## 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!

## Life-Size Sine Curve

Last weekend, there was some twitter chatter about making life-size graphs so students could explore points/transformations/what-have-you. That got my wheels turning, and after a quick trip to the dollar store and about an hour’s worth of work I ended up with:

Steps:

1) Buy four shower curtains at the dollar store. Bonus if they are prelined!

2) Tape together with packing tape. (hint: tape them down to the floor with washi tape to hold them down, then tape the seams on the front. When done with the whole thing, flip over and tape the back)

3) Use duct tape to mark the x-axis. Print labels (file here), cut out, measure your axis, and tape down with packing tape.

4) Apply colored masking tape for 1/2, root 2/2, root 2/3. (see next picture since the masking tape was at school)

Total cost depends on how much tape you have around the house. I used almost 3 full rolls of colored masking tape, but the good news is I bought it from naeir.org. Have you heard of this site? One of the teachers at school shared it with me. Basically companies donate overstock and you get to buy it for the cost of handling. You do have to spend at least \$25 and shipping takes about 2-3 weeks, but holy cow, can you get a lot of stuff for \$25!  My first shipment I got 8 rolls of patterned/colored masking tape, 2 packs of 12 small post-its pads, 8 post-it pop-up cubes, 12 correction tape thingies, a pack of sharpies, 3 sets of dividers, 2 packs of post-it labels, and I think some other things I’m forgetting. It’s crazy!

Anyway, in class, I handed out dry-erase pockets with a sheet that had an x value in it (0, pi/6, pi/4….2pi) (file here) and told the students to find sin x, 2 sin x, sin 2x, sin1/2x, and cos x. Holy moly. We could have easily spent the day doing that. No, if x = pi/3, sin2x does not equal 2pi/3. Once we got that sorted, we went out into the hall. I had all the students stand on their x-coordinate, then step to the y for the function I called out. It was very easy to find people who made wrong calculations! 🙂  Here’s what cosine looked like:

And sin 1/2x (with an outlier!)

By third time I ran through it, I had worked some of the kinks out:

1. In my first class, I had more people than x-coordinates, so I gave coordinates that were more than 2pi. This did not go well. They were way far down and we couldn’t really see the pattern continuing. The next class I handed out 2-3 points per group and had them work together to find the values, then as we graphed we substituted people in who hadn’t graphed yet.  (The class shown had just one person extra.)
2. I only did each graph once. We talked about max/mins, who didn’t move and why, how many cycles fit on the mat, etc. I think it would have been beneficial to do sin, then cos, then switch back and forth faster and faster. Then do the same for 2 sin x and sin x, sin 2x and sin 1/2x, etc. And also positive and negative. The last class we even tried sin x + 2 (“oh, that means we need to all step up 2!”)
3. Have them write down noticing/wonderings as we are doing it, or a quick sketch of the graph (maybe have some axes printed on the back of their point card?) to help solidify the concepts.

The next day, when we went to graph, I asked them if it helped to visualize what we did yesterday. Only a few raised their hands. I told this to Mr Craig, wondering if I would do it again or if it would be more efficient to just jump into graphing then practice. He said, “Hey, you helped those 5 kids see it better! Plus sometimes it’s about the experience, not about being efficient.”  Sometimes that Mr Craig can be pretty smart. (Don’t tell him I said that, his head is big enough already.)

In other news, this happened on Twitter the other night:

Do you think I will hit 1,000 followers or 10,000 tweets first?

Category: Precal, trig | 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!

## 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.

## Alg II Files: Polynomials

(see more files and FAQs here) This is one of my favorite chapters in Algebra II because it’s the first time we discover that:

(file here) and I just did the bottom part of the first page in class on the board. And during the same class period, we jump into this:

(this is the second page of the previous file). Then it’s time for some graphing!

(file here) but for the sake of time, I’ve been doing the same thing with this desmos file. Then we put all of our conjectures together and practice:

(file here) after teaching this about a bazillion times, I now really like how it goes.  The only thing I may change next year (and maybe more so in precal) is talking about how x^3 has three roots at 0, with (x-2)(x-3)(x+1) we just translate those three roots to 2, 3, and -1 just like we translated (x +2)^2.  Is this even a thing or am I just seeing transformations everywhere? Also, yes, we do call cubic functions and triple roots “John Travoltas” (I stole it from someone on the #MTBoS) because:

Then we spend a day practicing:

(file here).  I usually go around and stamp each row when they have completed it successfully, and then can only turn it in once it has all four stamps.

