# Category Archives: Uncategorized

## I Built a Crappy Digital Activity…Here’s How I Fixed It.

In the past 3 years, I have used Desmos Activity Builder in a number of different ways in my classroom: to introduce new ideas, as a formative assessment tool, and to allow students to “play” with mathematical ideas through Polygraph and Marbleslides. An activity I developed last year and re-built this year reminded me of two ideas I need to keep on my radar at all time. First, building an effective classroom activity is really, really hard.  Second, don’t be stubborn in evaluating an activity – it pays to be brutally honest.

For my 9th grade class last year, I wanted an activity which would cause students to think about variability in data distributions, and introduce standard deviation as a useful measure of variability. You can preview the activity here.  Take a few minutes, test drive it, and see if you can suss out the problems.

So, what went wrong.  Well, a number of things – but here are the two primary suspects.

1. It’s too damn long! It takes way too long to get to a working definition of standard deviation, and by screen 14 students are all over the place.  Using student pacing could help remedy some of this, but I found much of the class losing interest by the time we got there.
2. I was stubborn! I was looking for a “cute” visual way for students to think of standard deviation as “typical distance from the mean”.  In my zeal to hammer this working definition home, I tried to build slick graphs which lost many students.

How to fix it – last year, the Desmos teaching faculty developed the “Desmos Guide to Building Great Digital Activities“.  It’s worth a read (and a re-read) now and then to guide activity construction. In my variability activity, this bullet point from the guide resonated with me: Keep expository screens short, focused, and connected to existing student thinking.  In many of the screens, I over-explained things.  Students don’t want to read when they are completing a digital activity, they want to investigate, create, and explore.  I robbed them of that chance.

Today I tried my new, rebuilt variability activity with 2 classes (slimmed down to 12 screens from 19), and there was a vast improvement in class engagement. There were more opportunities for students to express their ideas regarding comparisons of distributions, and we had plenty of time to pause, recap, discuss and think about next steps.  A number of points from the Desmos Guide drove my thinking:

Ask for informal analysis before formal analysis.  While I kept in the “typical distance” definition of standard deviation, it was only in a small way – we’ll move on to a more formal definition next. Students were able to conceptualize standard deviation as a useful measure, and now can move on to a formal definition.  My old activity felt too “sledge-hammer-ish” and I knew it.

Incorporate a variety of verbs and nouns. I provided a number of ways for students to think about variability and distribution comparisons in the early screens, and strove to build different-looking screens.  This kept the ideas fresh, and students talked with their partners to assess these differences in different ways.

Create activities that are easy to start and difficult to finish.  In the last 3 screens, I ask students to extend their thinking, be brave, and apply new ideas.  For those students who got this far – most did, these screens elicited the loudest debates.  We ran out of time at the end, but we have some good stuff to build off of tomorrow morning.

I’ve learned that it’s important to be honest about an activity.  It’s easy to blame the students when something goes wrong, especially which class heads away from learning and towards frustration.  But performing an activity autopsy, focusing on clear goals, and keeping the design principles in mind is helpful to move an activity forward.

## Baseball, Brain Cancer and Relative Risk

The 1993 Phillies were the most fun team I have ever followed.  I was nearing the end of my college years, and I vividly remember the insane night when me and 3 buddies celebrated a win at 4:30 in the morning, and the exact spot I was sitting when our hearts were broken with Joe Carter’s home run (I still look away when it comes up on highlight reels).

This week the catcher of that team, Darren Daulton, died after a battle with brain cancer.  Newspapers have shared memories of “Dutch” and among the articles is one which reminds us of the surprising number of former Phillies who have passed away due to brain cancer (Tug McGraw, John Vukovich, and Johnny Oates). A revised 2013 article from the Philadelphia Inquirer analyzes the the unusual number of Phillies who have developed brain cancer, and contains many appropriate entry points for Statistics courses.  Some highlights from the article:

• A comparison of the observed effect to random chance – here a professor of epidemiology summarizes: “You can’t rule out the possibility that it’s random bad luck.”
• A summary of plausible variables which could lead to elevated levels of exposure, such as artificial turf (which may have contained lead) or anabolic steroids.
• An analysis of the increase rate of brain cancer among Phillies – here we are told that the Phillies’ rate is “about 3.1 times as high” as the national rate.  A confidence interval, along with an interpretation and associated cautions are also included.

Let’s explore that “3.1 times” statistic…time to break out the technology.

A few weeks back, I attended the BAPS (Beyond AP Statistics) workshop in Baltimore, as part of the Joint Statistics Meetings. Allan Rossman and Beth Chance shared ideas on using their applet collection to explore simulation (see my earlier post using the applets to Sample Stars) along with a “new” statistic we don’t often talk about in AP Stats – relative risk.

To start, I used the Analyzing 2-Way Tables applet and used the “sample data” feature.  Here I attempted to use the same numbers quoted in the article:

The national rate was 9.8 cases per 100,000 adult males per year, while the rate in the former Phillies was 30.1 cases per 100,000 – about 3.1 times as high.

