Tag Archives: quadratics

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:

Unit 4 1

Second part where we made sure everyone was on the same page:Unit 4 2

(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!

Unit 4 3

(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:

Unit 4 4

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:

Unit 4 5

(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:Unit 4 6

(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: Quadratics!

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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!

Unit 2 7

Thanks for the great example, Susan!

Hey, for the rest of y’all, want some homework?Unit 2 1

(file) Or maybe some group speed dating cards?

Unit 2 2

(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:

Unit 2 3

(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!

Unit 2 4

(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:

Unit 2 6

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

Unit 2 5

(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: ,

Parabolas Post Mortem

FINALLY.  I am FINALLY done with parabolas in Algebra II.

I spent most of last Sunday afternoon trying to take all the suggestions from my last post and put it together into some sort of lesson and this is what I came up with:

Parabolas from megcraig.org Parabolas from megcraig.org Parabolas from megcraig.org Parabolas from megcraig.org

(file here)

I went in on Monday feeling like Super Teacher.  I mean, I hate to brag, but check out #14. Taking a side!  And figuring out what the most important point of a parabola!  And all the other problems, where we find something in the graph and then relate it to the equation!

And then first period hit:



The thing was, 98% of them were tryingReally hard. But the questions!  I think when I had to answer “So it says find the value of y when x = 1. Should I plug in 1 for x or for y?” is when I had my complete George Michael collapse. I don’t know how to fix this. I can’t fix this AND teach one of the most packed curriculum in high school math. I was actually considering doing even more application problems without a graph the next day until my Best Teacher Friend (I hope everyone has a BTF as good as mine at their school) talked me out of it.  You have to meet them where they are, right? So, after finishing it up on Tuesday and discussing it, we went on to:


Parabolas from megcraig.orgParabolas from megcraig.org

(file here, with some bonus homework on pg 2)

I am not kidding about this exchange on the last problem:

Me: “OK, we found the x-coordinate of the vertex. How are we going to find the y-coordinate?”
At least three students: “PLUG IN ZERO!”
Me: “I’m glad you finally remembered that about finding the y-intercept, but now I need to find the how high the point will be on the axis of symmetry. So I know the x, but I need to find the y….how could I do this?”
Everyone: “….”
Me: “OK, well, guys, we’re going to plug it into the equation. Remember if we know one coordinate, we can always find the other by plugging it in?”
Student: “Whoa. That never would have occurred to me to do that.”

Wait, what? We’ve done this for a week and you just did a whole application worksheet where 1/2 of the questions were, “we know this x, let’s plug it in to find y” and it never would have occurred to you?!?!?!

I don’t want you to get the wrong impression; I’m not saying these kids are stupid or dumb. It’s just I don’t know how to get them to connect anything.

Ok, wait, I’m getting into a “Sometimes I Wish I Had Never Found the MTBoS Because I Used to Think I Was A Fairly Good Teacher and I went Home at a Normal Time and I Can’t Continue to Be Student-Centered if the Students Aren’t Prepared to Bring Anything to The Table” Funk so let’s focus on something that sort of worked!

We were still (!) struggling with characteristics of a graph. So I made this worksheet and put it into dry erase pockets:

Parabolas from megcraig.org

(file here) Here is one thing that I found that helped teach increasing/decreasing:

At what time does the parabola change direction? Draw a vertical line and label it with the x-value.” (also label +/- infinity)

Parabolas from megcraig.org

As you’re drawing from left to right, are you doing down or up? Ok, so we’re decreasing on this interval and let’s read it from left to right, (-infinity, 3).”

Repeat with the right side. This seemed to help (a) “but aren’t we started at the top which is positive infinity?” (b) “we’re decreasing to -5” (c) answering are we increasing/decreasing on a certain segment seemed better than where are we increasing/decreasing.

I did a similar thing for positive/negative, calling back to Dolphin Dave being underwater or above water and drawing the waves on the x-axis:

Parabolas from megcraig.org

The success rate on the quiz was lower than I expected after doing some formative assessment on the last two problems on the handout, but better than it was before this activity. I think doing this as a separate lesson on day 1 would have helped. Or just waiting until Precal, which is when we normally focus on this.