At this point I throw in solving sum/difference of cubics and quartic trinomials:

(file here)  S.O.A.P is a handy mnemonic that I learned from my coteacher. It tells you the signs of the sum/difference formula: Same, Opposite, Always Positive.  It becomes a bit of a chant: “Cube root; cube root, square, multiply, square; same sign, opposite sign, always positive.”

Some homework:

(file here) Because of scheduling, it was a good time to throw in complex numbers for a day or two:

(file here) Ugh, now there’s something that can be taken out of Algebra II if you ask me (but no one ever asks).

At this point I usually take a break and quiz:

(file here).  Yup, there’s a review powerpoint as well:

(file here).

Then it’s time to really get our hands dirty with some division:

But I really want to try the box method next year as promoted by @TypeAMathland (especially since I can probably get a tutoring session since Anna is going to be my #TMC15 roomie!).  But with just a bit of modification I can still use the same homework:

So the Algebra II book that we use likes to spend a section on “I give you a factor, you find all the rest” but that seemed like a waste of a day, instead I go with “I give you a factor, find all the zeros” as a lead-in for when “I give you no factor”:

I learned a while back that it’s handy to have them figure out how many answers there should be and write out that many blanks. Otherwise many would forget that the original given factor also told you about a zero.

Here’s another day that I’m not a fan of:

(file here).  I finally took a stand and stopped teaching the “what are the possible number of real/imaginary roots this could have?” because WHY?  I almost want to take a stand on “hey, I’m only going to give you 2/3 of the answers and one of them happens to be imaginary so do you think you could figure out the third?” because WHY? but I’m pretty sure that is specifically in our course of study. At least it’s a nice breather after all the heavy lifting we’ve been doing.

Then finally the moment we’ve all been waiting for!  Let’s solve some polynomials!

After doing a couple without the calculator, we start using the graphing calculator to find the first zero (or the first two if it’s a quartic).

Then let’s wrap it up:

(file here).  And of course a review powerpoint:

(file here).

Are polynomials one of your favorite things?  Do your kids know who John Travolta is or do you have to do the dance for them? Wait, am I the only one doing the dance?

## Algebra II Files: Systems

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tl;dr: notes, homework, and study guides for solving systems, graphing systems, and linear programming

Ok, I’m in the mood to knock some more of these posts out.  See more Algebra II files and FAQs here.

As teachers, we divvied up some chapters a couple years ago to try and fancy them up, so some of this was found by my co-teacher and not made by me.  We were really pressed for time this year, so I had to cut out this intro activity that I had success with in the past (and, yes, it’s for Algebra I…don’t tell!):

I like this idea of an introductory activity as well:

But I can’t find the source so I don’t have the worksheet with points (although I guess I could make my own.)  Anyone recognize it?!?!? Please??!!?

We then do some formal solve with graphing:

Ok, this is awkward…this file is so old, I don’t have a blank version on my computer!  But you get the idea. 🙂

No notetakermaker for solving systems by substitution or elimination (we take them on our own paper or else it gets a wee bit scrunched).  But here’s a tip: talk about substituting is just like substituting a player on a team because (1) some players are more beneficial are certain times in the game (ooh, I just thought of this…do the players have to also be “equivalent”?  As in, I assume you substitute a defensive player for another defensive player?) and (2) you can’t have both players on the field. That seemed to help so struggling students not substitute y = 7x + 2  into 3y + 9x = 8 as “3y(7x + 2) + 9x = 8.”  Also, it makes me seem like I know about sports. (Obviously false.)

Some homework:

File here. And in case you need some word problems:

File here. And some linear inequalities systems:

File here. This is a fun worksheet to assign for homework:

Let’s stop here and have a quiz, eh?

File here. Then Linear Programming, which, to be honest, there are 1,000 things out there that are better than what I have.  For example, Fawn’s Funky Furniture .  It seems Steve had a similar idea and made a worksheet. Let’s all say hi to Steve!

So here’s an idea that sprung from someone scheduling an IEP meeting during one of my Alg II classes one year. I certainly couldn’t waste a day (since I would be seeing all the rest of the classes) and I certainly couldn’t leave them to “discover” linear programming with a sub. So I made these notes instead and gave that period filled-in copies.

File here. I’ve kept doing this as a day’s worth of notes because it makes the next day of introducing linear programming much less stressful!  We’re not trying to graph more than two inequalities (new), finding possible max/min values (new), and plugging them in to find max/min (not new, but not common) AND read these really long word problems (scary), come up with constraints (new) and objective functions (new) all on the same day!

