A Math Teacher Trying to Have More Wags and Less Bark
Hey, it’s two posts in one day! And three in one week! What is happening here?!?!
Just wanted to point y’all to my new page of Tech Tips where I’ve listed all of my posts containing, um, you know….tech tips. Lots of goodies in there! And now you will always be able to find it at the top of my blog.
Happy typing!
Meg
Astute readers of my blog may remember my ramblings of the “flowchart method” last summer (and my use of it at the start of the year in Algebra II). After not focusing on it for quite a while, I brought it back for solving logs and exponentials, and it helped so much, all the way from struggling Algebra II students to PreAP Precal rock stars. Next year I really want to focus more on carrying it through all that we do, but baby steps first.
My introduction to composition of functions remained the same, next year I want them to really focus on writing the steps of each function in order. I changed up how I introduced inverse functions, but I’m not sure if it went better than the previous year or not. I need to work on melding the two together (I also think a couple more days for this unit would have been reallllly helpful, but Spring Break!).
I started by having them do the first row on the NTM below as a bellringer. Whoa, that’s weird, 3 and 4 have the same answer! And it’s what we started with! Then we did the next row, whoa, so 7 & 8 have an output of x! That means anything we put into it will come out the same!
(file here) (If you’re like my students, you may not get the ServPro reference: they are a disaster cleanup company with the tagline “Like it never even happened,” which became our tagline for inverse functions.) After explaining the joke, we worked on the chart, determining inverse functions and checking with whatever number they desired (We don’t make a big deal about 1:1 functions until Precal, although we did talk it about the next day a wee bit). And this is where building up the function machine the day before would have been super handy! Let’s just reverse the machine! (I also wanted to do Bob’s Inverse Function Partner Activity, but again, time!) Then the next day I felt I had to discuss some more properties of inverse functions, but again, not the greatest:
(file here) Trying to do too much at once, so we didn’t get to focus on the chart at the bottom: “Oh, so we’re really using inverse function machines when we solve equations!”
Then the next day we did exponential equations:
(file here) I really wanted to do the Zombies! Activity but, again, time! So at the end of the day we did #14 with our calculator, leaving the last two for homework. So much frustration! “Really, there’s no other way to solve these?”
Well, maybe there is….
(file here) Thanks to Kate for the fill-in-the-blank problems at the bottom. I would probably save the beginning part for the next day, when we are actually solving equations. But look at that glorious chart! Oh, so you’re telling me that logs and exponentials are inverses?
The next day is when we REALLY focused on the inverses idea:
(file here) I wish I would have had them write down the actual flowchart on these, though, just to reinforce the fact that log base 6 in the inverse of 6^x. As in, the number matters! You wouldn’t undo +3 by -4! So you wouldn’t undo 7^x by taking just log!
I also always have this conflict with myself, as illustrated here:
Teach “undoing a log” by converting to an exponential equation or as exponentiating both sides? I usually stick with option 1 in Alg II, then bring in option 2 in Precal. But maybe with the flowchart option 2 would make more sense?
Anyway, speaking of the flowchart, here is where it gets super useful:
(file here) Look how beautiful that is. No one thought to make #12 into 5 ln x! And it really focuses on ln and e being inverses of each other. We held off on the homework until we had some group whiteboard practice the following day, using slides like these:
(file here) Now, don’t get me wrong, we still struggled. We spent two days on the study guide (file here, with video key part I and part II) after this and I still had students try to undo a log by using a log. However, I also saw a lot of students that have been struggling do really well on this test–because they had a strategy they could use to attack each problem. (We also talked about how hard it is to intuitively feel like your answers is correct, so let’s use the calculator to plug it back in–this was complete news to many of them that they could have been doing that for any equation we solve!) And it wasn’t just the struggling students who were fans–I overheard one top-notch student say to another, “Hey, did you write the flowchart? It really helps!”
As I mentioned, I also used it in Precal with great success:
(File here) We also used it for actually solving equations, but I can’t seem to find that file! Doh! But hopefully you get the idea: flowchart it!
I’m sure this has never happened to you, my wonderful smart readers, but sometimes I read/hear something about math and outside I’m:
But inside I’m really:
Case in point: Glenn’s (@gwaddellnvhs) obsession about how everything is just made up of lines (It’s somehow comforting to know that Dylan struggled with this idea as well.).
Well guess what, y’all? NOW I GET IT!
