This High School football coach plays “Would You Rather” Math, and so should you

Add “Would You Rather?” to your bookmarks. Phrasing math problems in terms of “Would You Rather” is simple and brilliant. I love this framework for three reasons:

1) It’s relatable. We’ve all wondered whether it’s more efficient to mow the lawn in concentric rectangles or in stripes. We’ve all run. We’ve all argued with other people.

2) It allows for immediate estimation. Students will immediately have a conjecture. As I’ve mentioned that Dan has mentioned before, that’s one of the best pound-for-pound ways of getting kids to learn math in new ways.

3) It hammers – I mean just crushes – CCSS Standard of Practice 3: Construct viable arguments and critique the arguments of others, which Steve Leinwald has called “the most important nine words in the common core”. *

So yeah, bookmark it, give it to your kids tomorrow. I’ll be sprinkling them throughout my curriculum maps very soon.


Speaking of a “Would You Rather” approach to math, I was watching a video on Grantland about a High School football coach in Little Rock, Arkansas who never punts and always onside kicks. While statheads have been clamoring for less punting for years, he (and the video produces) articulates the math quite clearly and attainable (a mathematical skill).

Would you rather go for it on 4th and 7 from your own five yard line (about a 50% success rate)….



punt and give the opponent the ball on the 45 yard line?


Here’s where it gets interesting though. By going for it on 4th down that close to his own goal line, the opponent would score 92% of the time. By punting to the 45 yard line, the opponent would score 77% of the time.


There you have it. Coach Kevin Kelly played “Would You Rather” Math and used some compound probabilities to determine that they’ll never punt. While there’s more to it (i.e. it’s not a singular event: once you get the first down, you have to get the next first down and so on), just imagine if you could get kids to think mathematically this way under the Friday Night Lights. Coach Kelly uses similar logic regarding always onside kicking.


Go check out the full video and full article if you like. Shoot, try to get your class to convince your schools’ football coach they should never punt. It probably won’t work but it’s worth a shot. Wouldn’t you rather give it a shot?

*Thanks to Chris Robinson for helping me track down the author of that quote.

Update 11/20/13: Really wonderful interview with Kelly on this weeks’ Slate: Hang Up and Listen podcast. You can also find lots of bunny-trails in their links section to further elucidate the topic of never punting in football.

Who doesn’t want to relive the 2000 election? (Stats problem)

We’ll take a slight detour from my college readiness manifesto (that hasn’t even really started yet) to bring you the following election-related problem. Then again, this problem was lifted directly from a graduate level Statistics class, so this might give some insight into what college readiness could potentially look like. Hadn’t thought of that. Enjoy!


Here’s a (non-abridged) problem I received in my graduate level stats class last week (due tomorrow! hope it’s ok that I’m posting it!). I think it’s a great problem and one that’s certainly prevalent around this time:

from The Statistical Sleuth, Ramsey & Schafer, Ed. 2)

1. (SS#8.25) Presidential Election of 2000 

The US presidential election of November 7, 2000, was one of the closest in history. As returns were counted on election night it became clear that the outcome in the state of Florida would determine the next president. At one point in the evening, television networks projected that the state was carried by the Democratic nominee, Al Gore, but a retraction of the projection followed a few hours later. Then, early in the morning of November 8, the networks projected that the Republican nominee, George W. Bush, had carried Florida and won the presidency. Gore called Bush to concede. On the way to his concession speech, Gore then called Bush to retract that concession. When the roughly 6 million Florida votes had been counted, Bush was shown to be leading by only 1,738, and the narrow margin triggered an automatic recount. The recount, completed in the evening of November 9, showed Bush’s lead to be less than 400.

Meanwhile, angry Democratic voters in Palm Beach County complained that a confusing “butterfly” ballot in their county caused them to accidentally vote for the Reform Party candidate Pat Buchanan instead of Gore. See the ballot below. You might understand how one could accidentally vote for Buchanan instead of Gore because Gore’s name is the second listed on the left side, but his “bubble” is the third one. Two pieces of evidence supported the claim of voter confusion. First, Buchanan had an unusually high percentage of the vote in that county. Second, there were also an unusually large number of ballots discarded during counting because voters had marked two circles (possibly by inadvertently voting for Buchanan and then trying to correct the mistake by then voting for Gore).

