This activity, inspired by a similar activity in a Texas Instrument guidebook, has served to be very useful in helping students develop concepts of slope and y-intercept.
Before the Activity
This activity was inspired by an example in a Texas Instrument guidebook. I remember the activity, but I do not recall the name of the TI resource book that contained it. I have embellished this activity.
Before class, hand out index cards to each student. Each card should have "a =" and a number on it. I picked out a variety of decimals and fractions, between -8 and 8. I decided not to include zero, and I did not include any opposites, because of some of the activities that I do.
This activity assumes that students are just beginning to explore linear equations.
You can load the activity settings in YequalsAX.act. This is set up with a Standard [-10, 10] by [-10, 10] window and set up for receiving equations. Check the box to allow students to see their own graphs.
During the Activity
Begin the class and ask students to log in to the Activity Center. Don't project the image for the class yet. Tell students that the number on their index card is their "a" value.
Ask the students to type in the equation y = ax, where a is their "a" value. I showed my students how to submit their equation by using my "a" value of -1/3, since I wasn't sure if they would know how to enter fractional values, and I wanted to reinforce the - vs (-) comparison of calculator keys.
I then asked the class to predict what my equation would look like if I pressed the PLOT soft key. I asked them to predict what their own graph would look like, and then press PLOT to verify their conjecture.
I then asked the class to press SEND and send in their equations. I asked student volunteers to predict what the collection of everyone's graphs would look like. I turned on the projector and showed them the graph.
I then asked general questions like "Why do all of the graphs pass through the origin?" (Actually, a couple of them didn't. We looked at those equations, and saw that the students had forgotten the "x". It was a good introduction to talking about the equations of horizontal lines.)
Since we had talked about y = mx + b as an equation where "b" was the "begin" number, and "m" was the "move" number, I could highlight one line, ask about it's 'm' number, and then ask the students to predict the 'm' number for a nearby line. We developed the concept of a "bigger" m number meaning a "steeper" slope, and reinforced the concept of the difference between positive and negative slope.
I then asked my students to graph y = |a| x. I asked them to predict what the graph generated by the class would look like. We repeated with y = -|a| x.
After the Activity
There are many variations of this activity, such as working with y = 2x + a, and working with y = ax^2.
I changed the configuration of the Activity Center so students could submit two equations. I asked them to let y1 = ax, and y2 = -ax. I asked them to press PLOT; then I used the TI-Navigator™'s Screen Capture feature. Students enjoyed seeing all of the different graphs.
I have worked with y = ax + a. Again, have students predict what the graph will look like before they press PLOT. Then have them compare their plot with another students plot. Is there a point that the two lines both contain?
If everyone sent their graphs into the TI-Navigator system, what would it look like? Please SEND your graph now.
Why did all of the lines pass through (-1, 0)? Notice that we can plug in (-1, 0) into the general equation y = ax + a, and get a true equation. This is one of the benefits of algebra: We can prove that every equation in the room, even though they are all different, passes through the point (-1, 0).