## Hey, Make This Exponentially More Awesome for Me, Okay?

Monday I spent a couple hours falling into the #MTBoS trap of lesson planning: having so many shiny pretty ideas that I can’t decide what to do! I was trying to figure out how I wanted to start my exponential unit for PreAP Precal. Yes, they’ve seen it before, but I didn’t think they had a concept of EXPONENTIAL BEATS EVERYTHING (I know I really didn’t until my [REDACTED] year of teaching math.)  So this is what I came up with:

You just started a new job. Pick the best salary option and be prepared to support your opinion:
Option 1: \$50,000 a week, increased by \$5000 each week.
Option 2: \$100 times the square of the number of weeks you have been working (I didn’t really know a better way to describe this?)
Option 3: Start at \$10 a week, increase by 10% each week.

It was a weird day yesterday with some classes half-full due to class meetings, plus I had some tests to go over, so we only had about 25 minutes to work on it. I let them have 10-15 minute to discuss in their groups. Many people started with a table, which is quite deceiving at the start. A few groups finally asked, “how long are you working there?” To which I replied, “That’s a good question, how long are you working there?”

Most decided to focus on one year, and thus chose option 1. A few ventured out further and chose option 2. (Maybe next year I’ll make so option 2 overtakes option 1 just before year one instead of just after?) Most could not figure out an equation for option 3 (which didn’t bother me, especially when they haven’t seen exponentials in over a year), so just crossed that one out immediately based on the first few weeks.

After each group gave their reasoning, we worked out a table on the board, starting with 1, 2, 3, 4, 5 weeks, then figuring out the equation for each week. I then gave each group a time frame-1, 2, 3, 5, 10, or 20 years and had them figure out the weekly salary for each option.  The bell rang right in the middle of posting the results, but we still had fun talking about making \$10^22!

Today I showed them the graphs in desmos and we talked a bit about them:

(Desmos file here) But I think we FINALLY got the power of the exponential when we put it in table form:

I mean, look at how slow both the linear and quadratic are growing. 10^8 after 40 years? That’s not even worth getting up in the morning for! I also wrote out the final numbers on the board while they were working on the next task, using all 87 zeros.

Yup, I think they will say that exponential will win every time now. 🙂

As we were working on it, I thought of a lot more things we could extend with:

• Figuring out the time to switch by solving a quadratic (option 1 & 2) or by using technology–either Desmos or using Excel?
• Have them write an actual recommendation of which salary to chose and why.
• For the first couple of years, students were wondering if making so much at the beginning would make you have more money at the end of the year with the linear. What a perfect way to bring in area under the curve! Especially because they could actually calculate the linear function’s area with just one trapezoid, then I was thinking just to use the integral function on the TI for the other two.
• Of course it could also be a nice lead-in to logs: when will each salary hit \$100,000? \$1,000,000? \$10,000,000?  (Also nice to look at graphically!)

So this post is serving as my reminder to myself to devote some days to this next year, and try to do some of these extensions. But if any of y’all want to try it out and make it awesome as you are wont to do, please do and report back! I’ll be chillin’ with my cool \$10^87.

## 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:

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:

OKAY I WILL SHOW YOU AGAIN STEP BY STEP HOW TO DO EACH OF THESE.

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

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:

(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)

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:

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:

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:

(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?”
“….”
one meek voice: “(0, 8)?”
“Yes! So what should the x-coordinate be?”
“0?”
“Yes! So if I know the x is zero, how could I find y?”
“0?”
“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!)

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 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.

## Desmos Activity Builder Success

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This summer I was so excited when Activity Builder was revealed at TMC. About the second week of school, I made a really quick one that involved the students graphing their names with lines. I reserved the mobile laptop cart that our school has just for Math and Science to use, wheeled it into my room, and was ready for the excitement to begin.

