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.

Activity Builder – Classroom Design Considerations

This past summer, our forward-thinking math-teacher-centric friends at Desmos released Activity Builder into the wild, and the collective creativity of the math world has been evident as teachers work to find exciting classroom uses for the new interface. Many of these activities are now searchable at teacher.desmos.com – you’re welcome to leave now and check them out – but come back…please?

Its easy to get sucked in to a new, shiny tech tool and want to jump in headfirst with a class. I’ve now created a few lessons and tried them with classes which range from the “top” in achievement, to my freshmen Algebra 1 students. In both cases, I’ve settled upon a set of guiding principles which drive how I build a lesson.

  • What do I want students to know?
  • What path do I want them to take to get there?
  • How will my lesson encourage proper usage of math vocabulary?
  • What will I do with the data I collect?
  • How does this improve upon my usual delivery?

It’s the last question which I often come back to. If making a lesson using Activity Builder (or incorporating any technology, for that matter) doesn’t improve my existing lesson, then why am I doing it?


One recent lesson I built for my algebra 1 class asked students to make discoveries regarding slopes and equations of parallel and perpendicular lines. Before I used it with my class, a quick tweet 2 days before the lesson provided a valuable peer-review from my online PLC.  It’s easy to miss the small things, and some valuable advice regarding order of slides came through, along with some mis-types. The link is provided here in the tweet if you want to play along:

The class I tried this with is not always the most persistent when it comes to math tasks, but I was mostly pleased with their effort. Certainly, the active nature of the activity trumped my usual “here are bunch of lines to draw – I sure hope they find some parallel ones” lesson.

As the class finished, I called them into a small huddle to recap what we did. This is the second lesson using Activity Builder we have done together.  In the first, the students didn’t know I can see their responses, nor understand why it might be valuable.  In this second go-round, the conversation was much deeper, and with more participation than usual. In one slide, the overlay feature allowed us to view all of our equations for lines parallel to the red line:

parallels

We could clearly see not only our class successes, but examine deeper some misunderstandings.  What’s happening with some of those non-parallel lines?  Let’s take a closer look at Kim’s work:

Parallel 2

What’s going on here? A mis-type of the slope? The students were quite helpful towards each other, and if nothing else I’m thrilled the small group conversation yielded productive ideas in a non-threatening manner – it’s OK to make errors, we just strive to move on and be great next time.  The mantra “parallel lines have the same slope” quickly became embedded.

The second half of the lesson was a little bumpier, but that’s OK.  Before questions regarding slope presented themselves in the lesson, storm clouds were evident when the activity asked students to drag a slider to build a sequence of lines perpendicular to the blue line.  Observe the collective responses:

perp2

So, before we even talk about opposite reciprocal slopes, we seem to have a conceptual misunderstanding of perpendicular lines.  I’m glad this came up during the activity and not later after much disconnected practice had taken place.  In retrospect, I wish I had put this discussion away for the day and come up with a good activity for the next day to make sure were all on board with what perpendicular lines even look like, but I pressed ahead.  We did find one student who could successfully generate a pair of perpendicular lines, and I know Alexys enjoyed her moment in the sun.

perp1

What guiding principles guide you as you build activities using technology? How do they shape what you do?  I’m eager to hear your ideas!

 

 

How Do We Assess Efficiency? Or Do We?

A problem on a recent assessment I gave to my 9th graders caused me to reflect upon the role of efficiency in mathematical problem solving. In particular, how much value is there in asking students to be efficient with their approaches, if all paths lead to a similar solution?  And should / could we assess efficiency?

The scene: this particular 9th grade class took algebra 1 in 7th grade, then geometry in 8th.  As such, I find I need to embed some algebra refreshing through the semester to dust off cobwebs and set expectations for honors high school work. For this assessment, we reviewed linear functions from soup to nuts. My observation is that these students often have had slope-intercept form burned into their memory, but that the link between this and standard form is weak or non-existant.  Eventually, the link between standard form and slope ( -A/B ) is developed in class, and we extend this to understanding to think about parallel and perpendicular lines.  It’s often refreshing to see the class see something new in the standard form structure which they hadn’t considered before.

The problem: on the unit quiz, I gave a problem which asked students to find the equation of a line parallel to a given line, passing through a point.  Both problem and solution are given in standard form.  Here is an example of student work (actually, it’s my re-creation of their work)….

linear problem

So, what’s wrong with the solution?  Nothing, nothing at all.

Everything here answers the problem as stated, and there are no errors in the work. But am I worried that a student took 5 minutes to complete a problem which takes 30 seconds if standard dorm structure is understood?…just a little bit.  Sharing this work with the class, many agreed that the only required “work” here is the answer…maybe just a “plug in the point” line.

My twitter friends provided some awesome feedback….

