Seeing Stars with Random Sampling

Adapted from Introduction to Statistical Investigations, AP Version, by Tintle, Chance, Cobb, Rossman, Roy, Swanson and VanderStoep

Before the Thanksgiving break, I started the sampling chapter in AP Statistics.  This is a unit filled with new vocabulary and many, many class activities.  To get students thinking about random sampling, I have used the “famous” Random Rectangles activity (Google it…you’ll find it) and it’s cousin – Jelly Blubbers. These activities are effective in causing students to think about the importance of choosing a random sample from a population, and considering communication of procedures. But a new activity I first heard about at a summer session on simulation-based inference, and later explained by Ruth Carver at a recent PASTA meeting, has added some welcome wrinkles to this unit.  The unit uses the one-variable sampling applet from the Rossman-Chance applet collection, and is ideal for 1-1 classrooms, or even students working in tech teams.  Also, Beth Chance is wonderful…and you should all know that!

starsIn my classroom notes, students first encounter the “sky”, which has been broken into 100 squares. To start, teams work to define procedures for selecting a random sample of 10 squares, using both the “hat” (non-technology) method, and a method using technology (usually a graphing calculator). Before we draw the samples however, I want students to think about the population – specifically, will a random sample do a “good job” with providing estimates? Groups were asked to discuss what they notice about the sky.  My classes immediately sensed something worth noting:

There are some squares where there are many stars (we end up calling these “dense” squares) and some where there are not so many.

Before we even drew our first sample, we are talking about the need to consider both dense and non-dense areas in our sample, and the possibility that our sample will overestimate or underestimate the population, even in random sampling.  There’s a lot of stats goodness in all of this, and the conversation felt natural and accessible to the students.

Studestars1nts then used their technology-based procedure to actually draw a random sample of 10 squares, marking off the squares.  But counting the actual stars is not reasonable, given their quantity – so it’s Beth Chance to the rescue!  Make sure you click the “stars” population to get started.  Beth has provided the number of stars in each square, and information regarding density, row and column to think about later.

But before we start clicking blindly, let’s describe that population.   The class quickly agrees that we have a skewed-right distribution, and take note of the population mean – we’ll need it to discuss bias later.

Click “show sampling options” on the top of the screen and we can now simulate random samples.  First, students each drew a sample of size 10 – the bottom of the screen shows the sample, summary statistics, and a visual of the 10 squares chosen from the population.

stars2.JPG

Groups were asked to look at their sample means, share them with neighbors, and think about how close these samples generally come to hitting their target.  Find a neighbor where few “dense” area were selected , or where many “dense” squares made the cut, how much confidence do we have in using this procedure to estimate the population mean?

Eventually I unleashed the sampling power of the applet and let students draw more and more samples.  And while a formal discussion of sampling distributions is a few chapters away, we can make observations about the distributions of these sample means.

stars3

And I knew the discussion was heading in the right direction when a student observed:

Hey, the population is definitely skewed, but the means are approximately normal.  That’s odd…

Yep, it sure is…and more seeds have been planted for later sampling distribution discussions. But what about those dense and non-dense areas the students noticed earlier?  Sure, our random samples seem to provide an unbiased estimator of the population mean, but can we do better?  This is where Beth’s applet is so wonderful, and where this activity separates itself from Random Rectangles.  On the top of the applet, we can stratify our sample by density, ensuring that an appropriate ratio of dense / non-dense areas (here, 20%) is maintained in the sample.  The applet then uses color to make this distinction clear: here, green dots represent dense-area squares.

stars4

Finally, note the reduced variability in the distribution from stratified samples, as opposed to random samples. The payoff is here!

Later, we will look at samples stratified by row and/or column.  And cluster samples by row or column will also make an appearance.  There’s so much to talk about with this one activity, and I appreciate Ruth and Beth for sharing!

Compute Expected Value, Pass GO, Collect $200

Photo Oct 11, 7 08 49 AM.jpgExpected Value – such a great time to talk about games, probability, and decision making!  Today’s lesson started with a Monopoly board in the center of the room. I had populated the “high end” and brown properties with houses and hotels.  Here’s the challenge:

When I play Monopoly, my strategy is often to buy and build on the cheaper properties.  This leaves me somewhat scared when I head towards the “high rent” area if my opponents built there.  It is now my turn to roll the dice.  Taking a look at the board, and assuming that my opponents own all of the houses and hotels you see, what would be the WORST square for me to be on right now?  What would be the BEST square?

