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?

Math Picture Book Post #6: Fika

For fans of arrays (and those with OCD), there’s much to like about Fika, the Ikea cookbook. Each recipe spans two pages: the ingredients on the first, the finished product on the second.

A sample:

Fika 1

Fika 2

My daughters and I have been talking skip counting, equal grouping, repeated addition, arrays, multiplication, etc. “How many? How do you know?”

We got in on the act:

Cookies 1

Cookies 2

Our “family recipe”

Pythagorean Exploration

I don’t love this textbook task.

Too many substeps before students return to the question: what’s the relationship between the length of the sides of a right triangle?

“For each right triangle, write an addition statement…”? C’mon!

But I’m hesitant to join the down with textbooks revolution; I don’t want to associate myself with the back to basics movement. So in conversations where the suggested alternative is more worked examples, I soften my criticism.

Besides, it gives me something to modify. Instead of completing the table, I could challenge students to find right triangles and ask “What do you notice?”

One problem: this requires “attend to precision” to do some heavy lifting.


The 4-7-8 Right Triangle

This leads to some truly awkward feedback: “Are you sure it’s a right triangle? You might want to measure again.”

GeoGebra may provide a solution.

4-7-8 GGB
Click to view on GeoGebraTube

Pythagorean Mistakes

Consider the math mistakes below. Not real samples of student work (for that, go here), but real mistakes. I’ve seen each one. I think you’ll recognize them.



Answer questions 1 and 2.

1. What math mistake did each student make?

2. What are some implications for our work?

Good. Now answer questions 3 and 4.

3. What role did memorization of the times table play?

4. What are some implications for the conversations we could be having?

[Misleading Graph] Peyton Manning vs. Russell Wilson

Does the graph create the impression that Peyton Manning has about 10 times as many pass attempts as Russell Wilson?

What can you do with this?

One approach would be to show students the graph and ask how this visual representation could be misleading. Point to the sizes of the circles.

A different approach could be to remove information (and add perplexity). Show them this:

PMvsRW (w: perplexity)Have students estimate Peyton Manning’s career pass attempts. I’m anticating many students will compare the sizes of the circles. They’ll think about how many green circles could fit in the orange circle. They may not think 100, but I’m confident they’ll think much more than 10. They may have other strategies. Have students share them.

Give students rulers (and the formula A = πr² if they ask for it). Ask them if they’d like to revise their estimate.

Reveal this:

PMvsRW (w:o perplexity)Were students misled? I’m anticipating some will compare the diameters. Take advantage of that. If not, challenge them to find out why the circles are the sizes they are.

Given Manning’s circle, have students draw Wilson’s circle to the correct size. Again, have students share strategies.

(I’ve created this applet in GeoGebra. Not sure what, if anything, it gets me.)

Screen shot 2014-01-29 at 11.41.35 AM

Allowing students to possibly be misled by a misleading graph… should’ve thought of that earlier.

I don’t think @ESPNStatsInfo is trying to suggest a much wider experience gap. Seahawks fans may disagree, but the tweet backs me up. This is accidental: the result of focussing on graphic, not info, in infographic.

World’s Worst Person In Sports

Last week, Keith Olbermann named the Canucks’ Tom Sestito “World’s Worst Person In Sports.” In a game against the Kings, Sestito racked up 27 penalty minutes. His total ice time for the night? One second.

27:00 to 0:01 is an impressive stat. It’s hard to imagine this being surpassed. Sure, twenty-seven minutes can be topped. Randy Holt holds the NHL record for most penalty minutes in one game (67). The NHL record for most penalties in one game (10) belongs to Chris Nilan. But to do so in one second?! Inconceivable.

“I’d describe [Sestito] as a hockey player except he’s not,” Olbermann says. To make this point, he goes on to compare Sestito to Gretzky. That’s right: “The Great One” is his hockey player/”boxing hobo on skates” referent. In 101 games, Sestito had scored 9 goals, 885 shy of Gretzky’s record. Olbermann notes that Sestito would have to play about 10 000 games, or 123 seasons, to break the NHL record. Well, yeah, assuming he can keep up this pace.

I considered giving this the three-act treatment and bleeping Olbermann. But “When will Sestito break Gretzky’s record?” is not the first question that comes to your mind, is it? A more natural question re: Sestito might be “How many seasons would Sestito have to play to break Dave “Tiger” Williams’ record of 3966 career PIMs?” Apples to apples.

Olbermann, 54, followed this up by feuding with Tom Sestito’s sister, 13, on Twitter. Nice use of a unit rate by the kid:

Math Picture Book Post #5: 100 Snowmen

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. Here, students can double five and add one more to determine five plus six.


5 + 6 = (5 + 5) + 1 = 11

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


5 + 4 = (4 + 4) + 1 = 9

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?”

Doubles Plus One Cards

To practice this strategy, students can play a game.

Taking turns:

  • Roll a ten-sided die
  • Build the number
  • Build one more than the number
  • Cover the sum with a transparent counter

The first player to cover all of the sums wins.

Doubles Plus One Game

Snowmen Doubles Plus One

On the last page, every single snowmen is added.


This suggests a different mental math strategy: making tens.

doubles plus one

(1 + 2) + (3 + 4) + (5 + 6) + (7 + 8) + (9 + 10) + (9 + 8) + (7 + 6) + (5 + 4) + (3 + 2) + 1

make tens

(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + (5 + 5) + (6 + 4) + (7 + 3) + (8 + 2) + (9 + 1) + 10

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