“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 is… 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:

This Krispy Kreme three-act task above–from Graham Fletcher or YummyMath–cries out for partial products.

But more than once, the partial product strategies and models that I anticipated did not emerge. Not even close. 5 Practices-induced flop sweats. More on that in a future post. First, a progression of partial products across the grades, beginning with the basic multiplication facts:

Some students will see four rows of seven doughnuts and know that 4 ⨉ 7 = 28. Great. For students who haven’t yet mastered the basic multiplication facts, partial products are helpful. Have students use what they know. For example, they might break apart seven as five and two and then find the sum of two familiar products: 4 ⨉ 7 = 4 ⨉ (5 + 2) = (4 ⨉ 5) + (4 ⨉ 2) = 20 + 8 = 28. Or, they might double a double: 4 ⨉ 7 = (2 ⨉ 2) ⨉ 7 = 2 ⨉ (2 ⨉ 7) = 2 ⨉ 14 = 28. They might do both. They might even break a factor into more than two addends: 4 ⨉ 7 = 4 ⨉ (3 + 3 + 1) = (4 ⨉ 3) + (4 ⨉ 3) + (4 ⨉ 1) = 12 + 12 + 4 = 28. (Admittedly not the most useful relationship to help students derive this fact.) Mastery of the basic multiplication facts aside, playing with partial products–and open arrays–reinforces the big idea that numbers can be broken apart–or decomposed–in flexible ways to make calculations easier.

This idea extends to multiplying two-digit numbers by one-digit numbers:

Some students will understand that breaking apart by place value makes calculations easier: 5 ⨉ 12 = 5 ⨉ (10 + 2) = (5 ⨉ 10) + (5 ⨉ 2) = 50 + 10 = 60. Others might use doubles and double-doubles. Note that a factor can be broken into addends or smaller factors: 5 ⨉ 12 = 5(3 + 3 + 3 + 3) or 5 ⨉ 12 = 5(3 ⨉ 4). How students choose to express this will provide insight into their thinking.

Again, decomposing numbers in flexible ways extends to larger numbers:

Breaking apart both factors by place value is a common approach: 25 ⨉ 32 = (20 + 5) ⨉ (30 + 2) = (20 ⨉ 30) + (20 ⨉ 2) + (5 ⨉ 30) + (5 ⨉ 2) = 600 + 40 + 150 + 10 = 800. This approach might be too common if reduced to a procedure (i.e., the box method or FOIL). Again, it’s about flexible ways. Breaking apart just one factor by place value is an efficient mental math strategy: 25 ⨉ 32 = 25 ⨉ (30 + 2) = (25 ⨉ 30) + (25 ⨉ 2) = 750 + 50 = 800. A student who inefficiently decomposes 32 as 10 + 10 + 10 + 2 could be nudged towards 32 as 30 + 2. Or, a factor of 25 might spark thinking about 25 ⨉ 4 = 100, a familiar product.

The different varieties of doughnuts illustrate some helpful ways of partitioning the arrays. But each of these slides draws attention to a specific way of seeing the array. My preference would be to show the slides where all the doughnuts are the same. (Same goes for visual patterns.) Ask students how they see them. If students do not see a helpful way of partitioning the arrays, then corresponding slides with different varieties of doughnuts could be displayed. In a number string, 52 – 40 leads students to think about adjusting 39 in 52 – 39 to make the calculation easier. Similarly, a purposely crafted string of images could lead students to see fives, doubles, or place value–all useful relationships–in an original (glazed) array.

Determine an equation of a quadratic function with vertex at (-5, 3), passing through the point (-7, 15).

Lately I’ve been looking for activities that address this sort of naked math yet engage learners in processes similar to those in a mathematical modelling cycle.

