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.

Precalculus:

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)!

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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.

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.

Day 1: Variables, Solving Equations and DMS

This year I’ll be teaching AP Statistics, Algebra II Honors and Precalculus.  I want transparency and an open dialogue about how to transform my curriculum in these classes.  Two of the classes, I’ve taught for a few years.  One of the classes is brand new to me.  I won’t tip you off to which is which, but feel free to comment freely.

My hope is to post something about each class every day…so here we go…

AP Statistics:

Today’s focus was on types of variables.  Mainly the difference between quantitative and categorical variables.

We started with this warm up:

Chapter 1 Introduction_1

The hope was that they would talk to each other about the questions, and I was curious who would ask me about question #1 (I have three cats…just so you know).

Without answering any of the questions, I showed this slide:

Chapter 1 Introduction_2

I loved that most of the students grouped the questions in two important ways. 1) They grouped the questions based on whether there was one distinct answer versus the answers having possible variability (which showed that they knew there was a difference with a statistical question versus a non-statistical question).  2) They grouped the questions based on whether the possible answers were numerical or non-numerical.

At this point they’ve stated that they know of quantitative versus categorical variables.  So, I wanted to dive into the grey areas and get a really good definition of quantitative and categorical.  I asked them to classify the following situations:

Chapter 1 Introduction_5

Through working on classifying these variables, my students came to some interesting conclusions.  First, they were able to better state that quantitative variables require measurement, not just a numerical response.  Second, and most interestingly, there was an in depth conversation (student started) that quantitative variables could be treated categorically; they concluded that someone has to be clear in stating their expectations for measurement.

Algebra II Honors:

To ease into the new year, I wanted to make sure my algebra II students were able to solve one variable linear equations.  Sounds easy enough (and they thought it was too), but I wanted to give a different spin on it.

Unit 1 Intro (Solving Equations)_1

This question required that they keep everything in balance.  I wanted to encourage my students to work together and be self-sufficient in checking whether their solution is correct.  After having a consensus on a solution, they were asked to find the value of the missing shape in each of the following mobiles:

Unit 1 Intro (Solving Equations)_2 Unit 1 Intro (Solving Equations)_3

The first one was to build confidence; no problems…the triangle is worth 10.  In the second one, there were a bunch of methods used.  Some guessed and checked.  Others set up an equation.  A few noticed that a triangle and square cancelled each other out on the two sides and they were really solving 2 squares equals 1 triangle…which someone quickly noticed that they did the exact same thing if they set up an equation.  Amazing conversations.  Then we solved some equations.

Precalculus:

We started the period with practice using dimensional analysis to convert value.

1.1 Measuring in Degrees_1

The purpose of this was to put a seed in their mind for figuring out how to convert angle measurements from degrees into degree-minutes-seconds form.  After defining how many minutes are in a degree (60) and how many seconds are in a minute (60) (then explaining how the Sumerians worked in a base 60 number system…which is why we measure time the way we do), I had them complete the following conversion:

1.1 Measuring in Degrees_5

I question whether I made this too leading, but everyone came up with a method that worked for them.  I challenged them for homework to come up with a method to convert an angle measured in DMS back to degrees.  We’ll see how they did tomorrow.