This weekend’s big college football game on TV featured LSU. They do something at their stadium which is a bit unusual. Can you detect it?
Did you notice it? LSU is one of just a handful of schools who paint their yard lines every 5 yards, rather than 10.
Is one method, counting by 5’s rather than by 10’s, “better” than the other? Both methods communicate the field position effectively. I would argue that counting by 5’s causes the field to look more cluttered, and no doubt effects the paint budget. But there is no clear advantage to either method.
This is the same discussion we have with students when working to establish a scale for statistical displays like box-and-whisker and histogram. Should we count by 5’s? By 10’s? Does it really matter, if the communication is clear?
In every class, I always seem to have a student who wants to propose a non-traditional scale. Could we have our axis count by 7’s? Sure, but would the communication suffer as a result? It would probably suffer in the same manner as football fans scratching their heads if we painted the field lines every 7 yards….
Or what about the student who wants to use the 5-number summary as their axis markings? Let’s hire them to line the football field next time….
It’s all about the communication, and details matter. If we don’t pay attention to details, then we get un-desirable results, like this logo painted on a Minnesota football field.
Use football field photos to discuss scale, and discuss the pros and cons of 5’s vs 10’s!
Nothing drives me nuttier than boring probability units. Endless worksheets filled with tales of balls in urns and cards being drawn from decks. Nothing screams for fun and games more than a probability unit. Here’s some ideas for you to try:
DRAW THE CUBES, WIN A JOLLY RANCHER!
This is a good opener for teaching dependent events. Take a brown paper lunch bag, and place inside 20 colored cubes. In the past, I have used 10 white, 6 red and 4 blue. Do NOT let the students know what chips are in the bag. Shake the bag, and travel from student to student, having each student draw 3 cubes from the bag (without replacement, or all 3 at once, doesn’t matter). If they are able to draw exactly 2 whites and a red from the bag, they receive a Jolly Rancher reward.
After every student has had a chance to play, I add an additional challenge to the game. Letting the class know that there are exactly 20 cubes in the bag, can they predict the distribution of colors in the bag, using the information gathered from the pulls they observed as evidence? Students who can guess the exact distribution also win the coveted Jolly Ranchers.
Next, dump out the cubes to show the true distribution. Was the 2 white, 1 red game “winnable”? Was it possible to win? Was it plausible that one would win? I usually try to rig the color distribution so that one can win about 25% of the time. The 10 white, 6 red, 4 blue configuration will provide a 23.6% probability of winning.
STATISTICAL TALES OF THE IMPROBABLE
Always remind your students that probability is a long-term ratio. The 1/6 probability of rolling a five on a die does not mean that every 6th roll will be a five. In the short term, strange things can happen.
The CBS show “The Amazing Race” provided a great probability teaching moment in its 6th season. In a “Pit Stop”, teams must complete a challenge in order to earn a clue and move on in the race. Here’s a summary of the challenge:
In a field are 270 large hay bales. 20 of the bales contain clues. You must continue to roll out hay bales until you locate one with a clue, at which time you can move on.
Think about these openers:
What is the probability you select a bale with a clue on your first attempt?
What is the probability you select a bale with a clue on your second attempt? Third attempt?
What is the probability it takes you 5 or more bales to locate a clue?
What is the probability you roll out 20 bales, and do not find a clue?
The clip from this show appears below. You’ll love the reactions of your students!
One last example tale comes from the world of gambling. While I try to stay away from too much gambling talk in classes, this die-rolling activity leads to a neat backstory. Have your students roll pairs of dice, and record the number of rolls needed to roll a sum of 7. After a 7 is rolled, start a new count. Plotting the results in a class dotplot gives a nice example of a skewed-right distribution, which is often a new shape for our students.
In class, what was the most number of rolls needed? I have had students get into the 20’s or 30’s. Is it possible that it could take 50 rolls to get a sum of 7? Is it plausible? Note the subtle language lesson happening.
Now, on to craps. Craps is a fairly complex game, which boils down to this for our example: once a player rolls, they continue winning until a sum of 7 is rolled, at which point the round ends.
Meet Pat DeMauro:
Pat visited the Borgata casino in Atlantic City in May, 2009 and set the world record for a game of caps, by tossing the dice 154 times without rolling a 7!
Pat’s winning streak was featured in Time Magazine, and features some of the details of the record run.
A recent twitter post by Math Coach Geoff (@emergentmath) from New York asked for help in quantifying a poster which lists pros and cons of using paper towels versus electric hand dryers. The task reminded me of a quantitative data collection method I have used with Advanced Placement Statistics students. After the AP exams in May, my students were required to develop and complete a project which demonstrated some application of what they had learned during the year, which were then presented at a Stats Fair at the beginning of June.
Without fail, a few groups would propose some sort of a taste test: cookies, ice cream, soda, coffee. Ignoring the fact that taste tests always take 10 times longer to actually carry-out than students think, the inherent problem with taste tests is that the data you get isn’t all that exciting: “we did a taste test….39 people liked Coke, 45 people liked Pepsi”…yawn…. We can do some hypothesis tests for proportions, maybe break it up by gender, and that’s about it.
Using number lines, and having volunteers record their opinions by placing a dot on the lines, allows for more interesting data. Consider a test between two colas, Fizz and Shanta. Make a 10cm number line for each, and let each end represent the extreme opinions:
Invite volunteers to record their opinions by placing a mark anywhere on the number line:
Then, measure the distance from the left side in centimeters. The distance represent the taste score for each cola.
Think about how much more rich the data is that we get from this method. Not only do I have a record of which cola the volunteer prefers, but also a measure of the magnitude of the preference. We could do a two-sample t-test of the means, do a one-sample test for the differences, and even look at a scatterplot of the score pairs.
We can use a similar approach to look at Geoff’s request. In the pic he shared, the pros and cons of choosing between paper towels and electric hand dryers are listed:
The electric dryer seems to have many more pros than paper towels, but just how compelling are these pros? Perhaps one of the cons outweighs all of the pros? How can we measure the magnitude of each of the arguments? The number line method provide a means for us to quantify each of these items. Create number lines from -10 to 10, attaching each to a statement from the poster. Have students not only identify the pros and cons, but also judge how large or a pro or con each feature is by placing a dot on the number line.
In a pre-algebra class, students can discuss how to combine all of the data points, and we have a natural “in” for wanting to add positive and negative integers. “What’s the total con score?”, “What’s the total pro score?” and “What’s the overall score?” Have students use the data and measurements they collect to defend their opinion, and use the natural opening to encourage writing in math class.
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