Day 5: A Quiz, Special Right Triangles and Cups

Day five.  The first Friday.  My students were tired.  I was tired.  The first week was successful; at least, I felt that it was.

AP Statistics: The first week culminated in a quiz.  I did learn that we need to work on communication through writing.  I feel like they have a good grasp on what we’re learning, but they’re having trouble communicating it.


In precalculus, today’s lesson is to connect calculating trigonometric function values with their knowledge of special right triangles.  I was a bit concerned about their memory of special right triangles, so their warm up had them working with 45-45-90 triangles:

1.2 Trig Functions and Special Right Triangles_1

My hope was that they’d quickly realize the pattern that the length of the hypotenuse is the leg times the square root of two.  They did, and they had the “oh yeah” moment.  I regret not proving this fact to them.  I need to make sure when we’re coming up with rules and theorems that I’m more formal about it; those students who go on to do higher mathematics need that exposure.

After spending time with 30-60-90 triangles, the end of class culminated in:

1.2 Trig Functions and Special Right Triangles_6

The part my students are struggling with the most is the simplification of fractions with square roots.  I’m trying to convince them that they’ll become comfortable with them; they just need to be willing to practice.

Algebra II Honors:

I’m a bit concerned that my students don’t know what makes a linear function linear.  We took a day out to discover what linear really means.  I’ve told them that as we look at functions, we’re going to study them in three ways: a graph, a set of coordinates, and an equation.  They knew what makes a graph of a function linear, but not the other two representations.  So we explored:

1.3 Classifying Patterns_1

Each pair of students received three cups and a ruler.  From that information, they needed to make their estimation.

In case you want to play along:

1.3 Classifying Patterns_2

Most groups realized that there is a constant growth in height caused by the lip of the cup.  From that measurement (which is really 1.5ish centimeters…even though it looks like 2 centimeters in the picture) they extrapolated their guesses.

1.3 Classifying Patterns_4

A few groups got to 116 cups.  Most were within 10 cups.

From their realization that the height of the stack grows at a constant rate in relation to the number of cups, we had the discussion of what linear means.  Ideas about rate of change, y-intercepts, and functions were all discussed.  The hope was this concrete example would let them see that constant change (slope) is what makes a function linear.  This led to other discussion and playing with geogebra.  Fun was had by all (or at least me)!

Day 4: More Function Transformations, Percentiles, and a Golf Match

I know this is many days late, but this post refers to last Thursday (eep).  I’m going to do my best to get caught up this week.  Day 4 was a bit less eventful than most.  We spent a lot of time defining and transforming.

Precalculus:  This class was definitely the least eventful of the day.  I had to leave school early to go to a golf match.  So, we debriefed homework, and then I went on my way (while they did some conversion practice).  Unfortunately, this will happen more than I’d hope for this nine weeks.

AP Statistics:  We started the class with a warm-up that I modified from FSU’s Learning Systems Institute MFAS project (I think…I mean…I’m pretty sure).

1.1 Frequencies, Percentiles, and Ogives_1

We used this set of questions to emphasize details and clarity in their writing.  Defining the variable as clearly as possible, how they calculated certain values, etc.

The rest of class discuss percentiles and their usage.  We started the conversation with the least imaginative example I could think of:

1.1 Frequencies, Percentiles, and Ogives_6

Then came the hard-hitting question: “What does the national percentile represent?”

We spent the rest of the class calculating frequencies, relative frequencies (percents), and cumulative relative frequencies (percentiles).  Then we learned how to create and interpret ogives (they show percentiles versus variable values).  Nothing too spectacular, but necessary.

Algebra II:

We continued our conversation about transformations of functions.

1.2 Transforming Functions Day 2_2

This is the style of question I want them to be able to answer.  General trends for general functions.  Not memorizing rules, but knowing the 3 affects the input and the 2 affects the output.  Understanding that changing inputs affects the x-direction and changing the outputs affects the y-direction.  We’re really going to develop function ideas throughout the course of the year.

The rest of the activity was:

1.2 Transforming Functions Day 2_4 1.2 Transforming Functions Day 2_5 1.2 Transforming Functions Day 2_6 1.2 Transforming Functions Day 2_7

I could hear them starting to hypothesize what would happen (and justify their reasoning).  They’re still a bit afraid to be wrong.  I’m trying to convince them that they learn more from being wrong than being right all of the time, but they’re still a bit apprehensive.  It’s getting better though; we’ll stick with it.

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!


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.