Oh, wait, except my classroom is in some sort of wifi blackhole, so it was taking students 10-15 minutes to be able to log in to the computer and open the activity, and even then only about 4 kids got to that step. It was the Friday before Labor Day; I didn’t have any backup plans, so I just threw in the towel and they had a chill day. I, on the other hand was not chill.  I was Over Technology. I was at Unstackable-Cups-Otter level of Not Chill.

It took me two months to venture into trying it again. This time I reserved the computer lab, where the computers are wired to the Internet. It went much better. Here are the two activity builders that I made:

Algebra II:

Writing Equations of Transformed Parent Functions

We are graphing 5 different functions (lines, absolute value, quadratics, cubics, and square roots) and the students were having a lot of trouble writing equations from given graphs. Enter in Match My Functions Activity Builder!

Issues: Students still do not know how to find the slope of line. This makes me sad. So they just tried decimals and then were upset because it seemed “like random decimals for the slope.” At least they were trying something, I guess? Also a couple students said it didn’t help them because they could just guess until they got it right. But all in all I would probably do it again; they actually did very well on their test writing the equations and recognizing the functions.  Also, a student who is in two math classes bragged about how fun/cool it was to his other math teacher. 🙂 Plus when it came to study guide day, at least a few students broke out the app on their phone to check their equations!

PreAP Precal

Introduction to Polar Graphs

I had actually done this activity last year as just a regular Desmos file paired with a worksheet that asked them questions about the graphs. It didn’t take too long to create an activity builder based off of it. We had graphed polar points and graphed a couple circles by plugging in values of theta, but they had not seen any other type of polar graph. I set them loose (in pairs) to work on Polar Graph Exploration.

I had them play around with sliders for each graph; then submit “I notice…” and “I wonder…” On the more complex ones, we then looked at each part individually (like the slide above that just focused on the n value). Trying to learn from the feedback from Algebra II, I made a few “quick check” text questions where they couldn’t just guess (although some just went back a screen and played around with the sliders until it matched)

The fact that, with no direct instruction, most of them realized there had to be a 3, 6, and sin involved in the graph above is pretty amazing. I really loved the notice/wonder part as well (I’ve actually never used it before in class #MTBoSDirtySecret).  I always had an issue with the petals on a sine rose alternating between positive and negative y-axis. One student noticed that if it is positive, more petals will be above the x-axis. So clever!

I did get a few complaints on the feedback, “why didn’t you teach us?” but I’ve come to expect that.  The other complaint was not being able to know if their work was correct on the quick check. Maybe I should add a screen after that with the graph that says, “check your work” or “how close were you?” (but then I also don’t want to discourage them if they were wrong–I only wanted to see how much they had learned from the activity).  The best part was the last screen where I told them to have fun!  They made so many cool graphs, and then I was able to show them off the next day in class.

Now I know Desmos gets a lot of love around here, but let’s not forget about Geogebra, which is powerful and useful in its own right. For example, this beautiful, awe-inspiring, oh-so-that’s-why file that compares the two versions of a trig equation (made by Mark Fowler):

(Which I found the morning after we had done the Desmos activity, and now I’m debating if I want the Ss to play with this instead of the Desmos one! Or if Desmos could make this into a split screen activity with sliders that controlled both graphs?!?!) I used it as we talked about each graph, and was able to reference their responses from the day before, “Some of you were wondering what causes the inner loop…”

All in all, a fun week using technology in class. 🙂

## Hold Onto Your Hats: Algebra II Unit One

At the end of last year, my fellow Algebra II teachers agreed with me to try a modified version of Jonathan’s Pivot Algebra II. Basically turning a class I’ve taught some version of about 10 different years into an entirely new prep. Just like a new prep, I’ve had a lot of grand ideas that have resulted in some successes and A LOT of notes of what to modify for next year.