Anyway, we started the study guide and worked on it Thursday:

Parabolas from megcraig.org

(file here video here)

The quiz grades were actually really good–lots of As and Bs, only a smattering of Ds and Fs. But the level of the quiz was definitely lower than what I’ve given in the past. I don’t know what to do about that.

I would also like to apologize to the 10% of my students that got this on the first day. I actually had two of them say that this was so easy, why were taking a quiz on just this? I’m sorry we had to spend two weeks on this. I know you’ve been bored out of your mind but you’ve still been working hard and helping your friends and thanks. (I did tell them this, but in a nice way about “some of us found it really hard…”). I don’t know what to do about them either (and please don’t tell me “find some differentiated activities for them to do” because I just cannot handle one more thing at this moment in my teaching.  I’m really just saying I don’t have the answers to anything.)

But I do know one thing.

I am done with parabolas.


Category: Alg II | Tags:

When to Throw in the Towel

tl;dr: What should one do if, after a week of lessons, a majority of your students are still struggling on a concept that you’re not even sure is all that important?

This week was a really rough week in Algebra II. My goal was to have them be able to graph a parabola given in 3 different forms, then at the end of the week, also show how we can use completing the square to transform standard into vertex form, maybe do some applications of quadratics as well. This is what happened:

Day one: I want to update the chart I used last year, so I made this NTM:

Parabolas 1

(here’s the file, although after reading this post you probably won’t want it!) Ok, so this took us the entire class period. And it was horrible.
“hey guys, to find the y-intercept, what do we know about a point on the y-axis?”
“ok, well, let’s think about this point on the y-axis, what would its coordinates be?”
one meek voice: “(0, 8)?”
“Yes! So what should the x-coordinate be?”
“Yes! So if I know the x is zero, how could I find y?”
“Ok, that’s our x, what should we do with that to find y?”
“Ok, if I give you any x value, how could we find the y-value?”
(barely containing my frustration at both the students for not knowing and myself for being such a crap teacher that I haven’t even gotten this point across in the first semester of Alg II) “Well, we put it in for x. Because see how the equation equals y? This is how we can find any ordered pair if we know one of the coordinates! So plugging in zero for x in any equation always gives us our y-intercept!”

5 minutes later, for the next form:
“So, how would we find the y-intercept?”

I kid you not. Every. single. class. period. In retrospect, I think I should have had one example of each form under each category and worked through it, rather than just doing it as scratch work on the board. But I thought seeing it all together with the graph would make more sense. I was wrong.

Also, I thought this was going to be so easy that I might as well talk about where the graph is increasing/decreasing and positive/negative to fill in time. (Note: we had already done positive/negative when talking about quadratic inequalities).  In case you didn’t know it already, talking about where a graph is increasing in terms of x and not y is one of the hardest concepts in math. Ask me how I know.

Day 2:

Onto the back (which I had planned to have finished yesterday! Ha!)

parabolas 2

They were in partners and I let them work on each section for 3-5 minutes, then we talked about it. They were doing not horrible, but some didn’t know where to start. Again, in retrospect, I should have put “how to find the vertex” on the front, not “axis of symmetry” because they would tell me there were no instructions for the vertex. (I mean, yeah, except for the box at the bottom of the graph that had “vertex” and an example of each equation, but c’mon, that was all the way at the bottom.)  We had about 7 minutes left at the end of class, just enough time for them to cut out their dominoes for tomorrow. The right side was homework.  (The bottom was homework the previous night, because they didn’t do so hot on their unit circle test, either. So far this semester has not been off to a great start.)

