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!

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:

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.

HERSHEY’S TORTE! (and Alg II Unit 1)

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Yes, I am going to make you scroll through all of my NoteTakerMakers to get to the most delicious cake recipe in the entire world.   And, boy, do I have a lot a lot to share for this unit: number sets, order of operations, equations, inequalities, and absolute value equations/inequalities.  (Want more? Visit my new Alg II files page!)

Let’s begin with this HORRIBLE HORRIBLE first day of Algebra II notes. Hey, did someone say lots o’ definitions and then stupid review problems?  I’m interested! The good news is I’ve scrapped it for better things (e.g. Desmos Function Carnival), but you never know when you need a good number sets graphic organizer.  I usually just casually mention the sets as needed, but I think maybe this year, I’ll give each student a number and have them group themselves however they want. Maybe they’ll do even/odd, positive/negative, fractions/whole numbers, but then we could divide into even more specific groups?  And then reverse it: start engulfing each other until we are all one big happy Real Numbers Family?   Anyway, here’s the NTM:

(Ok, less talk, more files, then cake!) Order of Operations:

File here.

I even tried doing a Talking Points for Order of Operations!

Then I don’t even want to talk about my actual solving equations notes. I’m going to totally change them up next time. But for now, how about some Find The Mistake?

I only used Comic Sans to make my fellow teacher happy. Plus it can be counted as one of the mistakes to find!

File here.

And did someone say word problems?

I do so enjoy the travel chart, however, next year I’ll have them figure out what to do for each case instead of telling them.

File here.

And here’s the homework for these two sections:

Now it’s time for some inequalities!  Algebra II is the first time the students see interval notation, so this year we played a match game:

Word file here.  PDF File here-print 2 per page to get it all on the front of one sheet.

Full disclosure: it was a bit of a struggle and not sure if it was worth the time. I may revert back to this instead, or do a mixture next year.

File here.

I made a powerpoint for #1-14 so we could quickly go through them the next day in class, but you may want to use it for whiteboarding or review (each problem is animated step-by-step)

These are the notes we did this year, jumping right into compound inequalities the same day:

Next up, absolute value. I almost feel bad sharing this because I cannot teach this well.

File here. Here’s how I fill in the top boxes:

Moving right along…

Some homework:

And some practice powerpoint for the whole chapter, heavy on absolute value:

And the study guide, which as promised in my #MTBoSDirtySecrets, is very similar to my test.

And now the moment you’ve all been waiting for!!!

My Thing

My thing today is Hershey Torte. This is the yummiest cake in the entire world. On Mother’s Day I tried to make a chocolate cake with peanut butter icing. The cake part went horribly, so I rescued it by making the cake portion of this with the peanut butter icing. Then, upon tasting the final result, questioned why I didn’t just make this to start with.  Warning: I cannot be left alone with this cake.

Hershey’s Torte

CAKE:

1 box German Chocolate cake mix
1 small package instant vanilla pudding
4 eggs
1 1/2 cup milk
1 cup vegetable oil

Mix well. Pour into 3 round (greased) pans and bake for 30 minutes at 325 degrees. Cool completely before frosting.

FROSTING:

Cream together:
8 oz cream cheese
1 cup confectioner’s sugar
1/2 cup granulated sugar

8 oz Cool Whip
1/2 cup pecans (optional if your husband is anti-nut)
6 Hershey’s plain chocolate bars, chopped fine

After you have mixed everything together, frost between layers and on top. Refrigerate.

One taste and then you’ll want to:

Sunday Summary

It’s time for the weekly (ok, technically bi-weekly since I was at the beach last week for a very rejuvenating weekend with the parentals) #mtboschallenge 3-2-1 Sunday Summary!

3 Things I’ve Done Recently That Were Not Half-Bad.

1. Question Brainstorming

…Except for “why is -1 used for inverse notation?” (unless you count “because it is” as an answer).  I did a little googling this weekend and all I was able to come up with was if you want to perform f on x, you write it f(x).  If you want to perform it twice, you could write it as f(f(x)), or f²(x).  Following that same notation, undoing f once could then be written as  f‾¹(x).  Anyone have anything better?

2. Using Graphing Calculators to Graph Absolute Value

I found great discovery activity worksheet online and actually used the entire thing with limited modifications (I took out graphing the piecewise lines because we hadn’t discussed piecewise yet and one battle at a time, amirite?)  It went over great!  The kids were enthralled at using their calculator to do this.  (Actually, they were more amazed that it could graph y = |x + 2| than the fact it could give you a line of best fit when given 15 different ordered pairs.)   Hint: Have them turn their grid on (under “format”-the top middle button on TIs) for better transferring of graphs to paper.

3. Linear Modeling in Algebra II

Totally stealing from Mimi’s (@untilnextstop) fabulous worksheet, I made some math libs of my own.

(doc file here)We did these by hand (well, as in putting equation in point-slope form and then converting to slope-intercept).  Things I need to change: kids have no idea about having to pay a fare just for getting in a taxi nor do they understand #4 at all.  I should throw something in there about having to pay for both AOL and internet service just to complete date myself.  Also it’s best not to have a question about a rod expanding or growing if you’re teaching high school boys.

Then I made another worksheet to use the linear regression on the calculator.

(doc file here) As I said, I did steal the first two questions directly from Mimi.  One thing that I didn’t notice until a kid pointed it out is that the answers to 4D don’t correspond very well to the chart, which led to a great discussion of what the line of best fit can and can’t do for us.

You can’t see it in the picture, but the next problem has the y-scale in billions which was something good to discuss as well.

We then did Mathalicious’s Reel Deal lesson, which was using a scatterplot and line of best fit to determine the movie lengths.  Maybe it was because I was pressed for time, but it did not go over as well as I thought it would.  I think next time I will just add it as another example to discuss in class.

Two Things That Are Not So Great

1. Even after all the work we did with the linear equation mad libs (and we did more work in bellringers and study guides) some students still had trouble telling me what the slope or intercept meant on the quiz.  (As in being able to fill in the blanks: for every _____, the something will increase _____.  At the beginning, there was _____.)  Maybe I should have done Mathalicious’s Domino’s activity instead of Reel Deal.  I also waver back and forth between “they should know this already!” and “they need to know this and they don’t so I need to spend a lot of time on it!”

2. Time is still a huge stress, since we are also nearing the end of the nine weeks and I am struggling to get in 6 major test grades.  Do I just have one test on a single topic to get the required six?  Or cram a lot of material in to have the test cover more than one topic?  Ugh.

One Thing I’m Looking Forward to This Week

Trig Identities!!!!!!!!!!!!!!!!!!!!!  I could do those all day.

Ok, so maybe I just like them for the puns.  🙂