## 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: , ,

## Precal Files: Quads and Polys

Ok, technically, “Quadrilaterals and Polynomials,” but doesn’t “Quads and Polys” sound more fun? Also, this is my second-to-last unit for my Precal Files so my goal of having them up before school starts may actually happen! (See more of the files and FAQs here).

So most of this should be a review for Precal students so we booked through quite a lot of it. Starting with a quick review of parabolas:

(file here, modified from unknown source)

Some homework for the chapter:

Then we did a really cool NMSI activity about concavity. I added this to the end of it for a little derivative preview:

Review graphing polys:

And dividing polys:

And solving polys!

And solving polynomial/rational inequalities:

(file here) I did a factor-sign-row chart and we also did a mini-graph on some to determine the signs. If you’d like to see more about the factor-cool-way to do sign charts, here’s a showme video of me doing a quick explanation.

A pretty intense group-work day on these inequalities:

(file here) And then wham, bam, time for the study guide!

(file here) And if you’re superinterested (or want to use the study guide and not make a video yourself), here is the showme video key.

Only one more unit to go!  Woot woot!

## Epiphany Part II: The Return of the Tranformation

So the perfect storm happened at TMC. In case you missed it (because it was only in my head), here’s what happened:

1) I had an algebraic epiphany last month about using flowcharts to solve equations:

At the very end, I had an inkling of an idea to tie it into function transformations:

but I didn’t really know where to go from there or if it was a viable way to think about it.

2) I had a conversation with two people about how confusing it must be for kids to do transformations in Geometry where x + 2 means “move two to the right,” then move to algebraic transformations where seeing f(x +2) means “move two to the left.” We wondered how we could make the transition easier for students, but came up with nothing. (Side note: I can picture sitting at a table with a boy and a girl having this conversation, but I cannot picture the boy and the girl. I’m now thinking it was not at TMC, but maybe another workshop I was at this summer? But if you are reading this and it was you, let me know so it doesn’t keep me up nights anymore.) (Further side note: Um, somehow I never taught transformations in Geometry? Is that weird?)

3) I worked with Sheri Walker (@sheriwalker72) in the Going Deeper with Desmos morning session. We were tasked with making a new lesson using Desmos and she immediately turned to me and came up with an awesome idea because she knew that we both loved function transformations (who doesn’t, amirite?).  I also brought in a copy of my handouts from my session to share with her and then she just casually mentions how she approached function transformations and jots down something like this for the equation y = 5|2x – 6| + 7:

(I know, it’s weird that her handwriting is the same as the Chowderhead font, but that’s just how cool she is.)

Ok, so at this point I begin freaking out because Sheri can obviously read minds. I mean, I just had this epiphany three weeks ago. So I was so amazed at her mind-reading ability that it was not until later that I realized the elegance and awesomeness of her next step.

I really think you should be sitting down.

If you’re reading this on a tablet, make sure you have a soft spot for it to land when you drop it.

Here it is…

Ok, she called it something other than “fancy form” (programming form? transformation form? input/output form?) but you get the idea. Or maybe you’re like me and you think it’s pretty but “getting the idea” will suddenly hit you three hours later. SHE JUST TIED ALGEBRAIC AND GEOMETRIC TRANSFORMATIONS TOGETHER.

Added bonus? No more worrying about whether the form is in f(bx – h) or f(b(x -h), because this takes care of that. Want to see it action? Let’s take that same equation, y = 5|2x – 6| + 7. Except this time, let’s just find (x, y) pairs from the original absolute value function and use the fancy form to transform them:

What? That just happened. Are you as amazed as I was?  Now pondering how to fit this in to my already over-long function transformations unit….

## Precal Files: Function Transformations, Compositions, and Inverses

See more precal files and FAQs here!

As you may have guess from my TMC presentation, I LOVE function transformations. LOVE LOVE LOVE. So let’s get started with a foldable of parent functions:

(File with instructions and these pictures here)

Homework for the next 3+ days of transformations: (Could someone tell me if that second part is from your blog?!?!)

(File here). After the first day they have a quiz of sketching the parent functions. I think I may add writing the t-table out as well.

Then let’s start transforming!

(File here) Also see a more in-depth explanation in this post. And a great post from Shelley! And a great Geogebra app from Jed!  SO MUCH AWESOMENESS!

