“It’s fun to sing about the Central Limit Theorem!”

So, I’ll start with the good stuff. I’m fairly proud of the lyrics that I’ve written for the Central Limit Theorem Song set to the tune: YMCA. We sing it in class, but I wasn’t able to show a recording of my students singing the song since I’d need to have parental release forms signed, oh well. I’m a bit pitchy, but I think I my head-bobbing is legendary.

I usually teach the song after I have go through the “YMCA example” with the students. The population distribution (which is totally made up) shows the ages of people going to a YMCA during the middle of the day (9 AM to 3PM) over the course of a week. There are very few children of school age. Most people are either babies/toddlers, 20-40 year-olds (presumably care-takers or people who don’t have a day job), and retirees (60-80 year-olds).

CLT Song LyricsDownload

Ages of all of the attendees to the YMCA from 9:00 AM to 4:00 PM during the weekdays of a particular week.

Once we’ve looked at the population, we begin taking samples of size 2 being very clear on what each dot is and where it came from. Then we speed up the simulation to get 1000 samples of size 2.

Then, we do samples of size 5, and then samples of size 30. To get the respective approximate sampling distributions. It’s beautiful how it demonstrates how sampling variability is reduced when the sample size is increased and the sampling distribution also begins to look more normal as the sample size increases. We connect it to the formula for standard error (standard deviation of the sampling distribution) to see how close the simulation was to the true value. I particularly love unpacking why we get the peaks that we do in the sampling distribution for samples of size 2: either the people in the sample were both babies, both retirees, both middle-aged, or combinations of each. (Here’s a link to the CODAP workspace).

Points of Emphasis

In teaching statistics (regular, dual credit or AP), I have found that the transition from descriptive statistics to inferential statistics is a huge hurdle without a clear understanding of sampling distributions. As a result, we spend a lot of time developing vocabulary, symbols and being clear about what the distributions we are looking at actually represent. I frequently ask questions like:

Q: “What is this dot?” A: “It’s the height of a single student”

Q: “What’s this dot?” A: “It’s the mean of the heights of a group of 10 students”

Q: “What’s this value and what does it mean?” A: “It’s the standard error and it represents the how far off my sample means will typically be from the true mean.”

The Beginning – Rethinking the German Tank Problem

I have always started the unit with the German Tank problem, but I’ve had to “prime” the students by playing a very similar, simpler game: “Guess how many numbers are in the bag” I have slips of paper from 1 to 33 in a paper bag. I shake the bag up and pull out 5 numbers. I tell the students that the numbers in the bag are consecutive numbers from 1 to something, and they have to guess how many numbers are in the bag. First, they use their gut. Let’s say I pull out 3, 8, 11, 24, 27; they might guess that the largest number in the bag is 30. I ask them why, and I get answers like:

“It’s unlikely that we would have pulled the highest number, so I added a little bit to the top.”

I ask: “How did you know how much to add?” They say things like: “Since there was a space of 3 between the lowest number and zero, I added that to the top as well”, and I clarify: “You added the minimum to the maximum” It’s through this process that we get a few decent formulas:

Max + Min

1.2 * Max (20% more than the max)

Mean + 2 Std Dev

We then use these formulas to crunch the numbers and make predictions. That way when they get into the German Tank scenario (which is essentially the same thing, just larger numbers) they have a clue of what it means to create their own predictive formulas.

My MacOS no longer supports Fathom, so I’m using CODAP, which has been awesome. I like it better for somethings and not as much for other things. Here’s a link to the CODAP version of the German Tank Problem.

Heights Labs – Developing the Idea of a Sampling Distribution

I have created several “labs” for the students to do to learn about Sampling Distributions. First, I have the students sample the heights of students from the class by pulling slips of paper from a paper bag. I think I got this idea from the Starnes/Tabor textbook, but I’ve created a little guide for the students to follow that highlights the key points.

Heights Lab 1 (Word Doc)Download

I’ve also created a follow-up lab that get the the key idea about getting closer to the actual sampling distribution. Here’s a link to the CODAP lab.

Heights Lab 2 (Word Doc)Download

I hope you find these resources helpful. Please let me know what you think!

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“Students, please don’t gamble!”

How I teach random variables and expected value

Games of chance are a great way to engage students in probability and statistics. The topic of expected value is a great way to teach a life lesson: don’t gamble. Here’s the low-down on my unit on probability distributions and expected value.