Here’s the next day:

And some more practice. I usually have them do this in pairs.

File here. Warning: #4 is a doozy!  Sometimes I count this as a quiz (but I assist and they work together and I don’t tell them until the end), other times we solve some  three-variable equations and have a bigger quiz.

I’d really like to find some linear programming problems where the answer isn’t just where the two slanted lines intersect.  And by “I’d really like to find” I mean “does anyone want to provide me with.”

I feel this chapter is kind of meh. The first half they’ve already seen before and about half are great once we refresh their memories and half consistently struggle. Maybe this year it will improve because we’re going to do it at the end of all the different types of equations and focus on the graphing aspect a bit more.  Basically I want to do what Jonathan did.  Any other suggestions would be more than welcome.  🙂

## Precal Files: Dude, I Could Trig All Day.

tl;dr: Files for unit circle, graphing trig, and inverse trig functions.

So I’m going to post my precal files in the order that I taught them (see more of my precal files and FAQs here).  I met with a PreAP curriculum committee at the beginning of last school year, and they suggested that we do all the trig stuff in the fall, then go all the way from functions -> quads -> polys -> exponentials -> rational -> limits -> derivatives in the spring. It did work really well in the spring, but I need to do better at spiraling back to trig–I have a fear they won’t know what sin of pi is next August!

Ok, are you ready?  Here we go!

Starting with trig values at a point:

Then angles review, but I think I like the worksheet from Algebra II better.

Then the unit circle review:

File here. We also talk about the hand trick.  The hotmath at the bottom is for one of the better trig value flashcards website I’ve found.

The next day we expand past 0 and 360:

I use a worksheet from an Algebra II/Precal joke book for homework (which I just learned is frowned upon? I must say that these are usually well done and have some good questions that catch conceptual errors).

Then it’s time for one of my favorite group work worksheets, (that I already wrote about here):

At this point we stopped, reviewed, and took a small quiz.

Then it’s onto graphing. This is about the time I first learned about the windowpane method, so I taught some classes one way, some the other, and some both. This shows the window pane.

File here. This should have gone faster, but took over a day. The graphing from scratch at the top was like pulling teeth.

This is their practice/hw, which shows the old way of marking the graph into “exciting points”

Then we did a real life sine problem from Math Teacher Mambo.

Here’s her post on it. Be prepared: it looks like a cosine graph so they all wrote cosine equations because who reads directions?  Then I had to tell them to actually read #7.  Next year, I may have them choose whichever function they want, then make the last question be “convert from sin to cos or cos to sin.”

Next, cosecant and secant:

File here. I teach cosecant and secant graphs using a suggestion from a student: we sing “The Grand Old Duke of York,” since when you’re up, you’re up, when you’re down, you’re down, and when you’re only halfway up, you’re neither up nor down (asymptote!).

Ugh, tangent graphs.

File here. This is another example of the “exciting point/pattern” method of graphing, which looking back, I think I like better. Or maybe I need to come up with some hybrid.

Then, because it ties in so well with graphing, we did inverse trig functions in this unit.

File here. Even if you’re not a homework gal or guy, you may still want to use those last 3 problems as a lead-in for the next section

File here. Although next year I want to spend more time on the even/odd/unit circle-ness of sin/cos to discuss, “ok, well, we can’t use 4p/3 in the allowable region for cosine, but what angle in the allowed region should have the same cosine value?”

File here. *Note! The answer to #17 should be pi/3, not 2pi/3! It should be fixed in the file. Thanks to Chikae for spotting that!

Study guide time!

File here.  And, yes, it comes complete with review powerpoints (that could also be used for whiteboard practice).  And they come in both exciting points and windowpane varieties–choose one or both!

But wait there’s more!  If you act in the next 20 minutes (just like the real commercials, the 20 minutes starts whenever you read this 🙂  ), you can get a video of me working out some of the study guide problems!

I post these the night before the test and the students who watch them are very appreciative.

So, be honest: am I the only one who could Trig all day?  (Except for tangent graphs, obvs!)

## Lines in Algebra II: SRSLY, You Should Know This By Now.

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(The continuation of my posting all of my resources for Algebra II.  See more files and FAQs here.)

So the students have seen lines in middle school, Algebra I, and Geometry, so this should be a nice easy review, right?  “Hey guys, let’s graph a line! And then let’s start in point-slope form and write an equation of a line!”