Day 3: Check homework; at least a quarter of them just wrote down the points under each equation instead of actually showing how we got them (like, you know, the example right next to it). Went through the homework problem and had them work with the Math Shell quadratic dominoes.  Based on my previous experience, I had them cut out the graphs and match them to the equations (playing them as dominoes led to lots of silly looking-at-the-wrong-graph mistakes last year when we went to fill in the blanks). Then we rotated partners and checked. Then I revealed the correct answers and maybe 1/4 – 1/3 got them all correct, or had only one pair switched. Then I had them work on filling in the blanks. This seemed to go OK.  At the end of the day, I gave them a worksheet that had three equations, asking them to find the important information and graph. In my extra long period, they had time to start on it and there was so much struggle. Like not even knowing where to begin. Like not even knowing that this was the same thing we had been doing for the last three days. Like this was the third day I just wanted to cry at the end of the period.

Day 4: “Ok, I think we maybe need to see the big picture of all these equations again.” So on the back of our worksheet, we did an example together of each of the three types. Then I set them loose on the front again. It was horrible. We have no idea what these numbers are. The x-intercepts are (-1, 0) and (4, 0)? That must mean the vertex is at (-1, 4)! Oh, the vertex is x = 5 and y = 3? Oh, that must mean the x-intercept is 5 and y-intercept is 3. Or the vertex is just x = 5. I went around putting out fires and got most of them through it, but I doubt they would be able to do it again on their own (ok, maybe 15% could).

So this bring us to where we are today: Not much farther in our knowledge of parabolas than we were on Tuesday, perhaps even less (or at least more confused). Also, may I remind you that this is Algebra II, so we’ve already seen these in Algebra I, and I have many, many, many more topics to cover this semester. These extra forms aren’t even in the course of study; I just thought it would be a nice review of parabolas by themselves (we’ve already had a whole chapter on transforming famous parent functions) and practice in seeing the same thing different ways and being able to see why some forms may be better than others.

My question is: do I throw in the towel?

I honestly think it would take at least two more days of practice for them to even be able to attempt a quiz on this. And what could I do differently during those two days that would help them improve? Other than, here is yet another example step-by-step. And I still want to cover completing the square and maybe some applications (as in, where is the ball at its maximum, how long in flight, etc…stuff they’ll see on tests and in other courses). Should I just say, this just wasn’t a good week of lessons. Let’s work on changing standard to vertex because we like vertex form and that’s the only type of graphing that will be on the quiz. Should I try the applications on Monday, which would give them more practice about finding y-intercepts, vertices, and x-intercepts? Should I just forget that last week happened entirely and start the polynomials chapter?

Any advice would be greatly appreciated. Leave it in the comments, or better yet, tweet me (@mathymeg07) so I can pick your brain even more.

All The Cool Kids Are Guessing and Checking

Last night I landed in the middle of this discussion with Julie:

Until we get Glenn to share his ideas (C’MON GLENN I DON’T WANT TO HEAR YOUR PhD EXCUSES),UPDATE: Glenn has shared part one of his ONE MATHS blog posts here! these are my big three goals for Algebra II this year:

  1. Simplify stuff. Also, I want to check our simplifying.  Let’s plug 7 into (x + 8)(x + 3) and see if we get the same thing when we plug 7 into x^2 + 11x + 24.  Let’s graph it on desmos and see if we get the same graph! Oh, look, we’ll get the same output for any x!  (I think this would also help with the “plug in any number” method for the ACT, which totally blew my mind when I first read about it. “No, wait, it can’t work for any number I pick, can it? How does it know which number I’m going to choose?”)
  2. Solve stuff.  We will do this by legally undoing things. And also check by plugging in the answer and by graphing.
  3. Graph stuff.  90% of which we can graph by using the (h, k) method.

Our department is also planning on using Jonathan Claydon’s layout for Algebra II that hopefully will foster more connections as well!

Back to last night: as things often do in the #MTBoS, the discussion turned to factoring:

So if factoring by grouping or box or slide and divide or airplane or bottoms up is not working for you or your students (i.e. do they all remember the “slide” but forget the “divide” part?), join the cool kids and go back to Guess and Check.

“But wait, Meg, how can you be all hip and cool and use ‘guess and check’?  I mean, that doesn’t sound very mathy at all. I bet you don’t even wear pink on Wednesdays.”

