Always, Sometimes, Never

This week should have been my first official – third unofficial – week back. Instead, I’m starting this school year as I ended the last – walking the picket line. I haven’t been up to blogging since this started. (No “Nearer, My God, to Thee” here.) Below is a draft from June. I never got around to finishing it. The ending has a “pack up your personal belongings” feel. I left it as-is; seems fitting that this post should come up short… I mean, 10% of my pay – and my colleague’s – was being deducted at the time. 

Recently, I invited myself to a colleague’s Math 8 class to try out Always, Sometimes, Never. In this formative assessment lesson – originally by Swan & Ridgway, I think – students classify statements as always, sometimes, or never true and explain their reasoning.


Because it’s June, we created a set of statements that spanned topics students encountered throughout the course. Mostly, this involved rephrasing questions from a textbook, Math Makes Sense 8, as well as from Marian Small’s More Good Questions, as statements. That, and stealing from Fawn Nguyen.

.doc .pdf

To introduce this activity, I displayed the following statement: When you add three consecutive numbers, your answer is a multiple of three.

Pairs of students began crunching numbers. “It works!”

“You’ve shown me it’s true for a few values. Is there a counterexample? What about negative numbers? Does it always work? How do you know? Convince me.”

Some students noticed that their calculators kept spitting out the middle number, e.g, (17 + 18 + 19)/3 = 18. This observation lead to a proof: take one away from the largest number, which is one more than the middle number, and give it to the smallest number, which is one less than the middle number; each number is now the same as the middle number; there are three of them. For example, 17 + 18 + 19 = (17 + 1) + 18 + (19 – 1) = 18 + 18 + 18 = 3(18).

I avoided explaining my proof: x + (x + 1) + (x + 2) = 3x + 3 = 3(x + 1). This may have been a missed opportunity to connect the two methods, but I didn’t want to send the message that my algebraic reasoning trumped their approach. “Convince me,” I said. And they did.

To encourage students to consider different types of examples, I displayed a ‘sometimes’ statement: When you divide a whole number by a fraction, the quotient is greater than the whole number. Students were quick to pick up on proper vs. improper fractions.

Next, students were given eight mathematical statements. We discussed some of the statements as a whole-class. Some highlights:

A whole number has an odd number of factors. It is a perfect square.

I called on a student who categorized the statement as always true because “not all of the factors are doubled.” We challenged doubled before she landed on square roots being their own factor pair. For example, 1 & 36, 2 & 18, 3 & 12, 4 & 9 are each counted as factors of 36, but 6 in 6 × 6 is counted only once.

The price of an item is decreased by 25%. After a couple of weeks, it is increased by 25%. The final price is the same as the original price.

Like the three consecutive numbers statement above, students began playing with numbers – an original price of $100 being the most popular choice. I anticipated this as well as the conceptual explanation that followed: “The percent of the increase is the same, but it’s of a smaller amount.” I love having students futz around with numbers; it’s so much more empowering than having them “complete the table.”

The number 25 was chosen carefully in hopes that some students might think fractions: (1 + 1/4)(1 − 1/4) = (5/4)(3/4) = 15/16. None did. There’s a connection to algebra here, too: (1 − x)(1 + x) = 1 − x². Again, I didn’t bring these up. Same reason as above.

One side of a right triangle is 5 cm and another side is 12 cm. The third side is 13 cm.

All but one pair of students classified this as always true. That somewhat surprised us. More surprising was how this one pair of students came to realize

BTW, the blog-less Tracy Zager has a crowd-sourced a set of elementary Always, Sometimes, Never statements.

Update: I stand corrected.

Math Picture Book Post #7: Bean Thirteen

A few ago, I was invited to teach a lesson on division (Grade 3). First, I read Bean Thirteen aloud – once just for fun. About Bean Thirteen, from the author:

Ralph warns Flora not to pick that thirteenth bean. Everyone knows it’s unlucky. Now that they’re stuck with it, how can they make it disappear? If they each eat half the beans, there’s still one left over. And if they invite a friend over, they each eat four beans, but there’s still one left over! And four friends could each eat three beans, but there’s still one left over! How will they escape the curse of Bean Thirteen?

