On Pace

Act 1

Steph Curry On Pace Headline Retouch

The retouched headline is designed to have students ask “How many 3-pointers will Stephen Curry make this season?” There are related questions: “At what pace (rate) is Curry making 3-pointers? What makes this pace historically ridiculous? What’s the difference between a historically ridiculous pace and a ridiculously historic pace?”

Here’s the thing about historic paces: historically, they happen weekly.

history-making historically dominant

Act 2

I retouched the first sentence in the article to open things up a bit. Pre-edit: “We’re nearly through 20 percent of the 2015-16 season…” Only the number of 3-pointers made to date (74) is needed. We don’t need to know the number of games played to date (15) or the number of games played in an NBA season (82). That’s the point of percent–fanatical comparison to 100. (I wonder if students will ask for this superfluous information anyway.) Post-edit, this information might, in fact, be useful to know. And help draw out multiple strategies. Perhaps students will ask for a fraction, rather than a percent, to fill in the blank. Games played and 3-pointers made to date can be determined from the following graph:

Steph Curry On Pace Graph 1:2I cropped the infographic because it resolves an extension (see it from the waist down below). And because it’s too damn long.

Act 3

Steph Curry On Pace Headline

The article suggests two possible extensions: “How many 3-pointers does Steph Curry need per game remaining to reach 300? How many games will this take?”

Steph Curry On Pace Graph 2:2

Source: http://www.cbssports.com/nba/eye-on-basketball/25386027/steph-curry-3-point-tracker-on-pace-for-404-makes-in-2015-16


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.

Fair Share Pair

A couple weeks ago, I was discussing ratio tasks, including Sharing Costs: Travelling to School from MARS, with a colleague who reminded me of a numeracy task from Peter Liljedahl. Here’s my take on Peter’s Payless problem:

Three friends, Chris, Jeff, and Marc, go shopping for shoes. The store is having a buy two pairs, get one pair free sale.

 Chris opts for a pair of high tops for $75, Jeff picks out a pair of low tops for $60, and Marc settles on a pair of slip-ons for $45.

The cashier rings them up; the bill is $135.

How much should each friend pay? Try to find the fairest way possible. Justify your reasoning.

Sharing Pairs.pdf

I had a chance to test drive this task in a Math 9 class. I asked students to solve the problem in small groups and record their possible solutions on large whiteboards. Later, each student recorded his or her fairest share of them all on a piece of paper. If you’re more interested in sample student responses than my reflections, scroll down.

The most common initial approach was to divide the bill by three; each person pays $45. What’s more fair than same? I poked holes in their reasoning: “Is it fair for Marc to pay the same as Chris? Why? Why not?” Students notice that Chris is getting more shoe for his buck. Also, Marc is being cheated of any discount, as described by Student A. (This wasn’t a happy accident; it’s the reason why I chose the ratio 5:4:3.)

Next, most groups landed on $60-$45-$30. Some, like Student A, shifted from equal shares of the cost to equal shares of the discount; from ($180 − $45)/3 to $45/3. Others, like Students B, C, and D, arrived there via a common difference; in both $75, $60, $45 and $60, $45, $30, the amounts differ by $15. This approach surprised me. Additive, rather than multiplicative, thinking.

Student C noticed that this discount of $15 represented different fractions of the original prices; $15/$75 = 1/5, $15/$60 = 1/4, $15/$45 = 1/3. He applied a discount of 1/4 to all three because “it’s the middle fraction.” Likely, this is a misconception that didn’t get in the way of a reasonable solution.

Student D presented similar amounts. Note the interplay of additive and multiplicative thinking. She wants to keep a common difference, but changes it to $10 to better match the friends’ discounts as percents.

Student E applies each friend’s percent of the original price to the sale price. This approach came closest to my intended learning outcome: “Solve problems that involve rates, ratios and proportional reasoning.”

In spite of not reaching my learning goal, I think that this lesson was a success. The task was accessible yet challenging, allowed students to make and justify decisions, and promoted mathematical discourse.

Still, to increase the future likelihood that students solve this problem using ratios, I’m wondering about changes I could make. Multiples of 20 ($100-$80-$60) rather than 15 ($75-$60-$45)? Different ratios, like 4:3:2 or 5:3:2, might help; the doubles/halves could kickstart multiplicative thinking. (Also, 5:3:2 breaks that arithmetic sequence.)

Or, I could make changes to my questioning.

Sharing PairsWhen I asked “What do you notice?” students said:

  • the prices of the shoes are different
  • Chris’ shoes are the most expensive
  • Marc’s shoes are the cheapest
  • Chris’ shoes are $15 more than Jeff’s, which are $15 more than Marc’s
  • Jeff’s shoes are the fugliest

Maybe I could ask “What else could you say about the prices of Chris’ shoes compared to Marc’s?” etc. to prompt comparisons involving ratios. If that fails, I’m more comfortable connecting ratios to the approaches taken by students themselves than I am forcing it.