Here’s a practice worksheet:

I actually had students ASK to make a table like the day before because they could see the transformation easier. I also added these type of questions this year:

There is also a GREAT activity I used that is a bit copyrighted. If you are part of a NMSI/LTF school, look for the “Graphing Transformations” activity. Basically it gave the students a graph in the first quadrant. Then it asked them how the domain/range/max/min/x-values of max/min/x-intercepts/y-intercepts/AROC/area under the curve change based on different transformations. (They told them what the area under the curve was.) It would be really easy to recreate and there was a lot of great thinking and previewing of Calculus in it.

Also STAY TUNED TO THIS BLOG for another great activity to practice writing equations of transformations.

Next up, let’s do transform our parent functions!

(File here) Read more about this method at the end of this post. The big idea is that we move the ORIGIN (not the “vertex” since not every graph has a vertex) and count our stretched/shrunk graph from our new origin. So easy and beautiful! Works great for conics and trig functions, too!

We did some speed dating practice with it:

(file here) The first pages are the questions, the second set are the answers. I may change some of them up to make the difficulty more equitable. Some people had really quick graphs and others took a bit longer. Maybe making it so there’s just one hard one, but two easy ones? I’ll let y’all sort that out and get back to me.

So after what seems like forever (yet not enough time), we move onto function compositions:

(file here) Things to notice: I write the outside function first, putting (            ) wherever there is an x. Then plug in the inside function into those parentheses, leaving a (       ) wherever there’s an x in that function. Then plug in the value. This seemed to go a lot smoother than finding g(5), then plugging that into f, especially if you have a composition of more than 2 functions, or if you have 2 x’s in the “outside” function.  Also, notice that cool way of simplifying the complex fractions on #4. Read more about it here.

Homework: (file here)

Then some inverses. I want to do A LOT more with them this year and start talking about them WAY EARLIER (See my flowchart epiphany here). But here’s what I did last year:

(file here) (yes, even though it says 1.7 instead of 1.8 at the top. Numbers are hard.)

And a really good in-class sheet with some practice Free Response Questions:

And then it’s study guide day!

Now go forth and transform.

## 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:

## Precal files: Conics

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First of all, thanks for all the great comments/tweets/retweets about yesterday’s post. I want to be clear that I love the #MTBoS and I know everyone in it genuinely cares about improving math education everywhere, just we sometimes have a tendency to get a little bit evangelical about our beliefs.

Ok, now back to some boring ol’ Precal files! (See more files and FAQs here) How about some conics, eh?  This chapter really shows that I’m a visual learner/teacher at heart!

First, circles:

(File here) With some homework:

(file here) Then graphing parabolas using the focus and latus rectum (obvious proof that whoever thought up these labels did not think they would be taught in high school).  We started the day by making patty paper parabolas.

(file here) Instead of trying to memorize 1,000 different formulas in order to write an equation, we just draw a picture and find the important stuff from there:

(file here) Let them loose on #8 and see how many will tell you that the focus is at the origin.  (Spoiler alert: all of them).  At some point during this chapter, we also derive the formula for a parabola.  Hint: derive it with the vertex at (0, 0) and then just talk about how we can move it around with (h, k).

Then ellipses:

(File here) I need to update this answer key because instead of memorizing the focal length formula, we now draw the right triangle and label.  Then it’s time for writing equations:

(same file as above)  Homework:

(file here) And last, but not least, hyperbolas!

(file here)  Oooh, and this one is updated to show the right triangles.

The next day is a fun one.  We start off with this desmos file, which is totally mesmerizing (I had to use “p” instead of “e” because turns out desmos thinks e is a number!  Silly desmos, how could a letter be a number?)  We also talk about degenerate conics for about 2 minutes.  Then we talk about decision tree flowcharts and I usually show this one:

And then I tell them that secretly this one from buzzfeed is my favorite and should be dropped from the skies:

Then I tell them that they need to make their own flow chart to determine conic sections given an equation.  Then we go through this powerpoint and check that the flow chart works, making modifications as needed.

(file here) Ok, most of them are more straight-forward than this one, but I gotta challenge them sometimes, right?

Homework:

I also love to spend some time with conic cards during this chapter–if you haven’t tried them, you need to!  I’m planning on using them almost exclusively to teach the chapter next year in Algebra II.  But for now, let’s wrap it up with a study guide!

And a review powerpoint:

Now if you want to get super extra teacher bonus points, may I direct you to Julie’s Rice Krispie Treat conics?

