Transformations of Functions (Lesson 1.5 Day 2)
Unit 1 - Day 6
Explore how the order in which transformations occur affects the final graph
Reason abstractly and quantitatively
Quick Lesson Plan
Today’s lesson is a rich task that is designed to solidify and deepen students’ understanding of transformations of functions. It is based off of an NRICH task titled “Surprising Transformations.” Students explore how the sequence of transformations affects the final result, connecting graphical, analytical, and symbolic representations of functions.
To prepare for this lesson, it is important to anticipate student responses and strategies. This will help you ask more purposeful questions as you monitor groups. We use Peg Smith and Margaret Stein’s 5 Practices for Orchestrating Productive Discussions as a framework for facilitating this lesson.
In this low-floor, high-ceiling task, all students should find an entry point into the activity. Students might simply try performing the four transformations in various orders to see if they arrive at the final result. When students adjust their approach based on the results of what they tried, we consider this a strategic guess and check approach. The individual think time is critical for allowing students to reason about the problem before consulting with other people. Less confident students tend to simply adopt the ideas of their peers if they are not given the space and direction to use their own strategies and sense-making first.
As the activity progresses, students should start to notice patterns about which of the transformations are more “order-sensitive”. Perhaps the easiest way to see this is to compare the result of reflecting, then shifting vertically with shifting vertically, and then reflecting. Offering graph paper for students to make visual and graphical connections to their algebraic equations is a great idea.
When monitoring students, be listening for groups that can articulate key ideas, for example, why the horizontal shift can occur at any point because it is the only transformation that affects the independent variable. Some students may have ideas around why reflections and stretches are similar in that they both include multiplying by a scalar, but in this case the order in which they are applied does matter! Had they been consecutive transformations, then the order would not matter. Allow students plenty of time to make conjectures and test them.
When selecting and sequencing students responses for the discussion, one strategy is to show less “clever” and yet highly accessible strategies like guess and check first, before sharing less common strategies that make deeper connections. Be sure to demonstrate curiosity about student ideas rather than offering opinions about which strategy is better or more sophisticated.
There are many options for differentiating this task. To decrease the complexity, you can offer only three transformations and adjust the resulting quadratic appropriately. You might consider eliminating either the vertical stretch or the reflection across the x-axis. You could also change the original function to be the parent function, f(x)=x^2 or use a linear function, much like the original task. To increase the complexity, you can rewrite the resulting quadratic in standard form.
For students that are ready for an extension, have them create their own transformation mix-up by creating an original function, applying a set of transformations to produce a resultant equation, and then jumbling the transformations and having a partner figure out the order in which the transformations occurred.