Until they find a cure for Math Amnesia, I guess we’ll start from scratch!  Starting with functions:

File here. (sorry for the some wonkiness in the scans… I use a roller scanner. If my phone had any sort of storage/wifi/3Gsignal in school, I’d use CamScanner instead, but alas..) Next year I’ll be sure to use the vending machine analogy that Justin (@JustinAion) shared on twitter:

And some homework:

What is it about domain and range that students have such a hard time with it? As you can see, I try both the “flatten the graph to the x or y axis” method and the “box in your graph” method.

File here.  (Legal, but it’s sized to shrink to 2/letter sized).

File here (not my original idea, I just made it into a worksheet instead of card sort).  Also, yeah, #6 always gets me. I want to say it’s the curved line, but only by process of elimination. I’ll have to do some filming of a bathtub one day.

However, I think I’d like to replace it with this graph matching instead:

File here. Again, not mine and I haven’t tried it yet, but it seems a little bit higher level.

Ok, now it’s time to get started with those lines!

File here. I used HOYVUX this year, but not sure if it’s my favorite. I usually go with “x(or y) = #” means we cross the x(y)-axis at that number. Also, note the hearts around #5) y = x. I tell them it is my favorite graph of all time, the graph all others graphs originate from and aspire to.  And because it’s my favorite, it will be on every single test until every single student gets it right, which usually means it is on at least 4 tests.

This year, instead of my normal writing equation notes, I did this Translating Lines discovery instead (yes, it’s very similar to the Precal one I shared because it’s awesome)

File here. Next year, I’m going to add some more practice like the first 11, but wait on parallel/perpendicular/two points until the next day to reinforce the “new” point-slope form (they all learned it as y – y1 = m(x – x1) instead of y = m(x – h) + k) and also work a little on getting it into slope-intercept and standard (since that is what a lot of standardized tests use).

However, even without that, most did well on the in-class practice:

Or maybe you’d like a scavenger hunt with graphing, functions, and equations in slope-intercept and standard form?

This is just the first 2 pages; it goes all the way to X and takes most of a period to finish.

File here.

Or maybe you’d like to stop here and give a test?  Well, here’s a study guide

Now let’s actually use these lines!  Next year, I think I’d like to start with Mathalicious Domino Effect, or at least make that the first type of problem on the notes.  Actually, I need to change a lot of the problems on here. I teach in a suburb, so my students have no idea about the ride fare of a taxi. Also, don’t set yourself up to talk about “expanding rods” in high school.  And look how quaint #4 is–a toll phone call!

If you look below, I did take someone’s (??) suggestion about the new way of finding slope with a table and labeling the slope.  Also note the mad-libs portion of the worksheet describing what the slope and y-intercept tell us, an idea I got from Mimi (@untilnextstop) (side note: I miss regular posts from Mimi! If you haven’t read her entire blog, you are missing out on some AWESOME activities and teaching ideas. I’d say I get at least 1/3 of my ideas from her. Also, she lives the most adventurous life!)

Then we did some linear regressions on the calculator (of course you could also use desmos), again practicing some “Math-Libs”  on what part of the equation tells us.  Note: next year, I need to add a negative correlation example.

File here.  I must say I like the two part version of #5, where we find more data = more accurate (or at least a better picture).  Which always reminds me of this xkcd comic:

This year I did Mathalicious Reel Deal (members only), which talked about movie length over the years.  It didn’t go as well as I had hoped; there was a lot of handholding throughout and little “oh, I get it now!” moments.  Maybe because it was the first time I’d done a Mathalicious lesson?

Then it was time for some absolute value, a discovery lesson that actually went well!

I typed up the first part of the first sheet, file here.  It looks like the rest came from the Louisiana Comprehensive  Curriculum, the pdf file is here. The homework file is here.

Alternatively, if you’d just like some notes:

File here. Also, check out that nice vertical stretch work on #4.  I’d almost like them to do the chart and change the y-values instead of thinking of it as slope, since that won’t work for any other function.  But the discovery activity was also really nice…hmmm….decisions, decisions.

Well, at least I know how I like to teach graphing inequalities:

Hey, look, it’s my favorite graph again!

File here. (The shading on the last row usually prints nicely from the printer–I think it was a copy of a copy that I was using, so you couldn’t see it very well.) Also, no, we don’t have time for test points, we just go above and below.  Hint: make them put their pencil on the line and then move it above (or below).  That seems to help for when they secretly want to go left/right.

And finally it’s a study guide!

