Lesson 5 of 12
Objective: SWBAT evaluate the effect on a graph of replacing f(x), by f(x)+k, k f(x), f(kx), and f(x+k) for specific values of k using technology. SWBAT explain the effects described above using technology.
Set the Stage
Your students will need graphing calculators for this lesson - I have a set of TI 84s we use. I begin with this challenge question on the board, "What does the graph of f(x) = 2^x look like?" Most of my students will ask if they can use a graphing calculator to answer this and I tell them they can use whatever tools they want as long as they can identify key points to the graph and explain what they represent. (MP2, MP5) After a minute or two I ask for volunteers to identify key points and explain the graph. Acceptable responses would include saying that the graph curves up to the right, crosses the y-axis at (0, 1) and has a horizontal asymptote at x = 0. In addition I expect some comment about the f(x) values all being 2 to some power because that's what the original function is. It's surprising the number of students who haven't really made the connection between the function and the actual pairs of points on the graph, so I make sure to emphasize this aspect of what they're seeing.
The next challenge is for my students to use their calculator to try to duplicate a graph projected on the board of the same parent function [but they don't know that yet - simply translated]. For example I might use f(x)= 2^x - 3 making sure to label the y-intercept (0, -2) as well as a few additional points like (1, -1), (2, 1) and (3, 5). (MP4) I randomly select three or four students [because that's how many fit well] to post the equation they found on the board then have the class try these in their calculators to see which, if any, fit the challenge graph until we find one that works. (MP1) To conclude this activity I remind them of previous lessons where we've looked at transformations of functions and how those are reflected in the graphs, then ask what the parent function and transformation were for this problem.
Put It Into Action
My students like competition so I incorporate it into lessons on a regular basis. They will work in randomly assigned teams for today's competition. Each team has a mini-whiteboard to write their "final answer" on, but you can use plain paper. I've included a copy of the rules and problems, but encourage you to add to or adjust these to fit your students. After I explain the rules and check for understanding with a quick "fist-to-five", I'll do a ready-set-go style encouragement them to get their calculators and brains ready then I read the first problem. Students might ask me to post the equations on the board but I prefer to hone their listening skills* by just reading them. This also challenges and reinforces students' understanding of the rules for order of operations based on whether or not there are grouping symbols in a given problem. (MP6) I allow up to two minutes per problem giving students time to discuss and write on the whiteboard. When everyone is done or after two minutes I call time and have them to display their answers. (MP1, MP3) I am fortunate to have a teaching assistant recording team points on the board. We continue through the problems until completed or until about five minutes left in class. If we get through all the competition problems with more than ten minutes left, I ask each team to create a challenge problem for the rest of the class. (MP2)They write their problem on their whiteboard and I post all of these in the front of the room with the challenge to get as many correct as possible in the remaining time. I have enough mini-whiteboards for each team to post their problem at the front and also have a whiteboard for their answers. With about eight minutes remaining I call for final answers and we tally up points. This extension might also be used as a warm-up activity in the next lesson.
*Common Core includes K-12 ELA Speaking and Listening Standards
Wrap It Up
To wrap up this lesson I ask my students to pair-share what they think about using graphing calculators with a focus on when they are helpful and when they aren't. (MP5) I then ask each student to write a brief compare/contrast piece about when to use graphing calculators and when other tools or strategies are more appropriate and why. I tell them they can use examples but need to write out their ideas and explanations in complete sentences. If you want a more focused ticket-out-the-door you might have them respond to some specific questions like:
- How did the graphing calculator help you with this lesson?
- What things did you have to do without the graphing calculator?
- For what other kinds of problems would it be helpful to have a graphing calculator?
- What kinds of problems are easier to solve without a graphing calculator?