Consider the exercise above. What questions could you ask? If I were to ask a student about their equation, I’m likely to hear play-by-play, not colour commentary: “… and then I plugged -7 and 15 in y = a(x + 5)² + 3. Negative seven plus five is two…”

Instead, I could have students try to figure out a quadratic function that satisfies a set of criteria, gradually revealed to them as “clues.” Throughout, students would check their quadratic functions and make changes when necessary. This is the gist of Wanted Parabola, my adaptation of Cathy Marks Krpan’s Wanted Number:

I started with a very general clue: the direction of opening. I anticipated a variety of parabolas, which I got when I tried this activity out with math teachers in my district. When I tried this activity out in the MathTwitterBlogoSphere (#MTBoS), I got a bunch of y = x²s. The biggest difference was that my colleagues were invited to draw a parabola (on whiteboards) whereas my tweeps were asked to write an equation (in a Desmos activity). It’s interesting to think about this activity in terms of freedom and constraints. When I revealed the next clue, it pushed my colleagues’ thinking together. However, from my tweeps, it triggered new and diverse ideas, simulated here:

I like this as a blank-page (or whiteboard) activity but a Desmos activity (1, 2, 3) does provide the opportunity to talk about some interesting overlays. If using vertical non-permanent surfaces (#VNPS), I’d stop partway through to hold a “board meeting” where students would share possible parabolas.

In general, I progressed from providing more general to more specific clues. For example, “vertex in QII” divulges p < 0 before “axis of symmetry x = -5″ gives away p = -5. Most clues add new information and move students closer to the Wanted Parabola. Some confirm earlier decisions. For example, “vertex (-5, 3)” before “axis of symmetry x = -5″ and “minimum value of 3.” This last clue is anticlimactic. An earlier clue, “never enters QIII,” is much more interesting. It might feel like new information. But it must be true given preceding clues; a parabola that opens up and has no x-intercepts cannot contain points in QIII (or QIV).

You can play with the order of the clues. A second Wanted Parabola:

Here, the direction of opening clue is revealed midway through the set. It doesn’t add new information but is reasoned to through “two x-intercepts” and “vertex in QI.” I meant to delay students determining the direction of opening a bit, hoping to surprise them after a few clues. In a third Wanted Parabola, “passes through” is the first clue; I anticipate that some students will place the vertex at this point.

Instead of “How did you find a?” you could ask “Which clues were helpful? Which clues were necessary?” In my mind, helpful ≠ necessary. A clue might be helpful if it pushes students in the direction of the Wanted Parabola despite not providing the values of a, p, or q. Or a clue might be helpful if it tells students that they’re on the right track. In the way that the first Wanted Parabola plays out, three pieces of information are necessary (to determine three unknowns): “minimum value of 3,” “axis of symmetry x = -5,” and “passes through (-7, 15).” If some students don’t argue that only two clues are necessary — “vertex (-5, 3)” and “passes through (-7, 15)” — you could ask “What is the fewest number of clues you need?”

This activity helps students develop an understanding of the different attributes parabolas can have. It provides an opportunity for students to solve problems, reason, explain, justify, and connect mathematical ideas in ways that “Determine an equation…” does not.

In June, a colleague invited me into his classroom to teach a Desmos modelling task — Predicting Movie Ticket Prices — in his Math 12 class. Students experienced exponential functions earlier in the course. We were curious about whether his students would apply what they knew about exponential functions to a task situated outside of an exponential functions unit — a task not having to do with textbook contexts of half-life, bacteria, or compound interest. They did. And they deepened their understanding of how change by a common ratio appears in exponential equations (vs. change by a common difference in linear equations). They did this within 45 minutes of a 75-minute class. So my colleague let me try out another, less sexy, task — one adapted from MARS. This task, like much of Math 12, is about naked functions; no real-world context here. Nat Banting’s closing keynote at #NWmath reminded me of it. Watch Nat’s talk; view his slides.

The original MARS task above is closed: two functions, one linear and one quadratic, each passing through four points. I wanted to open it up so I changed the prompt: “A set of functions pass through the points shown. What could the equations for the functions be?” Also, I removed one of the points — (5, 3) — to allow for different solutions of two functions. The thinking is that open questions encourage a variety of approaches. And then, from fifteen pairs of students:

I anticipated this. The points scream linear and quadratic. They are sources of coherence. I had lowered the floor but no Rileys entered y = 5, y = 7, y = 8, y = 9. The problem wasn’t problematic. I had raised the ceiling but no one wrestled with equations for sinusoidal or polynomial or radical or rational functions. The freedom within my open question didn’t bring about new and diverse ideas. To support creativity — mathematical creativity! — I had to introduce a source of disruption, a constraint: “A set of nonlinear functions pass through the points shown. What could the equations for the functions be?”