There are two issues here: first, to perform a simulation we need counts, so numbers like 9.9 and 30.1 just don’t play nice.  I’ll use 10 and 30.  Also, I wasn’t surprised that this site was not real happy with my using a population of 100,000 for simulation.  Here, I am going with 1,000 for convenience and to make the computer processor gods happy – we can debate the appropriateness of this down the road.

The applet will then simulate the random assignment of the 2,000 subjects here to the two treatment groups (group A: being a Phillie, group B: not being a Phillie). How likely is it that we will observe 30 or more “successes” (which here represent those who develop brain cancer) in one of the two groups?  In the applet, we can see how the “successes” have been randomly assigned from their original spots in the 2-way table to new groupings.

AT BAPS, Allan Rossman then explained how we can summarize these two groups using Relative Risk, which is listed under the “Statistic” menu on the applet. In general, relative risk is the proportion of success in one group divided by the proportion of success in a second group.  If we have proportions in two groups which are equal, then the relative risk would be 1.  We can then link to the newspaper article which claims a 3.1 “relative risk”, simulate many times with the applet (below we see the results of 10,001 simulations), and compare to the reported statistic.

According to the simulation, we should only expect to see a relative risk of 3 or above about 0.08% of the time – clearly an “unusual” result.

But the article does not claim a significant difference, and cautioned against doing so as a number of assumptions were made which could alter conditions.  This would be an opportunity to discuss some of these design assumptions and how they could change the outcome.

Rest in Peace Dutch!

## Area Models and Completing the Square

I’m nearing the end of my time with my 9th graders, with this week dedicated to moving beyond factoring as the sole method for solving quadratic equations and towards more general methods like completing the square.

Late in May, David Wees shared materials which challenge students to investigate the relationships between “standard form” and “completing the square” form (aside – does anyone agree on proper terms for these?) using area models to build representations.  Given that I use area models often to introduce polynomial multiplication, I was eager to maintain consistency in the student understanding.

But before we dove into David’s lesson, I wanted students to revisit their understanding of area models.  In this Desmos Activity Builder lesson I created, students shared their interpretations of area models and worked in pairs to investigate non-square models.  In one of the final screens, students argued for the “correct” interpretation of a model.

Using the Desmos teacher dashboard, we could see clear visual arguments for both representations.  This was valuable as we ended the lesson for the day, and tucked that nugget away for Monday, when we would begin to formalize these equivalencies.

After the weekend, students worked independently through David’s Completing the Square lesson. Not only did students quickly move through the area models and the dual representations, the debates between students to explain how to move from one representation to the other were loud and pervasive.  I’m also loving how many of my students have started to use color as an effective tool in our OneNote-taking (below).

At the end of the sheet, all students completed problems which translate standard form to vertex form with no support from me (“no fuss…no muss”).  It dawned on me that something amazing had happened….my students had figured out completing the square without my ever talking about completing the square.

Tomorrow we’ll tackle those pesky odd-number “b” terms, but my students own this already!

## Another Pascal’s Triangle Gem

At a recent local conference, Jim Rubillo, Annie Fetter and I were saying our good-byes at the end of a fun evening, when Jim’s puzzly side emerged…

What proportion of the numbers in Pascal’s Triangle are even?

Every time I talk to Jim, he’s bound to have a neat problem for me to chew on.  The last time, he shared a fun task involving the harmonic series. Take a few minutes and think about this Pascal’s Triangle scenario…I’ll even leave you some spoiler space.

At the ATMOPAV Spring Conference last month, Jim shared an entertaining talk titled “Gambling, Risk, Alcohol, Poisons and Manure – an Unfinished Life Story”.  The talk led the attendees on a journey through the history of statistics, starting with games of chance and the meeting of Chevelier du Mere and Blaise Pascal, through the introduction of formal inference procedures developed at the Guinness brewery, and to identifying statistical abuses in the present day.

Jim is a life-ling educator and former Executive Director of NCTM who happens to live quite close to me.  It was a thrill having him share his ideas with the group.

So, which rows are in Pascal’s Triangle are we talking about here?

In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!).  But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle.  You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1.  What a neat result!  Below is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows.

For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area.

Already looking forward to my next encounter with Jim!

## Drinking the Statistical Power Kool-Aid

For my colleagues who teach AP Stats, there are few phrases more terrifying:

Today I am teaching Power.

Power: a deep statistical concept, but one which often gets moved towards the back of the AP Stats junk drawer.  The only mention of power in the AP Stats course description comes under Tests of Significance:

Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power

So, students need to understand the concept of power, but not actually compute it (which is itself not an easy task).  Floyd Bullard’s article “On Power” from the AP Central website provides solid starting points for teachers struggling with this concept; specifically, I appreciate his many ways of considering power:

• Power is the probability of rejecting the null hypothesis when in fact it is false.
• Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.
• Power is the probability that a test of significance will pick up on an effect that is present.
• Power is the probability that a test of significance will detect a deviation from the null hypothesis, should such a deviation exist.
• Power is the probability of avoiding a Type II error.

