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

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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!

## Things I Thought I Knew

Some days this job can be a bit ho-hum. Teach a lesson you’ve taught 37 times before, grade a few tests, make some copies, go home. And then some days you get to learn math that you totally thought you knew, but you had no idea. Friday was one of those days.  I’ve been teaching long enough that I probably should have known this stuff, but I didn’t. So here I am to enlighten you as well. Just be prepared for the following to occur:

##### Topic 1: Classification of Systems

Am I the only one that is really bad at reading directions? But then the students actually read it word for word and ask questions that I didn’t see coming because knew what the question was really asking without having to closely read it. This week, we had a question about making an independent and inconsistent system, which I read as “inconsistent” and knew what the goal was, but then that pesky “independent” thing was brought up by students.

So I always thought there were three types of systems:
Independent: One Solution
Dependent: Infinite Solutions
Inconsistent:  No Solutions

TURNS OUT I was mushing two different categories together:
Category One: Is there at least one solution?
Yes: Consistent  No: Inconsistent
Category Two: Are they the same line? (Technically, “can one be formed from the other with algebraic manipulation?”)
Yes: Dependent No: Independent

So the correct categorization would be:
Independent and Consistent: One Solution
Dependent and Consistent: Infinite Solutions
Independent and Inconsistent: No Solutions

When you realize you’ve been teaching something wrong for your entire teaching career:

###### Topic 2: Logs grow really really slowly.

“Yes, Meg, we know that.” No, I don’t think you do. The wonderful MTBoS Search Engine led me to this GREAT activity from @Johnberray. I modified it a bit, first of all by making some pretty 1-inch graph paper:

(file here) Each group got a sheet in a dry erase pocket and used their calculator to find the decimal values to graph y = log x. Then I asked them to calculate how many inches it would take for it to function to reach 2, 4, 6, and 8 inches high. (At least one group each period said 20, 40, 60, 80 so we may not have the idea of logs down pat just yet). Then I asked them to describe each distance in some way–is it the size of a pencil? classroom? from here to the front office? as tall as basketball goal?  I let them use the internet and a student showed me that if you put a measurement into Wolfram Alpha, it will give you real world comparisons! Pretty neat!

You really need to try it for yourself!  What fun we had when we started talking about how long the paper would be for a height 6, 7, and 8 inches (oh, plus go ahead and try 9!) using this draw-a-circle-on-a-map site that John had linked to. Then it turned into, how high will it be if the paper reaches the moon? The sun? Pluto? I won’t give it away, but guys, logs grow really really really really really slowly.

Yet they still go to infinity.

###### One raised to any power is equal to one, oh, unless that power is infinity.

So I’m pretty good with limits. Then my Calculus teacher friend mentioned, hey, isn’t it weird that the limit of (1 + 1/r) ^ r is e, not 1?  Wait, that doesn’t make sense.  But it turns out 1^infinity is indeterminate. WHAT? But I thought 1 to any power is 1!  WHAT IS HAPPENING TO EVERYTHING I THOUGHT I KNEW?  The best answer I saw was that “we don’t know if the one is really a one”, which ok, I guess, since 1 + 1/r is never exactly one. But what if the number IS actually 1? Not like “kind sorta” close to one but ONE. If 1^infinity isn’t one, at what number does it STOP being equal to one?

I was discussing all this mind-blowing math with my friend right as class was starting, complete with dramatic re-enactments of how high the graph of log will be on Pluto and what was happening to my brain. When I came into class, a student asked what we were talking about and I said “Math!” and she said, “You really do like math a lot, don’t you?”

Yes, yes I do. 🙂

Category: Alg II, Precal | Tags: , ,

## Precal Files: Logs

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Yes!  It’s my final unit for my Precal files!  See my entire year’s worth of stuff (and FAQs) on this page.

In my regular Precal classes, I normally start with an exponent review:

(doc file here-requires Running for a Cause font) (pdf file here)

Then we played a grudge match:

For my honors classes, we did exponent review during bellringers the week before and jumped right into graphing exponentials.

and solving exponentials:

Here’s the homework for the chapter:

Then it was time to break out the logs!!!

(file here) This year I want to be more explicit about how a log is the inverse/can undo an exponent. I think some of them still weren’t clear on that and what that meant for us. But meanwhile, we did some log graphs:

Then some log properties.

(file here) We did a nice worksheet using log properties to solve equations from a “Calculaughs” joke worksheet book for Algebra II/Precal.

