First Homework Assignment

Alright.  We’re back online.  I know it’s been a while.

This year, it’s time to really revamp Algebra II.  I have to do a better job of advocating for thinking mathematically and persevering through problems.  So we’re going to start day one.

Here’s their homework assignment:

Mr. Cloud tries to be a healthy person and takes multi-vitamins daily.  He absolutely LOVES the gummy ones!

Here they are:


As he was finishing this bottle, Mr. Cloud thinks he was ripped off (he didn’t get what he paid for).  Here’s what he saw:


Circle One:

Did Mr. Cloud get ripped off?     YES         NO         Can’t Tell

Provide a well written justification for your answer:



So that’s it.  That’s the whole assignment.  And I’m going to leave this right here.  Feel free to comment and argue your point!



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.

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.


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.


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: Extrema of Polynomial Functions)

A couple of months ago, I ran across a lecture that featured this graph:

Gobal and Local Extrema Lesson_1

This graph shows the water consumption on February 28, 2014 by the city of Edmonton during the gold medal men’s hockey game during the last Winter Olympics (in which Canada defeated Sweden 3-0). My students will try to answer one simple question: What can this graph tell us?

I’m expecting that the conversation will drift toward the fact that there are some weird peaks and valleys. Then, we can figure out what those peaks and valleys are.

In case you’re curious:

Gobal and Local Extrema Lesson_2

My hope is that this conversation can create a need for the importance of local and global maxima and minima. We’ll do the typical defining and practicing:

Gobal and Local Extrema Lesson_3

Gobal and Local Extrema Lesson_4

Gobal and Local Extrema Lesson_5

Gobal and Local Extrema Lesson_6

My hope is that the students get a sense of an example where maxima and minima mean something.

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!

Summer Lesson Building (Algebra II Honors: Solving One Variable Equations)

The first day of Algebra II is a chance to get my students up and running with something they know how to do, but may not remember the meaning behind it. They’ll walk in the door to the following warm-up:

Solving Equations Lesson_1

For the first ten minutes or so of class, we’ll focus on the idea of the solution is the value that balances each mobile:

Solving Equations Lesson_2

Solving Equations Lesson_3

I expect my students to bring equations into this with some sort of notation (whether it’s using squares and triangles, or variables). We’ll discuss the connection between the solution to an equation and it being the value that balances the two sides.

Next, we’ll practice solving one variable equations:

Solving Equations Lesson_4

Solving Equations Lesson_5

My students will solve them and explain the process. We’ll start the process of talking about inverses during the discussion about “the reverse order of operations.”

This lesson is meant to be a review of something they’re already comfortable with and a nice way to begin the year. The second half or so of class will focus on procedures and expectations.


Statistics Summer Institute Day 1 (Mostly Nerves and Fear)

I have the wonderful opportunity to facilitate a two week long summer institute on using statistics in an Algebra classroom. I am working with two other facilitators to help 25 classroom teachers learn everything from what a statistical question is to the basics of probability and statistical inference. Our goals as facilitators is to give these teachers the confidence and ability to teach statistics at a meaningful level to their students next year and beyond.

Day 1: The goals of day 1 include familiarizing the teachers with the common core…er…I mean mathematics florida standards, and have them start tackling the idea of what statistics is. One of the other facilitators is tasked with starting the conversation about what statistics is and how its used.

My task on day 1 is to familiarize the teachers with the standards for mathematical practice. Preparing for this hour long presentation I figured I had two options. I could do every presentation about the math practices that I’ve seen and dryly lecture about what the are and what they should look like…or I could throw them into a lesson that I’ve done in the past and force them to live the math practices. I decided to use the car crash problem. Here’s the link if you missed it:

We started with:
5.3 Intro to Radical Graphs Car Accident Problem_1 - Copy

The teachers decided they couldn’t answer who was at fault for the accident. So, I gave them:

5.3 Intro to Radical Graphs Car Accident Problem_2 - Copy

5.3 Intro to Radical Graphs Car Accident Problem_3 - Copy

They decided to check to see if the relationship between stopping distance and speed was proportional. After a few minutes they decided it wasn’t. Eventually, with some probing, they decided the relationship wasn’t linear:

skid mark scatterplot

skid mark scatterplot linear

skid mark scatterplot square root

They wanted to know the stopping distance of the white car:

5.3 Intro to Radical Graphs Car Accident Problem_5 - Copy

They surmised that the speed of the white car was about 68mph and that that car was at fault for the accident.

That part of the lesson went as planned. Some people were very comfortable with using the math practices in the classroom, some people weren’t comfortable at all, and some people decided not to participate too much in the lesson. It was your typical professional development.

There was one conversation that interested me. A few people “liked the idea” of what we did but concluded that it wasn’t realistic in a middle/high school classroom. Their argument was based on the fact that they had an end of course exam to prepare for. They needed to teach to the test. There’s no way they could do this style of lesson and get anything accomplished in time for the EOC. I tried to assure them that what we did is possible. Coincidentally enough, my geometry EOC scores came in yesterday. Geometry is the class that I’ve spent the most time with creating these styles of lessons and 88% of my class passed the EOC. I really want these teachers to buy into using these math practices for the sake of their students’ learning and loving of math. I’m going to spend every opportunity that I get to present to show them how this can look in a classroom. Hopefully, for the sake of my profession, they decide that this is how a math classroom should look.