(A funny story about beans, that may secretly be about . . . math!)

Bean Thirteen

Next, we revisited several of the pages. I asked students to write an equation to match the picture. I modelled this using magnetic “bean counters.” For the page above, students suggested 2 × 6 + 1 = 13; I introduced 13 ÷ 2 = 6 R 1. We discussed and recorded the meaning of this:

13 divided by 2

In pairs, students then chose their own number of beans (counters) and built different division as sharing stories for this number. They recorded (.doc) their stories using pictures, numbers, and words:



I called on students to share their stories with the class. They observed that some numbers gave remainders more so than others; Bean Thirteen can also be used to explore even/odd and prime/composite numbers.

This lesson served as the students’ introduction to division. I wrestled with the decision to introduce remainders at this time. An alternative problem – one consistent with both the prescribed learning outcomes and recommended learning resources – might be to start with 18 beans – a “nice” dividend – and share equally among 2, 3, 6, and 9 bugs – “nice” divisors. Note 15 ÷ 3 = 4 R 3 (and 15 ÷ 2 = 6 R 3) above. This mistake would not have happened had I not introduced remainders. I wonder if including remainders makes it more difficult for students to understand division and relate division to multiplication.

Then again, children will have already experienced remainders in everyday contexts.

What say you?

“Selfiest” Cities

Last week, I came across TIME’s ranking of the “selfiest” cities in the world and created the math task below.

Show the first slide–”sharing learning intentions” and all that.

The Selfiest Cities in the World.001

Display and discuss the selfies in Slides 2–8 to provide the context. Also, this will be helpful in the event that some don’t know what a selfie is. Hey, it happens. Each year I’d have to explain rock-paper-scissors to at least one of my Math 12 students (tree diagrams & experimental vs. theoretical probability & law of large numbers).

This slideshow requires JavaScript.

Show the map below and ask students what the data might represent. Given the previous discussion, I’m confident they’ll infer that each yellow dot represents one selfie taken in Anaheim. Yep, that’s Disneyland–Las Vegas for kids!

The Selfiest Cities in the World.009

Provide more details: each dot represents one Instagram photo taken during a 24-hour period tagged selfie that included geographic coordinates.

Put up this picture…

The Selfiest Cities in the World.010…which should lead to this question.

The Selfiest Cities in the World.011

Hand out cards for Anaheim, Milan, and six other cities to pairs of students.


Present the task.

The Selfiest Cities in the World.018

Call on students to share and justify their rankings. Note that there is (probably) more than one possible ranking.

(Slide 19 is just a placeholder. So is Slide 30, 41, and 43. These signal the end of one part of the task–and stop me from clicking through to reveal too much too soon.)

Provide more information: the number of selfies and the number of selfie-takers. Have students write this information on their cards.

The Selfiest Cities in the World.020

Ask students to revisit their rankings.

The Selfiest Cities in the World.029Call on students to answer the questions above.

Provide still more information: the population of each city. Again, ask students to revisit their rankings.

The Selfiest Cities in the World.031

Fingers crossed, some students will divide the number of selfies or number of selfie-takers by the population. Maybe ask, “Is it fair to compare Anaheim and Milan, knowing what you know about the two cities? What about Manhattan and Miami?”? Again, call on students to share and justify their rankings.

Using the data below, have students compare cities of interest or try on/poke holes in classmates’ measures of “selfie-ness.” Ask, “What’s London’s story?” and “What do the ‘selfiest’ cities have in common?”

The Selfiest Cities in the World.042

Share TIME’s ranking and methodology and have students critique the reasoning of others.

The Selfiest Cities in the World.045

Remember: the required Instagram-hashtag-GPS combination means TIME’s data accounts for a fraction of all selfies taken during a 24-hour period. Also, how many selfies taken in Anaheim (or Manhattan or Miami or San Francisco or Honolulu or Paris or …) were taken by tourists? Is it fair to divide by that city’s population?

For my first attempt, students ranked the data once, armed with all of the data–map, number of selfies/selfie-takers, and population. I wasn’t confident that students would use the population. Number of selfies per capita is a bigger leap than, say, party over business. Marc suggested the three successive rankings. I think this will work better.

