by Matt Owen, Algebra 1 and Physics Teacher at Lusher Charter School, New Orleans, LA
Here are just a few of the ways I’ve used Desmos in my class along with the CCSS Math standards that apply.
1) Recognizing that graphing the functions f(x) = 3x - 4 and g(x) = 6 gives you the solution to 3x - 4 = 6. I used to think that this was a waste of time, since solving this algebraically seems easier. But having the resource of Desmos there (that kids like using and see the value in), makes it simple for them to see the connection. Then when we go to solve |x-3| = 7, they can find the solution graphically first and work up to an algebraic solution. Same goes for exponential, quadratic, anything! I never placed much emphasize on graphical solutions until we starting using Desmos regularly. But it’s even a CCSS! Check it:
A-REI.11 states: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using DESMOS to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
(I may have made a small edit there) ;)
2) Finding a fit line and then adjusting a fit line (Are you doing Barbie Bungee this year? Me too!) This one is so easy to do with Desmos. The one I did last year was ok; I had the kids measure their arm length (from elbow to fingertip) and their foot length (no shoes, heel to big toe). Once each group was done measuring, they came up and entered their data in Desmos on my computer which was projected on the board. It looked like this. Then we discussed the best model to use here (linear was easy to agree on) and used that. Then it looked like this. So then I just shared that graph with every group (I used email to send it out, but Google drive may be an easier way to share), and had them write the equation of the fit line as they saw it. The great part here is that each group comes up with a different equation. Then, I had each group make a prediction about my arm length given my foot length and we could see who was closest. It’s a nice way to add a little friendly competition and get to discuss why the best prediction was the best. This is also a good place to discuss the meaning of slope and y-intercept. Should the y-intercept be zero? Does the slope have units? There’s really a bunch of CCSS that go with this one, but mainly I was thinking:
S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
Informally assess the fit of a function by plotting and analyzing residuals.
Fit a linear function for a scatter plot that suggests a linear association.
3) Exploring function graphs. Check out Building Functions 3:
F-BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
OMG! That’s so complicated sounding! But if you have students build on a parent function, they’ll start to be able to tell the difference between f(x) + k and f(x+k) even if that’s not how they’d explain it. Here’s the handout I used to do vertex form quadratics last year (needs a little work, but I think you’ll get the idea). Once they’d built up some rules for how to shift the graph around, I could show them the general vertex form graph and they could make some predictions about what the a, h, and k would do. Bam! Function built!! (of course that’s not the end of our quadratics explorations – it’s the beginning!)
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