BTW, “buy one, get one 50% off” vs. “buy two, get one free” would make a decent “Would you rather?” math task.

h/t Cam Joyce, Carley Brockway

Sharing Pairs - Sample Student Response A

Sharing Pairs – Sample Student Response A

Sharing Pairs - Sample Student Response B

Sharing Pairs – Sample Student Response B

Sharing Pairs - Sample Student Response C

Sharing Pairs – Sample Student Response C

Sharing Pairs - Sample Student Response D

Sharing Pairs – Sample Student Response D

Sharing Pairs - Sample Student Response E

Sharing Pairs – Sample Student Response E

“They’ll Need It for High School” (Part 4)

In Part 1, I catalogued and critiqued common uses of “They’ll need it for high school.” Samuel Otten’s take on “When am I ever going to use this?” was an obvious influence. “Poor Pedagogy Preparation” was one of my categories: “I want them to get used to it” as an indefensible defence for the mad minute. Since that post, I’ve wondered about putting a positive spin on “I want them to get used to it.”

As a K-12 Numeracy Helping Teacher, I have the opportunity to teach in elementary and secondary classrooms alike. Often, I’m struck by how pedagogical similarities overwhelm any differences. Malcolm Swan’s ordering decimals lesson — a current favourite of mine — and my sinusoidal sort illustrate this.

In Grade 5, students are asked to put the decimal cards in order of size, smallest to largest.

Card Set A - Decimals

The most common response is 0.4, 0.8, 0.04, 0.25, 0.75, 0.125, 0.375. This can be explained by how students compare whole numbers. Misconceptions are revealed, but not corrected. (The rest of the activity will take care of that.)

In Grade 12, students are asked to put the equation cards in piles.

Sinusoidal Sort - Equations

The most common response is to make two piles: sine and cosine. This can be explained by how students were introduced to y = sin x and y = cos x. This isn’t a misconception; there is no “right” sort. But it is unsophisticated. And it goes unchallenged. (The rest of the activity will take care of that.)

In Grade 5, students match hundred grids and number lines to their decimals. They explain how they know that the cards make a set, building connections between decimals and their understanding of fractions and place value. They argue. When there’s consensus, students are again asked to put the cards in order. I note their strategies that emerge (benchmarks, place value, equivalent decimals) for a class discussion.

In Grade 12, students match graphs and characteristics to their equations. They make conjectures. They explain how they know that the cards make a set, building connections between transformations of trigonometric functions and their understanding of transformations of other functions. When there’s consensus, students are again asked to put the cards in piles. I note their sorting rules that emerge (amplitude, period, phase shift, vertical displacement, range, maximum/minimum values) for a class discussion.

Elementary and secondary lessons need not be as closely aligned as above. It’s not about matching card matching activities. Or parallel “Which one doesn’t belong?” prompts. Or three-act math tasks for K-5. More generally, exploring and discussing ideas, working collaboratively in pairs/small groups, problem solving (and posing!), … you could make a case for students needing these experiences in elementary school because… wait for it… they’ll need it for high school. We want them to get used to it. I hope.

A nagging thought… I’m not convinced that pedagogy preparation is ever a satisfactory response to “So, what will they need?” Even if that pedagogy is positive. If we’re going to follow that pathway, then this question demands an answer not in terms of teaching methods but in terms of the mathematical habits of mind (or processes or practices or competencies) that these teaching methods aim to promote. (See my “Affective Domain” concerns in Part 1.)

That’s not to say that I wasted my time with this post. For me, it’s always helpful to think about and present examples of secondary mathematics education as something other than passive. Maybe that — reframing those pedagogical “TNIFHS” conversations with colleagues into something more promising — is the real value here.

Which One Doesn’t Belong?

As part of an upcoming “TNIFHS” post, I wanted to include one example of what Christopher Danielson’s approach to a better shapes book might look like in a high school math class. But then I had some fun with this and created a few more. In each set, a reason can be given for each of the four options being the odd one out. I’ve done this type of thing with numbers and equations before. Worthwhile, but not what I’m going for here. Each set below is pictorial. Also, I went naked; I stripped the graphs of grid lines and ordered pairs. More noticing properties, less determining equations. Aside from graphs, where else in secondary mathematics might this fit? The last two images below are my attempts at answering this question.

Download the pdf.


