Halving & Doubling: Very Fun to Play With

On last week’s Last Week Tonight with John Oliver, John Oliver used the mental math/computation strategy of halving and doubling as a punchline to a news story on nuclear waste.

The graphics nicely–and quickly!–illustrate why this strategy works. Starting with 1 × 20 (one football field twenty feet tall), if we double the first factor (area in football fields) and halve the second factor (height in feet), the product (volume in piles of nuclear waste), expressed as 2 × 10, remains the same. Similarly, we can halve and double to visualize that 1 × 20 is equivalent to ½ × 40. (Oliver also throws in the commutative property at the end–twenty football fields one foot tall.)

This reminded me of a video clip from Sherry Parrish’s Number Talks. In it, the teacher poses the problem 16 × 35. The fifth graders share several strategies: partial products (10 × 30 + 10 × 5 + 6 × 30 + 6 × 5); making friendly numbers (20 × 35 − 4 × 35); halving and doubling (8 × 70); and prime factors (ultimately unhelpful here).

I’ve probably shared this video in about a dozen workshops. There are some predictable responses from attendees. Often “not my kids” is the first reaction. I remind teachers that the teacher in this video has implemented this routine three to five times a week in her classroom. This isn’t her kids’ first number talk. Pose 16 × 35 in your fifth–or ninth!–grade classroom tomorrow and, yeah, the conversation will probably fall flat. Also, this teacher is part of a schoolwide effort (seen in other videos shared at these workshops).

Teachers are always amazed by Molly’s halving and doubling strategy. Every. Single. Time. I ask attendees to anticipate strategies but they don’t see this one coming. I note that doubling and halving wasn’t introduced through 16 × 35. I would introduce this through a string of computation problems (e.g., 1 × 12,  2 × 6, 4 × 3). “What do you notice? What patterns do you see? Does it always work? Why?” We can answer this by calling on the associative property: 16 × 35 = (8 × 2) × 35 = 8 × (2 × 35) = 8 × 70 above. Better yet, having students play with cutting and rearranging arrays provides another (connected) explanation.

Rather than playing with virtual piles of nuclear waste, I had fun with arrays of candy buttons:

Number Talks (pdf)




From The Blacklist

Act 1

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

Act 2

Students may ask to see the four digits.

Keypad - Act 2

Keypad – Act 2

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.

¹I know, I know… permutations.

Survivor: 100 Chart Challenge

I don’t watch Survivor. Stopped watching after Richard Hatch, often competing naked, won the first season.

Channel surfing last week, this grabbed my attention:

Host Jeff Probst:

“Alright, let’s get to today’s duel. For today’s duel you’re gonna race across a balance beam, collecting bags of numbered tiles. You must then place the tiles in order, one to one hundred.”

(Aside: If there are three opponents, is it still called a duel?)

The reaction online was swift and harsh:

“It is seriously the most idiot-proof puzzle in the history of puzzles. You basically have to know how to count and that’s pretty much it.” (source)

But that’s not pretty much it. I mean, it is counting from one to one hundred (and that is how the contestants solved the “puzzle”), but it could be more than that. A better strategy involves comparing numbers, understanding place value, and identifying patterns found in tables.

At 1:49 and 2:02, we see two contestants, Laura and Brad, respectively, place 25 from the second bag (11 to 30).

26A literal translation of “You gotta put ’em in order”? Each competitor places 25 only after placing 24. Then, he/she tries to find 26 in his/her pile o’ tiles. Some tiles are facing down. Suppose a player turns over a tile and finds 28 rather than 26. He or she should take advantage of another pattern and place it under 18.

At 3:11, it’s down to Brad and John for the last spot. At 3:18, Brad places 87 after 86.


He could have caught John if he had an understanding of place value. Suppose Brad turns over 94 before finding 87. Should he drop 94 and continue looking for 87 or just place 94 in the 9th row (9 tens) and 4th column (4 ones)?

This challenge reminds me of an activity I’ve used in Grade 3 classrooms. Take some 100 charts. Cut each chart into “puzzle” pieces. Place in a Ziploc bag. In pairs, have students reassemble. Ask students to describe how they solved their puzzle. This activity is much more engaging (and puzzling) than it has a right to be.