Side note: Because of some weird geometry class issues, my students are much more all over the board than previous years. I normally have about 90% juniors with 10% sophomores (who should be in PreAP). This year I have about 70% juniors, 20% sophomores and 10% seniors. And compared to previous years when 27 or 28 was a big class, these are 29-30. Also, we have 47 minute classes three times a week and 52 minutes two times a week.

Ok, let’s get started. The first real day of teaching, I thought I’d hit on all the mistakes students make, “Hey, if we learn this now we won’t make them later!”  Um, yeah, what that turned into was “Hey, these are all the things you struggle with and are hard to understand and remember, so let’s talk about them all on the first day!”

Starting with this card sort. Determine if these are true or false.

Oh holy cow. This went horribly. They had no idea how to determine if they were true or not. I suggested using a calculator to check, but imagine them trying to put some of these in a calculator. Frustration mounted. Then we checked and I was trying to use this as a big “ta-da” moment to introduce the GERMDAS house that Kent Haines suggested (more about this below).  “See, we don’t have to memorize all these different rules!  We can just use this one big rule!”

The worst part was, after a few example problems, I never really went back and used those rules again with the types of problems they saw on the cards. I don’t know if I’d just scrap the entire idea next year or spend a day just focusing on those types of problems, or what. I’m also getting really frustrated with the fact that it’s like we’re starting from scratch every year in math. Should I really have to be covering order of operations again?

But do take a moment to embrace the beauty and simplicity of the GERMDAS house. I’ll let Kent explain it:

I showed it to the other teachers at school and one of them who works with lower level kids said the kids really liked it and understood it. Another teacher is planning to show it to her Calculus students (and she showed it to her 5th grade son as well with a really good explanation–when you clean a floor of a house, you don’t just jump from room to room to room, you go in an order. So don’t jump around to addition and then subtraction, but just clean up in order!).  Thanks again, Kent!

And because throwing all that at them wasn’t enough, let’s practice plugging in values into the un-simplified and simplified version and see if we get the same thing!  Or next year, I’m thinking probably not because it was just too much going on in one problem. Instead we’re going to spend a day on the difference between 3 – (x + 7); -3(x + 7), 2(x + 4) – 3(x +7) and 2(x + 4) – (x + 7). BECAUSE SERIOUSLY WHY IS THIS SO HARD. DID I MENTION THIS IS ALGEBRA II? Then I would spend a day on the last third of the notetakermaker: with a caveat that after today, if anyone tells me *I’m* wrong because my answer doesn’t match theirs because you didn’t put parentheses around a negative when squaring, then you have to do ten pushups. Because I’m over it.

Ok, so now we’re onto day three in what actually happened. It’s like when I was planning this unit, I got caught up in all excitement and forgot everything I’ve learned about teaching math. Because I thought what would make things really exciting would be to introduce a new, somewhat confusing (it turns out) method to solve equations, as well as adding writing out tedious steps! It’s going to come as a shock, but it did not go over well. (Note: this NTM ended up taking two days)

(No file available and I don’t really want to talk about why.)

After the entire chapter, I am IN LOVE with the flowchart method. But a lot of kids struggled with it to begin with (I think it was “so easy” for me to see that I didn’t take into account that (a) they don’t know the order of operations (see day one) and (b) it is something completely new that needs some soak time).

Learn from my mistakes! Here’s how I’m going to go about it next year:

1. Save the variables on two sides for a different day.
2. Save writing the justification for another day.
3. Work on just the flowchart for a few problems.
4. THEN say, well, we’d like to make this a bit more “mathy” and “formalized” so here’s what we’re going to do: draw the top half (the building part) of the flow chart in short hand, with just the arrows and operations above them, and not each intermediate step. Now work with the actual equation algebraically, check off each arrow as you go. That epiphany came to me at the end of the week, and it seemed to make all the difference. They were a bit frustrated (rightly so) that I was making them solve the equations twice (oh, and write out each step). But they didn’t mind drawing a little flowchart to help them. And again, the flowchat is MAGICAL. Just wait until the next post when we use it for 5+3|x – 2| = 14. (Or read more about it here)

If you’re still not sold on the flowchart’s worth, just check out these literal equations:

I *almost* want to make them solve these with just the flowchart. I would say the students got them right 90% of the time with the flowchart. It was beautiful. One type of problem I need to add is something like x/b – a = c, where they can see the need for parentheses for the answer: c + ab was a common answer on the test instead of (c + a)b.

