## 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. 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. 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. 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.

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

## Mystery Transformation

A Principles of Math 12 learning outcome states “It is expected that students will describe and sketch 1/f(x) using the graph and/or the equation of f(x)”. In my classes, students were more likely to ask ‘the question’ at the beginning of this lesson than during any other lesson.

Over the years, I simplified my explanation. Three steps:

1. asymptotes
2. signs (i.e., if f(x) is greater than zero, then 1/f(x) will be greater than zero)
3. invariant points (and other important points)

Most of my students were able to successfully sketch reciprocal functions. I had successfully prepared them for the provincial exam. Still, I wasn’t satisfied with this lesson. My students weren’t learning mathematics, they were just following directions – following my steps. Together, Marc Garneau and I created the activity below, probably inspired by this book.

Warm-up: 1. How is the blue graph related to the red graph?
2. Write an expression to represent this transformation.
3. The point (5, 3) is on the graph of y = f(x).
What point must be on the graph of y = -f(x)?

Activity:

1. Have students work in groups of 3-4. Give each student in the group one of six cards.
2. Have students record any observations they make about the graphs that are on their cards.
3. Have students take turns sharing their observations. Encourage them to look for similarities and differences.
4. Ask students to describe, in words, how the blue graph is a transformation of the red graph. Ask students to write an expression that represents this transformation. (The extra 2-3 cards can be used to test and confirm ideas.)
5. Have students share their strategies with the class.

Discussion:

Sunita Punj invited us into her class to try out this activity. (Thanks again, Sunita!) Her students made some key observations, such as:

• “There’s an asymptote at the x-intercepts”
• “When the y is 1 or -1, it stays the same.”
• “Points on the red graph that are 2 spaces up become points on the blue graph that are 0.5 spaces up.”

Sunita’s students were then able to use this information to discover that the blue graph could be obtained from the red graph by taking the reciprocal of the y value. I enjoyed listening to, and participating in, the mathematical conversations that were happening at each table.

• “Is that always true?”
• “I have a theory…”
• “But why don’t the blue graphs touch the dotted lines?”

Students were not simply following directions. Nobody asked ‘the question’. (Still, if there is a real-world application here, I’d love to learn about it.)

There may be limited opportunities in Math 8-12 to have students identify a ‘mystery transformation’. However, I think it’s worth exploring the bigger idea – giving students questions and answers and then asking them to talk about how the answers may have been determined.