Make a scatterplot of the data, with X = # of votes for Bush and Y = # of votes for Buchanan. What evidence is there that Buchanan received more votes than expected in Palm Beach County? Analyze the data without Palm Beach County to obtain an appropriate regression model fit. Obtain a 95% prediction interval for the number of Buchanan votes in Palm Beach County from this fitted model (assuming that the relationship between X and Y is the same in this county as the others). If it is assumed that Buchanan’s actual count contains a number of votes intended for Gore, what can be said about the likely size of this number from the prediction interval?

Why couldn’t a similar problem be asked in a HS Stats class? Maybe modified, but seriously, why not? And especially why not now, in a year divisible by four (Summer Olympic and presidential election years)? The problems a bit wordy though. Let’s try this:

Artifact, reworked:

The US presidential election of November 7, 2000, was one of the closest in history. As returns were counted on election night it became clear that the outcome in the state of Florida would determine the next president. When the roughly 6 million Florida votes had been counted, Bush was shown to be leading by only 1,738, and the narrow margin triggered an automatic recount. The recount, completed in the evening of November 9, showed Bush’s lead to be less than 400.

Meanwhile, angry Democratic voters in Palm Beach County complained that a confusing “butterfly” ballot in their county caused them to accidentally vote for the Reform Party candidate Pat Buchanan instead of Gore. See the ballot below.

Guiding Questions

  • How could we use statistics to determine whether or not the “butterfly” ballot confused voters?
  • How big of an outlier was Palm Beach county?
  • Had the ballot been more traditional, can we predict the outcome of the Florida electoral votes (and presumably, the 2000 election?).
  • Is there a model of sorts we could employ to detect such anomalies in the future?
  • While we’re at it, what’s up with Dade County over there?

Suggested activities

  • Make a scatterplot and a linear fit and be, like, DUH, something was whack in Palm Beach county. (data of Bush votes and Buchanan votes by countyare at the bottom of this post)
  • Socratic discussion on outliers, not to be confused with Outliers by Malcolm Gladwell.
  • Workshops on confidence intervals, standard deviation and the like.
  • What does the line of best fit look like with and without Palm Beach? And what might that tell us about the voting discrepancies in Palm Beach?

Attempted solution

I’m not going to post my response to the problem prompt, because it may violate academic honesty or something. But I’ll post a scatterplot of Bush/Buchanan votes by county and leave it at that.

Data: election2000

The Minnesota Timberwolves are complaining about having the second pick in the NBA Draft. Do they have a point?

Lest this become a sports-draft blog, I feel contractually obligated to post about the NBA Draft Lottery, what with the Cavs winning and the statistical odds and all.

Last night, the Cleveland Cavaliers (my favorite team for you non-twitter followers) won the NBA Draft Lottery, meaning, by chance they were awarded the #1 overall pick in June. This, despite not having the worst record.

For the uninitiated, the NBA Draft is unique among leagues’ player selection drafts in that the top pick does not necessarily go to the worst team. You see, in order to prevent teams from “tanking” (that is, losing intentionally at the end of the season in order to obtain a higher draft pick) the NBA has instituted a lottery system. Teams with worse records have higher odds of having a higher pick, via ping-pong ball selections, but there’s not guarantee that the losingest team in the NBA will have the highest pick. The logic is that by not automatically awarding teams picks based on record, that will potentially prevent teams from throwing games at the end of the season. To be sure, it still happens, but probably not to the extent that it would otherwise.

The end result is that the worse a team does, the better chance they have to pick higher. And in the NBA, which only allows five players on the court at a time, having a single high pick is generally more valuable than in any other sport.

