Monthly Archives: October 2014

Class Opener – Day 40 – Candy Corn Samples

It’s Halloween – it’s also close to election day – let’s combine these events into one big super-terrific class opener!

I spent last night at Wegman’s assembling my candy corn population, which consists of lots of “regular” candy corn, and some apple-flavored candy.  Walking through the aisles as students got out their calculators and homework, of course they all wanted to know if the candy was destined for their bellies, or just another statistical tease.  But we can do both!  As the class worked through their entrance ticket, the bucket passed through the room, with instructions to pour 20 candies from the spout – without looking or choosing deliberately.  Our task: to estimate the proportion of candy corns which are apple.

After our warm-up, students then approached our class dotplot and contributed their result. A discussion of how this drawing on candies is similar / different that what happens in polling scnearios followed, and I have planted lots of seeds for margin of error, which we will study formally next week.

Class Opener – Day 39 – It’s a Heat (Map) Wave!

Finishing up discussions with scatterplots – today’s visual when students entered presented a new idea in scatterplots (from the awesome Plot.ly site) – a scatterplot representing the score of every NFL game ever played!

What’s the story here? So many great features of this plot to discuss including:

• It’s apparent symmetry
• The vertical and horizontal avoidance lines
• The colors – many students have never seen a heat map before
• The clustering in the center of the graph

This was a quick warm up as I wanted to get to the main event – scatterplot stations!  Students worked in teams to complete activities (in 15-minute intervals) designed to strengthen their understanding of many ideas surrounding scatterplots.

Station 1 – using graphing calculators to assess data sets, and writing clear summaries of the trends.

Station 2 – estimating best-fit lines given a scatterplot, and using their algebra skills to make good estimates.

Station 3 – netbooks! Play with the Rossman-Chance “Guess the Correlation Applet” and develop and understanding of “least squares” with this Geogebra applet.

Fun day today…..moving on to sampling tomorrow!

Class Opener – Day 38 – Are Any of My Students Compatible?

Today’s opener was inspired by a movie correlations activity I have used in AP Statistics, and Cathy Yenca’s awesome activity which brings this idea down to the Algebra level.

For my freshman class, I wanted to students to “discover” the role of the correlation coefficient r – how it acts as a measure of the strength of the relationship between two quantitative variables.  To begin, 10 potential vacation / off-day activities were listed on the board:

• Ski
• Go to Beach
• Amusement Park
• Baseball Game
• Camping
• Washington DC Tour
• Shopping Day
• Big Concert
• Cruise

Students were each asked to rank these activities from 1 to 10 (10 being most desirable) and using each number only once. The class then moved into partnerships with my suggestion that they work with someone they maybe did not know so well in class, and compared results.  With an odd number of students, I worked with a student to share interests.  Results for each activity were plotted as ordered pairs, with each partner contributing their number score.  Students plotted their points on graph paper, while my student partner and I used Desmos – and quickly discovered that we have little in common.

From there, students learned how to use graphing calculators to analyze the data – making the scatterplot and finding the best-fit line.  The partnerships also wrote this mysterious new statistic – r – on the bottom of the graph and shared their graph in the board.  Through a gallery walk, the class examined the graphs and tried to conjecture the meaning of r.

This worked better than planned, as the class quickly made some key observations:

• Pairs with stronger relationships have “higher” r values.
• There are no r-values greater than 1.
• r can be negative if people answer opposite each other.

Definitely will add this activity to my arsenal every year!

If you are interested in the activity for AP Stats, you can check out the Google Form we use, then some instructions for processing the data in this video:

Class Opener – Day 37 – Random Sampling

We’re into the home stretch of our stats unit and I am looking to reflect upon our study of the normal distribution, yet look ahead at what’s to come – sampling and margin of error.  The Rossman-Chance Applets provide some meaningful, interactive discussion starters in statistics.  Today, as students entered, the Reeses Pieces sampling applet was drawing random samples of size 25 – displaying the results in a dotplot.  Little by little, the dotplot took shape as more samples were drawn…until, eventually, an old friend made an appearance….

Hey, that’s the Normal Distribution!  Why yes, yes it is…..isn’t it great that random samping reveals such a powerful statistical concept?

But that’s not all.  With election day approaching next week, we can start to build connections between random sampling, the normal distribution, and political polling.  The parallels are strong, and we’ll talk about them within the next week.

• With sampling candy, there is a pre-existing proportion of “orange” candy.  In a voting population, there is an existing proportion of people who will vote for a certain candidate.
• With candy, we can draw a random sample of candies. For a poll, we contact a random sample of potential voters.
• When we sample candy, sometimes we might get 50% orange, or 45 % orange, or 60% orange – variability is part of the game. In political polling, we try to estimate the poroportion who will vote for a candidate, and we hope to get close to the target. Margin of error gives us an idea of how close we are.

There are many other interesting discussions to be had surrounding the applets – try some with your classes. They aren’t just for AP kids!