A student could have used the linear nature of absolute value functions to get around my nonlinear constraint — a bit of a Riley move? — but no one did.

Instead, some students picked up on the symmetry of two new possible parabolas:

Writing the equation of the second parabola — finding the parameters a and q — presented more of a problem.

Others bent the line; they saw the middle of its three points as the vertex of a cubic function that had been vertically stretched and reflected:

Some saw four compass points and wrote an equation of a circle. This led to a function vs. not a function conversation: “Does that count?” Others saw a sine function that passed through three of these four points. There were “close enough” solutions — great for Coin Capture but not quite passing through the given points:

I didn’t anticipate this. Students weren’t as constrained by “pass through” as I was. Also, they were motivated to capture the points using only two functions, as before.

With more time, I could have shifted constraints again: “A set of functions pass through the points shown. What could the equations for the functions be? (P.S. The graph of at least one of them has an asymptote.)” This would have triggered exponential and logarithmic or rational functions. (Even without introducing this constraint, we noticed at least one student playing with rational functions at the end of class.)

Above, there’s evidence to support Nat’s #NWmath conjecture: “Shifting constraints triggered new mathematical possibilities.” My (more) open question didn’t cut it. The student thinking — and conversations — that I had hoped for only emerged when freedom “sloshed against” constraints.

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. Now math happens to be… 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. 2004 to 2014 is another 10. That’s 30. You’re absolutely right. That’s 34 years. Good for you.

David: Now…

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 you… I guess the premise of this show, although who knows what this show is… you know, I don’t even know what I’m doing here, but I thought the premise was that… you’re choosing six people… 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 observations…

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? All of this is typical of a classroom number talk.

Howard uses an adding up (or add instead) strategy for 2018 − 1984. He moves forward from 1984 to reach 2018. The context implies distance–not removal–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 for 2018 − 1984 and consider how they would record them. I chose an open number line to model Howard’s adding up strategy:

Howard is confident: “Now math happens to be… 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 (see Alex & Jonathan).

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

Keira, Grade 4, asked me to show her “the nines trick” one morning last week before school.

If you don’t know it, watch Jaime Escalante/Edward James Olmos:

I did not show my daughter this trick. I am not the Finger Man. It’s like she doesn’t even know me!

Instead, we had a quick conversation. No time for manipulatives. Five minutes to brush her hair and pack her lunch before we had to hop in the car.

Me: You remember what a ten-frame looks like?

Keira: Yeah. Ten dots. Five and five. Array!

Me: Ok, what about nine? What does it look like?

Keira: One missing.

Me: What if there were two nines? How many?

Keira: Don’t ask me that one. I already know it’s eighteen.

Me: Ha! Ok, what about seven times nine?

Keira: I knew that you were going to ask me that one!

Me: What if you had seven ten-frames, each with nine dots? How many dots altogether?

Keira: Sixty… three?

Me: Why?

Keira: You start with seventy but you take seven away.

We did a few more together. Success!

Then she asked me to show her the nines trick.

For the purpose of this post, I quickly put together this slide (and video):

In the car, Keira asked me “Can you multiply decimals? Like seven times nine point five?” This reminded me of “I’m wondering if fractions only work with circles” from Annie Fetter’s#NoticeWonderIgnite talk. (We showed it at a workshop the night before.) This also reminded me of what I take for granted. Her sister and I did some explaining, but I’m wondering about a better (?) approach:

(Not my normal approach to multiplying decimals — the photo below probably had something to do with that.)

A few weeks ago, I took my daughter to the mall. Later, she complained that “Dad spent half the time taking math photos!” Five of one hundred twenty minutes is not half!¹

One of those photos:

I thought that this would make a great “Would You Rather…?” math task. I considered a few approaches. My preference is probably to just display the offer and have students make up their own prices and riff on “What if…?” That might be a tall order. I created a few combinations. (More on these in a sec.) But I wanted something more open.

The idea is that students would mix & match specific combinations of board games to justify their decisions.