Now, a bit about the mechanics. The Draft Lottery is conducted using an overly complicated method of ping-pong balls and number combinations. Ping pong balls numbered 1-14 are drawn at random, yielding a potential 1001 combinations (order is disregarded, that is 1-2-3-4 = 4-3-2-1). Each of the 14 teams that do not make the playoffs are assigned a certain number of number combinations. The worse a team’s record, the more number combinations they are awarded. (Note: the specific number combination of 11-12-13-14 is disregarded, yielding exactly 1000 assigned number combinations)

The team with the worst record is assigned 250 number combinations, good for a 25% chance at being the first team drawn.

The team with the second-worst record is awarded 199 number combinations (19.9% chance). The team with the third-worst record is awarded 156 (15.6%) and the team with the fourth worst record is assigned 119 (11.9%) combinations. This proceeds all the way down to the team with the highest record of all the non-playoff teams being awarded 5 number combinations, good for an unlikely 0.5% chance of being awarded the first overall pick.

There’s one more catch for the mechanics, then we’ll get to the actual driving question of this: the lottery is only utilized for the top 3 picks. After the top 3 picks are determined, the remaining draft picks are awarded based solely on record. So, in effect, if you have the worst record in the NBA, you will pick no worse than 4th. And that’s if your number combos (totally 25% of the combinations) are somehow not selected the in first three. Similarly, the second-worst team can slip no further than 5th. And so forth.

[Looks at teacher notes: “Check for understanding”]

Does this make sense?

[Sees lots of confused looks]

Here’s the long and short of it:

  • 1: 25%
  • 2: 19.9%
  • 3: 15.6%
  • 4: 11.9%
  • etc.

Now, on to the driving question.


Last night the Cavs were awarded the #1 overall pick. The Minnesota Timberwolves, who had the worst record in the NBA, were awarded the #2 overall pick. David Kahn, the General Manager of the Timberwolves went on record, complaining about how the lottery system is rigged.

“This league has a habit … of producing some pretty incredible story lines,” Kahn said, while smiling, on Tuesday. “Last year it was Abe Pollin’s widow and this year it was a 14-year-old boy… We were done. I told (Utah executive) Kevin (O’Connor): ‘We’re toast.’ This is not happening for us, and I was right.”

(Last year, they were beat out by a widow, representing the Wizards, this year by the 14-year-old son of the Cavs owner, who has a nerve disorder.)

Guiding Questions

  • Does Kahn have a point?
  • Should the Timberwolves be upset with the second pick, or happy that they didn’t slide all the way to fourth?
  • If you have the worst overall record, you have the best odds for the top pick, but that’s only 25%. So what’s your “expected” pick?

Potential Solution

It should be noted that the Timberwolves have had especially terrible “luck” with the NBA Draft Lottery system, at least according to them. They’ve been in the lottery 14 times and never been awarded a higher pick than their record should indicate. In fact they’ve ended up with a lower pick than their record would indicate 8 times. Some Timberwolves fans says this makes them 0-14 in the lottery. I’d say more like 0-8-6. That’s a whole other blog post. There’s also the subtext of Cleveland being awarded the top pick a year after losing LeBron James, but whatever. Our concern is with this particular draft.

Namely, if a team has a 25% chance of being selected (via 250 out of 1000 number combinations), should they expect to get the top pick? Should they be happy they didn’t get the 4th (or 3rd) pick? If not, what pick should they expect to get, and how often? (side note: mean, median, and mode anyone?)

Obviously, the team with the worst record should expect to pick 1st 25% of the time and either 2nd, 3rd, or 4th 75% of the time.

So we need to split up that 75%, right? Depending on who gets the #1 pick, that will affect the remaining odds. Last night, for instance, the Cavs were awarded the #1 pick via a trade with the Los Angeles Clippers, which only had a 2.8% chance of hitting. (Interestingly, the Cavs’ original pick was the “biggest loser”, slipping to 4th, despite having the second-most opportunities.)

The Timberwolves started with a 250/1000 chance of hitting, then after the first pick was awarded, if we eliminate the 28 combinations for the Clippers, they had only a 250/972 chance of being awarded the second pick, good for only 26%. 26% of 75% is 18.75%.

So the T-wolves had a 25% to get #1, a 18.75% chance to get #2, and, by subtracting from 100, a 56.25% chance of ending up with the 3rd or 4th pick.