I start with an introductory activity where the students learn how to do a weighted average (which is essentially what the calculation of expected value is). During the pandemic, I turned this into a pretty sweet Desmos activity (DESMOS: Weighted Average) where students estimate the final grades of 3 students in a math class before calculating the actual averages. I took a play out of Steve Phelps’ playbook to aggregate the students guesses on a dot plot. Here’s the google sheet for the students to calculate the weighted average.

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Next, I teach the students a little vocabulary and the key concepts through a short series of worked out examples (about 15 minutes). Here’s the note guide I use. (Download below)

Probability Distribution / Expected Value Note Guide – PDFDownload

Probability Distribution / Expected Value Note Guide – DOCXDownload

Here’s another Desmos activity (DESMOS: Expected Value Practice) that is self-checked which gives the students a little bit of practice on the concepts that we just learned. I love the feedback feature so that I can give the students feedback real-time.

Next, the students do a little probability experiment with three dice and do some more work calculating and interpreting expected value. Since we were remote for the last week, I had them use random.org to roll 3 dice to take their data. (DESMOS: Additional Expected Value Practice)

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We do several practice problems involving lotteries and casino games to show them that ALL lotteries and casino games have negative expected values.

GREED – The Expected Value Game

Finally, we play a game called GREED.

This game is one of most memorable activities of the year for some of the students. Each student stands next to a chair. I roll a six-sided die. If I roll a 5, students get five points. If I roll a 3, students get three points…or whatever number I roll as long as they are still standing. Since I have whiteboards around my room, the students tally their points at the board. They continue tallying points EXCEPT if I roll a one; then, any student who is still standing will lose all the points that they’ve tallied for that round. A student can save their points by taking a seat between rolls, but once they’ve sat down, they can’t stand back up that round. A round ends when a one is rolled or all students are seated. At the end of the round, all students count their tallies and save their points by adding it to the previously accumulated points. Don’t announce that it will be “last round” because desperate students will just stay standing until they win or they bust.

The trick of the game is for students to decide whether the cost of losing all the points they’ve accumulated during a round outweighs the benefit of gaining more points. This is a perfect way for students to get a sense for expected value. Obviously in this game, the expected value of your gain changes as you accumulate more points. So what is the “sweet spot”? When should you take a seat? It turns out that when you get to 20 points, the expected value of your gain is zero, so generally, you should take a seat once you’ve gotten to 20.

Expected value is a fun topic to teach. I hope some of the ideas and activities are helpful.

It’s About Culture: Celebrate What You Value!

This is a post for the Virtual Conference on Mathematical Flavors. Although my favorite flavor of ice cream is vanilla, I don’t think it’s an apt metaphor for the “flavor of my teaching”. If I had to pick a flavor that symbolized my teaching, it would be Moose Tracks. Let me explain by unpacking the constituent parts of the Moose Tracks mixture:

Vanilla – To me the vanilla base represents my classroom culture of kindness, mutual respect, and consistency. The classroom culture undergirds everything.

Fudge – The fudge ribbon represents the challenge that I swirl throughout the course. And just like every bite of moose tracks ice cream includes a bit of fudge, there’s not a day that goes by in my class that I don’t throw something at my students that’s meant to stretch them and puzzle them.

Peanut Butter Cups – The best part of the moose tracks ice cream, to me, is when you get a peanut butter cup. The PB cups represent the discoveries made by the students. I craft my lessons so that, each discovery is wrapped in a challenge of some sort (just like the PB is inside some chocolate).

Okay, enough of the ice cream metaphor. The key questions to be answered in the “presentation” are:

How does your class move the needle on what your kids think about the doing of math, or what counts as math, or what math feels like, or who can do math?

In short, any “needle moving” that occurs in my students perceptions about the doing of math, what counts as math, what math feels like, and who can do it, is rooted in my classroom culture.

Because my classroom is based on kindness and discovery, I emphasize from day one that mistakes are okay. Reasoning, not “answer getting” is celebrated. I constantly ask my students to make conjectures, and I require them to share their reasoning. I consistently follow-up with: “What makes you say that? Why do you think that?” I try not to approve or disapprove of correct or incorrect answers myself, but let the math and the reasoning be addressed by the other students in the room.