## Precal Files: What’s our Vector, Victor? (+ Parametrics)

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(More files and FAQs here) Because it’s required to watch and/or quote when discussing Vectors:

Ok, so even though we did a lot of talk about vectors at TMC last year, I still haven’t gotten full control of them and taught them in a way that shows how gosh darn handy and “easy” they are. This is the NoteTakerMaker I used this year:

File here. With this homework:

File here. The year before this, I thought because it was such an “easy” topic, that it would be a good chance for my Honors students to practice reading and understanding math without my “mama birding” it for them, which I think is a skill they need for college. So I made a worksheet that condensed the section from the book into one page:

File here.  Then the next day I had them read the actual textbook section on the dot product.  How did it go?  Well, let’s just say I didn’t try it again this year.  Maybe I should incorporate more of it throughout the year rather than just springing it on them?  Anyway, here’s this year’s dot product NTM:

File here. Then we did some bearings problems:

Oh, see my neat trick about remembering when to use Law of Cosines? C = if you know two sides and included angle and O = all three sides S = otherwise use Sines!

File here. We were told to teach it both ways (geometrically and component-wise). WARNING WARNING WARNING! Some of these have some ambiguous Law of Sines issues!  That I was completely unaware of when I pulled the questions because the key was also wrong!  It lead to a great discussion the following day when the geometric and component-wise answers didn’t match, but it would have been nice to have some warning, so I’m giving it to you now.

ANYWAY, then it was onto parametric equations.  I used this modified introductory activity:

File here. It was quick and went pretty well,  although I secretly like this introduction that I actually made way back when I was a student teacher, which introduces vector equations and ties it into parametric:

File here. and coordinating worksheet:

File here.  This is the NTM I used this year:

File here. Followed by day two:

File here.  Then some applications:

File here. For the first part, we watched three different dog jumping videos and tried to guess which velocity/angle matched with each one.  See them here, here, and here (!!!! OMG SERIOUSLY YOU HAVE TO WATCH THE LAST ONE, it’s the famous corgi flop:

Then it’s time for our study guide:

File here. Video key here and here.

Or maybe you’d like to use this one if you focused on vector equations as well as parametric:

My Thing
Hey, I haven’t done a “my thing” in a while!  With everyone packing for various conferences and vacations (and soon TMC!), I thought it might a good time to mention  ebags packing cubes:

These are the bestest!  You can fit so much in one bag, then stuff the bags into your suitcase. Everything is neat and organized when you get to your hotel, you can just put the bags directly into the closet or dresser drawers, then you don’t feel like you’re living out of a suitcase all week. I have the red, but I’m really digging this tropical blue, too!

Category: Precal | Tags: , ,

## Precal Files: Polar Coordinates and Complex Numbers

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A quick aside before I start sharing: in one of our tours in Iceland, the tour guide mentioned that 3 Miss Worlds have come from Iceland.  Since there’s about 320,000 citizens, and about half of those are female, it means that you have a 1 in 50,000 chance of meeting a Miss World in one of the bars (there was also a side note about the Vikings choosing all the pretty women from Ireland and Scotland to take and leaving them with….). Then the punchline: “Iceland: where everyone is statistically exceptional.” How could I not like this country? Plus over half the population does not discount the fact that there could be elves. Oh, and they eat ice cream all the time.

One thing they do not have in Iceland is polar bears. But if they did, wouldn’t that be an awesome segue into polar coordinates?

Alas, I guess we’ll just have to start with some Polar Coordinate Battleship then these notes:

And some equations, and a hint of graphing by hand:

The first year I did polar equations, we just did them all by hand and did some noticing. If you’d also like to, here’s a worksheet:

This year, I did a desmos exploration (read more about it here):

Sadly, the title is cut off of the next worksheet, it’s labeled as “The Greatest Polar Graph NoteTakerMaker Ever.”

Does the fact that a negative rotates a lemniscate 90 degrees make you freak out because HOLY CRAP THAT’S WHAT IMAGINARY NUMBERS DO ON A NUMBER LINE WHAT THE HELL, MATH??? or is that just me?

File here.

We did some practice with matching (from Mastermathmentor) and some practice sketching worksheets (from another teacher).  Also a wee bit of “where do two polar graphs intersect?” I’d also point you to Michael’s Reason and Wonder polar posts for some more ideas on introducing and graphing.

Then some complex numbers:

File here.  I did a lot of stuff in class (this actually took us two days) from the Better Explained website: here, here, and here.

Finally a study guide:

File here. And study guide videos: #1-16, #17-26, and #27-37 (even though it says 17 as the first problem-doh!)

Find more precal files and FAQs here. Hope you’re finding these helpful! 🙂

## Algebra II Files: Matrices

(See more Algebra II files and FAQs here)I’m sure this is post you’ve been waiting for!  This is totally how I feel after I typing up a matrices worksheet:

But have no fear, my little math darlings, I’m here to save the day….with handwritten notes:

Ok, but at least I typed up the homework for ya!  With graphs!

