Building Quadratic Functions: f(x), kf(x) and f(kx)
Lesson 6 of 13
Objective: SWBAT use technology to identify the effect on a graph of replacing f(x) with f(x) + k, k*f(x), and f(k*x).
This Entry Ticket- Manipulating Quadratics ties in directly to the standard IF-7a as I ask students to graph a quadratic function. This entry ticket task reinforces the ideas and skills around graphing quadratics covered in a previous lesson, and also sets up the beginning function we will be using to manipulate and change in different ways with the constant k.
Before beginning this lesson it will be important to determine the best way/place to give students access to a graphing calculator. I personally like having students use the website fooplot.com because we use a computer lab and provides a change of scenery for the class. The benefit of using the computer lab also is students get to use a larger screen that might make the ways the graphs change clearer over using a graphing calculator with a smaller screen. Either way, the activity works with any form of a graphing calculator.
The Activity: Manipulating Quadratics: f(x) + k, kf(x) and f(kx) of this lesson is having students work in pairs to complete an investigation into the effect of changing f(x) in different ways. Technology is a helpful friend in this lesson - I like to use either the TI NSpire app for ipads in my class or the Math Open Reference Quadratic Explorer (an excellent and clean applet, that allows students to change values of a, b and c for quadratics in standard form).
I typically think of this section as three mini-parts, with each part consisting of a 20 minute working period for students to grapple with creating different functions and generalizing their results. To begin students graph f(x) + k for different versions of k and record their results. After graphing a number of situations each pair of students must discuss and agree with one another on a general description of what happen to f(x) when we add a constant.
The process is then repeated for two other manipulations of f(x), namely kf(x) and f(kx).
After students complete the activity we have a class wide discussion where I give each group of students a chance to share out their thoughts (I typically go through each of the three examples and have at least a few pairs of students share so that by the end of reviewing all three examples each and every student has had a chance to contribute to the class conversation). We then shift to the exit ticket.
To close this lesson, I have each group complete a the Exit Ticket: Manipulating Quadratics: f(x) + k, kf(x), and f(kx)
I ask students to focus on the area of providing examples/evidence to support their ideas, as I am trying to focus on explicitly teaching, and providing practice opportunities, for one of the five main skills of academic conversations for each unit. During this time, students each have a copy of the graphic organizer and work on completing the template using their work from the class activity. One way to differentiate this section is to allow each group to generate one Idea Organizer to take the fine motor demands off of some students. Students could then take a picture of the group’s Idea Organizer so they had access to it at home and still be able to complete the homework for the lesson.
The prompt for the exit ticket is: Compare and Contrast the effect of the different ways we changed f(x) with k. The three ways were 1. Adding a constant: f(x) + k, 2. Multiplying the function by a constant kf(x), and 2. Multiplying x by a constant: f(kx).
With 5 minutes before the end of the lesson, I assess where the class as a whole is on the Idea Organizer and assign homework as completing the graphic organizer if they are not close to finishing it. If students have, for the most part, completed the task, then for homework I ask students to write a 1-2 paragraph written response based on their work. That way I am giving students a chance to practice their writing skills and also review the concepts from class with a high level of support. This exit ticket ties directly into the Math Practice standards of MP3 because I am asking students to develop arguments and also understand and integrate the perspective of their classmates to come to a deeper understanding of the concepts covered in class today.