Then we stepped up the solving logs a bit:

(file here) And did some group whiteboarding with these problems the next day:

Then some applications:

(file here) WARNING!!!  You see that nice pretty chart where we’re going to notice that as we compound more and more, it will equal the Pert formula?  Yeah, it breaks when you do the seconds one in a TI!  It looks like you actually make more than continuously compounding!  Wolfram Alpha saved the day, but it made for a great discussion! Just wanted to let you know ahead of time so you don’t freak out in the middle of class. 🙂

Then it was time for a study guide:

Because of some weird scheduling, after the test we spent a couple days on these advanced, precalculus-in-the-true-sense-of-the-word problems:

Well, that’s it! I’m done with my Precal files! Until I make something new when I start back next week. Stay tuned!

Category: Precal | Tags: , ,

## An Algebraic Epiphany

People, this post is why I love the #MTBoS.  You can’t read everything, learn everything, critically think about everything; but if you read blogs and tweets, then you can collect more of that knowledge than you would alone. So even though I am not participating in the #intenttalk book study/chat (Am I the only one who always thinks it’s Kimmie Schmidt on the cover?), I did see this tweet from Bridget:

I used that method a wee bit this year when I taught inverse functions and a few students really latched onto it. But now I’m thinking of starting this way on day one,  building on it, and tying it into Glenn’s three rules of mathematics. I sat down and played with it a bit for the last few days and all I can say is:

Are you ready for this?  Ok, let’s just dip our toes in:

The main idea being that we think through the equation “forwards” and then work back to the solution using inverses. Another easy one:

I like (a) completing the circle of life by checking our answer and (b) each column showing equal values.

How about we try out the shallow end:

Yeah, I’m totally digging the two arrows for square root, too.

So one place where this method has problems is if there are variables on both sides. But I want to use this more as an introduction in each section, not a method for solving each individual equation. However, we can use the fact that each column is equal to set up the rest of the problem and finish with quadratic formula.

Now I thought for sure this could not work with quadratics. OR COULD IT?

Ok, so the weird thing here is that (a) my new erasable markers don’t like it when you rewrite over something you just erased and (b) we have 2 places that x is involved, so 2 starting points. But then I don’t know how they are going to add to equal 6. But (spoiler alert!) we do know what has to happen if we’re going to multiply to equal zero…

Here the two back arrows from zero come from the fact we had two x inputs. Pretty powerful, eh?  Let’s try it on some other tricky problems, like rational exponents:

Ok, guys, we’re going to jump into the deep end now….ABSOLUTE VALUE!

Update: I was so excited about “un-absolute valuing” that I forgot to “un-multiply”. -6 should turn into 3, which would then turn into -3 and 3; and finally -6 and 0 as the answers. Which I probably would have noticed if I followed my own recommendation to circle back through.

Holy cow I’m in LOVE LOVE LOVE with having to “unabsolute value” as a step, because of course to “unabsolute value” you go back to positive or negative.

Ok, ok, a little tricky, but not undo-able.

Now I did have trouble with this problem:

I wasn’t sure if my beginning value should be x or 5. When I tried it with 5, I thought of it as “If I’m at 125, what root would I need to get to 5?  Oh, the third  root. That means the original operation in the top line needs to be the inverse of the third root, which is cubing, which means x = 3.”

But if I keep my beginning value as x, then it leads into a nice intro/need for logs:

And then I went crazy with the log problems!  (Although not pictured is two logs equal to each other, e.g. log (x + 7) = log (2x – 4). I’ll leave it as an exercise for the reader; it really is quite pretty.)

The last one being another case of, “Uh-oh, need to rewrite this as something isn’t so ambiguous.” Another case of that:

Ok, ok, I don’t know why I didn’t have two starting x’s and then divide them, but isn’t it just beautiful how it works out this way?  So I went some more down that path:

Then I thought of other problems that cause students anguish, and immediately thought of the difference between 2sin(x) and sin(2x):

After this, my brain was pretty much done for the day.  Or at least, I thought it was. Then I had a shower thought (where all problems are solved): hey, wonder if I could tie it to graphing transformations?

GAH!!!!!  So you go through all the steps, then find your parent function, in this case absolute value. You have to use inverses to get to x (minus three, or in this case three to the left) and OH I SHOULD HAVE PUT = Y AT THE VERY END BECAUSE THEN YOU TRAVEL “FORWARD” (stretch 2, down 4) FROM THE PARENT FUNCTION TO GET TO Y.

