Day 2: Displaying Categorical Data, Functions and Radians

It’s Day 2 and I’m tired.  I need to work on my conditioning.  My lessons were a bit dryer than I’d prefer today.  Here are some highlights from today.

AP Statistics:

The goal today was to be able to summarize univariate categorical data (and start the conversation about summarizing quantitative data).  The lesson started with having my students collect some categorical data and telling them organize/graph the outcomes.  Every student chose to make a bar graph.  We discussed features of a bar graph and its advantages over a pie chart.  The most interesting part of the lesson, however, happened with this slide:

1.1 Bar Graphs, Dot Plots, Stem Plots_6

We used remote responders so I could get instant feedback from the class and found that 80% of them missed this question (the answer is E by the way).  After discussion, we came to the consensus that there was a reading issue.  Whether they read too quickly, or not carefully enough, I need to keep an eye on this and help with their critical reading skills.

Algebra II Honors:

In class today, we had a crash course on everything they should know about functions.  Discussions included domain and range, with proper notation.  Interestingly enough, they did struggle with the domain and range of the triangle here:

1.1 Intro to Functions lesson_4

They wanted to tell me that the triangle had three points.  After quelling that misconception, we realized that we can write domain and range using inequalities.  I’m glad we came across that gap in their knowledge.

After domain and range, we discussed the input/output idea behind a function…and we got to watch one of my favorite educational videos.

The conversation we have during this video is really rich.  The nuggetizer really gets to the input/output idea without an equation.  good stuff.  and it’s entertaining.

Precalculus:

Today we explored radians.  The warm-up allowed them to review circumference and arc length.

1.1 Radians and Arc Length_1

I’m finding they’re a little rusty/apprehensive about fractions.  I need to make sure we get better quickly.

My hope today was that they would figure out the conversion to go from degrees to radians.  I found an intriguing activity in the textbook we currently use.  I modified to fit my style, but it has the same bones:

1.1 Radians and Arc Length_2

I was thrilled with how quickly they found that the s/r ratio is constant (in this case pi/3).  We defined that ratio as the number of radians, and my students decided that ratio measured the angle.  pi/3 is equivalent to 60 degrees.  Then we derived how to convert from one measurement to the other.

Algebra II Honors: Introducing Function Composition

In Algebra II, our second semester begins with function operations and inverses.  Since function composition is a very commonly used concept outside of the math classroom, I wanted to introduce the idea within a context.

So I opened the lesson with this picture:

6.2 Function Composition_4

The goal for my students was to determine how much I paid for gas the day before.  I gave them some information:

6.2 Function Composition_6

My students wanted to know how much gas was and then proceeded to spend 45 seconds commenting on how inexpensive gas is at the moment (I figured this would happen given that my a lot of my students are starting to drive).

Then I told them how many miles I had driven:

6.2 Function Composition_5

Now that they knew the price and the fact I had driven 117.2 miles, I wanted them to make a prediction.

I got anything from a $15 prediction to a $40 prediction.  I was surprised to see that when I probed them about why they predicted the price that they did, quite a few of my students used the fact that I was filling up about a half tank of gas.  Questions and comments started flying about how many gallons of gas my car’s tank would hold.  I didn’t have an answer for them, and I tried to question how that would be a useful piece of information.  They said that if they knew how many gallons the tank was in total, they could figure out how many gallons half a tank is and use that to determine how much I paid.

A conversation about how they didn’t know exactly what proportion of my gas tank was being filled.  They weren’t happy…they said it was about half of a tank…I said that using the word “about” makes that information unusable…one or two students still weren’t happy.

However, we did come to the conclusion that it would be nice to know how many gallons of gas my car needed…and I could help them with that:

6.2 Function Composition_7

Then, I sent them off in their groups to determine how much I paid.  Five minutes later, I saw a lot of work that looked like this:

6.2 Function Composition_9

This was typical of the work I saw from my students.  There were a lot of good conversations about keeping track of units and how the process worked.  After discussing with students the answer:

6.2 Function Composition_8

We had a conversation about what the process entailed.  They decided that they took a mileage and turned it into a number of gallons, then took those gallons and turned it into a price.  I then defined this as a composition of functions.  Then we decontextualized and practiced.

Summer Lesson Building (Algebra II Honors: An Introduction to Functions)

I know that they probably don’t need an “introduction” to functions. They’ve seen functions and they’ll be able to tell me all about the vertical line test (even if they don’t know why they use it). I see this lesson as a review of function notation, as well as a way for me to set the table for the relatively formal notation and understanding of functions they’ll use throughout the year. I will implement this lesson on the second day of class.

Their warm up will start here:

Intro to Functions lesson_1

We’ll start the conversation about finding a “rule” that related the x-value to the y-value. I want them to start to look at these relations and see patterns (eventually they’ll recognize the types of patterns…e.g. linear, quadratic, etc.).

We’ll discuss the three big types of ways we’ll look at relations…coordinates, tables and graphs. I’ve found that my students don’t see them as three ways to represent the same thing. This will be the first time we have a chat about how an equation, a graph, a table, and coordinates all show the same thing; that the graph of an equation is not “the answer” they’re trying to get to match the back of the book.

Intro to Functions lesson_2

We’ll then discuss the domain and range of a relation. They’ll be used to the idea, but I expect them to struggle with any representation that isn’t a set of coordinates:

Intro to Functions lesson_3

Intro to Functions lesson_4

Then comes the time to discuss a function. I found a video a few years ago that some group of teachers made and posted to youtube. It is by far the most entertaining input/output video I’ve ever seen:

I didn’t make that video, but I thoroughly enjoy watching my students experience it in class!

Next, we’ll define a function and work with that definition:

Intro to Functions lesson_6

We’ll refer back to the idea of domain and range so that my students can decide that a vertical line test will work on a graph to determine functionality. I’m going to post the slide that I currently have for this, but looking at it now, I’m going to change it.

Intro to Functions lesson_7

Then we’ll discuss function notation:

Intro to Functions lesson_8

Intro to Functions lesson_9

And finally, we’ll bring it all together with them creating their own function:

Intro to Functions lesson_10

This last slide is the most important. I will emphasize with them that a function relates two variables. The input gives you one value for an output. The number of hours gives you an amount in the bank account. My goodness, that’s such an important idea. An input gives you an output. We’ll be using that all year!