## Alg II Files: Matrix Multiplication Application #MTBoSBlaugust

I’ve already blogged most of my matrix notes on this post (and as always, you can find all of my Algebra II Files and FAQs here), but I did do a new introduction to matrix multiplication that I liked:

(NTM file here, practice file here) Sure, you have to do a little teacher manipulation to make sure that the second matrix on the calzone example is a column matrix, but I think it really helps to see why we multiply matrices like we do, and what the resultant matrix tells us.

It also gives us a reason to play this in class:

And as an added bonus, a pretty worksheet with a calculator picture and arrows! (Also I totally skipped finding determinants and inverse matrices by hand. Sorry, but sometimes you gotta ruthlessly cut stuff.)

(file here) Yes, it is required that you play Jackson 5 after the last problem. REQUIRED.

## Alg II Files: Inequalities

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(More files and FAQs here) Oh, holy moly. I thought it was a good plan to have all the inequalities in one chapter, like I did with equations and graphing. We could revisit all of our graphs and equations again, notice similarities/differences…yeah, not so much. Much like any traumatic event, I’ve blocked out many of the details. I just remember ending each period completely frazzled.

So, well, here you go. Do with it what you will.

(file) I tried doing the “test each region” method of compound inequalities. I’m pretty sure I don’t like it and prefer the draw-each-number-line-then-shade-overlap method.

Here’s some homework:

(file) Hey, you know what would be really cool?  If we did a notice and wonder with some pre-solved absolute value inequalities!

(blank file and file with solutions) Oh, wait, I meant “not cool.” How I was being tested this day, when after 10 minutes one group’s contribution was “they all had absolute value bars.” So we did some formalized notes the next day.

(file) I just really don’t want to talk about absolute value inequalities, okay? Let’s talk about something more pleasant, like two-variable inequalities. Happy sound of everyone shading!

(file and homework file). Then I tried combining a graphical approach with quadratic inequalities:

(file) No. Just no. To be honest, even though it’s in Algebra II, I’ve always kept these until Precal and taught it using sign charts. I think that’s a good place for it.

We did all love some systems of two-variable inequalities!

(file) Now would have been a great time for linear programming, but we were running into final exams, so I just did one more lesson: radical inequalities

(file) You can see here I went back to the draw-two-lines method of finding the solution.

Then this chapter was finally done and we had a study guide:

(file and video key part 1 and part 2) And I think that’s all I want to say about this chapter!!

Category: Alg II | Tags:

## Alg II Files: Systems

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(See more Alg II files and FAQs here)

So this was an interesting chapter…one that I think I improved from last year, but could still have used more connections and also some more activities (e.g. double stuf oreos wafers and creme).

It started with a great discussion:

(hey guys, hold out for a couple more chapters when I got a new phone that does pictures a whole lot better! Sorry for the random quality until then.)

(file) So, guess what? I had been teaching the types of systems wrong for a long, long time! To recap from that post, the correct way is:

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

Then we did some substitution, some new stuff here: being really conscientious on boxing equal things and also looky there in #6, doing substitution with quadratics!

(file with these questions and more practice)

and I even went a little crazy and did this:

(This may also be why so many of my students remembered how to expand (x – #)^2–we did them a lot of them throughout the year!)

Elimination.

I was pretty proud of the practice I prepared for the pupils:

(file) The last set was interesting when they didn’t choose the method I thought they would!

Did I hear someone say they’d like to see more textbook-like systems of equations word problems? Here you go!

(File with these & more practice problems) But I was able to add a pretty cool activity from Amy (@sqrt_1) where the students made their own word problems. The only thing I changed was condensing all the work onto one page for easier grading:

(instructions and worksheet) It was a nice day and a lot of the students had fun with it (how often do they get to break out the colored pencils and color? I also gave bonus points for the most creative one from each class and put my favorite one on the test!). I will say next time I will have them show their work that they tried it! (I said they had to do it on their notes, but not turn it in.)

Have I told you how much I like doing group speed dating?

(file) Then it was study guide time!

(file and video key part I and part II) As I said, a lot of room for extensions, activities, and connections in this chapter that I just didn’t have the time to incorporate. TBH, if it’s the night before the lesson, go ahead and use some of my stuff, but if you have time to plan, please go see all the more awesome things there are out there in MTBoSland!