Had the Cavs won the #1 overall pick with their original pick, with 199 out of 1000 combinations, the T-wolves would have had a 250/801 (31%) chance at the second pick. (for some mental gymnastics, we can say that the T-wolves had a 19.9% chance to have a 31% chance at the second pick)

For general purposes (i.e. no prior knowledge of who the #1 pick is), the NBA Draft Lottery wiki has done the calculations for us:

Seed    Chances    1st    2nd    3rd    4th
1           250         .250  .215   .178    .357

According to this, if you have the worst record in the NBA, the odds of you receiving the 3rd or 4th pick, worse than the Timberwolves landed at #2, total to 54%. The most common (mode) pick the worst team can expect is the worst possible pick at #4.

With odds like that, should the Timberwolves consider themselves lucky to only be picking as low as 2nd?

The Dallas Mavericks are 2-16 in playoff games officiated by Danny Crawford. Is this statistically significant?


This shocked me.

The Mavs have a 2-16 record in playoff games officiated by Crawford, including 16 losses in the last 17 games. Dallas is 48-41 in the rest of their playoff games during the ownership tenure of Mark Cuban, who has been fined millions of dollars in the last 11 years for publicly complaining about officiating.

First of all, is that right? That A) the Dallas Mavericks perform so poorly in Crawford-officiated games, and B) Crawford is still being allowed to referee them? Really? Wow.

And there’s this, which might even be more damming: The Mavs are 4-14 against the Vegas spread. ESPN provides a nice chart of the individual games.

I specifically remember those 2006 Finals games against Miami. By many accounts, those were two of the worst officiated games in NBA history, in which Heat guard Dwyane Wade got what seemed to be every favorable foul call. It pretty much ushered in the era of NBA ref scrutiny.

This has to be tested for statistical significance.

Guiding Questions

  • Really?
  • Is this just coincidence or is there something else going on here?
  • Does Crawford have a vendetta against the Mavericks for some reason?
  • Does Crawford have a suspicious record with any other team?
  • Is there a potential way, other than referee malfeasance, that we could explain away this alleged disparity?
  • Maybe the Mavericks are just playoff chokers?

Suggested activities

  • Obviously if this were a statistics course you could look at statistical significance, which we’ll do in a minute.
  • If students are really up for it, they could delve into the games themselves and look for disparities in “referee stuff” like fouls, technicals, travelling, etc. We’re not going to do that in a minute.
  • Homework: students watch tonight’s Mavs-Trailblazers game closely and look for anything fishy from Crawford (although, this might serve as its own lesson in confirmation bias).

Potential Solution

Let’s start with our null hypothesis:

H(o): Danny Crawford is NOT biased against the Mavericks. The Mavericks’ playoffs woes in games he’s officiated is due to random chance.

I suppose first we have to figure out what the Mavericks’ Crawford-officiated games “should be.” The Mavericks are 48-41 in playoff games not officiated by Crawford, good for a winning percentage of 54%. Although, if you’re like me, you believe more in random chance for sporting events, and the “true percentage” is probably pretty much 50% over the course of a decade. But that could be a fun debate point in your class.

We also need to decide on a significance/confidence level α, usually 0.05 or 0.01.

So what is the probability of a team that “should” win 50% of its games (debatable) ending up winning just 2 of 18 games at random? Or rather, that this team should lose 16 (or more) of 18 games by random chance?

A P-Test would could look like this,

Probability of 16 losses + Probability of 17 losses + Probability of 18 losses =



So no matter what confidence level we choose, this is, again, pretty damning. If we assign a 50% of the Mavericks winning (less than for their other playoff winning percentage) there is only a six hundredths of a percent chance of this being total flukiness.

Before we go nuts, though, let’s look back at that chart. Now, if you’re not familiar with Vegas lines, the negative sign in front of the “DAL Line” column indicates the Mavericks were favored that game. You’ll note that Dallas was only favored/expected to win 8 of those 18 games, and Vegas is usually pretty dead-on about these sorts of things. If we use that as a “true winning percentage” the Mavericks would only be expected to win a mere 44% (a losing percentage of 56%) of their games, not 50%. Let’s recalculate.