For example, consider Carcassonne ($43) and Blokus ($40). With “buy one, get a second 25% off” the discount is $10 (25% of $40). Add Othello ($35) and with “buy two, get a third 50% off” the discount is $17.50 (50% of $35). It looks like the second option is the clear winner. But if we think about the (total) percent discounts, we get about 12% ($10/$83) and 15% ($17.50/$118), respectively. Proportionally, the gap shrinks.

What if we replace Othello above with Spot it! ($20)? Again, the discount is $10 (50% of $20). But it’s not a tie. Saving $10 on $83 is better than saving $10 on $103 (about 12% vs. 10%).

There are a couple of combinations where we can’t justify the second option. For example, consider Catan ($63) and Pandemic ($60). With “buy one, get a second 25% off” the discount is $15. Add Rock Paper Scissors ($6) and with “buy two, get a third 50% off” the discount sinks to $3.

Beyond making and justifying a decision using mathematics, I’d push students to generalize: When would you rather…?

A couple more photos from the mall:

“Dad, stop taking photos of arrays! Are these like the paint splatter thing?” Yep. Partially covered arrays in the wild. Lack of fraction sense aside, it’s nice to know that she’s paying attention. And making connections.

¹BTW, I use Microsoft Office Lens to quickly crop, clean up, and colour these photos on the fly. An essential app for teachers using vertical non-permanent surfaces (#VNPS on twitter). Check it out.

At that moment, three students jumped to their feet and cheered. High fives may have even been shared. I asked them what was up. They asked if we could talk about it later. (Never press here, by the way. Rookie mistake. If kids give you an out, take it.) So we did. Each student had estimated how many times I would say “Okay, so, um…” during the lesson. Their earlier excitement? I hit the highest of the three estimates.

The Interpret process in this mathematical modelling cycle involves reading contextualized situations in order to identify real-world problems.

In this task, we can start with the following clip and ask “What do you notice?”

I noticed the sports clichés (NSFW). Brock Boeser’s “I just want to come here and help the team get a win” is damn close to “Nuke” Laloosh’s “I’m just happy to be here, hope I can help the ball club.” I also noticed that Boeser says “you know.” A lot. I wasn’t alone.

From here, we can develop a real-world problem by asking “What do you wonder?” or “What’s the first question that comes to mind?” My question: How many times does Brock Boeser say “you know” in the postgame interview?

Note: the starting point — in the diagram and in the video — is a situation, not a problem.

Apply (Mathematize)

The next process involves identifying and activating mathematical understanding in order to translate real-world problems into mathematical problems. The MoE calls this Apply, a misused and abused term in mathematics education. Thankfully, Mathematize immediately follows in brackets throughout the documents.

We can ask “What information would be helpful to know here?” Students might want to know:

the number of times that Boeser says “you know” in the clip (12)

the length, in seconds, of the clip (44)

the length of the entire interview (2:58)

the rate at which Boeser says “you know” (?)

the fraction of the time in which Boeser is speaking (?)

This process also involves — among other things — creating relationships to represent the real-world problems. Here, a proportional relationship. A simple approach might involve setting up 12/44 = x/178. A math problem.

Solve

At first glance, this looks trivial: simply cross-multiply and divide. But the Solve process involves using a variety of approaches and representations. For example, students might use scale factors or unit rates; bar models or ratio tables. Or, not proportions, but linear relations. Tables, equations, graphs. Does the solution make mathematical sense?

Analyze

Does the mathematical solution (x = 48.545454…) make sense within the contextualized situation? The Analyze process involves identifying possible limitations and improvements. Brock Boeser says “you know” 12 times in the 44 second Act 2 video. But he reaches this count at 33 seconds and finishes answering the reporter’s question at 40 seconds. Does any of this matter? Is my simple proportional approach still useful?

Communicate

Students communicate throughout the Interpret, Mathematize, Solve, and Analyze processes. This communication happens within their groups. The Communicate process in this mathematical modelling cycle involves clearly and logically defending, explaining, and presenting their thinking and solutions outside of their groups.