For this question, we assumed that my current location is between the B&O and the Short Line Railroads.  The conversation quickly went into overdrive – students debating their ideas, talking about strategy, and also helping explain the scenario to students not as familiar with the game (thankfully, it seems our tech-savvy kids still play Monopoly!).  Many students noted not only the awfulness of landing on Park Place or Boardwalk, but also how some common sums with two dice would make landing on undesirable squares more likely.

ANALYZING THE GAME

After our initial debates, I led students through an analysis, which eventually led to the introduction of Expected Value as a useful statistic to summarize the game.  Students could start on any square they wanted, and I challenged groups to each select a different square to analyze.  Here are the steps we followed.

photo-oct-11-8-14-56-am

First, we listed all the possible sums with 2 dice, from 2 to 12.

Next, we listed the Monopoly Board space each die roll would causes us to land on (abbreviated to make it easier).

Next, we looked at the dollar “value” of each space.  For example, landing on Boardwalk with a hotel has a value of -$2,000.  For convenience, we made squares like Chance worth $0.  Luxury Tax is worth -$100.  We agreed to make Railroads worth -$100 as an average.  Landing on Go was our only profitable outcome, worth +$200. Finally, “Go to Jail” was deemed worth $0, mostly out of convenience.

Finally, we listed the probability of each roll from 2 to 12.

Now for the tricky computations.  I moved away from Monopoly for a moment to introduce a basic example to support the computation of expected value.

I roll a die – if it comes out “6” you get 10 Jolly Ranchers, otherwise, you get 1.  What’s the average number of candies I give out each roll?

This was sufficient to develop need for multiplying in our Monopoly table – multiply each value by its probability, find the sum of these and we’ll have something called Expected Value.  For each initial square, students verified their solutions and we shared them on a class Monopoly board.

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The meaning of these numbers then held importance in the context of the problem – “I may land on Park Place, I may roll and hit nothing, but on average I will lose $588 from this position”.

HOMEWORK CHALLENGE: since this went so well as a lesson today, I held to the theme in providing an additional assignment:

Imagine my opponent starts on Free Parking.  I own all 3 yellow properties, but can only afford to purchase 8 houses total.  How should I arrange the houses in order to inflict the highest potential damage to my opponent?

monopoly-back-row

I’m looking forward to interesting work when we get back to school!

Note: I discussed my ideas about this topic in a previous post.  Enjoy!

Pulling In To the Station

My school isn’t 1-1 with technology yet, though there are rumblings we will get there next year….or the year after….or 2031…anyway, it’s time to get techy!  My new classroom features 4 computer stations in the back – nice to have, but not super-helpful with classes of about 24 each. Station-model classroom structure has been super-helpful in my pre-calculus class in the first month. Besides the chance for all students to participate in rich technology-based activities, I’ve had the opportunity to carve out valuable small-group time with students.  Here’s an example:

In our first pre-calc unit, we review functions and their shirts, folding in new ideas like the step function, piecewise and even/odd functions.  My objective for the class was for students to consider functions in varied forms.  As students entered class, playing cards were drawn to establish their groupings, so there were 3 groups of 7 or 8.  With 15 minutes on the classroom clock, students started on their first station:

  1. Group 1 gathered in a small group with me in a circle of desks, where we worked through proving functions even or odd, and sketching their graphs.
  2. marbleslideGroup 2 worked at the computer stations on a Desmos Marbleslides featuring quadratic functions, with many students pairing up to work together. If you have never tried a Marbleslides, run and play now – we’ll wait for you to come back…
  3. Group 3 worked out in the courtyard (hey, my new classroom leads outside – which is nice) on a group task involving a piecewise function.

After groups had rotated through all 3 activities, we had time to recap / share and assess our learning over the hour.  Here’s why I need to do this more:

  • The small group station let me touch base with every student, assess strengths, find out what we need to work on, and provide feedback to everyone.
  • Marbleslides is sneaky awesome! When students begin to obsess over function shifts and how to restrict domains and don’t want to peel away from their computer, you know something is going right.
  • Class went fast! It felt like the mixed practice from Let It Stick was now becoming part of my classroom culture.
  • My pre-calc is mostly 11th and 12th graders, who have had a pretty traditional classroom experience in their math lives.  I can sense they appreciate that something difference is happening.
  • All students are responsible for their learning.  Even the least-active task, the piecewise function, was used the next class for sharing out and a jumping-off point.

