About Chris Hunter

K-12 Numeracy Helping Teacher School District No. 36 (Surrey)

Timbits

I took this photo last summer.

Timbits

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.

4-7-8

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.

0001sW

Homework

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.

SDEC1-4124-13111414440

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

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

SDEC1-4124-13111414450

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.

SDEC1-4124-13111414480

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

Previous Math Picture Book Posts: 1 2 3 4

Math Scavenger Hunt

Yesterday’s post reminded me of an activity we created for a recent department head meeting/pro-d workshop.

In pairs, DHs were asked to take a photo (with an iPad — sorry, Timon) of each of the following:

  • a perfect square
  • the use of a referent to determine the linear measure of an object
  • a positive & negative slope
  • a non-linear relation
  • an irrational number
  • similar 2-D shapes or 3-D objects
  • angles formed by parallel lines and a transversal
  • a contextual problem that involves the sine law or cosine law
  • a z-score of ±2
  • elements in the complement, the intersect, or the union of two sets
  • a stranger engaged in math

Math Scavenger Hunt (Secondary)

A fun break from a wrapping our heads around a transformed competency-based curriculum.

In my classroom, I’d probably prefer a more narrow focus — a specific concept over a general math activity. For homework, have students take a photo of parallel lines and a transversal. In class, ask What do you notice? Christopher Danielson’s students — future elementary teachers — were asked to photograph a composed unit, which led to a lovely classroom discussion. Dan Meyer kicks the find a positive & negative slope challenge up a notch by holding a steepest stairs competition.

Any other ideas for challenges/activities?

DEC Stairs

Another DEC photo

Sort of Another Sort

The week before I was asking students in Pre-Calculus 12 to sort trig functions, I was asking students in Grade 6 to sort triangles.

I adapted a textbook task. One of these days, I’m going to finish my post “In Defence of Textbooks. Kinda.”

To get ready, I gave each pair of students a handful of SET cards to sort. Students shared their sorting rules. “There are many ways to sort” was the message. Did this in Pre-Calculus 12, too, by the way.

Next, I gave each pair of students this blackline master:

I asked students to sort the eight triangles into groups. That’s it. About two minutes later some asked, “Can we have rulers?” I gave some rulers.

I called on students to share their sorting rules. (The textbook just tells students to sort the triangles by the number of equal sides.) Order matters. The first pair of students classifed the triangles as small, medium, or large. This closely matched the groups made by the second pair who measured the perimeter of each triangle: something like, “shorter than x centimetres, between x and y centimetres, and longer than y centimetres.” The third pair sorted the triangles based on the length of the longest side. This set up the fourth pair who noticed that, for some triangles, the lengths of two or three sides were equal.

Then, and only then, I defined the terms scalene, isosceles, and equilateral.

In this post, Patrick Vennebush has his sons define arithmetic progression by giving them examples and non-examples.

Compare either approach with this. (Read the comments: Bowman nails it.)

After, students were sent to the hallways, library, gym, and playground with their iPads to take/make a photo of a each type of triangle. Students returned to share their favourite scalene, isosceles, and equilateral triangles through the Apple TV. This led to some fun conversations.

I didn’t collect students’ photos. My photos at DEC reception instead:

Sinusoidal Sort

On Monday, I was invited to Sandra Crawford’s Pre-Calculus 12 classes to try out an activity we created together. Thanks, Sandra!

Sandra’s students were familiar with how transformations of functions affect graphs and their related equations. They’ve stretched & shrunk (vertically & horizontally), flipped (in the x-axis & in the y-axis), & slid (up, down, left, & right) linear (& piecewise linear), quadratic, absolute value, reciprocal, & radical functions. These were topics in prior units. In this unit, students were previously introduced to radian measure, the unit circle, the six trig ratios, & the functions y = sin x, y = cos x, & y = tan x. Next up: determining how varying the values of a, b, c, & d affect the graphs of y = a sin b(x – c) + d & y = a cos b(x – c) + d.

Such was the case when I last taught trig functions (in Principles of Math 12). Back then, my approach was to provide clear and concise explanations, connecting these transformations to those transformations (or, better, transformations of these to transformations of those). But was this necessary? Shouldn’t students be able to make this connection? On. Their. Own.

In small groups, students were handed a set of equation cards to sort and were asked to explain their sorting rule. We designed the equations so that there were plenty of similarities and differences in terms of whether or not there were leading coefficients, coefficients of x, brackets, etc., as well as in terms of the values of a, b, c, & d themselves. After all that, most groups just sorted the equations into sine and cosine functions — to be expected, I guess, given the focus of the prior lesson.

sort1

Next, students were handed graph cards and were asked to match each to the corresponding equation card. We encouraged students to make predictions, then test these predictions using technology. Interestingly, few reached for their graphing calculators or phones. We asked students if, having seen the equations and their graphs together, they wanted to re-sort.

sort2

This process was repeated with characteristic cards. Note: The terms amplitude and period were introduced the lesson before; phase shift and vertical displacement were not. Hence, horizontal translational and vertical translation at this stage of the lesson.

For the most part, students were communicating and reasoning mathematically, making connections, and problem solving. They were engaged with mathematics. A minority probably would have preferred to be engaged with taking notes.

Groups shared their sorts the following day. In the end, the functions were sorted in a variety of ways, which allowed Sandra to highlight each transformation.

sort3

A few groups struggled with matching all of the cards. Therefore, I reduced the number of functions. If finished, some students could be given two additional functions. Each of these is actually a phase shift of one of the initial eight (e.g., y = cos x + 2 ↔ y = sin (x + 90°) + 2). I wonder what they’d do with that.

Sinusoidal Sort (doc)
Sinusoidal Sort (pdf)

(Note: I’ve triple-checked these. Still, no guarantees.)