WhichOneDoesn'tBelong?.002 WhichOneDoesn'tBelong?.003 WhichOneDoesn'tBelong?.004 WhichOneDoesn'tBelong?.006


WhichOneDoesn'tBelong?.008 WhichOneDoesn'tBelong?.009

“They’ll Need It for High School” (Part 3)

I’m picking “TNIFHS” back up. At the end of Part 1, I promised Part 2 would answer “What are the big ideas in elementary school mathematics that students will need for high school?” Instead, I talked times tables.

In this third half, I’ll refocus on these big ideas. Or one of them — one that came up in that initial “they’ll need long division for high school” conversation.

More than the standard algorithm, what students will need is an understanding of division as sharing (finding the number in each group) and measuring (finding the number of groups). More generally, what students will need is an understanding of the fundamental meanings of all four operations.

Here’s part of a task I presented to my secondary math colleagues:

Evaluate, or simplify, each set of expressions. Make as many connections as you can conceptually & procedurally, pictorially & symbolically.

fundamental meanings of all four operations

Sticking with division, in this task (−6) ÷ (+3) was chosen to bring to mind sharing (3 groups, −2 in each group) whereas 6/5 ÷ 3/5 was chosen to evoke measuring (3/5 in each group, 2 groups). (This often leads teachers themselves to revisit 6 ÷ 3.)


(−6) ÷ (+3) as sharing (top) and 6/5 ÷ 3/5 as measuring (bottom)

Flexibility is key. Consider (−6) ÷ (−3), 6/5 ÷ 3, 6 ÷ 0.3, 0.6 ÷ 3, 6x ÷ 3, 6x ÷ 3x, etc. (Note: division of fractions & integers are high school topics in Western Canada.)

(I’m not saying that dividing by a fraction — or decimal fraction — always means measuring. You can think sharing, which can be challenging. Andrew Stadel’s estimation jams are my favourite examples of this. How long is “All Along the Watchtower”?


Did you see 2/3 in the picture? Did you divide by two, then multiply by three? In other words, did you invert and multiply? What’s the meaning of 2:40 ÷ 2/3 in this context?)

The subtraction set above is interesting. Teachers pick up that the expressions are variations on a theme: five “take away” two. Their pictorial representations tend to show subtraction as removal: “if you have five apples/quarters/x‘s/square root of two’s and I take away two…”


Pictorial representations that show subtraction as comparison (the “difference”!) are less frequent, but maybe more helpful.


5/4 − 2/4 as removal (top number line) and comparison (bottom number line)

Consider (+5) − (−2). To “take away” negative two from positive five means introducing zero pairs whereas the “difference” between positive five and negative two means understanding that positive five is seven greater than negative two.

This second meaning is probably more meaningful in high school. For example, subtraction as removal reduces (1.89t + 15) − (1.49t + 12) to an exercise in collecting like terms whereas subtraction as comparison has students contrasting rates of change (e.g., cost per additional pizza topping) and initial amounts (e.g., the price of a plain cheese pizza). Similarly, if F_1(C) = 9/5C + 32 and F_2(C) = 2C + 30, then (F_2 − F_1)(C) compares conversions given by an estimate and the formula. When solving |x − 5| = 2, it’s more helpful to ask “What numbers differ from five by two?” than think missing minuends in take away problems.

Addition and multiplication — as well as other big ideas needed for high school such as proportional reasoning (or “multiplicative thinking”?) — will be addressed in future posts.

Cola Comparison

Coke is now sold in 20, not 24, packs!

Coke 20 (2)Coke 12 (2)Pepsi 24 (2)So to determine the best buy, I couldn’t just double. I use that strategy all the time; it’s my Frank’s RedHot. The exclamation point is there because I think that 20 leads to more strategies than 24. (Some of) these strategies are listed in my 5 Practices monitoring tool below. I’m curious if you think that I have anticipated likely student responses correctly. What incorrect strategy could I have anticipated? I wonder how you’d purposefully sequence these responses during the discussion.


More than SWBAT solve problems using unit rates, I want my students to recognize that there are many ways to solve rate problems and understand that we can easily compare rates with one term the same. This big ideas connects the strategies. In the fourth strategy above, we can think of 24 cans as a unit. Call it a “two-four” (Is that just a Canadian convention?) or a “flat” (Are we cool with calling the Pepsi cube a flat?). In fact, Save-On-Foods wants us to think of 24 as one; we’re encouraged to buy two packs of 12, a composed unit. For this task, I’d prefer that they didn’t, so I went back to the store and found this:

Cola 12

Comparing 20 packs with 15 packs is more likely to lead to common multiples than comparing 20 packs with 24 packs as above. Numbers matter. There’s this, but it doesn’t get us a clear winner:

Pepsi 15

Recommended: Dan knocks motivating unit rates out of the park; Christopher asks “What is one?”