This was my go-to review activity. I picked it up at an un-unconference as a student teacher.
First, have students get in groups of four. This is their home group. Have students number themselves from one to four.
Have students move and form groups so that each student in the group has the same number. This is their expert group. Each expert group is responsible for one part of a review assignment, such as this. For example, the 1’s (Adele, Ellen, Lea, and Oprah) may be responsible for becoming experts on solving quadratic equations by factoring, the 2’s on solving using the square root method, the 3’s on solving using the quadratic formula, and the 4’s on the nature of the roots. Emphasize that each member of the group must understand and be able to explain the solution to each question in this part of the assignment. I play up that I will only help students while they are in their expert groups.
(Classroom Management Tips: Ask just the first four home groups to move and form their expert groups. Have the remaining home groups remain seated until this is complete. You will be able to see if each student is moving to the correct group. I’ve used this activity in classes of 24 to 32 students. Plan for this. For example, 12 students will form three home groups of three and will move to form four expert groups of three.)
Have students return to their home groups to complete the assignment. If any student needs help with any question, he or she is sitting with an expert. For example,
Ashton needs help factoring when the leading coefficient is not equal to one?
Adele’s an expert.
Barack has difficulty using the square root method when there are brackets?
Beyonce struggles with simplifying expressions when using the quadratic formula?
Adele can’t remember which condition results in two equal real roots?
Beyonce can help.
I may have gone to the well one too many times with this as a review activity. Time to try Kate Nowak’s speed dating activity. Also, I’d like to use expert groups to have students learn, rather than review, concepts. I’ve used this activity, with some success, to teach exponent laws in Math 9.
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.