The full slideshow:

.key .ppt

“They’ll Need It for High School” (Part 2)

So Part 2 was supposed to be about the big ideas in K-7 mathematics that students will need for high school. But that’ll have to wait for Part 3. Instead, more on times tables.

Three oft-used arguments for the importance of memorizing times tables:

  1. When learning higher levels of math, there just isn’t time to use calculators or strategies to determine basic facts.
  2. Besides, thinking taxes working memory which means by the time you’ve worked out the first part of the question, you will have forgotten the… Where am I?
  3. Because factoring.

1 & 2 are gospel. Well, so is 3; nevertheless, it’s the focus of this post. I have a couple of thoughts on times tables and factoring trinomials.

The reason some students struggle with factoring trinomials is not because they haven’t memorized products to 10 × 10. I can get away with this if we’re talkin’ Pythagoras. But factoring?! I mean, that’s all it is, right? To factor x² + 7x + 10, you just have to ask yourself, “What two numbers multiply to 10 and add to 7?”

HS math teachers, try this: give your students a quiz on factoring. Include both x² + 10x + 24 and x² + 25x + 24. Get back to me. For extra credit (yours, not theirs), throw x² + 6x + 5 in there. If your students are anything like mine, I bet x² + 25x + 24 gives them at least as much difficulty as x² + 10x + 24. What does this mean for these students? More practice multiplying by one?!

Of course, 1 × 24 falls outside most times tables. Recall of products to 10 × 10 gets us the factors of x² + bx + 60 – if b = 16. But x² + 17x + 60, x² + 19x + 60, x² + 23x + 60, and x² + 32x + 60 are fair game, right? Try c = 48. Or 72. Or 96. Or 100. What role does memorizing times tables play? What role does being flexible with numbers play?

My point, I think, is that these are different, albeit related, skills. In other words, the “it” they’ll need for factoring (trinomials) is factoring (numbers). And number sense. This has some implications for K-7: not necessarily more “What’s 4 × 6?” but more “A rectangle has an area of about 24 square units. What could its length and width be?” or even “The answer is 24. What’s the question?”; not thinking digits/standard algorithm but thinking – and talking! – factors/mental math strategies, e.g. 16 × 25 = (4 × 4) × 25 = 4 × (4 × 25) = 4 × 100 = 400 (via Sherry Parrish).

origami by @Mythagon nothing to do with post @k8nowak says put pictures in posts

origami by @Mythagon
nothing to do with post but @k8nowak says put pictures in posts

Say you’re still asking, “How am I supposed to teach them factoring when they don’t even know their multiplication facts?” When I introduced polynomial division in Math 10, some of my high school students didn’t even know long division. So I taught division of numbers and polynomials side-by-side, highlighting connections. Can the same miiindset (channeling my inner Leinwand) be applied to factoring trinomials and times tables?

And what about something like x² − 2x − 24? If that – asking yourself, “What two numbers multiply to -24 and add to -2?” – is all it is, why not factoring trinomials to teach multiplication (and addition) of integers?

Part One

[TMWYKS] Rainbow Loom

Christopher Danielson brought you #tmwyk, or talking math with your kids. I bring you #tmwyks, or talking math with your kid sister.

It happens to every parent, I think: the kid says something and nobody has to ask “Where’d she hear that?” Maybe it’s the kid’s choice of words. Or maybe it’s the tone, pitch, or rhythm that gives you away. Rare is it for me that these are proud parenting moments.

A recent exception:

Gwyneth (9 years old): What patterns do you see?

Rainbow Loom

Keira (6 years old): Red, white, yellow, red, white, yellow, red, white, yellow.

Gwyneth: Great! Can you find another pattern?

“They’ll Need It for High School” (Part 1)

“They’ll need it for high school.” I hear that. A lot. From elementary and secondary alike. I’ve been doing the K-12 Numeracy Helping Teacher thing (think “Math Coach”) for four years now. Previously, I taught Math 8 to 12. Twelve years. In Part 1, I’m going to look at math topics, teaching practices, and other things related to readiness where this phrase is used.

