Day 3: Histograms, Trigonometric Functions, and Function Transformations

Three days into the school year, my students and I are still getting used to each other.  Some classes have figured out that I prefer dialogue between teacher, student and other students.  Other classes have not started to trust that idea.  I’ll continue to encourage them to work together.

AP Statistics:

Today’s goal was to appropriately display quantitative data.  Most of the conversation focused on making histograms, stemplots and dotplots.  We did collect some interesting data though:

1.1 Histograms_3 1.1 Histograms_4

All of the data points were between 52 and 70 seconds.  Not too shabby.  Today was also the first day of AP Exam prep (even though they didn’t realize it).  Once we listed all 20 of the times from the class, I stepped back and said “now describe the data.” They had to work together to get all of the aspects they needed to describe: shape, center, spread, outliers/gaps/clusters.  They had some great conversations too!

Precalculus:

We started class connecting what they learned about converting from degrees to radians and how to calculate arc length (with angles in both degrees and radians).  Man, the connections that were made and the speed at which they picked it up was impressive.  I’m really excited about their potential.

Speaking of their quickness, we defined three new trig functions they hadn’t worked with before: secant, cosecant and tangent.  Rather than drill methods to solving problems, I threw this with them without any hints:

1.2 Trigonometric Functions Lesson_5

The ease in which they figured out to draw a right triangle, and use the Pythagorean Theorem was great!  I figured this would challenge them at first, but I was definitely wrong.  I definitely need to step up my game.

Algebra II:

We’ve begun to go deep into the world of functions.  I’m noticing that some of my students are struggling to see the big picture.  In today’s activity the focus was supposed to be on how changes in a function affects its graph:

1.2 Transforming Functions Day 1_2 1.2 Transforming Functions Day 1_3 1.2 Transforming Functions Day 1_4 1.2 Transforming Functions Day 1_5

We ended up so bogged down in the details of the order of operations and plotting points that we lost sight of the big picture.  Hopefully their homework tonight can help re-focus them.

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How Far Does Mr. Cloud Have to Run

I wanted to give my students a way to define an angle of depression for themselves. I found an interesting context involving finding a vertical distance given a horizontal distance and an angle of depression. I found it easier to switch which piece of information I gave them, but it’s practically the same idea.

The weather here today is pretty terrible. It’s been raining, is raining and is supposed to continue to rain. So I used that as the opener:

7.1 How Far Will Mr. Cloud Have to Run_1

Then, I gave them the situation they were to work through:

7.1 How Far Will Mr. Cloud Have to Run_2

7.1 How Far Will Mr. Cloud Have to Run_3

Then came the question:

7.1 How Far Will Mr. Cloud Have to Run_4

The students gave their guesses and we were off. A student pretty quickly gave the idea that a triangle would come in handy and two other students asked for my height and one of the angles (you’d think we’d been doing trig ratios in class all week).

7.1 How Far Will Mr. Cloud Have to Run_5

Getting the students to figure out what the angle I gave them represented in the picture was harder than I anticipated. I had a student tell me it was the angle from the other side of the courtyard to my eye line was the angle (which is equivalent) but couldn’t tell me why.

Once we figured out what the angle was on the picture, we ended up here:

7.1 How Far Will Mr. Cloud Have to Run_6

Overall, we came to the definition I wanted and each group did well in working the problem out, but this problem too quickly turned into an exercise with pretty pictures rather than a great discussion about angles of depression. I’ll have to tweak this one next time.