Monthly Archives: January 2016

When Binomial Distributions Appear Normal

We’re working through binomial and geometric distributions this week in AP Stats, and there are many, many seeds which get planted in this chapter which we hope will yield bumper crops down the road. In particular, normal estimates of a binomial distribution – which later become conditions in hypothesis testing – are valuable to think about now and tuck firmly into our toolkit.  This year, a Desmos exploration provided rich discussion and hopefully helped students make sense of these “rules of thumb”.

gifsmosEach group was equipped with a netbook, and some students chose to use their phones. A Desmos binomial distribution explorer I had pre-made was linked on Edmodo. The explorer allows students to set the paremeters of a binomial distribution, n and p, and view the resulting probability distribution. After a few minutes of playing, I asked students what they noticed about these distributions.

A lot of them look normal.

Yup. And now the hook has been cast.  Which of these distributions “appear” normal, and under what conditions?  In their teams, students adjusted the parameters and assessed the normality.  In the expressions, the normal overlay provides a theoretical normal curve, based on the binomial mean and standard deviation, along with error dots. This provides more evidence as students debate normal-looking conditions.

bins1

Each group was then asked to summarize their findings:

  • Provide 2 settings (n and p) which provide firm normality.
  • Provide 2 settings (n and p) which provide a clearly non-normal distribution.
  • Optional: provide settings which have you “on the fence”

My student volunteer (I pay in Jolly Ranchers) recorded our “yes, it’s normal!” data, using a second Desmos parameter tracker.  What do we see in these results?

bins2.PNG

Students quickly agreed that higher sample sizes were more likely to associate with a normal approximation. Now let’s add in some clearly non-normal data dots. After a few dots were contributed, I gave an additional challenge – provide parameters with a larger sample size which seem anti-normal. Hers’s what we saw:

 

bins3

The discussion became quite spirited: we want larger sample sizes, but extreme p’s are problematic – we need to consider sample size and probability of success together!  Yes, we are there!  The rules of thumb for a normal approximation to a binomial had been given in a flipped video lecture given earlier, but now the interplay between sample size and probability of success was clear:

And what happens when we overlay these two inequalities over our observations?

bins4

Awesomeness!  And having our high sample sizes clearly outside of the solution region made this all the more effective.

Really looking forward to bringing this graph back when we discuss hypothesis testing for proportions.

“The 35 Game” for Compound Inequalities

This week in Algebra 1, my students completed the first part of their inequalities unit with much success, but now storm clouds appear on the horizon – compound inequalities, where english class meets math class with talk of conjunctions – those pesky and’s and or’s. A dice game helped my students make sense of these compound ideas.

The 35 game: 3 students, 3 dice, 3 rounds.

mickeyIn each round, a player rolls the 3 dice and records their sum. The goal: by the end of 3 rounds, to get as close to a total of 35, without going over. After round 2, each player has the choice to stop if they like, but highest score, closest to 35, wins the game. To help students understand the game, I gave the class time to play in their groups, record results, and think about strategy. The next day in class, we selected 3 students to play in front of the class. Players took turns rolling, and results were recorded on the board after each roll.  After round 1, here is how a game between Mickey, Sam and Kim was shaping up:

score1

Kim has taken a small lead. Round 2 rolls then go in order. We record them, then look at the round 2 sums.

score2

Still pretty close, Sam now leading. It’s Mickey’s turn to roll. Mickey probably needs to roll in round 3, but what is he hoping for?  Some rolls will cause him to go over. Will any rolls cause him not to take the lead?  All students in class were equipped with number lines going from 3-18 which I made using Number Line Generator.  Class discussion quickly yielded consensus that 3 was the lowest roll for Mickey, and 14 was the highest. How do we write these as inequalities, and how do these inequalities “play” together. The key word here is “and”, and all students recorded the possibilities:

numline1.jpg

After we agreed on the interval of possible “safe” values, Mickey made his round 3 roll – and was safe!

score3

A total of 31 – not bad, but 2 other players yet to go. Moving on to Sam, students discussed her possible “safe” rolls, and I was surprised how quickly we were able to generate the inequality. Note, for ease of discussion, we made ties “safe”, as a tie would keep a player in the game (we’d do a new game after to break any ties).

numline2.jpg

How did Sam do in round 3?

score4.jpg

Too much! And Mickey is still in the lead.

Moving to Kim’s turn, I changed the focus from the player to her opponent.  Rather than find rolls which are advantageous to Kim, I asked students to think about Mickey: what is HE hoping for her to roll?  Which rolls would cause him to win the game?  This small twist took a bit more time in groups, and provided rich discussion of the difference between the conjunctions AND and OR.  In this case, Mickey would be happy if Kim rolled less than 11 OR if she rolled more than 15. Shading these on the number line revealed a solutions set which looked different from the previous 2:

numline3.jpg

In the end, this simple game allowed for group discussion and a natural discussion of the conjunctions. In class the following day, we started once more with the game, and I stopped the game now and then to have students sketch solution sets of the game from differing perspectives.

One last note: there is a clear discussion of discrete vs continuous variables to be had here, and I brought it in when it seemed like the class could handle it. In our game, it’s not possible to roll a sum of 9.5, yet we shaded values between integers on the number line. A chance to bring in domain discussion here, where the domain of the game is limited to integers between 3 and 18 versus the real number line we often use, is welcome here – grab the opportunity to highlight the precise mathematical language.