File here. Is that not the prettiest worksheet ever?  Now let’s try some multiplying:

File here. And some practice:

File here. What’s that you say? You’re totally awed and inspired by my ability to put a fraction inside a matrix?  Aw, it was nothing.  If you’d like to learn how to do it yourself, may I recommend my equation editor post?

Then, yup, it’s back to handwritten again:

I’m taking a stand on not teaching finding 3×3 determinants by hand.  I almost want to take a stand on not solving matrix equations by hand, too, I mean it’s pretty much just witchcraft at this point and it’s not like we don’t have calculators, which we’ll learn to use the next day to solve some systems:

File here. Then some more applications practice (after the first four, we did the rest in groups on the whiteboards) and the study guide:

In the same file here.  Added bonus: Study Guide Key videos!  #1-12 here and #13-25 here.

Does your school teach matrices in Algebra II? (We moved it to Precal for a year which was great, but then they decided to move it back)  Do you do them all by hand or all technology or a mix?  Anything to make them more exciting?

Category: Alg II, Precal | Tags: ,

## Precal Files: Dude, I told you I love Trig.

tl;dr: Notetakermakers, homework, and study guides for trig sum/difference/double/half angles, trig identities, and solving trig equations.  Part of my ongoing series of posting all of my files; see more and FAQs on this page.  Plus I tell you about an awesome book at the end!

Yes, I love trig. I love that I there’s always new ways to think about and teach it. I love that it’s so elegant. And I love that it’s one of those topics that looks scary and is scary and new but eventually most kids get it and feel so smart about it.

Now, check out the middle box of the “three fraction hints” above. If it’s the first time you’ve seen this multiply-by-the-common-denominator-of-the-small-denominators, then be sure to read this post about it.  It’s totally awesome and is so handy in Precal and Calculus!

Now, don’t worry, we don’t do all of those in identities in one day!  We do the first six together:

This is also the first time I talk about Q.E.D and I tell them they could use any symbol to show “YES! I DID IT!” such as a check, smiley face, corgi, or unicorn.

Then I have them work on the rest of the first column for homework with the rule: if you’ve been on a problem for more than 5 minutes without getting anywhere, stop and move on. Since I teach honors, I know some of them would get trapped in a problem for 20 minutes and then just get frustrated with the whole thing. Then we work on the others in class on group whiteboards for a day, and finish them up whenever we have a few minutes throughout the week.

Another reason I like trig is because there’s ACTING! involved. Sure, you could just show the powerpoint of Sinbad and Cosette when you teach the sum and difference formulas. But why tell it, when you can get four chairs at the front of the room, make some nametags (write the names really big, put them in page protectors, then tie some string through the holes of the page protector), and then act out the whole thing with 3 volunteers?  I even bring in a tie and a scarf for when I’m playing each driver. And yes, as Cosette, I wear sunglasses so I can do this move:

and say, “we do not have the same sign.”  Although, confession: I have no idea how the story is supposed to help memorize the tangent sum/difference formula–please let me in on the secret if you know it!  Another confession: Crazy Stupid Love is one of my favorite movies of all time.

FOCUS!  Back to sum and difference:

File here. We also decided this year not to do the problems like 7 and 8 so I will allow you to skip those as well.  You’re welcome.

File here.  If you do skip 7 and 8, also skip 12, 13, 18-20 on the homework.

Some double/half angles:

File here. Fun tip: have them derive the double angle formula of sin and cos from the sum formulas and then everybody gets to feel smart.

File here.  Now’s a good time to consolidate all our knowledge:

File here.  (omit #19 if you’ve been omitting stuff) And then begin solving trig equations!

Check out that awesomeness about sin 2x having twice as many answers, but 1/2x could have the same number of answers or even no answers between 0 and 2pi.

Yeah, I was really clip-art happy when I was making all these.

File here.

And then some quadratic and mixed equations!

Cute and cuddly, boys, cute and cuddly….

File here.

It’s only one section, but worthy of its own study guide and test.

File here.  There’s even a couple showme videos for the study guide: #1-9 and #10-16

My Thing

My thing this week is Simon vs. The Homo Sapiens Agenda. I read about 50 pages of it the night before last, then spent all afternoon yesterday finishing it because I HAD TO KNOW WHAT WAS GOING TO HAPPEN TELL ME TELL ME TELL ME.  And it’s obvious that the author works with teenagers because the dialogue is spot-on.  And they’re normal teenagers doing normal teenager-y things which is a rarity in YA. And it’s just a nice pleasant story where no one dies, not even the dog. 🙂