Yep, we would all prefer efficiency (maybe except Jason). Thinking that I am headed towards an important math practice here:

CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

It may be unreasonable for me to expect absolute efficiency after one assessment, but let’s see what happens if I ask a similar question down the road.

Confession, I really had no idea what #CThenC was before this tweet.  Some digging found the “Contemplate then Calculate” framework from Amy Lucenta and Grace Kelemanik, which at first glance seems perfect for encoruaging the appreciation for structure I was looking for here.  Thanks for the share Andrew!

Yes, yes!  Love this idea.  The beauty of sticking to standard form in the originial problem is that it avoids all of the fraction messiness of finding the y-intercept, which is really not germane to the problem anyway. Enjoy having students share out their methods and make them their own.

What do you do to encrouage efficiency in mathematical reasoning?  Share your ideas or war stories.

Activity Builder Reflections

The super-awesome Desmos folks set Activity Builder into the wild this past summer, and it’s been exciting to see the creativity gushing from my math teaching colleagues as they build activities.  So far, I have used Activities with 2 of my classes, with mixed success.

In my 9th grade Prob/Stat class, I built an Activity to assess student understanding of scatterplots and lines of best fit.  You can play along with the activity if you like: go to student.desmos.com, and enter the code T7TP.  I am most excitied by the formative assessment opportunities an activity can provide – here are 3 places where I was able to assess class understanding.

In one slide, students were shown a scatterplot, and asked to slide a point along a number line to a “reasonable” value for the correlation coefficeient, r.  The overlay feature on the teacher dashboard allowed me to review responses with the class and consider the collective class wisdom.

r overlay

In another slide, students were again given a scatterplot and asked to set sliders for slope and y-intercept to build a best-fit line.  Again, the overlay feature was helpful, though it was also great to look at individual responses.  This led to a discussion of that pesky outlier on the right – just how much could it influence the line?

LSRL overlay

Finally, question slides were perfect for allowing students to communicate their ideas, and focus on vocabulary. In our class debrief, we discussed the meaning of slope in a best-fit line, and its role in making predictions about the overall pattern.

 question roll

But all has not been totally sunny with Activity Builder.  In my Algebra 1 class, I built an Activity to use as a station during class.  Splitting the class in half, one group worked with me on problems, while the rest worked through the activity, then flipping roles halfway through class.  You can try this activity at student.desmos.com, code 3FGM.

Storm clouds approached early, when a student complained that they didn’t know what to do – even though the first slide offered instructions to “Drag the points…”.  Quickly my “I’m an awesome teacher who uses stations” fuzziness turned into saltiness as students clearly were not following the activity faithfully.  Here’s what I learned:

Leading class through an activity beforehand would have been helpful. In the future, I’m going to make a vanilla lesson which walks students through simple tasks – dragging points, answering questions, entering equations, adjusting sliders – and let them see how I can view and use their responses.  Just setting a class into the wild, especially a class which often struggles with instructions, didn’t work so hot.


Last Saturday, I led a group of about 20 teachers in an Activity Builder workshp at the ATMOPAV Fall Conference at Ursinus. I had 3 goals for the assembled teachers for the hour:

  • Experience activities through a student perspective.
  • Experience the teacher dashboard.
  • Start building their own activities.

Some have asked for my materials, and I can’t say I have too much to share.  Check out my Slides and feel free to contact me with questions about the hour. Some highlights of the group discussions:

  • When is the best time in a unit to use an Activity?  So far, I have used it as an intro to a unit, and also as a summary of a unit.  The difference is in the approach to task.  An intro activity should invite students to explore and play, and think about generalizations – include lots of “what do you think?” opportunities.  In my summary activity, I asked specific questions to see if students could communicate ideas based on what we had learned.
  • Think about how you will leverage to teacher dashboard to collect and view ideas.  How does the overlay feature let all students contribute and build class generalizations in a new way?  How will you highlight individual student responses to generate class conversation?
  • Ask efficient questions.  There’s really not a lot of room in the text for long-ish tasks.  Keep things short, sweet, and focused.
  • Many teachers wanted to know more about building draggable points.  The way I do this is to create a table, enter some points, and use the Edit feature to make the points draggable.  Your best bet may be to take an already existing activity and pore through its engine, which reminds me….

Desmos is now assembling an searchable archive of vetted activities.  Go to teacher.desmos.com, and use the search bar at the top-left.  I highly recommend any creations by Jon Orr, Michael Fenton and Christopher Danielson.

copyAnd finally, an exciting new feature to Activity Builder just appeared today – you can now copy slides within an Activity.  Click the 3 dots to duplicate a slide and use it again, or edit a graph to use later.

Residuals from the Past Month

It’s been a busy month of October. I don’t have a new lesson or resource to share this time – instead, here are some things which have been rattling around my brain.

