This is the first in what will likely be a series of posts about classroom moves which I have adapted for remote learning. I hope you enjoy them!

In my freshman-year Prob/Stat course, students experience a probability lesson featuring the game “Egg Roulette”, based on a bit from Jimmy Fallon’s Tonight Show. Here is a summary of the “live” lesson: https://mathcoachblog.com/2015/09/20/an-egg-cellent-simulation/. This year, there were two considerations for how I would have students investigate the game: conducting the simulation and collecting the results.

CONDUCTING THE SIMULATION

The first class simulation involves two unsuspecting volunteers and my actual container of 12 “eggs” – filled with little fuzzballs. Click the link in the last paragraph to see a video of how it works. In the main simulation, students use decks of cards to play the game repeatedly. Give pairs of students 13 cards all of the same suit. Discard the ace. Then, the 10, jack, queen and king represent “raw” eggs. The other cards represent the “hard-boiled” eggs. In a remote environment I could have used a site like random.org to draw cards, but I also saw an opportunity to build a simulation students could use to quickly analyze repeatedly. This Desmos link allows students to play the many times: https://www.desmos.com/calculator/2b7f6p4r3o. Click the “rerandomize” button to generate repeated plays of the game. Online, we talked through a few of the simulations and I found the students quickly understood the format.

COLLECTING THE RESULTS

I have used a number of methods for collecting class results over the year: sticky dots on a poster, Post-It Notes on a wall, digital data collection. Clearly this year we had to go digital, and the site http://stapplet.com came to the rescue. New this year, teachers have a “collaborative” option – this feature generates a class code from which students can submit their data to the class (thanks Josh Tabor and Luke Wilcox!). The results update in real time. Each student then pasted the class graph into OneNote and a discussion of Jimmy Fallon’s “meanness” – is he being nice to his guests by letting them draw first? – followed.

The rest of the lesson and discussion felt similar to previous years. I challenge small groups to find the probability of a player losing in round 3. This leads us to probability ideas of independence / dependence and the multiplication rule. The engagement remained high and the conversation was on par with previous years!

Today our friends at Desmos released an update of their Activity Builder editor. You can head to teacher.desmos.com now and explore the changes – edit a previous activity, create a new one OR copy screens from activities in the library of Desmos activities.

The first thing I noticed about the updated editor was the increased freedom in screen design. Previously, elements like Graph, Note and Input were limited in their placement and number. There is more freedom now to move elements around, order them as you like, and include more of them in a single screen.

Immediately I wanted to explore this new freedom and think about intentionality in my design process for an activity I wrote and used in my AP Stats class this spring – Is My Die Fair. In this activity students “roll” both a virtual hand-made die and a virtual real die. The activity allows students to discover the chi-squared statistic as a reasonable measure of variability in a categorical distribution. Here are two ways I changed this activity with the new editor, with the intent that students will be able to follow the narrative with less arrowing through the activity.

ORIGINAL VERSION:

Screen 2: students roll the hand-made die 60 times

Screen 3: results are copied from screen 2 and students make observations.

NEW VERSION:

Students roll the die AND make a conjecture on the same screen.

Another place I was able to leverage the new block placement freedom occurred later when students begin to think about the computation of the chi-squared statistic.

ORIGINAL VERSION:

Screen 8: the new statistic is explained, and students complete a table for the homemade die only.

Screen 9: a summary statistic is shown, and students now complete the table for the real die.

Screen 10: both summary statistics are shown and students make a final conjecture about the dice.

NEW VERSION:

Screen 7: the new statistic is explained, and students complete tables for both types of dice on the same screen.

Screen 8: both summary statistics are shown and students make a final conjecture about the dice.

Share your ideas for altering your previous activities to leverage the new design freedoms. Below, you can test-drive both activities and see how I altered them. Your ideas are always appreciated…now get building!

Last spring, the awesome folks at Desmos released a slew of slick (but easy-to-use) statistics features. Here is a brief video I made which walks through a few of the new features. With a new academic year beginning, I’m looking forward to changing some of my classroom moves in AP Stats to leverage the new features and build understanding. Here are 3 moves I’m planning to try this year:

ASSESSING NORMALITY (Here is a previous post on this topic)

Pop quiz! Below you see 6 boxplots. Each boxplot represents a random sample of size 20, each drawn from a large population. Which of the underlying populations have an approximately normal shape? Take a moment to think how you…and your students…might answer…

Have your answers ready? Here comes the reveal…..

Not only do each of the samples above come from normal populations, they each come from the same theoretical population! This year in class I plan to walk students through how to build their own random sampler on Desmos, which takes only a few intuitive commands. When the “random” command is used, we now get a re-randomize” button which allows students to cycle through many random samples and assess the shapes. You can toy with my graph here.

Often students look for strict symmetry or place too much stock in different-sized tails. This is a great opportunity to have students explore and understand the variability in sampling. Teach your students to widen their nets when trying to assess normality and remember – our job is usually not to “prove” normality; instead, these samples show that the assumption of population normality is often safe and reasonable, especially with small samples.

LINEAR TRANSFORMATIONS OF DATA

Analyzing univariate data using Desmos is now quite easy. Let your students build and explore their own data sets. Data can be either typed in as a list or imported from a spreadsheet using copy/paste. The command “Stats” provides the 5-number summary, and commands for mean and standard deviation are also available. You can play around with my dataset here.

Next, I want my students to consider transformations to the data set. In my example I have provided a list of test scores and summary statistics are provided. Let’s think about a “what if”. In the next lines I provide 2 boxplot commands, but I have intentionally ruined the command by placing an apostrophe before the command (thanks Christopher Danielson for this powerful move!). What will happen if every student is given 5 “bonus” points? What if I feel generous and add 10% to everyone’s grade?

What will happen when I remove those apostrophes? Think about the center, shape and spread of the resulting boxplots? How will these new boxplots be similar to and different from the original?

Compute new summary statistics. Which stats change…by now much…and what stays the same? Why? I’m looking forward to having students build their own linear transformation graphs, investigating and summarizing their findings! Here is a graph you can use with your classes to explore these linear transformations with sliders.

COMBINATIONS OF DISTRIBUTIONS

An important topic later in AP Stats – what happens when we combine distributions by adding or subtracting? Often I will use SAT scores as a context to introduce this topic because there are two sections (verbal and math) and a built-in need to add them – What are the total scores? On which section do students tend to do “better”…and by how much? To build a Desmos interactive here, I start with a theoretical normal distribution with mean 500 and standard deviation 100 to represent both mean and verbal score distributions. Next, taking 2 random samples of size 1000 and building commands to add and subtract them allows us to look at distributions of sums and differences and compare their center, shape and spread.

The most important take-away for students here should be that distributions of sums and differences have similar variability. This is a tricky, yet vital, idea for students as they begin to think about hypothesis tests for 2 samples. You can use my graph, or build your own. Note – in my graph the slider is used to generate repeated random samples.