File here (print it out on legal, then copy two-sided and cut in half).

My thing

Ok, this is going to seem like a weird thing, but have y’all tried the Command adhesive shower products? They are seriously awesome. I hate the suction cup caddies that either (a) slowly slide down the wall or (b) quickly crash to floor (usually in the middle of night).  We’ve had these in our shower for almost two years now and they haven’t slipped a bit!  So treat your shower to a makeover this summer and install some of these. You can thank me later when you’re not woken up in terror at the sound of a burglar shower caddy falling.

## All aboard! Destination Function Junction!

(This post is part of my attempt to get all of my resources online for y’all. See more files and FAQs here.)  Are you ready to get this Precal train rolling???

I used to start Precal with the “Chapter P” in my book: basically, hey, remember this from Algebra II? No, ok, let’s teach it again. Because there’s no better way to start off a course than with absolute value inequalities, amirite?

Right….so I changed it up this year because (1) I had only PreAP Precal and I do expect them to be a little more prepared that regular and (2) the first full week of school we had a week of grade level meetings and picture days, which meant I saw some periods all five days and some three and some kids in and out throughout. So we did a review of equations instead. I just gave them the sheet and said “have at it! See what you remember and what your group can figure out, then we’ll follow up on the rest!”

A former coteacher made this, which is why it’s written with the new equation editor (HATE) and also why it says “Pre-Cal” instead of “Precal”.

File here (4 to a page!).  I think I remember it taking about 3-4 days for them to finish.

Then we started with the “Chapter 1” business: lots of definitions and descriptions and probably stuff they should know from Algebra II but don’t, e.g. functions.

File here. I really need to start using NAGS (Numerically, Algebraically, Graphically, Sentence) more throughout the year.

The homework for the chapter:

File here. (I print it 2 to a page)

Then everything you could possible want to know from a graph:

Oh, wait, actually that wasn’t everything you could want to know about a graph!  How about relative extrema and even/oddness?

File here.  I use a powerpoint to introduce even and odd graphs:

File here.  The animation on this is actually pretty neat. Once you download it, be sure to watch the actual slideshow (not just scroll through the slides) so you can see it.

Then we spend some time playing “Math Pictionary,” where we break out the whiteboards and make graphs that meet different conditions:

File here. If time allows, I also show a funky graph and have groups come up with as many descriptors as possible.  Then we go around and each group shares one, no repeats. The last group to have one to share wins!

By this time, everyone should have had a chance to get a graphing calculator, so we start using it:

File here.  Next time I may put some graphs on there that require changing the window to see everything.

I know I said I wasn’t going to reteach Algebra II (or, ahem, Algebra I) topics, but being able to write the equation of a line is just too important a topic not to spend a day on. Also, using this new way of graphing-by-translating and writing point-slope form is a nice (re)intro to (h, k).

Now I’m going to stop here because I think you may have just breezed by this without trying it, thinking “oh, just another graphing/writing equations worksheet” but it’s not! I promise!  Try this first problem:

1) Graph y = 2x. (it’s ok if you do it in your mind, but feel free to get paper. I’ll wait.)

2) Translate to the right 4 units by moving each point 4 to the right.

3) What is your new y-intercept?  Write your equation in slope-intercept form.

4) Now factor out the GCF.

WHAT?!?!?!?!?!?!?!?  Yeah, that just happened.  The mystery of why h is negative in y = a(x – h) + k is solved with a simple four-step line problem. (Ok, maybe not “solved” because that will come a little bit later, but for now it’s pretty cool, eh?)

Ok, enough amazement, back to work with piecewise functions (we haven’t started doing them in Algebra II yet, so this is the first time they’ve seen them):

File here. Regular Precal file (more graph practice, no writing equations) here.

Yeah, check out those first graphs…the gray graphs are already on there for them so we can focus on the restricting the domain of each one before we pull it all together and graph from scratch (which I do by graphing all the functions with dotted lines, then filling in the parts that I need, so it’s very similar to the first examples).

Then it’s time for Average Rate of Change, which didn’t go so well this year (I was also out for a meeting on the second day so that didn’t help).  And it started so well when I let them loose on this:

I print 2 to a page so this was all on the front.  And they were rocking and rolling. Then we flipped to the back:

(file here) I HAVE NO IDEA HOW TO DO ANY OF THIS BECAUSE IT’S USING DIFFERENT WORDS THAN THE FRONT!!!!!  WHAT IS THIS MADNESS?????? And now you want us to do these practice problems?