Last night at the weekly Global Math Department online meet, NCTM President-Elect Matt Larson shared new and “in the works” resources for members, and a revised vision for PD in the coming years.  In the past few months, I have been fortunate to speak personally with both Matt and current President Diane Briars, and I am happy to hear that NCTM not only values the work of the Math-Twitter-Blog O-Sphere, but are now beginning to take lessons from the successes of ourline community and bring it to the national organization.

With regional conferences starting this week, I am most excited to see a new website NCTM has established to encourage ongoing dialogue: http://regionals.nctm.org/.  I won’t be able to make any of the regionals, but will be checking in from afar on this new site. I’m already enjoying the sharing from presenters, and the sense of ongoing discussion.

I re-arranged my bedroom furniture this summer, and I’m not sure I like it better.

This afternoon, I took one of my daily walks through the neighborhood, with the Bill Simmons podcast as my companion. His guest was Judd Apatow, and the conversation turned towards the negative aspects of celebrity.  Judd mentioned Eddie Murphy who started as observational comic, then became hugely famous, as someone whose work was altered by the seclusion of celebrity status. No longer able to make the every-day connection to his audience, the observational aspect of Eddie’s career withered away, and was replaced by other things.  Judd Apatow, sensing a need to re-visit his comedic roots for inspiration, dusted himself off to do stand-up and has caught his comedic second wind.

Is there a lesson here for teachers who leave the classroom to become administrators? How long does it take for separation from the classroom to take root – and can (and should) it be re-visited now and again?

Sometimes I wonder why nobody has been arrested yet for assaulting one of the Impractical Jokers

I have 3 quite different preps this semester, and I am professionally miserable because of it.  With block scheduling here, teachers have 3 courses each semester.  Now and then, 3 preps is not a big deal.  But I teach each course with someone different (or a different group) from the department, and I rarely share prep with any of them.  I’m also the only member of my department to have 3 preps, and this is the second semester in a row this has happened.  OK…I’m getting real close to my whining quota here, but I don’t think I am doing a good job right now.  Instead of having laser-focus on my courses, I find myself all over the place.  This is not helping my students and I am worried.

Some of my AP students report that they will go trick-or-treating next week.  For me, high school age is when you are out of the candy loop.  Am I right?

My Math Club kids are the most enthusiastic bunch I have “coached” in recent memory.  And the weekly Math Madness contests have been great for getting kids to talk about problem solving approaches.  I don’t usually enjoy doing math for competition’s sake, but we have been holding weekly de-briefs after each contest and the conversations have been informal, spirited and genuine.  I’m lucky to work with such a great group of kids!

Today is “Back to the Future” day – October 21, 2015.  The day Marty McFly visited the future on the big screen.  And I passed a DeLorean on the way home from work (no lie, this really happened!)

My new local hero is a colleague of mine at my school who teachers Anatomy and Physiology, Chris Baker.  In addition to being an awesome role model for kids, and someone passionate about his craft, he has jumped deeper into the Twitter pool and has embraced 20% time as part of his classroom culture.  Consider giving him a follow – he’s a good egg!

Making It Stick…With Beanbags

The book Making It Stick – the Science of Succesful Learning has caused me to consider how I approach practice and assessment in my math classroom. The section “Mix Up Your Practice”, in particular, provides ideas for considering why spaced practice, rather than massed practice, should be considered in all courses.

But it was an anecdote which began the chapter on spaced practice which led to an interesting experiment for stats class.  The author presents a scenario where eight-year-olds practiced tossing bean bags at a bucket.  One group practiced by tossing from 3 feet away; in the other group, tosses were made at two buckets located two feet and four feet away.  Later, all students were tested on their ability to toss at a three-foot bucket.  Surprisingly, “the kids who did best by far were those who’d practiced on two and four-foot buckets, but never on three foot buckets.”

Wow!

Let’s do it.

My colleague and I teach the same course, but on different floors of the building during different periods. Each class was given bean bags to toss, but with different practice targets to attempt to reach.

  • In my class, lines were taped on the floor 10 and 20 feet from the toss line.
  • For Mr. Kurek’s class, one target was placed 15 feet from the toss line.

Photo Oct 05, 9 33 54 AMAfter every student had a chance to practice (and some juggling of beanbags was demonstrated by the goofy….), I picked up my tape lines, and placed a new, single line 15 feet from the toss line.  Each student then took two tosses at the target, and distances were recorded (in cms).

We then analyzed the data, and compared the two groups (the green lines are the means):

bean bags

I love when a plan comes together!  The students, who did not know they were part of a secret experiment, were surprised by the results – and this led to a fun class discussion of mixed practice.  Here, the mixed practice group was associated with better performance on the tossing task. Totally a “wow” moment for the class, and a teachable moment on experimental design.