100 Chart Puzzle

Don’t be surprised if you see some completed 100 charts that look like this:

100 Chart Puzzle 2

Parts Unknown

Last night, I caught a recent episode of “Anthony Bourdain: Parts Unknown.”

My first thought, “Ten-frame!” My second, “A possible three-act math task?”

Act One

I wrestled with including the first fifteen seconds of the clip. Will students ask their own questions if they suspect they’re going to answer one of Bourdain’s? Does the remainder of the clip make sense without this? Or, are the first fifteen seconds the first act, the remainder the second? By the way, Bourdain does a pretty good job on his blog of tossing out questions students may have:

Was I doing a good thing? Is it OK to be in the chocolate business? I don’t have any problem with wealthy people who can afford making impulse buys in expensive gourmet shops spending a lot of money on my chocolate. But where does the money go? In fact, where does this chocolate come from anyway? Just about everybody loves the stuff. It’s everywhere. A fundamental element of gastronomy. But I knew so little about it. Where does it come from? How is it made? Most importantly, who does it come from? And are they getting a good piece of the action? Or are the producers, as in so many cases, getting screwed over? I very much hoped to find that whoever was growing our cacao was, at the end of the day, happy about the enterprise — that life after Eric and Tony’s Excellent Chocolate Adventure was, on balance, better than life before.

Act Two

What information would be good to know? I wanted to know, what is a “nosebleed price”? From the man himself:

Thing is, it’s a very boutique-y, very high end, screamingly expensive end of the biz. One of the only 7,000 bars we were able to produce (the whole year’s supply sold off in just a few months) cost the nosebleed price of $18. Even reflecting the remote location, the rarity of the raw ingredient, the long trip from the mountains to the city to Switzerland and then to the States — the whole artisanal process — that’s still a f**k of a lot of money for a chocolate bar.

It looks to me like the producers get 15% of each chocolate ten-frame for the raw cacao, labour another 2.5%. For comparison, the three investors get 5% each.

Act Three

Raw Cacao: $2.70/bar; $18 900 in total
Labour: 45¢/bar; $3150 in total

Doesn’t exactly answer “Are they doing a good thing?” does it? And is it even possible to “show the answer” to this question? Can we adapt this task so that students use proportional reasoning to make a case for our cacao growers rather than just perform a couple of quick calculations? That is, can students use math to answer “How fair?” rather than “How much?” Differences in purchasing power and cost of living between nations now come into play.

Maybe this just doesn’t fit the three-act framework. Too bad. I kinda liked this sequel: How long would a Peruvian cacao grower have to work to purchase a luxury chocolate bar in Manhattan?


Teacher of Interest: Episode 1

I’m a fan of Person of Interest, a TV series about a software genius and an ex-CIA agent who work together, in secret, to prevent violent crimes before they can happen. In a recent episode Mr. Finch goes undercover as a substitute teacher to protect a high school student.

There are ≥ 3 clips of interest to (math) teachers. The first:

First off, I am well aware that this is fiction. The teacher receiving a last-minute opportunity to attend an all expenses paid teaching seminar in Maui is a dead giveaway. Still, part of this depiction of mathematics teaching may painfully ring true.

“Math is not punishment,” Mr. Finch/Swift says when a student explains that the classroom teacher has left busywork. Often, tedious problems are used as classroom management. Students are assigned one to fifty-nine odd only because there are forty-five minutes left in a seventy-seven minute period. I’ve been an eyewitness to teachers using math as punishment. They play good cop/bad cop (“You guys have worked hard today, so no homework”/”Get to work, or I’ll assign the evens”). I, too, may have been guilty of this. The message is undeniable: math is unpleasant. Behave, or do math.

Mr. Finch/Swift is surprised and disappointed to learn that he has been left to teach addition. “That can’t be right.” It isn’t right. But it isn’t uncommon. He feels this is below his students. He wants to elevate the problem from arithmetic to mathematics: “Who’d like to take a crack at working out Gauss’ equation?” Finch/Swift provides a hint: 100(100 + 1). Like most math teachers, he means to be helpful. However, by trying to be helpful, he may have scaffolded problem-solving out of the problem for his students. At least he would have, if more than one of them were actually listening to him. The solution is 100(100 + 1)/2. Dividing by two. That is all that is left for his students to figure out. The rest is ‘rithmetic.

A sneak peek at Episodes 2 & 3: A Statistically Improbable Score & What It’s Good For.