Another one?  ANOTHER ONE!

I don’t know why you would want it, but if you did want all of these examples in one pdf, here you go. Now there are some drawbacks as I’ve mentioned: things need to be simplified first, somethings get a little wonky, how will this work for trickier equations; but I think Kayne sums it up pretty nicely:

## More Logs!

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Logs: the chapter that will never end.

See blog post 1 and blog post 2.

As I said at the end of blog post 2, I tried a new way of teaching log equations, using the question, “Hey, what power of ___ gives you ____?”

I decided to jump in with both feet and use the same question for logs on both sides.  Normally, I would just cancel them:

But then we’d end up thinking that works for this problem, too:

Ugh.

So I went back to the question:

(Pssst…drawing the box around the right side does help!  Also lead in with a reminder of the “easy” ones from the first day of logs. And spend a bit of time talking about why they’re inverses and what inverses do.)  Yes, it does seem like we’re taking the long way around but wait until you get to here:

On the quiz, I’d say less than 5% tried to cancel all the logs at once.  Some students asked me that since we were doing the same thing to both sides, we know the arguments had to be equal so could they do it that way. I told them that was very neat that they noticed that but warned them against just randomly crossing out logs.

I also did the same with exponentials.

Old way:

New way:

Then everyone that doesn’t have math print on their TI got sad (psst..log with a modifiable base is towards the end of the Math menu), but with a few parentheses we were good to go using the change of base formula.

I must admit I’m not 100% sold on the new exponential way. I’m still having to reinforce that the log helps us answer an exponent question. I still have a few people just make log x = 60 into log 60 =x because aren’t we just randomly moving numbers around anyway? But don’t I have that every year?  And, as Conic Card Cindy famously said, they weren’t learning it the old way I  taught it, so what’s the worst that could happen?

And just because I like you, here are some more files for you.

Common Logs!  .doc file

Natural Logs! .doc file Yeah, I kind of glossed over the discovery of e. Don’t tell the math police.  Although they’re probably on their way after that cross-multiplication I did in #9 above. But the common log lesson is not the time to take away the one thing they remember how to do from middle school.  ANYWAY….

Bonus review powerpoint of log laws and common logs!  Great for whiteboard practice!  Each problem is worked out step by step.

I still have some applications to discuss and post (and graphing logs…now there’s something I don’t love to graph AT ALL.) but hope this keeps you busy for now. Thanks for reading!

Category: Alg II | Tags:

## Blogging Log Laws Blog

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Ah, Arrested Development, how I miss thee.

But onto some Log Laws….  When we last left our intrepid reporter, I had just introduced logs and was trying to figure out a way to start log laws. So I did what any self-respecting teacher would do and stole Kate’s idea.  But I was worried about my kids being able to make the leap to filling in the blanks, so I turned it into a match game:

File here (with the typos fixed.  This is why you don’t watch Gilmores while making a worksheet.  I can’t focus when there are 1,000 yellow daisies involved.)

The first row was a nice refresher after our snow day and then I gave them 2 – 3 minutes to work on each box, then we discussed the results and the rule.

Because it was a shortened day after a snow day, I had 3 central office people do a pop-in observation.  Of course they left right before we started discussing the addition/multiplication rule box when a student said, “Ms Craig, I think I found a shortcut…can’t we just multiply the arguments?” (ok, to be honest, he said “big numbers” instead of “arguments” but who can blame him?).

Also it’s fun to watch them all choose option A in the last box. Then tell them to go back and actually find the values and listen to the sound of erasing and/or “I TOLD YOU SO!”s.

In 30 minutes (minus bellringer/homework time), we got through #4 or 5 on the bottom practice.  Based on the one class in which we did finish,I think I need to go back to Kate’s and make 6-9 more scaffolded before jumping into pure craziness.

The next day was a full day and we used our laws to simplify/expand and solve equations:

File here.

See how I left that middle box blank?  I decided on a whim to try teaching logs on both sides the same way I taught log equations–logs ask “what power of ___ gives you ___?”  We did some preliminary work about inverses and b^(log x) [with the same base] = x then jumped into the problems. It took a little bit for them to get used to it, but I think it may just end up working.  And maybe stop them from just randomly crossing out any log they see anywhere in time.

Stay tuned for more log blogging with common logs, natural logs, and the debate over perhaps abandoning one of my favorite lessons ever.

Category: Alg II, Precal | Tags: ,