## Alg II Files: Systems

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(See more Alg II files and FAQs here)

So this was an interesting chapter…one that I think I improved from last year, but could still have used more connections and also some more activities (e.g. double stuf oreos wafers and creme).

It started with a great discussion:

(hey guys, hold out for a couple more chapters when I got a new phone that does pictures a whole lot better! Sorry for the random quality until then.)

(file) So, guess what? I had been teaching the types of systems wrong for a long, long time! To recap from that post, the correct way is:

Category One: Is there at least one solution?
Yes: Consistent  No: Inconsistent
Category Two: Are they the same line? (Technically, “can one be formed from the other with algebraic manipulation?”)
Yes: Dependent No: Independent

So the correct categorization would be:
Independent and Consistent: One Solution
Dependent and Consistent: Infinite Solutions
Independent and Inconsistent: No Solutions

Then we did some substitution, some new stuff here: being really conscientious on boxing equal things and also looky there in #6, doing substitution with quadratics!

(file with these questions and more practice)

and I even went a little crazy and did this:

(This may also be why so many of my students remembered how to expand (x – #)^2–we did them a lot of them throughout the year!)

Elimination.

I was pretty proud of the practice I prepared for the pupils:

(file) The last set was interesting when they didn’t choose the method I thought they would!

Did I hear someone say they’d like to see more textbook-like systems of equations word problems? Here you go!

(File with these & more practice problems) But I was able to add a pretty cool activity from Amy (@sqrt_1) where the students made their own word problems. The only thing I changed was condensing all the work onto one page for easier grading:

(instructions and worksheet) It was a nice day and a lot of the students had fun with it (how often do they get to break out the colored pencils and color? I also gave bonus points for the most creative one from each class and put my favorite one on the test!). I will say next time I will have them show their work that they tried it! (I said they had to do it on their notes, but not turn it in.)

Have I told you how much I like doing group speed dating?

(file) Then it was study guide time!

(file and video key part I and part II) As I said, a lot of room for extensions, activities, and connections in this chapter that I just didn’t have the time to incorporate. TBH, if it’s the night before the lesson, go ahead and use some of my stuff, but if you have time to plan, please go see all the more awesome things there are out there in MTBoSland!

Category: Alg II | Tags: ,

## Things I Thought I Knew

Some days this job can be a bit ho-hum. Teach a lesson you’ve taught 37 times before, grade a few tests, make some copies, go home. And then some days you get to learn math that you totally thought you knew, but you had no idea. Friday was one of those days.  I’ve been teaching long enough that I probably should have known this stuff, but I didn’t. So here I am to enlighten you as well. Just be prepared for the following to occur:

##### Topic 1: Classification of Systems

Am I the only one that is really bad at reading directions? But then the students actually read it word for word and ask questions that I didn’t see coming because knew what the question was really asking without having to closely read it. This week, we had a question about making an independent and inconsistent system, which I read as “inconsistent” and knew what the goal was, but then that pesky “independent” thing was brought up by students.

So I always thought there were three types of systems:
Independent: One Solution
Dependent: Infinite Solutions
Inconsistent:  No Solutions

TURNS OUT I was mushing two different categories together:
Category One: Is there at least one solution?
Yes: Consistent  No: Inconsistent
Category Two: Are they the same line? (Technically, “can one be formed from the other with algebraic manipulation?”)
Yes: Dependent No: Independent

So the correct categorization would be:
Independent and Consistent: One Solution
Dependent and Consistent: Infinite Solutions
Independent and Inconsistent: No Solutions

When you realize you’ve been teaching something wrong for your entire teaching career:

###### Topic 2: Logs grow really really slowly.

“Yes, Meg, we know that.” No, I don’t think you do. The wonderful MTBoS Search Engine led me to this GREAT activity from @Johnberray. I modified it a bit, first of all by making some pretty 1-inch graph paper:

(file here) Each group got a sheet in a dry erase pocket and used their calculator to find the decimal values to graph y = log x. Then I asked them to calculate how many inches it would take for it to function to reach 2, 4, 6, and 8 inches high. (At least one group each period said 20, 40, 60, 80 so we may not have the idea of logs down pat just yet). Then I asked them to describe each distance in some way–is it the size of a pencil? classroom? from here to the front office? as tall as basketball goal?  I let them use the internet and a student showed me that if you put a measurement into Wolfram Alpha, it will give you real world comparisons! Pretty neat!

You really need to try it for yourself!  What fun we had when we started talking about how long the paper would be for a height 6, 7, and 8 inches (oh, plus go ahead and try 9!) using this draw-a-circle-on-a-map site that John had linked to. Then it turned into, how high will it be if the paper reaches the moon? The sun? Pluto? I won’t give it away, but guys, logs grow really really really really really slowly.

Yet they still go to infinity.

###### One raised to any power is equal to one, oh, unless that power is infinity.

So I’m pretty good with limits. Then my Calculus teacher friend mentioned, hey, isn’t it weird that the limit of (1 + 1/r) ^ r is e, not 1?  Wait, that doesn’t make sense.  But it turns out 1^infinity is indeterminate. WHAT? But I thought 1 to any power is 1!  WHAT IS HAPPENING TO EVERYTHING I THOUGHT I KNEW?  The best answer I saw was that “we don’t know if the one is really a one”, which ok, I guess, since 1 + 1/r is never exactly one. But what if the number IS actually 1? Not like “kind sorta” close to one but ONE. If 1^infinity isn’t one, at what number does it STOP being equal to one?

I was discussing all this mind-blowing math with my friend right as class was starting, complete with dramatic re-enactments of how high the graph of log will be on Pluto and what was happening to my brain. When I came into class, a student asked what we were talking about and I said “Math!” and she said, “You really do like math a lot, don’t you?”

Yes, yes I do. 🙂

Category: Alg II, Precal | Tags: , ,

## Algebra II Files: Systems

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

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

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

I like this idea of an introductory activity as well:

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

We then do some formal solve with graphing:

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

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

Some homework:

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

File here. And some linear inequalities systems:

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

Let’s stop here and have a quiz, eh?

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

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

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

Here’s the next day:

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

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

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

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