My sincere hope is to have the students realize that: MATH IS NOT JUST A SET OF PROCEDURES, MATH IS REASONING AND PROBLEM SOLVING. 

Therefore, anyone can do math. It’s not about who’s quickest or who can memorize the stuff the best. Rather, it’s about: who can explain their reasoning clearly, who can persuade others with a compelling argument, who can persevere in solving a complex problem, who can notice a pattern, or who can recognize when certain mathematical tools can be of use. Whenever I ask my students to embark on a difficult task, I remind them of this aphorism: “How do you eat an elephant? One bite at a time.”

My hope is for my students to leave my class with the following impressions:

• I felt supported and valued
• I’m a thinker
• Complexity doesn’t scare me
• I’m capable of understanding
• Reasoning is much more valuable than an “answer getting”
• Reasoning is more fun and rewarding than mindless procedure following
• Getting started is often the hardest part.
• Making mistakes is okay, and it’s okay to start all over again.

That’s the flavor of my class…at least the flavor that I strive to serve up for my students. I can’t wait to break out the scoop and start serving up some math again this year!

Desmos Activity – The Sine and Cosine Ratios

I’ve was delighted to develop the idea of the tangent ratio in my Geometry classes using this activity called Slanty Hills created by the folks at Desmos. It approaches the tangent ratio from a very relevant perspective.  It focuses on the percent grade of roads which is really just the slope of the road…which is really just the tangent ratio!! I’m not sure why that didn’t ever occur to me before. In other words, if a road has a 10º angle of elevation, it has a slope of ≈0.176 (that’s the tan(10º)≈0.176). In other words, for every 1000 feet of horizontal distance covered by the road, there is a gain of 176 feet of elevation.

Screenshot of “Slanty Hills” Desmos activity

It has been so helpful to introduce the idea of the tangent ratio being the SLOPE of a line or the slope of the hypotenuse of a triangle. All of the students know that slope is rise over run, so it makes a beautiful connection between the world of algebra and the world of geometry. Thanks Desmos for the great activity!!

Although my activity is fairly rudimentary when compared to the “Slanty Hills” activity, I’m still very pleased with it. It sneakily introduces the unit circle definition of the sine and cosine (although that’s not a focus of my geometry students). I particularly like that students have a few opportunities to do a card sort to check their understanding.

My Desmos Activity – Sine and Cosine Ratios

Card Sort Activity

Unit Circle Definition of the Sine

I’m going to try this activity out with my students next week!  I’m excited, and I’ll let you know how it goes.

 

Polar Graph Investigation

Polar graphs can be tricky.  Though most students can plot polar points given a radius and an angle, it’s a very different proposition for students to really picture how the radius of a polar function varies as the angle increases.  Essentially, the students can create a table of values and plot those ordered pairs, but connecting the dots…that’s another story (it’s just not intuitive for most students).

This year, however, I think I taught it more effectively and assessed the students more effectively, so I’d like to share.

Here’s the activity that I have my students do. It connects to prior knowledge (graphs of sinusoidal functions): Graphing a Rose Curve – WHY?

I ask them to focus on specific critical points (mins, maxes, and zeros) that they can gather from a rectangular equation of the same form as the polar equation. that’s nothing new.

The new thing is this desmos graph (CHECK IT OUT!!) that helps them make sense of how to connect the dots, and how the sinusoidal function is connected to the polar graph.

On their quiz, I asked the students to explain to me how the radius changed over particular intervals of angle measures, and overall I was very impressed with the students’ ability to put their thinking into words.

Here’s a video that shows the full functionality of the desmos graph. It’s one of my favorite demos that I’ve ever created.

 

 

Polynomial Discovery!

I love teaching polynomials because I think they offer an opportunity for students to see the rule of 4 constantly (graphical, numerical, algebraic, and verbal). I’m including several links and documents that I use in developing the ideas during the unit.

End-Behavior Lab:

In Precalculus we start our polynomials unit with a short review of the rules of end-behavior in a lab format. I like this because it gets into an exploration of the “WHY” at the end.

end-behavior-lab-short-version

The first part is just sketching what the students notice when they change the leading coefficients for polynomials of different degrees. There are links to sketches of polynomials of different degrees like this one:

 

Repeated Roots Lab

Here we talk about multiplicity and how repeated roots affect the behavior of the function. I use the language that the function “bounces” off the axes at zeros with even multiplicity, and it “lags” through zeros with an odd multiplicity greater than 1.

repeated-roots-lab

In addition to students noticing the behavior of these functions, I ask them to see if they can determine the degree, the leading coefficient, and constant term without expanding the entire polynomial. They use wolframalpha.com to check their answers to that question.