Speaking of test, I decided to stop here, review and quiz:

Also, I’ve been trying mixed and lagging homework to mixed results. The biggest problems are (a) at the start of the year there was only so much to mix in (b) making homework for the entire week, but getting behind so it’s not lagging anymore and (c) students really lagging their homework, i.e. only doing it the day before the quiz. But I guess it makes a good review that way! Here is the homework for the week:

Ok, so that was just the first half of Unit One but I think I’m all reflected out. As always, any comments, questions, or suggestions for improvement are welcome in the comments or on twitter!

Read about the awesomeness that is Radian Fraction Cutouts in Shireen’s original post (and my implementation). After using them in Algebra II last year, I made one little modification that made them even better (one of last year’s students that I have again even said so!):

File here.  The secret? Cut only on the dotted lines!!!  So easy to see the whole enchilada (or em”pi”nada as Shireen also says) divided up into sixths, thirds, fourths, etc, then seeing if you add one, take one away from one, or take one away from two.

Comments I heard from the first class I showed it to (the others will see it tomorrow):

“This is so easy!”

“OH THIS ACTUALLY MAKES SENSE NOW!”

“Why did no one teach it this way earlier?”

“These are really great!”

Not one person wanted to convert radians to degrees!  Woot woot!!!!!

Timing wise: In one 53 minute class, we cut them out, made a pocket (fold a half sheet into thirds [like a letter] and tape down the sides to the back of the NTM, leaving the top flap untaped), talked about degrees, talked about radians, and finished the front of this notetakermaker. The back has more practice that they had for homework:

DO NOT LET THEM LABEL THE UNIT CIRCLE!!!  Otherwise they won’t have to work at any of the rest of the worksheet. 🙂

And since it is the first day all week that I’m finished will all my homework before 9:00, I’m going to go read a real book!  That’s not about math! Or teaching! Take that, SCHOOL!!!!!

## First Two Days Reflection

I actually started on the 13th, but am just getting around to blogging about it because I spent all last weekend making lagging, spiraling homework as well as activities that were not worth the effort I put into them (but that’s a topic for another post).  I’m teaching Algebra II w/ Trig and PreAP Precal, but I did the same thing in both classes for the first two days. I’m also adding a rating system from 1 (that sucked) to 10 (that was awesome)

Thursday:

• High-fived all students and checked off names/got nicknames as they walked in. Rating: 9 due to difficulty in multitasking, also weird looks from students
• Had them fill out google survey. Besides name, nickname, class period, these questions are also on there:  Rating: 8 I thought these kids could text fast, but it always takes sooooo long.
• Told them a bit about myself. Rating: 10 because I’m awesome.
• Hit the high points of the syllabus, showed them how to get to my google doc that will have links to everything we do in class SO DON’T ASK ME WHAT YOU MISSED. Rating: 5 I mean, it’s a syllabus.
• Played the first day video from youcubed’s week of inspirational math. Rating: 7 Couldn’t get much discussion out of them after it, but it was first day.
• Continued with youcubed’s day one activity of writing group norms. Now I just need to make them into a poster. Rating: 8 Got everyone involved and talking. Got some good descriptors: “Open-mindedness” “Listening” “Optimistic”
• Spent the rest of the day with the Four Fours activity, also from youcubed’s suggested first day. Basic idea: use four fours and any math operation(s)/symbol(s) to make all the numbers from 1-20. Rating: 10 Almost 100% participation the whole time. This would also be a great starting activity for order of operations. Two classes had enough time to get all 20!
• Homework was reading the Make It Stick handout and also having parent fill out google survey, here’s the good part of it:Rating: 7 Only about 75% of parents have filled it out as of yesterday. On the other hand, one parent actually wrote in the additional comments section, “Thank you for a great and innovative syllabus experience.” The “I’m proud of my child because” question is great to refer to in parent meetings.

