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Power and Virtual Coins

This activity was inspired by the article “Innocent Until Proven Guilty”, by Catherine Case and Doug Whitaker. NCTM Mathematics Teacher, Volume 109, Issue 9 (May 2016)

Around February each year, the AP Statistics message boards come alive with new and veteran AP Statistics teachers seeking ideas to help students understand the concept of statistical power. While Power is a “minor league” topic in the AP Stats curriculum, a robust discussion of the concept can help tie together the logic of statistical inference: P-values, error and sampling variability. I’ve developed a few activities to try to bring Power to life (see here and here). And while each was satisfying in their own way, none of them really met one of my overarching classroom goals – to have students identify and express a new idea with their groups before I provide clarification. This year’s activity worked nicely as it allowed students to experience statistical power and generate meaningful conversation. Download the student version below, then read to learn how it works.

In this activity students will investigate the “fairness” of 3 virtual coins through a Desmos graph, using 3 different sample sizes to compile evidence. For each sample, students use their graphing calculator to compute a P-value and then reach a statistical conclusion. For coin A, I led students through the steps for n=10 and encouraged them to work through the next two sample sizes using their group-mates as a support system.

As students completed all three columns for coin A, I asked them to make a final decision regarding the fairness of coin A – is there convincing evidence that coin A is unfair? Students discussed findings with their groups and thoughts about how each column provided convincing evidence. Here is what the class-wide vote and conversation revealed:

• Of my 42 total students (2 classes), only 1 student concluded that coin A was unfair.
• All groups agreed that the larger sample size (n=100) was more useful in reaching a decision about the coin.

Spoiler alert: coin A is unfair! If you take a peek under the Desmos hood, you will find that coin A is “programmed” as 48% heads, 52% tails. I didn’t reveal the true proportion until the end, but we are off to a good start here: small differences between the null and “truth” are less likely to be detected.

Groups then tackled coin B with little assistance from me. Working through each column, then the follow-up conversation and decision, took about 5 minutes. This time about 60% of the students concluded that coin B was unfair.

Finally, coin C. Many students quickly concluded that coin C was unfair (it is!) but worked through each of the columns and sample sizes. In the end, there was class-wide agreement that coin C is an unfair coin.

At this point I revealed the truth about each coin:

So, what do our finding show us about hypothesis testing and decision-making as a whole? I was thrilled when one of my students who does not volunteer often raised his hand to offer the following: “If there is a big difference between the null and the truth, it’s easier to reject the null.”

Yes! That’s a big part of power. What else?

Larger sample sizes are more likely to detect a difference when one exists.

Yes! And now we have a nice framework for power. From here I shared a working definition of power and included thoughts on alpha, which are not part of this activity now but could be in a later version.

EmPower your students to develop statistical ideas!

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Adapting to Remote Learning – Egg Roulette

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!

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Let’s Explore the Activity Builder Space

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!

“Is My Die Fair” – original version – https://teacher.desmos.com/activitybuilder/custom/5e713346f6e84e7dd46727fd

“Is My Die Fair” – version 2 –https://teacher.desmos.com/activitybuilder/custom/5ef38c6cad9e310dc0a6ac23