Category: Alg II | Tags: ,

## Alg II Files: Let’s All Translate Some Graphs!

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[More files and FAQs on my Algebra II files page!] One big change I made in Algebra II was making an entire graphing chapter. Usually, we would learn a function, solve it, then graph it, repeat. Now, using Jonathan’s model, they all got mushed together in one unit, which actually really helped them with (a) things that are similar with all the graphs (shifting, stretching) and (b) things that are different. It also solved the issue that I had before where if we were in the quadratics chapter, they would just write y = (x – 2) + 6 for the equation, leaving off the most important part! Now they realized why that was so important! I was reeeeeealy pleased with how well the students did on this unit. I needed to keep spiraling back to these through the rest of the year, though, because when you graph all at once it’s a long time before you graph again!

When we last left our intrepid reporter, she had just finished translations of linear functions, so now we’re ready to jump into absolute value. The first part that they did mostly on their own:

Second part where we made sure everyone was on the same page:

(file and a practice WS file) I REALLY liked those questions on #1 that I stole from some worksheet; you’ll be seeing them for the rest of the chapter! Like on the quadratic NoteTakerMaker!

(file and homework) Again, we were just focusing on graphing by translating in this chapter. Let’s try translating some square root functions!

(file) And then it’s time for some John Travolta!

(file) This was the last function we were going to study, so we spent a day doing a Desmos “Match My Function” Activity Builder:

You can find it here. I think this was the first activity builder I made all by myself, so it’s not very elegant (it was before hidden folders so I had to monitor students not scrolling down to the answer!). I also had some students say it was too easy to just use sliders until it matched, so next time I would definitely add some Desmos-style questions like “Here is Addison’s (wrong) equation and graph. What would you tell her to correct so it matches?” and “How would you explain to your friend how to move a function left or right?”

Then a group speed dating day:

(file and yes the graph answers are included!) Some sort of dry-erase graph is a must for this activity so partners can see work! If you don’t have individual graphing whiteboards, take Tina’s (@TPalmer207) suggestion of buying a pack of job ticket holders and printing off graphs to put inside.

Then it was study guide day:

(file and video key part 1 and part 2) As I said at the beginning, for the most part the score were GREAT on this test! Was it because we ended up going pretty slow through this unit? Or because they had graphed most of these before in Algebra I? Or because all the graphs were together? I don’t know the reason, but I will definitely put this portion of restructuring Algebra II into the “win” column!

## Alg II Files: Functions & Graphs

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TMC has got me all mathy-feeling, so here’s another unit! Or at least, the first part of it!

I didn’t really change much to this chapter as from previous years, but I’ll go ahead and post everything so it’s all in one place. As usual, find more files and FAQs on my Algebra II Files page.

Let’s begin with what is and is not a function:

(file here) I took the NAGS from Sarah at mathequalslove and I think the rabbits came from Shireen at MathTeacherMambo. (Definitely 2 of my top five math blogs).

Here’s part II, but according to my calendar, I did part 1 and 2 the same day.

(file) I’m just going to warn you if it’s the first time teaching Algebra II, the struggle is real when trying to find function values from a table or graph. Just be prepared.

Also the magic parentheses for evaluating a function = amazing. We took the parenthetical promise (h/t mathequalslove again) in Unit 1 that said every time we substitute in a value, we put it in parentheses. And we’re going to be substituting for x, so let’s go ahead and put the parentheses first like:  3(       ) + 2 then INPUT the INPUT INTO the parentheses!

Here’s some homework:

(file) Then it’s the ever-popular graph stories!

(file) Then…it’s time for….domain and range!!!!  Y’all, I just totally had a genius idea: have them figure out what the scale/domain/range should be for the graph story graphs first! One of you try it out and let me know how it goes. But since I didn’t think of that until just now, here’s what I used:

(file) Since we’re doing this before we did inequalities, it’s domain and range PLUS learning interval notation. Note the color with a purpose! I have them “box in” the graph before they think about writing the interval. I also use Sam’s domain and range meter, sometimes breaking out the spaghetti to use if I feel like picking a million tiny pieces of spaghetti off the floor.

Here’s some slightly lagging homework for the chapter:

(file) Then it’s a graphing line “review.”

(file) As you can see, I always go back and forth on graphing standard by converting or by x and y intercepts. Here’s the boring homework file.