Hey, Mean Girl, it’s not just guess and check….it’s educated guess and check! It’s like finding factors of 111…you could try every number from 1 to 111, but if you’re smart about it, you know not to try 2, 4, 5,… so you can focus on the ones that at least stand a chance!

True story: One year I taught Algebra II both guess and check and slide/divide. Those that had the least math skills chose slide/divide, but then they would get a really big number for ac that had a bajillion factor pairs, and since they weren’t great at arithmetic to start with, they couldn’t choose good number pairs (“hmm, I wonder if 2 and whatever 168 divided by 2 is will add to equal 13? Better break out the calculator and punch. every. button. so. slowly. so. very. very. slowly. Huh, wow, didn’t work. Ok, what about 3 and…”) so it took them way longer to guess and check ac than educated guess and check process. AND THEN THEY STILL FORGOT TO DIVIDE!  Also, in Precal, I have kids that are in love with slide/divide and then slowly but surely, they’ll come in for some extra help….”so could you show me your method again?”

Ok, so here it is:

Factoring 6Factoring 2 Factoring 3And, no, it’s not like our first guess is right every time…Factoring at megcraig.orgbut it usually doesn’t take too long!

Here are the above charts in a handy word doc in case you want to discuss with your department. I actually don’t do a NoteTakerMaker for these, because another big secret is to USE WHITEBOARDS or some other dry erase surface. Unless you want to hear them complain about the guess part of it all day long.

Also, I think next year I’m going to teach a equal and not equal to 1 the same day. If it’s equal to one, awesome, I can lock that in!  Maybe that way they won’t freak out as much when a doesn’t equal 1?  Because full disclosure: yes, mine still complain when a doesn’t equal one.

Let’s round out this post with some more Quadratics (part of my summer goal to get all of my resources online, see more on this page!)

Factoring homework (hint: I sometimes use the first problems as examples in class, then tell them they get to start on #____ or assign just the odds, with evens for optional practice).

Factoring 4File here.

And my first day of factoring review:

Factoring from megcraig.orgFile here. Next year: save grouping for polynomial chapter.

Now let’s solve these puppies!

Factoring from megcraig.orgFile here. (I can’t seem to find my solving by square roots, but I do teach it! Promise!)

And now let’s solve some with complex answers (I usually wait and do complex number operations later–it’s a good “oh, here’s three days before break” section that can really go anywhere in the year)

Factoring from megcraig.org

My favorite annual quote: “So, what’s this backwards j?”

And then the quadratic formula (I save completing the square for converting to vertex form; see it in this post.).  And now let me reveal Jim’s (@mrdardy) awesome quadratic formula manipulation:

Factoring from megcraig.orgIt’s only 1,000 times easier to simplify AND you can still sing “Pop Goes the Weasel” because it is still “all over 2a” AND whoa look at that vertex just pop out!!

And here’s some practice (6 to a page!):

Factoring from megcraig.orgFile here

And the study guide.

Factoring from megcraig.org File here.

Now go forth and spread the news of educated guess and check throughout the land!!!

Like a Parabola…

Alternate title: A Week in the Life of an Algebra II Teacher

Like a parabola, my quadratics graphing unit started off going downhill fast, then made an unexpected upturn at the end.  Warning: this post is also to remind myself how to make it better next year, so there’s a lot of second-guessing and rambling.

I knew I wanted to talk about the three forms you see when graphing…standard, vertex, and intercept/factored form.  I also needed to  talk about completing the square and wanted to throw some applications in there.  So here’s how it went:

Monday: Gave them this investigation activity that I think I stole from a book.  Thought, “hey, they liked investigating absolute value graphs with the calculator, so this should go well.”  Out of the mouths of babes…

Quad patterns worksheet pg 1 quad patterns worksheet pg 2Files: First page, Second page

After handing back and going over tests, we had maybe about 30-35 minutes to work on this in class.  If I had planned this earlier than Sunday night, I could have tried to get the computer lab and used desmos, but as it was, it was kind of difficult for them to figure out which graph was which.  (yes, we talked about the trace button and then they all put the y-intercept as the vertex on #2 because that’s what x- and y-values came up on #1.).  They finished to the first chart on the second page and I told them to finish the rest for homework.  Well, since the rest involved doing something that was not specifically taught step-by-step in class, they all gave up.