Slightly less suspicious, but still grievously suspicious. It’s well below our 5% or even 1% confidence level.

Still, before I would get Ralph Nader involved, I would ask students, investigators, etc. to look for specific evidence pattern within the games themselves (as has done for one specific game). The original ESPN article that led to this investigation suggested Dallas had more fouls and less free throws in Crawford-officiated games than others. A next step would be to look at the beneficiaries of the suspect officiating, i.e. Dallas’ opponents for these games. Did they get an inordinate amount of free throws? Did they tend to overperform, just as Dallas underperformed in these games. Now that we have the statistical basis to be suspicious, we can start the investigation in full.

Couldn’t this work as a real nice Project Based Learning Unit for statistics? The Entry Event could be the ESPN article, or Game 5 of the 2006 NBA Finals. The summative event could be that students could present their findings, host a panel or debate, or write a letter to their congressperson. Or Ralph Nader.

Let’s crowd-source this ; School closure predictor

A couple weeks ago a fellow New Tech math guru and I travelled to Bowling Green, KY to observe some teacher activities. Sadly, school was iced out such that there was a round of early school closures on Day 1 and total shutdown on Day 2. We had some great collaboration and were able to hash out a bunch of ideas, but basically we could have done that in a more temperate climate.

Also, several other New Tech employees were on trips visiting schools whose school days were cancelled. Basically the last two weeks were a series of boondoggles.

Maybe if we had a tool like this, we could have saved ourselves a trip, and New Tech a lot of money.

Predicting the weather is hard enough. Predicting the reaction of humans to the weather is another level of complexity. Now, I’m not sure how accurate this tool is.

One person’s $100 Super Bowl Pick Pack

(note: that does not imply that he’s picking the Pack.)

Based on our discussion on Friday’s Super Bowl prop bet post , we thought a fun, and possibly marginally educational, activity would be to let students (and teachers!) have $100 of “money” to bet on the myriad of Super Bowl odds, with justification.

Here’s a set sent in by Drew.


Ok so here is my $100 Super Bowl Pick Pack.  I made these picks with the following criteria:

  1. Only had $100.
  2. More fun to cheer FOR things than against – willing to acknowledge an entertainment value.
  3. If I win a bet, I want to come out ahead overall.
  4. I have no freakin’ idea what will happen, so I need to do some hedging.

I am going a somewhat strange direction with these picks.  The most interesting bets for me are the MVP best.  Now historically, QB is the MVP a disproportionate % of the time – and this Super Bowl seems particularly quarterback driven.   However, to bet either Rodgers (7/4) or Roethlisberger (7/2) as part of a hedge ties up too much money to do much else.

The Super Bowl is 100/1 odds to be in the 71+ point range; is that a good bet? columnist Bill Simmons and fellow sports gambling addict Cousin Sal had their annual Super Bowl prop bets podcast where they discussed the best gambling deals of this Sunday’s Super Bowl (if gambling were legal, *ahem*). Aside: if you haven’t checked out the list of potential bets for the Super Bowl, you should (how long will Christina Aguilera hold the word “brave” in the National Anthem? over/under is at 6 seconds).

Anyway, one of Simmons’ prime suggestions was the bet of 100/1 for the number of total points scored being in the 71-75 range and 150/1 in the 76-80 range. So you plunk 10 bucks on each of those and a $20 bet could net you $1000 or more in the event it’s a high scoring game. The logic Simmons mentioned was this: “if these teams played 100 games, they would score over 70 points at least once.” That totally makes sense to me. In fact, the two teams playing in the Super Bowl, Green Bay and Pittsburgh, played in 2009 and scored a total of 83 points (which, strangely enough is too much for the bets he suggested). But if you wanted insurance against that you could put another $10 down on the higher point total range, but it’s starting to add up.

Again, this makes total sense to me. It seems like every other week in the NFL there’s some wacky game that sees both teams score over 35 points. Why, it happened just a couple weeks ago. And while it’s not likely to happen, it sometimes does, and at 100/1 odds that’s some pretty good winnings.

But is it good math?