There are better tasks that I could have picked to illustrate this mathematical modelling cycle. In fact, last year — in the absence of sample numeracy tasks from the MoE — my go-to here was Michael Fenton’s Charge. BC’s Graduation Numeracy Assessment aside, mathematical modelling with three-act math tasks (and the pedagogy around these tasks) has played an important role in my work with Surrey math teachers for several years. The MoE did release a sample numeracy assessment in late September; I am now able to include a Reasoned Estimates, Plan and Design, Fair Share, and Model task in these conversations with colleagues. For more numeracy tasks, see Peter Liljedahl’s site.

Okay, so, um, if I didn’t pick this Brock Boeser task because it, you know, epitomizes the mathematical modelling cycle, then why did I share it? Coming full circle to the story of my three students at the beginning of this post, there’s a missing piece. Yeah, we shared a laugh and I was more self-conscious of my verbal fillers for the rest of the year (2005 ± 3). But the most embarrassing part is that I have no idea how my students came up with their estimates. Because I didn’t ask. I mean, three girls spontaneously engaged in mathematical modelling — I promise there was more mathematical thinking here than in the task at hand — and not a single question from their math teacher! In my defence, it would be several years before mathematical modelling was on my radar — an unknown unknown. Still, what a complete lack of curiosity!

I probably painted myself into a corner. Adding a fourth shape/graph/number/etc. to a set often knocks down the reason why one of the other three doesn’t belong. Not all two-by-two arrays make good WODB? sets (i.e., a mathematical property that sets each element apart).

Still, there are similarities and differences among the four numbers above that are worth talking about. For example, the top right and bottom right are close to 100 (or 1); the top left and bottom right are greater than 100 (or 1); top left and top right have seven parts, or rods, of tens (or tenths); all involve seven parts in some way. There is an assumed answer to the question, “Which one is 1?,” in these noticings — a flat is 100 if we’re talking whole numbers and 1 if we’re talking decimals. But what if 1 is a flat in the top left and a rod in the bottom left? Now both represent 1.7. (This flexibility was front and centre in my mind when I created this set. The ten-frame sets, too.)

Last spring, Marc and I offered a series of workshops on instructional routines. “Alike and Different: Which One Doesn’t Belong? and More” was one of them. WODB? was a big part of this but the bigger theme was same and different (and justifying, communicating, arguing, etc.).

So rather than scrap the hundreds/hundredths grids, I can simplify them:

Another that elicits equivalent fractions and place value:

Another question that I like — from Marian Small — is “Which two __________ are most alike?” I like it because the focus is on sameness and, like WODB?, students must make and defend a decision. Also, this “solves” my painted-into-a-corner problem; there are three, not six, relationships between elements to consider.

The numbers in the left and right images are less than 100 (if a dot is 1); the numbers in the centre and right can be expressed with 3 in the tens place; the left and centre image can both represent 43, depending on how we define 1.

At the 2017 Northwest Mathematics Conference in Portland, my session was on operations across the grades. The big idea that ran through the workshop:

“The operations of addition, subtraction, multiplication, and division hold the same fundamental meanings no matter the domain in which they are applied.” – Marian Small

That big idea underlies the following slide:

At first glance, the second and third are most alike: because decimals. But the quotient in both the first and second is 20; in fact, if we multiply both 6 and 0.3 by 10 in the second, we get the first. The first and third involve a partitive (or sharing) interpretation of division¹: 3 groups, not groups of 3.

Similar connections can be made here:

This time, the first and second involve a quotative (or measurement) interpretation of division: groups of (−3) or 3x, not (−3) or 3x groups. (What’s the reason for the second and third? Maybe this isn’t a good “Which two are most alike?”?)

I created a few more of these in the style of Brian’s Same or Different?, including several variations on 5 − 2.

Note: this doesn’t work in classrooms where the focus is on “just invert and multiply” (or butterflies or “keep-change-change” or…).

And I still have no idea what I was going for with the Cuisenaire rods.

¹Likely. Context can determine meaning. My claim here is that for each of these two purposefully crafted combinations of naked numbers, division as sharing is the more intuitive meaning.

Updates:

An edited version of this post appeared in Vector.

From a recent Cuisenaire rods workshop…

Visual patterns…

Polynomials…

Odds and Evens…

If not WODB?, then Which two are most alike? or What is the same? What’s different?…