(file here)  We spent 3 days on this (again one of them I was out), and I can’t help but thinking if we started with this like I usually do:

(file here) and then spent two days applying it, they’d have a better feel for what AROC is than they did. Or maybe some hybrid of the front of the first worksheet, this, then the back?

As a side note, using The Biggest Loser as an example of AROC is great. It always hurt my head if the graph went up and down, but the AROC was zero. “It can’t be ‘no change’!  We were totally changing the whole time!!” But if you think of it as a contestant gaining weight and then losing the same amount of weight in a week, at the weigh-in she’ll have 0 change.  (This is also a good time to do a PSA to your students about not being obsessed with your weight every day.)

Finally (finally!) it’s review:

and the study guide:

Whew! I think I’m going to get off the train at Function Junction and take a break, but stay tuned for the next installment of “Wow, Meg Wasn’t Lying When She Said She Killed a Lot of Trees.”

## The Things I Do For MTBoS (Posters & Function Transformations)

Casey (@cmmteach) says, “MAKE ME A POSTER!”  and I say, “HOW SPACEY?”

.pdf  and .doc and yeah, because that would use a crapton of ink, bwpdf and bwdoc.

Then Julie (@jreulbach) says, “I NEED THAT IN A POSTER!” and I say, “I FEEL VAN GOGH IS SORT OF PICKY ABOUT USING HIS STUFF”

.pdf and .doc

Then no one says “MAKE ME A POSTER OF THIS QUOTE, TOO!” but I think they just didn’t realize their classroom was incomplete without it so YOU’RE WELCOME, EVERYONE.

.pdf and .doc

Updated: I also made a color version of the above poster: .pdf

Then Friday night on twitter we got into quite the discussion of (h, k) and function transformations, because that’s what cool kids do on Friday nights.  Julie was lamenting at kids not sure what to do first…reflect, dilate, shift?  So I mentioned what I call the S.S. method of graphing after one of the best math teachers with whom I’ve ever had the privilege to work.  I’m not going to lie to you; I didn’t trust it for a while. But then after a couple of years (!) she finally convinced me and it is beauty and efficiency all at once.  Then Julie says “GIVE ME AN EXAMPLE” and I say, “WELL HERE’S A WHOLE BLOG POST ABOUT IT ARE YOU HAPPY NOW?”

So you spend time talking about what the various parts do, maybe with some discovery desmos, or some crazy function match game with worksheet.  (hint: don’t use the t-tables! it just confuses the issue!), or the beautiful HOLY COW SO THAT’S WHY f(2x) COMPRESSES activity (which yes, here’s a quick showme video because Elissa said “CAN ANYONE HELP?” and I said “I WILL SHOWME YOU” even though other tweeps answered her much better than I did).

After all of that, it finally gets down to the nitty gritty of graphing a real function with all sorts of exciting things happening. And yes, yes, you could do all the shifting and dilating and reflecting, and substituting (M, N), but then I don’t know how that would be any quicker than just plugging points into the original equations after all that work (don’t get me started on the people that have them draw a graph for EACH TRANSFORMATION like we’ve got all the time in the world) and the Cal teacher would like them to be quick and efficient at these graphs.

But the good news is I can multiply and divide really well and I’m also really good at counting from the origin.  And that’s all that’s needed.

1) Mark (h, k) with a small x.  This is your new origin.

2) Take your basic t-chart (-2 to 2 normally does it for me, unless there’s domain or excitement issues).  Multiply/divide x’s/y’s as needed.  Feel free to multiply/divide by negatives in the SAME STEP if a flip is involved because we are that CRAZY GOOD AT MATHING.

3) Graph your t-chart from your new origin.  SHAZAAM.  Feel free to label points counting from the original origin if your teacher’s into that sort of stuff.

Here are some examples:

The one catch–a negative INSIDE the function.  We just have to do some reorganizing:

(and some of us need to learn the difference between flipping horizontally and vertically. I know which way to flip; I just always call it the wrong thing. I’ve started going with “flip across x-axis” or “y-axis”).

Anyway, maybe this is what most of you do anyway?  But I always have some teachers that are freaking AMAZED by it when I start graphing these babies at workshops so maybe you were, too. If you want more, here is the notetakermaker and the filled-in version.

Full disclosure: Of course y’all know Casey, Julie, and Elissa are three of the sweetest people on twitter and would never ask for things in the manner in which I implied. So I hope y’all consider this post just a feeble start to paying back all that MTBoS has done for me!