“That’s a lot of smiles,” Keira (10) said as we waited for our Teen Burgers.

“Yeah. How many?” I asked. “A lot” wasn’t going to fly with a “real-world” number talk in front of us.

“Sixty-three and nineteen isâ€¦ hold on,” Keira said. She wanted to add tens and ones: three twenties is sixty and one and two make three. She knew that the nine in nineteen would make this strategy more challenging. So she took advantage of the associative property and (wisely) punted.

After a few moments Keira offered eighty-two. She explained that sixty-three and twenty make eighty-three so sixty-three and nineteen make eighty-two.

Her sister Gwyneth (13) used a different strategy. “I took one from the twenty-one and gave it to the nineteen,” she said. “That’s four twenties–ha!–and two more.”

At Graham Fletcher’s session at the Northwest Mathematics Conference in Whistler, he shared a story of one student using this strategy after engaging in his Bright Idea task: “Numbers are just Skittles now,” she said. Similarly, Gwyneth decomposed twenty-one, taking and giving one to create two landmark or friendly numbers. To Gwyneth, numbers are just smiles.

Earlier this year, I wanted to share student work on Graham Fletcher’s Krispy Kreme three-act task with a group of intermediate teachers. When I last facilitated this task, many students thought of multiplication as repeated addition (only). Others used the standard algorithm — few successfully. At that time, analyzing student work revealed what students really understood (or didn’t). Further, the teacher and I discussed implications on practice going forward. (This prompted my last post.) But with my group of teachers I wanted to talk partial product strategies and models and these samples weren’t helpful. So Marc and I faked it and created some possible approaches:

We shared the approaches with the group and after some noticing and wondering invited them to find as many connections as they could. Some intended connections:

Students 1 & 5 thought of multiplication as repeated addition

Students 2 & 4 & 7 think place value to decompose 32 into two (or more) addends

Student 2 “splits” 32 symbolically; Student 7 partitions an open array

The partial products in Student 3’s algorithm can be seen in Student 4’s open array

Students 1 & 8 make use of the fact that four 25s make 100

Students 4 & 8 make use of halves and doubles

Teachers then discussed the placement of these approaches within a learning progression and how they might “nudge” each student.

Analyzing student work has become my favourite professional development activity. Here, what is lost in terms of authenticity is gained in terms of diversity of thinking. Still, I was excited to see this from @misskwiatkaski5‘s real students:

David: You know how long you and I have known one another?

Howard: How long?

David: Well, it’s pretty much to the month since 1984.

Howard: Wow. Now I’m gonna do some quick math and figure out how long that is, if you don’t mind.Â Now math happens to beâ€¦ I’m good at it. This is how I do it. This is 2018. Right?

David: It’s 34.

Howard: Oh, you gave it away.

David: It’s 34 years.

Howard: Let me check your math.

David: Yeah.

Howard: The way I get to it is, you say 1984 and I add ten immediately.

David: Yeah.

Howard: That brings us to 1994.

David: That’s right.

Howard: That’s ten.

David: Yeah.

Howard: 1994, then 2004 is 20.

David: Yeah.

Howard: Now here’s tricky ’cause I get confused.Â 2004 to 2014 is another 10. That’s 30. You’re absolutely right. That’s 34 years. Good for you.

David: Nowâ€¦

Howard: I love to show how I do the math.

David: Speaking of which, you realize that all of that will be subtracted from the show?

Howard: Wow. But really for youâ€¦ I guess the premise of this show, although who knows what this show isâ€¦ you know, I don’t even know what I’m doing here, but I thought the premise was thatâ€¦ you’re choosing six peopleâ€¦ and I’m way more fun than Obama already, I’m sure. I mean, this is fun.

David: Really?

Howard: Oh, for God’s sake, yeah.

Lucky for us, Letterman didn’t subtract all of this from the show. Some observationsâ€¦

Despite David giving away the solution, Howard continues to share his strategy. David is not the ultimate authority; Howard is eager to prove this solution. Howard, at least, is interested in Howard’s reasoning. He’s focused on sense-making, not answer-getting; how?, not what?Â All of this is typical of a classroom number talk.

Howard uses an adding upÂ (or add instead) strategy for 2018Â âˆ’ 1984. He moves forward from 1984 to reach 2018.Â The context implies distanceâ€“not removalâ€“which lends itself to this strategy. Stern’s jumping by tens gives us an opportunity to discuss efficiency, e.g., one jump of thirty rather than three jumps of ten. For what it’s worth, I used an adding up strategy too. First I added 16 to 1984 to get to 2000 (or six and ten to get to 1990 and 2000), then I added 18 to get to 2018.

David, of course, does not record Howard’s thinking. I might use this video clip to have teachers anticipate possible strategies forÂ 2018Â âˆ’ 1984 and consider how they would record them. I chose an open number line to model Howard’s adding upÂ strategy:

Howard is confident: “Now math happens to beâ€¦ I’m good at it.” He is enthusiastic: “I love to show how I do the math.” He is joyful: “I mean, this is fun.” Over the last two years, it has been my privilege to work alongside Surrey teachers Alex Sabell and Jonathan Vervaet (and others) as they’ve incorporated number talks in their classrooms. These same positive attitudes towards mathematics come through in their students’ interviews (seeÂ Alex & Jonathan).

What did you notice in this clip? What did I miss?

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 Ã— 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 Ã— 10, remains the same. Similarly, we can halve and double to visualize thatÂ 1 Ã— 20Â is equivalent toÂ Â½Â Ã— 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 problemÂ 16 Ã— 35. The fifth graders share several strategies: partial products (10 Ã— 30 + 10Â Ã— 5 +Â 6 Ã— 30 +Â 6 Ã— 5); making friendly numbers (20Â Ã— 35Â âˆ’Â 4 Ã— 35); halving and doubling (8 Ã— 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. PoseÂ 16 Ã— 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 throughÂ 16 Ã— 35. I would introduce this through a string of computation problems (e.g.,Â 1 Ã— 12, Â 2 Ã— 6, 4 Ã— 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: 16Â Ã— 35 = (8 Ã— 2) Ã— 35 = 8Â Ã— (2 Ã— 35) = 8 Ã— 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:

I’m not usually a fan of equations in math picture books. But I like 100 Snowmen by Jennifer Arena and Stephen Gilpin. On each page, students can use the mental math strategy of adding one to a double to determine basic addition facts to 19. Each number is represented as both a number to be doubled and one more than a number to be doubled. Take five.Â Here, students can double five and add one more to determine five plus six.

Here, five is not doubled, but one more than four, which is doubled.

Dot cards can be used to draw attention to the doubles plus one strategy. Ask “How many do you see? How do you see them?”

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.)Â 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 questionâ€¦ How many Frankenkinis (sp?) are possible?

The website advertises it is possible to createÂ 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 ALERTâ€“ all does not end well. How did this happen? What went wrong?

My guess? The Sharks have number sense.Â 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 percentagesâ€“ 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.Â The bikini guys have this:

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