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:
- signs (i.e., if f(x) is greater than zero, then 1/f(x) will be greater than zero)
- 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.
- How is the blue graph related to the red graph?
- Write an expression to represent this transformation.
- The point (5, 3) is on the graph of y = f(x).
What point must be on the graph of y = -f(x)?
- Have students work in groups of 3-4. Give each student in the group one of six cards.
- Have students record any observations they make about the graphs that are on their cards.
- Have students take turns sharing their observations. Encourage them to look for similarities and differences.
- 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.)
- Have students share their strategies with the class.
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.