Now here’s some exciting stuff! This is a pretty magical activity that is a really good introduction to the (h, k) form. Just read question 1 and let that awesomeness just sink into your brain.

(file) And some practice:

(file) Then it’s time for a study guide:

(file) And of course a study guide video!

Off to make another post!

Category: Alg II | Tags: , , ,

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(If you want more Algebra II files and for FAQs (including fonts), look here!)

Part of switching up the order of Algebra II involved moving up solving quadratics (which used to be Chapter 5, after equations, lines, systems, and matrices). I like how nicely this worked with Unit 1, however, I definitely needed to spiral more through the year, I wasn’t very good at making sure they saw all these different types of equations again through the year. But here’s how solving quadratics went!

(file).  This is basically how every lesson went this year: “Oh, man, I’m going to make the COOLEST note-taker-maker and it’s going to have SO MANY CONNECTIONS and it’s going to be awesome!” and then about 20 minutes into class…

Usually it was a case of “Trying To Fit An Hour’s Worth of Class into Forty-Seven Minutes” but other times I think I aimed too high, or didn’t plan for so much review, or maybe it was just “First Time to Teach a Lesson Never Goes Well” issues.

Anyway, as you can see, I didn’t get finished with my connections the first day, but I think you can appreciate where I was going with it-my students always seem to think “factor” means “solve” and “simplify” means “factor” and such. And also they forget that they can solve quadratics without a linear term without factoring, just like they did last chapter! (Although I will admit I often get into factoring-robot-mode and forget that, too!) I think with three minutes left in the class, I decided to focus on factoring/simplifying for the moment, then go back to solving.

The next couple of days were spent factoring trinomials by guess and check. (Hey, spell-check, “trinomials” is a word and I did not mean “binomials”). We used whiteboard for easy guessing and checking. You can read all about it here! If you’ve always been wary about guess and check, I ask you to go give it a read and maybe try it out next year. It’s not your momma’s guess and check!

[Updated 7/9 to add the following picture] Ok, so I bet you DIDN’T go and read all about the cool guess and check method, did you? Well fortunately for you, Susan (@Dsrussosusan) tweeted me a concise picture of it!

Thanks for the great example, Susan!

Hey, for the rest of y’all, want some homework?

(file) Or maybe some group speed dating cards?

(File). As mentioned, I may have gotten lazy and just handwrote the answers on the back of each. Perfect is the enemy of the good, eh?

Ok, now back to solving quadratics!  News flash: I use a lot of terms that I just think students should know by now. But they don’t! Even if I didn’t do it formally each time, I started each equation/function intro with “how do you spot it in the wild?”

Those weird symbols meant we asked ourselves HOW we were going to solve it before we actually started solving.

(file). Make sure you play the “Guess Which 2 Numbers I Multiplied Together” game that I stole from someone on the #MTBoS (please tell me if it’s you!). Start with 10. They’ll chose 5 and 2 and then 10 and 1. Nope, it was 20 and 1/2! Tell them you’ll make it even easier…they just have to name ONE of the numbers. Give them 20. Someone will say 1/2 and 40. Nope! 1/5 and 100! Then say maybe it will be easier with a smaller number, like -1. Nope, it was 1/pi and -pi!  Then finally give them one more chance…name one number out of 2 that I multiplied to get zero. TA-DA! And that’s why it’s the Zero Product Property, kids, and not the “0 or 1 or 10 product property.”

And why we’re blowing minds, let’s talk about imaginary numbers!

(file) Truth: I just introduce imaginary numbers as a way to solve a problem we couldn’t before (I start with the story about the caveman owing more sheep than he had, so he tried splitting it (fractions), but then he still owed more sheep (negatives) then bring in Pythagoras and the madness of irrationals and how we end up making/discovering new categories of numbers in order to solve previous unsolvable problems. If I teach this again, I’m also going to use the tidbit from The Thrilling Adventures of Lovelace and Babbage about how one of her tutors didn’t believe in negative numbers (and this was in the 1800s! Not that long ago!)). I don’t get into doing operations with them, or rationalizing them, or even the cool power pattern unless I have a few random days free later in the year.

