Tag Archives: inverse functions

The Flowchart Method: Learn It, Love it, Log it.

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!

from megcraig.org

(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:

from megcraig.org

(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:

from megcraig.org

(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….

from megcraig.org

(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:

from megcraig.org

(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:

from megcraig.org

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:

from megcraig.org

(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:

from megcraig.orgFlowchart 10

(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:

from megcraig.org

(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!

Algebra II Files: Functions & Radicals

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News flash: I secretly love making math powerpoints. I need to find a job that is just making them (and NoteTakerMakers) all day. Or maybe half a day because, ok, it would probably get old after a while. But for now, enjoy the bounty of my obsession.

Our textbook starts the radicals chapter by doing composition and inverse of functions, so that’s where I start as well:

Function Files from megcraig.org

(file here).  I have to spotlight my two favorite slides:

Function Files from megcraig.org

Yes, “pig squared” gets a laugh every time.

Function Files from megcraig.orgFunny story: On one of my student’s review of a Vi Hart video, the student said that Vi talked about doing some operation with dolphins. The student said that didn’t bother her because “my teacher does math with corgis and unicorns.” Awesome!

And yes, there is an NTM to go with it:

Function Files from megcraig.org(file here) For the past [redacted] years, I’ve always worked from the inside out on functions, then I had an epiphany last year…try working from the outside in!  For example, on that first problem, it’s j(h(1/2)).  Let’s start with j, which is 6x, but we know we’re going to replace x, so we’ll write 6(              ). What are we filling that with? Oh, h! So now we have 6(2(     )+5) and what do we want to put in there? oh, 1/2! 6(2(1/2)+5)!  I found it really helpful for when there’s more than one x that you have to plug in for, like #4.  Anyway, just thought I’d mention it since it’d hard to tell what order I’m doing things on the key. We need magical time-telling paper. Get on that, people.

Here’s the homework (I found that finding function values from a chart or graph is something that Precal students struggled with, so I tried to add some practice)

function files from megcraig.org(file here)

Ok, now inverses!

Function Files from megcraig.org(file here) To find out the 5 things to know about inverses, you’ll have to view the powerpoint (clickbait!):

Function Files from megcraig.org(file here)  Let’s zoom in on my favorite question from the homework that was posted above:

Function Files from megcraig.org(although I guess I should make it a 1:1 function?) Discussing this problem the next day is a great way to reinforce the idea of inverses!

Then it’s time to graph some radicals:

Radical files from megcraig.org(file here) and review for a quiz:

Radical files from megcraig.org(file here)

Then it’s time for the phrase that strikes fear in teachers, students, puppies and unicorns: EXPONENT RULES.

Radical Files from megcraig.org(file here) To clarify some stuff, the PMA/RDS at the top is from someone in the MTBoS. Exponent rules follow the pattern of doing operation “below” it: power means you multiply, multiply means you add, and you can’t do anything with addition since there is not a function lower than it.  Then the same thing is true for roots/division/subtraction. I really wish I could find the original post because that person explained it a lot better than I can right now.

If you’re not aware of the Dead Puppy Theorem, go visit Bowman immediately!  I made my own corollary which is “Every time you say a negative exponent makes the number negative, a unicorn dies.”  “But Ms Craig, there’s not any unicorns left!”  “EXACTLY.  That’s how many students have made this mistake.  There are actually 4 of them left in a secluded meadow in Ireland; it is up to you to make sure they do not go extinct.”

Homework that we do in class:

Radical Files from megcraig.org(file here) I obviously typed this right after reading a tweet about allowing students to make choices in problems to do.  It actually worked out better than I had planned because they would say stuff like, “oh, wait, this has a zero exponent, that one’s going to be easy!” As in, they were actually looking at all the problems and evaluating how they would be solving them. (Although some of them just did the first 10).  I did the same thing throughout the chapter, but I just gave the instructions verbally.

Next up, let’s work with radicals!

Radical Files from megcraig.org

Radical Files from megcraig.org(file here)  I also changed this up this year.  Instead of spending one day where all the radicals were perfect, then another day when they weren’t, I started with perfect radicals but then gave them a tricky problem at the end of their practice row (#17-24).  Then we discussed how we would go about simplifying them. I think it worked out pretty well. This took us most of two days to finish front and back, then we did some practice:

Radical Files from megcraig.org

(file here) which pulled questions from this homework:  (I think I called it homework because a lot of students were absent for some reason? Then they felt like they should do it rather than, “Oh we just practiced in class, nothing I need to make up.”)

Radical Files from megcraig.org(file here) Now it’s time for some binomials, again, I mixed everything together (and this was before I read Make it Stick about varied practice!):

Radical Files from megcraig.org(file here) And homework:

Radical Files from megcraig.org(file here)  And a review:

Radical Files from megcraig.org(file here)

Ok, we’re almost there, guys!  We need to talk about rational exponents:

Radical files from megcraig.org

(file here) and homework:

Radical files from megcraig.org(file here)  And then solving!

Radical files from megcraig.org Radical files from megcraig.orgDay 2:

Radical files from megcraig.org Radical files from megcraig.orgFinally it’s time for the last quiz of the chapter!  Review:

Radical files from megcraig.org

(Due to weird scheduling issues this year, we started the next chapter before we quizzed.)

(File here)

Of course there’s a powerpoint!  It’s more of an overview (i.e. not the same probs as study guide).

Radical files from megcraig.org(file here)

So, holy cow, I have a lot of stuff for radicals. Kudos for you to reading til the very end!