 

A Bulleted Assemblage of Items for the New School Year (but not a list)

The “list” article is a popular device, and one which often draws the eyeballs. Lists are also, often, a cop-out – a way to express many ideas without having to dig too deeply.  I hate lists….

As I start my new school year tomorrow, I give you this bulleted assemblage of items which are on my mind as I look forward to our first day.

  • Fawn Nguyen’s 7 Deadly Sins of Teaching Math is required reading for all professionals. In particular, I strive to pay more attention to my (teacher talking / student talking ) ratio.  I like to think I am strong in this area, but I need to do better. Before the end of the last school year, our district screened the movie “Most Likely to Succeed” to all professional staff.  In an opening scene, the teacher provides first-day freshmen with an opening day task – and then leaves the room.  The students struggle, the teacher eventually intervenes, but a powerful classroom culture is established.  I want to provide more tasks to my students where I’m simply not needed.
  • I have used a number of opening-day activities for AP Statistics over 14 years. Distracted Driving and the Henrico hiring case are two I used most often. But I think Doug Tyson’s Smelling Parkinson’s activity could be my new favorite. It’s a powerful premise which gets kids talking about the possible vs the plausible on day 1, with a hint of simulation thrown in for good measure. I show the video below to the class and right away the statistical importance of what we do for the entire school year is established.
  • Desmos Activity Builder will take on a much bigger role in my classroom.  I’ve created activities for both my Pre-Calculus and my freshman Prob/Stat class to review their understanding, and also to serve as my “getting to know you” opportunity.  Look forward to sharing out how it goes.
  • Shoes.  I hate new shoes. They’re tight and often rip apart the back of my ankle until I break them in.  If we can have pre-washed jeans, then we can have pre-worn shoes.  We need our best people on this.
  • Who knew a cute Pythagorean triple generator could be of interest to so many. After I posted about an interesting share from Ken Sullins at the PCTM summer conference, so many folks chimed in with their ideas.  Thanks especially to Joel Bezaire who shared additional ideas from Twitter Math Camp.  I’m using this in my pre-calc class on day 1.

  • I’ve given the same probability problem to my freshmen for the last few years. I love everything about this problem on day 1: it gets kids talking, it gets kids struggling, and it tells me much about their problem solving background.

 

OK, maybe this was a list after all.  I need to do some last-minute ironing.

Who Assessed it Better? AP Stats Inference Edition

Free-response questions and exam information in this post freely available on the College Board – AP Statistics website

Today I am stealing a concept from Dan Meyer’s task comparison series “Who Wore it Best”, and bringing it to the AP Statistics exam world. In the series of 6 free-response questions on the AP Stats exam, it is not unusual for one question to focus solely on inference. Compare these two questions, which each deal with inference for proportions.

From 2012:

2012q

From 2016:

2016q

I read (graded) the question from 2012 as an AP Exam reader, and observed a variety of approaches. I find that while many students understand the structure of a hypothesis test, it’s the nuance – the rationales for steps – which are often lost in the communication. In the 2012 question, students were expected to do the following:

  1. Identify appropriate parameters
  2. State null and alternate hypotheses
  3. Identify conditions
    • Independent, random samples and normality of sampling distribution
  4. Name the correct hypothesis procedure
  5. Compute / communicate test statistic and p-value
  6. Compare the p-value to an alpha level
  7. Make an appropriate conclusion in context of the problem

It’s quite a list.  And given that individual AP exam problems are worth a total of 4 points, steps are often combined into one scoring element.  Here, naming the test and checking conditions were bundled – as such, precision in providing a rationale for conditions was often forgiven.  For example, if students identified the large sample sizes as a necessary condition, this was sufficient, even if there was no recognition of a link to normality of sampling distributions.  Understanding the structure of a hypothesis test – with appropriate communication – was clearly the star of the show.  While inference is one the “big ideas” in AP Statistics, my view is that questions like this from 2012 encourage cookbook statistics, where memorized structure take the place of deeper understanding.

