Categories
Algebra Middle School Statistics Technology

Bob’s Favorite Things

This year, as I have moved from a role as classroom teacher to a math coach, I find myself sharing some of favorite lessons and approaches with my colleagues.  Taking a cue from Oprah, this is the first of a series of my “Favorite Things”, hopefully without the rampant screaming or random things hidden under your seat.

The Monty Hall Problem on NLVM

The famous “stick or switch” problem is one which will generate tremendous discuss in a classroom.  The premise is simple: you are offered the choice of one of three doors, behind one of which there is a great prize.  After the host reveals a non-winning door, you are offered the chance to switch your guess.  Should you stick or switch?  The problem rose to a next level of fame with fierce debate in the “Ask Marilyn” article by Marilyn Vos Savant in Parade Magazine.  After having included the problem in my prob/stat course for any years, the discussions rose to a crest as the problem was featured in the movie “21”, with even students I did not know stopping me in the hallways to ask me about its logic (note, this video has a number of ads on it):

As an emerging stats teacher, I would play the game using cardboard doors and hand-drawn pictures of cars and goats.  In later years, I used an applet at the National Library of Virtual Manipulatives to play the game with my class.  Look under Data Analysis and Probability for “Stick or Switch”.  What I really like about this applet is that it will play the game 100 times quickly and display the results.

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There are many ways to explain this game to your students.  One straight-forward approach is to have students attempt to write out the sample space (all possible outcomes) for this game.  For example:

Car behind 1.  Pick 1.  Shown 2.  Stay.  WIN

Car behind 1.  Pick 1.  Shown 2.  Switch.  LOSE

Car behind 1.  Pick 1.  Shown 3.  Stay.  WIN

Car behind 1.  Pick 1.  Shown 3.  Switch.  LOSE

Patterns in Pascal’s Triangle

In one of my favorite lessons each year, the many, many patterns in Pascal’s Triangle were discussed.  This would come after an investigation of combinations, and students  would often recognize that each entry in the triangle is actually a combination, along with other patterns like the counting numbers or perhaps the triangular numbers.  But how about the Fibonacci sequence and the “square” numbers?  What’s the “hockey stick” theorem?

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Over the years, I have used a number of sites to display the triangle and invite students to share their ideas.  One nice resource which explains patterns in the triangle can be found here.

The cherry on the Pascal sundae would come when I would invite students to approach the board and circle all of the even numbers.  Often, only 15 rows or so would be visible, so the chance to make a generalization is rich and there to discuss.  How would this look if we colored 100 rows, 200 rows, 10,000 rows?  And do we only get patterns for even numbers?  How about multiples of 3, or 5, or 24?  Check out this site from Jill Britton, which includes an applet that will color multiples in Pascal’s Triangle.  Watch the expressions from your students when you reveal multiples of 5 in the first 128 rows of the triangle:

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Try different multiples and observe the great symmetries.  Prime numbers like 17 and 23 provide the most surprising results.  Digging deeper, start a discussion of the modulus function with your class (it’s just the remainder, no big deal….).  What happens when we not only color multiples, but also color the remainders similarly.  From the Centre for Experimental and Constructive Mathematics, we get this great applet, which will color remainders based on specifications you provide:

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Electric Teaching

While I have been using the Monty Hall problem and Pascal’s Triangle in my classrooms for many years, this last resource is fairly new to me.  Electric Teaching by David Johns is an excellent site with an effective interface which is ideal for SMART Boards.  The site challenges students to match-up equations with data tables and their graphs.

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The site has problems for linear, quadratic and trigonometric functions, along with conics and even problems which include derivatives.  What a great way to not only have students participate, but verbalize their thoughts as they identify the matches.  The YouTube channel has additional resources from Dave, including a tutorial of the Electric Teaching site.

Categories
Middle School Statistics

Even Great Presentations Have Their Moments….

Recently, I attended a talk where the circle graph below was used to help emphasize the many online tools our students utilize.  To be fair, the presentation was otherwise fantastic, but sometimes my stats-abuse-radar is on full alert.  Use it as an opener for class discussion, and see if your students notice the inherent problem with this graph:

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Some questions for a class discussion:

  • Does this graph portray the data accurately?
  • Is a circle graph appropriate here? Why or why not?
  • How can we re-display the same information effectively using a new circle grpah, or a different type of graph?

In moments like this, sometimes it is best to draw energy from inspirational quotes.  I leave you with this, from the Simpsons:

Hypnotist: You are all very good players
Team: We are all very good players.
Hypnotist: You will beat Shelbyville.
Team: We will beat Shelbyville.
Hypnotist: You will give 110 percent.
Team: That’s impossible no one can give more than 100 percent. By definition that’s the most any one can give.
Categories
Statistics

Explorations in Polling

Primary election season is here, and news reports are filled with sound bites from candidates, their supporters, and pundits all trying to get the edge by being the first with breaking news.  It’s also polling season, as every news organization seems to have their own poll, all designed to project the winners.  This provides a great opportunity to talk about some statistics concepts which often get buried in the high school curriculum: sampling, surveys, margin of error and confidence intervals.

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One nice resource I have used in my classes before is the site pollingreport.com. This site collects polls from many sources: news agencies, university organizations and polling companies. Students can search from a long menu of topics and examine the careful wording of survey questions, time-progression data and information on sample size and margin of error.
Having students select their own survey, and interpret the results, can lead to interesting class discussions. One problem with polls is that the results are often taken as absolute, rather than an estimate of a population. An interval plot can help remedy this, and get students thinking about that pesky margin of error, which is often buried, italicized, or shown in a smaller font than the rest of a poll’s results. Here’s an example of an interval plot, using the results of a poll from pollingreport.com:

Quinnipiac University Poll. Feb. 14-20, 2012. N=1,124 Republican and Republican-leaning registered voters nationwide. Margin of error ± 2.9.
“If the Republican primary for president were being held today, and the candidates were Newt Gingrich, Mitt Romney, Rick Santorum, and Ron Paul, for whom would you vote?”

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Some questions for discussion can then include:

  • How can these results be used?
  • What do you think would happen if we asked more people? Or if the election were held today?
  • What would it mean if intervals over-lapped each other?
  • How likely is it that nation-wide support for Rick Santorum is within the interval?

While confidence intervals don’t need to be defined formally, the concept of these intervals indicating plausible values for the population parameter can be discussed. The New York Times, in particular, does an excellent job of providing an accessible explanation for margin of error, such as this excerpt from a telephone poll summary:

In theory, in 19 cases out of 20, overall results based on such samples will differ by no more than three percentage points in either direction from what would have been obtained by seeking to interview all American adults. For smaller subgroups, the margin of sampling error is larger. Shifts in results between polls over time also have a larger sampling error.

Next, we can take a look at formulas for margin of error. One convenient formula found in some textbooks links margin of error directly to the sample size:

By going by to pollingreport.com, and pulling a sample of polls with different sample sizes, we can examine the accuracy of this short and snappy formula.  The scatterplot below uses sample size as the independent variable, and reported margin of error as the dependent variable.
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The formula seems to be a nice guide, and some polls clearly use more sophisticated formulas which generate more conservative margins of error.

Classes who wish to explore polling further can check out the New York Time polling blog, FiveThirtyEight, which provides more detailed analyses of polls and their historical accuracy.