It was all thanks to a presentation by John Abby Khalilian that I attended at the Alabama CTM Fall Forum, where he not only presented an activity that involved multiplying lines, but actually had us do the activity. And I was fortunate enough to be seated next to another attendee (but I don’t remember his name!) who was super excited about joining me in a conversation about extending it past the Algebra I level at which it was intended.
I modified the original file slightly (with permission from Dr Khalilian):
(File here) I urge you to work through the first page–I promise you this whole made-up-of-lines-thing will finally make sense! Note: I would make some modifications for next year. First, Dr Khalilian gave us two versions and I used the one we did in class. However, his first version used two positive-sloped lines, which I think would work better as the first example (Just use the lines from #10) Second, instead of giving them the equations of the lines in #10 and 11, I would have them already graphed and ask them to sketch the result?
Also, I know what you’re thinking, “Man, those are some Algebra I level questions on the first sheet.” I agree, and yet it was definitely a review that my Precal students needed.
After the worked on this, we then reviewed graphing polynomials, tying it together with the idea that we were just adding another line and what that would do. I was also made sure to focus on the fact that, “Oh, we have two negative lines and a positive, that would make it positive” because I knew what was upcoming:
Polynomial and Rational Inequalities.
Well, it went stunningly this year. All we had to talk about was the fact that sometimes drawing all those lines could be tedious and messy, when all we really care about are the places where a line is positive and negative. And if the line has a positive slope, it will be negative before its x-intercept and positive after. Like most math things, easier to see than to tell:
The red represents the sign of the red line, blue of the blue line, and black is the final result. So now sign charts are SUPER EASY. We don’t plug things in! Who wants to do that? MATH BABIES, that’s who. We are going to use our BRAINS instead:
note the two rows needed for 0 in #5, since it is a double root, meaning there were TWO lines that had zero as an intercept
(file here) Note: I started them on the first page with making sure that we always got rid of a negative leading coefficient (you have to “flip the sign”–get it??). This made it easier on the back when we had -x-5 hanging around; yes we could have pictured a negative line and write the chart as + – – but I felt like they would forget that one was negative. Easier just to take care of it than remember it! They all did fantastic with this on their test!
I know, could this whole “everything is lines” thing get any more exciting and awesome? Yes. Hold on to your hats, I’m going to change your Rational Function Graphing World.
Maybe your world use to be like this: Find your horizontal and vertical asymptotes (and holes) using these rules that I told you. Then make a t-chart of all the values on either side of the vertical asymptotes and plug in to find points to graph:
See, I hated doing the arithmetic so much I made it into a powerpoint. (But the powerpoint is pretty awesome for introducing horizontal asymptotes).
But then I used Kate’s wonderful introduction to rational functions worksheet that I always admired but never quite saw how it fit we how I graph rationals, but NOW I GET IT! It’s just like polynomials, except dividing! (modified only because I did not want to pay scribd to download it, plus one page, amirite?)
(File here) I’m going to modify some more for next year by (a) replacing trash panda (I have a serious problem where I can’t leave blank space) with the questions “what is the equation/zeros of f(x)/g(x)?” and (b) adding some more graphs, mostly stealing from being inspired by Sam’s worksheet. Instead we spent a few minutes the next day talking about what would happen if we were dividing a line by itself in order to lead into the idea of a hole.
So NOW! we can graph rationals THE SUPER EASY AWESOME WAY.
H: Find the horizontal aysmptote
F: Factor and find holes
Z: Find zeros of the numerator (x-intercepts) and denominator (vertical asymptotes)
S: Make a sign chart with all the zeros you found in the previous step and then graph!
When I asked the students to come up with a mnemonic, the most memorable was, “Horses Fart, Zebras Smell.”
So here it is in all its glory:
(file here, with more practice on page 2!) More modifications to make: Now that I don’t need all that space for a sign chart, I can add another example to the front! The one we did in class involved two factors in both the numerator and denominator. Also, I would change #2 so the horizontal asymptote isn’t 1/3! If you notice, in #1, I did the sign chart below the graph, but that got super confusing in #2 with everything so close together, so we started drawing it separately. We almost finished up the front page (including introducing horizontal asymptotes with the above-mentioned powerpoint) in one day, then did a couple more examples and practice on the back the following day. When we get back from spring break, we’ll do another day involving Weird Things That Can Happen like 1/x^2, but I think the sign chart is going to work nicely again!
I hope this helped you figure out the power of understanding the whole “everything is lines” idea and how it can carry through to so many things!