This year, I tried an activity which used the third bullet above, picking up on effects, as a basis for making decisions.

HEY KOOL-AID MAN!

Arriving at school early, I got to work making 3 batches of Kool Aid.  During class, all students would receive samples of the 3 juices to try.  Students were not told about the task beforehand, or where this was headed. Up to now, we had discussed type I and type II error, so this served as a transition to the next idea.

THE BASELINE SAMPLE:

All students received cups and as they worked on a practice problem I circulated, serving tasty Kool Aid – don’t forget to tip your server!  I told students to savor the juice, but to pay attention: I promised them that this first batch was made using strict Kool Aid instructions.  Think about the taste of the juice.

SAMPLE A:

Next, students received a drink from “Sample A”.  Their job – to assess if this new sample was made using LESS drink mix than the baseline batch.  Also, I varied the amounts of juice students received: while some students were poured full cups, some received just a few dribbles.  To collect responses, all students approached the board to contribute a point to a Sample A scatterplot, using the following criteria:

Sample size: how much juice you were given

Evidence: how much evidence do you feel you have to support our alternate hypothesis – that Sample A was made with LESS mix than the baseline?

As you can see, the responses were all over the place – a mixture of “we’re not quite sure” to “these are strange directions” to “I just don’t trust Lochel – something’s up”.  But the table has been set for the next sample.

Sample A: it was made with just a smidge less mix than the baseline.  So I wasn’t totally surprised to see dots all over.

SAMPLE B:

I poured drinks again from this new sample, and again varied the sample sizes.  I asked all students to think about their evidence in favor of the alternate, and wait until everyone tasted their juice before submitting a dot.

And check out those results!  Except for a few kids (who admitted they stink at telling apart tastes), we have universal support in favor of the alternate hypothesis.

Sample B: this was made with 1/2 the suggested amount of drink mix.  Much weaker!

FOLLOW-UP DISCUSSION:

This activity made the discussion of power much more natural.  In particular, what could occur during a study which would make it more likely to reject the null hypothesis, if it deserves rejecting?

Larger sample size: smaller samples make it tough to detect differences

Effect size: how far away from the null is the “truth”.  If the “truth” as just a bit less than the null, it could be difficult to detect this effect.

In terms of AP Stats “concepts of power”, this covers much of what we need.  Next, I used an applet to walk students through examples and show power as a probability.  And like most years, this was met with googly eyes by many, but the foundation of conditions which would be ripe for rejecting the null was built, and I was happy with this day!

Suggested reading: Statistics Done Wrong by Alex Reinhart contains compelling, clear examples for teachers who look to lead discussions regarding P-value and Power.  I recommend it highly!

## 4 Activity Builder Formative Assessment Ideas

The creative team at Desmos continue to develop engaging lessons using their Activity Builder interface, found at teacher.desmos.com. While teachers I encounter have their own favorite activity, many desire to dive in and create their own. But building your own activity, testing it, and hoping it works with your class can be an intimidating task (pro-tip: making your own activity is really hard!).  But there are a few simple ways teachers can use Activity Builder as a mechanism for formative assessment.  Here. I share 4 quick and easy ideas – you can check them out and observe their structure at this link.

SELF-CHECKING GRAPH MATCHING

I used this often with my Pre-Calculus class in the fall, and the concept works equally well with younger students.  Simply start a new Activity Builder screen, and enter the equation you’d like students to provide.  Place the equation in a folder, which you can hide so students won’t see it when they encounter the screen.  Finally, by making the graph with dashed lines, students can easily see if their submission matches the requested graph, and can adjust accordingly.

GALLERY WALK

Here’s a neat Activity Builder hack you may not know about.  If you have an existing Desmos graph, copy the URL from your graph to the clipboard.  Then, in an Activity Builder screen click the “Graph” button and paste the URL into the first expression line – and PRESTO, the graph is imported into an Activity Builder screen.  I often collect student work by simply having them submit a Desmos URL.  Consider taking samples of student works and create a virtual gallery walk.  Let students view each other’s ideas, comment and make suggestions. Thanks to my colleague DJ for providing neat student graphs!

SELF-ASSESSMENT SLIDERS…AND OVERLAY

Have students assess their own learning with a moveable point. Provide an “I can…” prompt and let students consider where they fall in the learning progression.  Hold a class-wide discussion of unit skills by anonymizing student names and using the overlay feature to take the class pulse on skills.

MY FAVORITE DISTRACTOR

Activity Builder allows teachers to build their own multiple-choice questions, with the option of having students provide an explanation for the choice they make.  In “My Favorite Distractor”, students select an answer they KNOW is wrong, and explain how they know.  This may not work for many multiple-choice type questions, but consider using this idea in situations where the distractors have clear, interesting rationales for elimination.

Have your own quick formative assessment ideas?  Share it here!

## 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.