Tag Archives: linear functions

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:

exp 1

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.

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

I usually try not to reward promoted links on twitter, but this article about students asking questions was too tempting not to click.  I read it Thursday night and then tried it out Friday morning in PreAP Precal where we were preparing for a test on Monday.  I handed each student a piece of scrap paper and let them have 3 minutes to jot down any question they had–it could be general (“how do you write an equation for a tangent graph?”) or specific (“how do you do #11 on the study guide?”).  Then I gave them 6-8 minutes to try to answer the questions in their groups–I saw A LOT of good peer helping at this stage!  Then I collect any remaining unanswered questions and answered them

…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.

Capture1(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.

Capture4 Capture3(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.  🙂