Tuesday: Hey guys, let’s talk about what a college-ready course is and is not.  I’m sorry you’ve been spoonfed math for the last ten years.  I’m sorry you think you can’t do this.  I’m sorry you’re so scared to fail you won’t even try something.  I’m sorry you didn’t even realize that just like you did for the last FIFTEEN graphs, you could have put your equations in your calculator and checked them.  I’m sorry you think I’m the worst math teacher in the world, but you’re stuck with me until May so deal.  I discussed finding the a value given a graph a little bit and repeated this for at least two questions on the bellringer for the rest of the week*.  About 25% could write the equation correctly on the quiz.

*Side note: Maybe I’m doing that part wrong. I used to do the ol’ plug in the vertex and another point and solve for a, but that always turned into:

y = a(x – h)^2 + k

3 = a(5 – 2)^2 + 6

3 = a9 + 6

3 = 15a!!!!!!!

Plus I wanted to make a connection about how the a affects the graph, because I don’t think they see the connection between “solving this equation for a” and “what does that mean about our graph?”

So I tried this way:

To graph, we make our standard t-chart and multiply the y column by a:

3xSo let’s solve this as a puzzle, let’s say in my graph I can tell that when I move over 2 from the vertex, I go down 8:

findaWhat would we need to multiply the y (or x squared) column by to get that value?  I personally loved it, as you can tell by the % correct, the students not so much.  I think maybe everything was a bit rushed and we should have worked on it after the following instead.

End side note, returning to Tuesday’s lesson…

We formalized graphing in vertex form and completing the square using this note-taker-maker.

graph quads vertex formgraph quads vertex form pg 2

(file here) I was impressed with how well they were doing.  Probably 97% got vertex form right (2 or 3 still multiply the x’s by the a value instead of the y/x squared), about 80% got completing the square completely right, 17% silly errors, 3% well, that’s an interesting take on completing the square.

Wednesday: Oh, so it’s going well?  Hahaha, let’s fix that.  Applications!

quad apps

(file here) Wednesday is our short period (47 min versus 52 min on T/R) so I didn’t get to show the Vomit Comet video (especially when you factor in the 5 minute lead time I need to open a web browser on my computer).  One kid was really into it and started asking all sorts of science-y questions that I had to come home and ask Mr Craig about, so that was kind of fun. We had time to do get through problem #1 in most classes.

Thursday: Finished up #2 as a class, omitted #3, then let them loose on their VNPS (vertical non-permanent surfaces AKA whiteboards) for the rest.  And every single team found the vertex when they asked for the number of tickets sold in 2005.

Me: Was that when the maximum number of tickets were sold?
S: Yes?
Me: How do you know?
S: Because we used -b/2a
Me: What does that tell us?
S: The x-coordinate
Me: Of what?
S: The vertex
Me: So what does it tell us?
S: When the max was sold
Me: Did you get 2005 for the x-coordinate?
S: No
Me: So do you think the max was in 2005?
S: Is it because we forgot parentheses in the calculator?

Even though it was a long period day, some students still had to finish a couple for homework, including the mixed graphing at the bottom.  Let’s just say my idea to throw in y = 2x-1 was not met with enthusiasm.  Or knowledge of a line. (I may need to use @stoodle’s function spotting guide with them!)

On the test, most students were able to find the maximum and when the object landed.  However, on the test the x’s represented feet away instead of seconds and that threw a lot of kids (as in, many students told me the max was the x value and it occurred at the y value).  Something to focus on for next year–maybe try it without numbers or solving first? Maybe talking about what f(6) = 15 would mean in context?  I also found myself using the reverse of Sam’s phrase of “turn what you don’t know into something you do know”: “how can you use what you do know to find what you don’t know?” (Another side note: we were doing the same idea last week in precal with parametric equations and that idea came in very handy!  Know the x?  You can find t!  Then you can use t to find y!)