Friday:

• High-fived everyone again. Rating: 10 Pro tip: If you’re setting stuff up between classes as kids are coming in, just go around the room quickly and high-five them before you stand outside the door.
• I had them fill out their math goal and find accountability buddies. Rating: 3 Almost every single student just wrote “make an A.” I also haven’t had time for them to check in with their accountability buddies. Would not do again, or maybe wait until a few weeks into class.
• Up next was paper folding from youcubed.org.  Rating: 2  I would not do this activity again. After the quick success of the first two (fold a square into a square that is 1/4 of the original; fold a square into a triangle that is 1/4 of the original), the next ones amp up the difficulty by quite a bit (fold a triangle that is 1/4 of original square but not congruent to first triangle; fold 2 different squares that are 1/2 of the original). Also, when students thought they got it, instead of convincing their partner, they would call me over and ask me if it was right. Quite a few students embraced the challenge and kept on folding, but for many of them, the frustration (and maybe pointless-ness?) was just too great and they quit. I’d be interested to hear if those teaching younger students have more success. And I still don’t know how to do the second square oriented differently from the first that has 1/2 the area of the original. I thought quite a few of them had it but upon trying to convince me, they didn’t.

I hope to back soon with a recap of the first real week of teaching, but after working on math for the last two weeks straight I need an afternoon of not thinking. At all.

My thing: It has been a while since I share a favorite thing, so today I’m going to absolutely amaze you with CamelCamelCamel.  This is a price tracker for Amazon and works in three different ways: 1) Connect it to your Amazon wishlist. It will automatically alert you via email when the price of something you want has gone down. You can also add individual items on the website (great for tracking stuff from other people’s wishlists for gifts) 2) Use the browser plug-in or copy the amazon URL into its webpage to see the price history. Great to know if that \$40 price is just a high mark, or if it’s been \$40 for the last three months, or if it wavers between \$30 and \$50. 3) Browse their list of popular products for “good deals” and “best prices.” Two things it doesn’t do: it does not alert you to lightening deals (but I usually get an alert if its a an-day daily deal) and it also cannot track kindle book prices. But I still think it’s a great tool!  And you could probably do something really mathy with the historic price charts, too.

## Alg II Files: Rationals

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My second-to-last post for my Algebra II files! (See more files and FAQs here)You can tell by how long I’ve procrastinated that I’m not really a fan of what I do for this chapter. Each year seems to get a little bit better, but I would never classify put it in the “win” column. In other words, feel free to take any of this and make it awesome. Then let me use it. 🙂

The first few sections don’t have note taker makers, because the problems need space to be neat and tidy:

Some notes on my notes: At the very top, it’s hard to see, but we start off talking about how we reduce 12/15. We don’t subtract 10 from each and say it’s 2/5! We talk about invisible ones.  This year, I want to focus on the fact that in the expressions, I can substitute any value for x and you will get the same value in the beginning expression and in the final expression. Magic!

Also, I talk about restrictions as “warning the villagers.” What if someone came along and put 5 into x/(x-5)? THE WORLD WOULD ERUPT INTO FLAMES.  And sure, I know *you* wouldn’t do that, but what if some innocent villager came in off the street and started putting numbers in? We need to warn them! This year, I also want to pull in the graph and talk about how the original has a hole at 5, so in order to be equivalent, the reduced one also needs a hole, which we make by excluding 5 from the domain. (I *just* realized this fact in the last few weeks.)

Some homework:

Then…can you hear the jaws music….