So, it was with much excitement that I saw question 5 from 2016.  Here, the interpretation of a confidence interval was preserved. But I appreciate the work of the test development committee in parts b and c; rather than have students simply list and confirm conditions for inference, the exam challenges students to be quite specific about their rationales. With parts b and c, students certainly struggled more than with the conditions in 2012, but I hope their inclusion causes statistics teachers to consider their approach to hypothesis conditions. The mean scores for each question speak to struggles students on this question, compared to traditional hypothesis testing structure.

  • 2012: 1.56 (out of 4)
  • 2016: 1.27 (out of 4)

The inclusion of part b of question 5 this year, where students were asked to defend the np > 10 condition, was perfect timing for my classes.  This year, I tried a new approach to help develop student understanding of the binomial distribution / sampling distribution relationship. I found that while many students will continue to resort to the “short cut” – memorizing conditions – a higher proportion of students were able to provide clear communication of this inference condition.

The AP Statistics reading features “Best Practices Night” – where classroom ideas are shared.  You can find resources from the last few years at Jason Molesky’s APStatsMonkey site. I shared my np > 10 ideas with the group, and received many positive comments about it.  Enjoy my slides here, and feel free to contact me with questions regarding this lesson:

Finally, I can’t express how wonderfully rich a professional-development experience the AP reading is.  I always find myself with a basket of new classroom ideas and contacts to share with – it’s stats-geek Christmas.  For me, 2016 is the year the #MTBoS started to make its mark at the Stats reading – I met so many folks from Twitter, and we held our first-ever tweet-up!

Also, the vibrant Philadelphia-area stats community was active as always.  We meet as a group a few times a year to share ideas and lessons; seeing so many from this area participate in the reading makes us all better with what we do for our students.

 

The People in My Math Neighborhood

Oh, who are the people in your neighborhood?
In your neighborhood?
In your neighborhood?
Say, who are the people in your neighborhood?
The people that you meet each day

I work with an awesome group of people at a high school outside of Philadelphia.  They are my colleagues, the people I share ideas with on a daily basis, and some of my closest friends.

But in recent years, my math neighborhood has grown considerably.  I suppose I discovered the power of the online neighborhood 4 or 5 years ago, developing and growing a wonderful network of professional colleagues through the #MTBoS. And my relationship with this neighborhood has grown from a mechanism for sharing ideas, to a source of inspiration, positive thought, discussion and reflection.

We are now 3 weeks after the NCTM Annual Conference in San Francisco. It’s easy to forgot the little things which occur in a big conference, and I hopefully will find time to reflect and utilize new ideas later. NCTM this year has done a wonderful job of providing a means to continue reflections and growth outside of the conference, along with archiving session resources.  Here, I highlight 4 sources of inspiration, and friends in my math neighborhood, as I look back on my San Francisco experiences.

GRAHAM FLETCHER – Graham, an elementary specialist from Georgia (or is he Canadian? such a chameleon), challenged teachers to consider the mathematical story we share with students in his ShadowCon talk. How is your story different than the one being told by your colleague teaching the same material just across the hall?

grahamHigh school teachers may be intrigued by Graham’s discussion of fractions, reducing and equivalence and the role of “simplifying”. His talk has caused me to think about the many odd restrictions we place on student work: i.e. “write the equation of your line in standard form”, and their necessity in my math story. Graham’s call to action – challenging teachers to identify their own “simplifying fractions” (something they teach not currently in the standards) – is an appropriate task for all grades.

ROBERT KAPLINKSY – Robert was featured on the MathEd Out podcast last summer, and I recall taking a walk, listening last year when it occurred to me that Robert’s path to becoming a math teacher was eerily similar to mine. His ShadowCon talk, “Empower”, reminds me that no matter how top-down our education world may feel, we all have a role to empower others and become influential in our math neighborhoods. I appreciate the multiple mechanisms Robert suggests for fostering empowerment, and his call to action that we thank a colleague who helped us feel empowered is a wonderful way to close out a school year – and look forward to new things.

PEG CAGLE – I have admired Peg’s ideas for some time now, and was thrilled to meet and chat with her last year at Twitter Math Camp. Even though I rarely teach geometry, I felt pulled to Peg’s session “Paper Cup + Gust of Wind”, and was awed by the simplicity, engagement, theme-building in this simple task. By rolling a paper Dixie cup along a surface, Peg develops a lesson which extends through the school year, building complexity each time.

Day 1:Explain what happens when we roll out the cup

Day 40: Convince a skeptic of the shape it makes. Find its area.