The Chestnuts

Long division and times tables.

Teaching long division may be the greatest time suck in all of elementary mathematics education. When I was new to this gig, I asked an intermediate teacher “Why the em⋅PHA⋅sis on long division?” “TNIFHS,” she answered. Having taught HS, her answer surprised me. A HS student will spend 5 years × 90 classes/year = 450 classes, give or take, in math. She will not need long division in 449 of them. HS math teachers, back me up here — one lesson: polynomial division. That’s it. Her turn to be surprised. But don’t blame her: this idea gets a lot of play in the media.

Over lunch at a recent pro-d workshop — the tortelloni was lovely — a mathematics professor from a local university complained that her Calculus students struggled with long division. How could she know? What’s long division got to do with Calculus? Finger Pointing 101.

This is not a call for scrapping the standard long division algorithm in K-7. We need more history of mathematics in math class, not less. Wanna argue dividing multi-digit dividends by multi-digit divisors without using technology is an important life skill? Fine. But don’t point to HS math.

“How can I teach them when they haven’t even memorized their times tables?” is my Groundhog Day conversation. Granted, recall of the multiplication facts is important. And overblown; it’s no silver bullet.

Worse still is “they need to quickly recall the basic facts for high school.” How fast? Faster. But “faster equals smarter” is not a productive belief for learning mathematics at any level. And we know Mad Minutes cause math anxiety. This bleeds into the next category…

Poor Pedagogy Preparation

“They’ll be lectured to at high school.” Often, this is an assumption, one many HS teachers I know take issue with. And, even if it is true, “I want to get them used to it” is not much of a defence. The same holds true of assessment and homework. Future poor practice should never be the reason for current poor practice. High school math teachers are guilty of making assumptions and justifications looking ahead, too.

They’ll Need High School Math for High School Math

Michael Pershan posted a few calculus readiness tests on his blog. One question jumped out at me:

Let f\left( x\right) =2x^{2}-2x. Simplify \dfrac {f\left( x+h\right)-f\left( x\right) } {h}.

If this isn’t calculus, it’s damn close. I can’t think of a conceptual context outside of calculus in which there’s a need for the difference quotient. (Compare this with what they’ll really need for calculus from Christopher Danielson’s NCTM session from a year ago.)

I wonder what this looks like at HS. Maybe SWBAT simplify \dfrac {-\left( -7\right) \pm \sqrt {\left( -7\right) ^{2}-4\left( 2\right) \left( 4\right) }} {2\left( 2\right) } as readiness for quadratics? I should stop, lest my HS brethren get any ideas.

This is silly, but it does illustrate one problem I have with TNIFHS: we meet students where they’re at, not where we want them to be.

The Affective Domain

So, what will they need? “Give me a student with a positive attitude towards mathematics, who’s persistent, who’s curious, etc. and she will be successful in high school,” I’ve answered in the past. I stand by this.

But there’s a problem with this answer. Implied in “they’ll need it for high school” is “they’ll need it before high school” (see times tables). I’ve met HS math teachers waiting for these curious, persistent students to one day show up at their classroom doors.

Another problem: there are big ideas, or enduring understandings, or key concepts, or whatever you want to call them, in mathematics that students will need for high school and this answer gives them short shrift. These will be discussed in Part 2.


I took this photo last summer.


Didn’t know what to do with it. Still don’t. Not enough there for a rich task. A warm-up?

My first question: Suppose Tim Horton’s offers the next size. How much should they charge?

First, students will identify a geometric sequence in the number of Timbit. The common ratio, r, is 2. The next size is an 80 pack.

Students will also need to think about unit prices. And ignore the price-ending-in-nine nonsense. The unit prices are 20¢, 18¢, 16¢. An arithmetic sequence! The common difference, d, is 2¢. The next unit price is 14¢.

Students will solve a problem that involves both — both! — a geometric and an arithmetic sequence. Rare in the textbook, rarer still in the real-world. Okay, this may excite math teachers more than their students.

My follow-up question: Suppose Tim Horton’s continues this pricing. How many Timbits should you get for free?