Then some practice with every operation as either a worksheet

or as a step-by-step powerpoint (great for whiteboard practice)

Let’s solve some now!

I’m not *in love* with just crossing those denominators out, but haven’t figured out anything better yet.

Some practice:

(file here) And a study guide for these three sections:

(file here)

Some inverse graphing:

(file here). Hint: make a t-chart by listing pairs that will equal k. Super easy!

Inverse/Direct/Compound variation:

Powerpoint to go along with NTM:

(file here)  Yes, I’d like to do a lot more with this, but this chapter is always rushed since it’s the last one before we have to start trig at the end of the year.

Homework:

Then it’s time for graphing rational functions. I like what I do, but I think I want to streamline the process a bit–we don’t have to do all of these by hand!- and bring in some of Sam’s stuff.

We spend the first day talking about end behavior and points of discontinuity:

(file here) Why, of course there’s a powerpoint!

Then the next day we review the first two steps, then jump into graphing them. This goes a lot better than just throwing everything at them in the first problem!

And yes, another powerpoint:

(File here) There’s even a powerpoint to check the homework that’s at the bottom of the NTM-file here!

Then it’s finally time for the study guide (note: this does repeat problems from the first study guide!)

## Alg II Unit One #SMPTargets

At the end of last school year, I convinced my fellow Algebra II teachers to write out a pacing guide following Jonathan’s (@rawrdimus) “pivoted” Algebra II sequence. I must admit I’ve been teaching a version of Algebra II for so long that I’m often leave that on autopilot while focusing on newer courses. With this new sequence and some new goals for the year, I’m hoping to change it up a bit this year while still not going crazy.

With that in mind, I met with our math coach and the other regular Algebra II teacher yesterday (he’s a brand new teacher so I hope to mold him into #MTBoS ways) to write some learning targets. Added goal/twist: make sure to cover an Standard Mathematical Practice within each one, a la Chris Shore.

If you haven’t downloaded Chris’s SMP posters, you need to do so now. They are listed under “my stuff” in the left column of his Math Projects Journal. I’m going to copy them two to a page and hand them out to students. I think it will be four pages that are going to get a lot of use during the year! My favorite part about them is the questions that he listed under each practice–really helps to clarify them for students (and, ahem, some teachers that couldn’t even list all 8 before his session).

So now that you have your practices ready, let’s put them into action with the first unit of Algebra II:

Alg II Unit 1 Learning Targets

1. I can persevere in evaluating numerical and algebraic expressions using order of operations.  Ms Craig will persevere in not losing it when told for the 4,793rd time that “my calculator says -3 squared is -9!”
2. I can reason abstractly and quantitatively when translating verbal and algebraic expressions.  Who is up for some contextualization and decontextualization? And by that I mean, who has a good activity for Alg II students about translating back and forth?
3. I can construct viable arguments and critique the reasoning of others when solving linear equations.  My plan with this is first day have them write out each step they are doing and/or do a flowchart equation. Then the second day do a “find and correct any mistakes with an explanation.”
4. I can model with mathematics when solving absolute value functions. I really want to bring in Kate’s exploration of absolute value.
5. I can look for and make use of structure to solve power equations with inverse operations. (i.e. solve an equation that has just an x squared or cubed)
6. I can attend to precision while solving equations with roots on both sides. I thought that would be good to talk about square roots in calculator and checking for extraneous answers.

So now I have a really stupid question that I will probably post on twitter as well: what can I do so I don’t have to write these out on the board every day? I really don’t have the space or the motivation to do so. Should I print them out and post them? Even if that means 180 sheets of paper? Times two classes? Give them to students as a typed worksheet so they can calibrate their learning as we go through the unit? Type them at the top of the day’s handout (if there is one)?

Category: Alg II | Tags: ,

## Epiphany Part II: The Return of the Tranformation

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

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

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

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

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

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

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

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

I really think you should be sitting down.

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