Day 105: find area of shape based on dimensions

Day 140: How can you build a cup from a single sheet (with base) of 8.5 x 11 paper to trace out the maximum area as it rolls?

Day 175 (after trig ratios): how do you NOW find the area of the shape, given its dimensions

This session has caused me to think about other simple tasks which could become full-course themes. Peg’s inspiration came from a cup blowing in the breeze – you never know where the next fun math idea will come from!

CHRISTINE FRANKLIN – Why was I so nervous and awe-struck to meet Christine at the AP Stats forum in San Francisco? Because she is so awesome – and was the inspiration for my NCTM talk on Variability and Inference, geared towards the middle school community. It was at Professional Night at the AP Stats reading 2 years ago where Christine diagrammed the historical path stats has taken in K-12 curriculum, and the parallels between AP and middle school descriptions. Christine was recently named the K-12 statistical ambassador by the ASA, and a sweeter person could not fill the job.

Hoping I never move out of the neighborhood!

Activity Builder Reflections

We’re now about 9 months into the Desmos Activity Builder Era (9 AAB – after activity-builder). It’s an exciting time to be a math teacher, and I have learned a great deal from peeling apart activities and conversing with my #MTBoS friends (run to teacher.desmos.com to start peeling on your own – we’ll wait…). In the last few weeks, I have used Activities multiple times with my 9th graders.  To assess the “success” of these activities, I want to go back to 2 questions I posed in my previous post on classroom design considerations, specifically:

  • What path do I want them (students) to take to get there?
  • How does this improve upon my usual delivery?

 

AN INTRODUCTION TO ARITHMETIC SERIES (click here to check out the activity)

My unit or arithmetic sequences and series often became buried near the end of the year, at the mercy of “do we have time for this” and featuring weird notation and formulas which confused the kids. I never felt quite satisfied by what I was doing here.  I ripped apart my approach this year, hoping to leverage what students knew about linear functions to develop an experience which made sense. After a draft activity which still left me cold, awesome advice by Bowen Kerins and Nathan Kraft inspired some positive edits.

seatsIn the activity, students first consider seats in a theater, which leads to a review of linear function ideas. Vocabulary for arithmetic sequences is introduced, followed by a formal function for finding terms in a sequence. It’s this last piece, moving to a general rule, which worried me the most.  Was this too fast?  Was I beating kids over the head with a formula they weren’t ready for? Would the notation scare them off?

plotsThe path – having students move from a context, to prediction, to generalization, to application – was navigated cleanly by most of my students.  The important role of the common difference in building equations was evident in the conversations, and many were able to complete my final application challenge.  The next day, students were able to quickly generate functions which represent arithmetic sequences, and with less notational confusion than the past.  It certainly wasn’t all a smooth ride, but the improvement, and lack of tooth-pulling, made this a vast improvement over my previous delivery.

DID IT HIT THE HOOP? (check out the activity)

DAN.PNGDan Meyer’s “Did It Hit the Hoop” 3-act Activity probably sits on the Mount Rushmore of math goodness, and Dan’s recent share of an Activity Builder makes it all the more easy to engage your classes with this premise. In class, we are working through polynomial operations, with factoring looming large on the horizon.  My 9th graders have little experience with anything non-linear, so this seemed a perfect time to toss them into the deep end of the pool.  The students worked in partnerships, and kept track of their shot predictions with dry-erase markers on their desks. Conversations regarding parabola behavior were abundant, and I kept mental notes to work their ideas into our formal conversations the next day.  What I appreciate most about this activity is that students explore quadratic functions, but don’t need to know a lick about them to have fun with it – nor do we scare them off by demanding high-level language or intimidating equations right away.

The next day, we explored parabolas more before factoring, and developed links between standard form of a quadratic and its factored form. Specifically, what information does one form provide which the other doesn’t, and why do we care?  The path here feels less intimidating, and we always have the chance to circle back to Dan’s shots if we need to re-center discussion.  And while the jury is out on whether this improves my unit as a whole, not one person has complained about “why”…yet.

MORE ACTIVITY BUILDER GOODNESS

Last night, the Global Math Department hosted a well-attended webinar featuring Shelley Carranza, who is the newest Desmos Teaching Faculty member (congrats Shelley!).  It was an exciting night of sharing – if you missed it, you can replay the session on the Bigmarker GMD site.