Friday: One day left before study guide day.  I wavered back and forth a lot on what to cover.  The other Alg II teacher wasn’t going to be able to cover factored form, so I thought maybe just focusing on standard.  But we already covered finding the vertex and converting to vertex form.  So I did a little searching and found this awesome foldable from High Heels in High School (oh, I remember when I was young and could wear high heels every day.  Those were good times.).  I ended up modifying it a bit by (a) not making it foldable and (b) changing up “axis of symmetry” to “how to graph.”  Here’s how mine ended up looking:

Then I found this TOTALLY AWESOME activity from the Mathematics Assessment Project that involved matching equations in the three forms to graphs (I skipped all the rest of the stuff in there and focused on the dominoes).  The dominoes print nicely 2 to a page, but next time I think I will make a matching worksheet–have some of the graphs/equations already printed, you have cut out equations/graphs to match.  Reason being when the students (and myself!) started to fill in the blanks for the equations, it is tempting to write for the graph that is on the card, not the matching one, which led to unnecessary frustration.  All the students had them matched before they left class, some had completed the filling in the blanks as well.  They were really engaged and I heard some great discussions.  The best thing I did was NOT print off the answer key, so I could not answer their questions with anything other than, “why do you think that?”

Which also led to this conversation in our study hall period:
S: I know I’ve made a mistake because my two ends don’t match.
Me: Oh, that’s a good catch.  Did you work on trying to find your mistake?
S: No, I’m asking you.
Me: Oh, see, I don’t have the key yet, so maybe you can go through them again and see which aren’t matched up correctly.
S: I’ve already matched them but then ends don’t work.
Me: Right, so that means you must have incorrectly matched some in the middle.
S: I know. That’s why I’m asking you.
Me: Yes, but I don’t know the right order either. So did you go through them again?
S: I’ve already matched them.
Me: And we’ve decided you must have missed one. I bet if you sat down and went through them you could find your mistake.

That needed a quadruple facepalm. 

Monday: I quickly went over the answers and blanks but they didn’t see anything more about it on their study guide (since the other teacher didn’t use this activity or get to discuss factored form).  They did have to graph one in standard form without converting to vertex form (well, I didn’t explicitly say that so I didn’t count off if they did) and a majority did well finding the vertex and y-intercept. I also put one fill-in-the-equation-blanks-based-on-the-graph like the dominos had as a bonus.  About 50% tried it (there was not a time crunch at all) and maybe about 15% got every form right.  This makes me sad; I KNOW more of them could do this because they were doing it in class in front of me!!  Why won’t they even try?  I also feel as soon as I said (on study guide day) that these types of problems wouldn’t be on the test, they completely cleared all that information from their memory.

Tuesday: 50 point quiz.  About 1/4 of the kids were out in the two of three classes (although many stayed just to take the quiz and then checked out; I even had one girl check in 8th period to take it!).  The grades were at either end of spectrum–either A/B or F, although none of the Fs were horrible.

I think next year I would like to (a) use Desmos to investigate graphs and have them practice writing equations to match graphs on there.  (oh, could I take a picture of a quadratic on desmos, then import the picture so they couldn’t see the equation?  Then have them try to match it with their own equations?) (b) do a real-world quadratic modeling activity (basketball three-act?  Mathalicious?)  (c) spend more time on the dominoes activity (d) maybe start the week with the foldable and fill is as we go. But does all that turn what used to be a week lesson into two weeks and can I afford that in my lesson plans?

Other questions I have:
Why do students hate graphing as much as Dracula hates tanning beds?
Am I trying to do too much in this unit?  Too little?
Should I expect them to remember more from Algebra I instead of starting from scratch?
Did I really just type over 1,500 on graphing parabolas?
Did you really just read over 1,500 words on parabolas?
What else would you like to read about on here?