Class Opener – Day 36 – Introverts vs Extroverts

We’re about half-way through a basic stats unit in my 9th grade class, with a quiz tomorrow on standard deviation and the normal distribution.  I need one last class example to have students compare and contrast data sets by looking at their centers and variability.  A morning brainstorm turned into a fun exploration of my students’ personalities.  3 groupings were shown my back whiteboard:

• EXTROVERTS
• MIDDLE
• INTROVERTS

After a brief discussion of what it means to be introverted or extroverted, and doing my best to steer discussion away from any negative connotations, I asked students to self-identify and move to a corner of the room based on where they see themselves.  To clean up things some, I told them to arrange themselves so that we had exactly 8 introverts and 8 extroverts, with everyone else in the middle.  Some adjusting then took place, as we agreed on who belonged in which group.

Now for the data collection aspect.  I had each student approach the back board and write their signature in the appropriate column.  This is where the fun began – as my introverts calmly waited for their peers to write their names and move away, the extroverts fought over markers and board space.  As students sat down after contrubiting their signature, some noticed immediately what was happening :

After all names were written, and we had a good laugh over the clear differences in the categories, we needed some data.  Each student approached the board and measured the height of a name at its tallest point, recording to the nearest tenth of a centimeter.  Tonight’s homework is then to compute the standard devation “by hand” for one of the groups, and comment on differences.  My old friend the Nspire App is helpful here to show the clear difference between the introverts and the extroverts:

Using authentic data in class matters, as kids more readily discuss what they see and are generally more eager to dig deeper into a problem.  This was a fun way to culminate the first half of our stats unit.

A Surprising Harmonic Series Result

If you are a loyal followed of the blog (and I know you are!), you’ll notice there was no class opener this past Friday.  This is because I was not in school, and instead traveled about an hour north of my home to New Brunswick, NJ. for the Association of Math Teachers of New Jersey annual conference. I facilitated a fun hour of tech tools for math class (Padlet, Poll Everywhere, Answer Garden, etc…) and couldn’t help myself but to unveil Desmos regressions to an appreciative group of Desmos lovers.

But the highlight of the day was my participation in an Ignite session, hosted by my friends at the Drexel Math Forum. 10 speakers each had 5 minutes to get their ideas across through 20 slides which changed every 15 seconds. It’s a fun an intimidating concept!  Some of those who shared during the Ignite were Max and Annie from the Math Forum, Phil Daro, past NCTM President Jim Rubillo….and me?  How the hell did I get invited to present with a group of people I admire so much?  I think my topic, language in math class (based on a past blog post on math phrases I’d like to see expunged) went over well, and I look forward to sharing the video when it becomes available.  It’s probably the most thrilling experience I’ve had as a speaker.

But the fun math times continued after the Ignite session, when Jim Rubillo and I shared a conversation about the exciting role of technology in math class, and its ability to allow students to investigate ideas efficiently.  It turns out that Jim lives about 10 minutes from me (small world) and he shared a math idea from his (I think) college math course where students used technology to pursue an idea.

This particular investigation involves the Harmonic Series, which I have shared with my freshmen classes recently:

This series diverges, heading off towards infinity – though quite slowly, and Jim’s students investigated when the series passed integer values. How many terms does it take for the series to go above 3, or 4, or 5, etc?  Some examples:

• When n=4, the sum climbs above 2
• When n=11, the sum climbs above 3
• When n=83, the sum climbs above 4
• When n=227, the sum climbs above 5

Technology was uber-useful here, as a Desmos graph (click the picture to the right to explore) allowed me to observe the sum, and use a Floor function to watch when the next integer value is reached.  A table was also useful to verify the changes in sum.

This was all done after I went home that day, and it was Jim’s next challenge when had me intrigued:

If you look at the n’s where the sum reaches a new integer value, and find the ratio of these consecutive n’s, you’ll never guess what the ratio approaches.

Well, in math, there are lots of surprising results, and many of them seem to involve a few “usual suspects”.  I really had no idea what those number might approach – but I had a good guess: “I bet it’s e”.

Yes! It’s e!!! Shared Jim excitedly, with a pretty good punch to the arm.

Some quick calculations involving these landmark n’s seem to lend some evidence towards Jim’s claim.  It’s an idea he said he had never seen before, and I don’t recall encountering it.  But Jim did find some documentation regarding a proof, but couldn’t recall where it was located.  If anyone out there has more info to share, please contribute in the comments!

In the end, Friday was just an awesome day of sharing, and fun to talk one-on-one with some legends in the math education community.

Class Opener – Day 35 – Tall and Short

We’re thinking about standard deviation in my 9th grade class, and the idea of variation and “unusual” data points. I think the picture which greeted students today says just about all which needs to be said on standard deviation, doesn’t it?

Later in class, I asked students to plot their heights on a number line I had drawn, with a low of 60 inches and a high of 74.  From here, I asked students to estimate what our class standard deviation might be.  Some interesting responses were generated:

• 10 – probably because 60 and 70 appeard on the line.
• 5 – because that would seem to cover the number line

When I reminded them that standard deviation can be thought of as “typical distance from the mean”, the responses evolved and eventually we settled on between 2 and 3, where travelling 2 standard deviations in each direction would cover everyone in the class.  Next, when I told them that the World’s Tallest Man had a height over 8 standard deviations from the mean, meaningful gasps were shared, and we could move on to notes onvolving the normal distribution.

Short post today as I am about to start 23 parent conferences over 6 hours….wish me luck!