Part 1: The topic I chose was horizontal and vertical shifts of functions. I think it’s important to really take some time with this topic so that students get really comfortable with graphs of common functions and how algebraic transformations are related to graphical transformations. I used WolframAlpha to graph the familiar function f(x)=x^2 on the same graph as f(x)=x^2-2 and f(x)=(x-2)^2. The results can be found here.
My initial graph showed x and y values that I thought were a little strange so I had to play around with the syntax so that it displayed a graph with a domain and range that give a better picture of the graphs. I thought this was a good example of how it is important to use technology to enhance understanding about graphs of functions and not as a replacement for knowing the shapes of common functions, learning how to find intercepts by hand, etc. If students used WolframAlpha to graph functions but did not have the basic understanding of what a quadratic (or cubic, square root, etc.) function should look like, they would not know if a particular set of x and y values on the graph was giving an inaccurate picture of the function.
Part 2: I found this video that discusses vertical and horizontal transformations. My comment: I really like how you approached horizontal shifts. A lot of the videos I've watched on this topic just gloss over it as being "backwards" but you provided a nice explanation.
Part 3: I found this interactive graph that would be useful for this topic. Although it has the limitation of only being able to graph two functions at once, I thought it had some user-friendly features that made it easier to use than WolframAlpha. For example, you can easily shift the viewing window around by re-centering and zooming in and out, and you can get back to the basic window with the click of a single button. There are also some clearly written instructions directly underneath the applet that make it easy to use. Like WolframAlpha, you can save and copy a url with a specific graph. Here is a graph showing f(x)=x^2 and f(x)=(x+2)^2.
Part 4: Here are two higher order questions about this topic:
Why is it important to understand the basic shapes and features of functions when we have so many ways to graph using technology?
Why does the graph of f(x)+a move the graph of f(x) in the positive direction for positive values of a, and the graph of f(x+a) move f(x) in the negative direction for positive values of a?
Part 5: I talked about question 1 in part 1 above, and I think this is something that students could discover for themselves very easily with the help of technology. For example, I could give them functions to work with that give an incomplete picture of the function if graphed on a standard viewing window. Rather than me telling them why it’s important, they could discover for themselves that it’s important to know the graphical features of basic functions. Graphing tools such as a graphing calculator or the interactive tool I linked above would be good for this task. Although now that I’m thinking about it, WolframAlpha or graphing calculators might be a better choice for this task than the applet initially since they both require the user to input the graphing window, while the applet makes it easy for students to randomly click around and zoom in and out. I think it’s a great user-friendly feature to have, but it might make it too easy to see the whole graph without understanding the graphs conceptually!
With question 2, I feel like this is one of those things that is often glossed over so students end up memorizing that one transformation shifts the graph the “logical” way and the other is “backwards”. As I was browsing videos, I saw several that just kind of handwaved it away as an exception or counterintuitive, and I think that does a disservice to the topic because I believe it does make sense if you approach it the right way. I came across this blog post that has tips on how to present this topic to students, and I really like how it shifts (ha!) the focus to the intercepts. I would approach the idea of horizontal shifts as shifting x-intercepts, x-intercepts occur when y=0, what happens to the equation of the function when y is 0, and how do we solve for the x values (and the same idea for vertical shifts as shifting y-intercepts). I think the result becomes more intuitive if students do the algebraic manipulation to find intercepts to reinforce how when x is 0, the y-intercept is the constant term, and when y is 0, the constant term has to be moved to the other side of the equation to solve for x, and compare that result to the graphical shifts. For this question, I think that graphing tools would be a great enhancement to the material, but I think this is one of those times when students need to manipulate the equations algebraically by hand to get to the point where the direction of horizontal shifts is intuitive rather than “backwards”.
Part 6: I’ve been thinking about the difference between computer based and computer delivered math since our discussion on Wednesday and trying to get my head around the differences. Based on my understanding, I would say that the video in part 2 is computer delivered since the computer is only using technology to make the lesson more widely accessible. There is nothing in that lesson that can’t be taught with an old-fashioned chalkboard. My example in part 3 would be computer based, since the learning comes from using the mathematical capabilities of the computer: graphing functions, shifting viewing windows, etc.
After thinking my way through the two questions I discussed in part 5, I feel like even though my suggestions for how to use technology is similar for both questions (graphing functions) the first one is computer based and the second one is computer delivered. For my suggestions for the first question, the computer is where the learning is happening because students are using technology to manipulate graphical models. For my suggestions for the second question, the learning comes from the algebraic manipulation, which is outside the computer, and the technology aspect comes in as a way to quickly verify results. So for the first, the computer is where the significant learning happens, but for the second, the computer is used as a convenient way to check answers.