I maintain it is. It is best that I not elaborate.

Thing Two: The Pitch

“We are seeking fifty thousand dollars in exchange for five percent of our business,” says the first partner.

“That means that you’re saying the company is valued at one million dollars,” says Daymond, one of the Sharks.

“It was ten percent we were asking,” interrupts the second partner.

“So half a million dollars,” Daymond clarifies.

Uh-oh. The budding businessmen are confused. Mathematically disoriented. The Sharks smell blood. SPOILER ALERT– all does not end well. How did this happen? What went wrong?

My guess? The Sharks have number sense. They have mental math strategies. Daymond understands 5% is equal to 1/20. Therefore, if 1/20th of the business is valued at $50 000, then the total value of the company can be calculated by multiplying by 20 (or, more likely, by doubling and multiplying by 10). If $50 000 is 10%, or 1/10th, of the company, then the Sharks can multiply $50 000 by 10 (or, more likely, halve $1 000 000, the original evaluation).

In the “Shark Tank”, the Sharks often counter with benchmark percentages– 5%, 10%, 25%, 50%, 75%. I suspect the Sharks have strategies for other popular percentages (eg, for 40% they may halve, halve, and multiply by 10).

Our pitchmen, on the other hand, do not have number sense. They do not have mental math strategies. The bikini guys have procedures. The bikini guys have this:

BTW, if you’re looking for a lesson on combinations, check out Pair-alysis from Mathalicious.