Why the “flat butt” and “sharp arms”?

The next class, I have the students do a little more exploring graphically and numerically. The question is: When a factor is repeated more times, WHY does the graph get “flatter” on the bottom and “sharper” sides?

They explore using desmos, and they look at graphs like this:

Why is the “butt” flat and why are the “sides” sharp?

 

I also ask them to make a table of values for the final function, and do the calculations by hand (or at least x=2.8 and x=3.2).

I want them to remember that: repeated multiplication by a number whose magnitude is >1 will give a number whose magnitude is larger (larger than the original), whereas repeated multiplication by a number whose magnitude is <1 will be a number whose magnitude is smaller (smaller than the original).

So the major take-away is that: in the polynomial once I subtract 4, the difference will be repeatedly multiplied by itself, and if it’s taken to the 16th power it will behave much more dramatically than if it’s taken to the 2nd power. 1.2^16 is much larger than 1.2^2, and 0.8^16 is much smaller than 0.8^2.  The “flat region” corresponds to the x-values within 1 unit of the zero, and the “sharp arms” correspond to x-values beyond 1 unit of the zero.

Time to Problem Solve:

I am not big into teaching processes if I don’t have to. I’d rather have my students discover or wrestle with the math, so I have the following problem as a warm up on a particular day. The students are using what they know to write an equation for a particular polynomial:

polynomial-thinking-problem

The trick with this one is that most student can get the correct factors for the polynomial, but they need to do a little work to find the leading coefficient.

Creating Polynomials:

In this activity, I have the students reinforce what they know about polynomials so far, and they get to use Wolfram Alpha to expand polynomials in factored form to polynomials in standard form.

creating-polynomials-acitivity

 

What else do I do?

I have a lot of white board space, so we spend time practicing finding zeros, expanding polynomials, and drawing sketches of polynomials.We also spend time considering how many zeros and what type–those are fun problems because they require the students to explain verbally their reasoning.

I love this because students have to explain their reasoning.

I also require them to determine how many rational vs. irrational zeros there are. In this case, both zeros are irrational!! Even though you might sware that the polynomial crosses at 2. However, since 2 is not a factor of 9, 2 cannot be a zero (according to the RZT).

MOSTLY, I care about the students making the connection between the algebraic and the graphical representations. 

 

Sinusoidal Graphs and the Unit Circle

 

So, I was gone for a few days at a conference. You ever feel like it’s twice as much work to prepare for a sub as it is to just do a good lesson yourself?  In this case, it was worth the extra work to create a self-guided lesson for the students, and it was much appreciated too. I created a series of activities for the students to help them learn about the sinusoidal functions while I was gone. They watched videos, played around with desmos, and completed a series of exercises to learn what I wanted them to learn. It went pretty well.

I’d like to share the pieces that I am most proud of: the desmos sketch and the video tutorial that I created to go with it. This was the first part of their introduction to sinusoidal functions.

The main idea is to connect the unit circle to the graph of the sine function and the cosine function. I’d visualized this dynamic picture for years, and now, I’ve been able to turn it into a reality. Booyah!!

Transforming Functions

One of my favorite topics to teach is transformation of functions. The reason I like it is the same reason a 3-year-old likes pushing buttons and turning knobs: when you make a change you can see what happens–immediately.

I have labored to make the development of these ideas seamless in my classes. I think I found a really good way to teach it.

I often do a discovery exercise using all different types of functions (square-root, quadratic, cubic, etc.) to see what each parameter in an equation does to the graph of the equation. However, this time I opted for a piecewise function so that I could really make use of the function notation.

The point that I really wanted to emphasize is to answer the question: Why does a horizontal transformation behave “opposite” how I would expect? Why does f(x+2) shift the graph of f(x) to the left and not the right?

I am attaching a copy of the investigation worksheet. Also, here’s the link to the desmos graph that I created for my students to tinker with.

Here’s the questions that I had them answer as they explored:

Word Document: desmos-exploration-transforming-a-piecewise-function

PDF: desmos-exploration-transforming-a-piecewise-function