Then it’s time for a big ol’ bag of practice:

(file)  Another truth: Not every single practice activity has to be filled with razz-ma-tazz. This has some nice self-checking built in, and I just wander from group to group. I’m in the camp that sometimes you just have to do a lot of practice and I’d rather have them spend all period in groups working on all of these problems, then spend half the time trying to do the practice and play a game, then not finishing, then just copying the rest from their partner. Not to say I don’t love a good activity, I’m just sticking up for the worksheets because somehow worksheets started to be shorthand for bad teaching and I don’t agree with that. And I’m giving you permission to NOT spend hours converting a perfectly passable worksheet into an (awesome) activity and maybe watch Halt and Catch Fire instead?

Ok, let’s get back to something we can all agree on: the quadratic formula!

(No file, but there is a page 2) Well, except we don’t all agree on how to write it. Notice the beautiful splitting into 2 fractions (and you can still sing “all over 2a!” during pop goes the weasel) which is a masterful tip I learned from Jim (@mrdardy). (We do have to talk about recognizing the right answer during multiple choice tests, though).

Am I the only one that talks about “pretty” numbers? And gross decimals?  I am? I’m cool with that.

Hey, remember what I said about worksheets? Yeah, here’s one that is based on Amy’s great activity. I feel so guilty-like I just undid millions of MTBoS tweets by turning an activity into a worksheet. Does it help if I told you they work on this in groups? Please don’t kick me out of the MTBoS; my TMC16 airfare is non-refundable!!  I mean, check out the directions…they still have to choose which 5 to use the quadratic formula on! There’s thinking, not just rote practice!  I promise!

(file) Hey, you know what this chapter needs? Some radical equations!

(No file). Ok if there is one thing my Algebra II students left knowing this year it’s that (x – 5)^2 means (x – 5)(x – 5). A few dramatic gasps and some fake tears over the dead puppies the first few times someone tries to distribute the exponent and they seemed to have remembered it. Of course, I’ve done the same thing in other years and it never worked, so YMMV.

You know, I think we may need some more group speed dating:

(file here AND it included answers!) Then it’s finally study guide time!!

(file) And if you’d like to hear the beautiful sound of my voice explaining 31 different quadratic problems, here are the showme videos of the study guide: #1-11 #12-27 #28-31.

Whew!  That’s a lot of quadratics, my friend. Hope you found something useful! As always, feel free to leave a comment or tweet me if you have any questions or found a matho!

Category: Alg II | Tags: ,

## Alg II Files: Equations Grab Bag!

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On September 12 of last year I posted about the first half of first unit of my Algebra II course redesign with the cliffhanger of “more to come later.” And technically, today is later!

To catch up: I tried pivoting Alg II a la Jonathon, but I’m not as awesome as he is at this long-term-everything-connects thinking so basically I just redid the first semester order. These are the materials I used for the first chapter, solving equations. I REALLY liked putting all of the equations together into an equations grab bag. I wish I would have slowed down here and spent more time on inverses and flowcharts. (Also see Julie’s better take on this).

Also, in case you’re new to my blog and/or just reading it in a blog reader, be sure to check out all of my Algebra II posts–organized here for all of your graphic organizer/homework/study guide needs. (I have them for Precal and Geometry, too! [Geometry is still a work in progress])

Also also, here is the google doc I use to post all of my notes and homework for the students. It may also be helpful to you for planning.

Ok, so we last left off at solving linear and literal equations.  We took a small quiz on that, then jumped into “let’s throw all the different equations together and see what happens!” Starting with absolute value:

(Blank file here) Nothing fancy, except for the fabulous flowchart action. To “unabsolute value” you have to make two equations! Here’s the homework for the next few sections:

(File here). Now should probably be the time I mention that I tried really hard to correct typos/mathos as I found them, but don’t be surprised if there’s still a few here and there. Please let me know if you find any so I can correct the file!

Then it’s time for powers!

(file here) I especially liked having square roots and cube roots together so we could discuss when we did or did not need two answers. Also look at that #10 just begging to be brought up again if you want to use completing the square to solve quadratics!

And if you’re going to talk about powers, you should probably talk about roots!

(file) What’s that you say? This would have been a perfect time to bring in some radical equations that didn’t work, and you could tie it in to yesterday?  That sounds like a great idea. But remember I had a wide range of abilities and this was still the first month of school. Sometimes you have to pick your battles.

We played a round of group speed dating, which you can read about (and get the file for) here.

Then study guide:

(file) While I still have some blogging mojo left, I’m going to try to get another unit up today!

Category: Alg II

## 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!

## It Really Is All About Lines.

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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!

## 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: , ,