# Math Digests April 2022

### Why the Home Field Advantage is on the Decline

Discover Magazine, April 7, 2022

If you’re a sports fan, you’ve likely witnessed the “home-field advantage”: Teams tend to win more often when they’re playing at their home venue. Statistical studies have shown that this phenomenon is real. As Cari Shane explains, the primary cause of the home-field advantage is that hometown fans exert a subconscious sway over referees, leading to a few more calls in favor of the home team. Travel-related factors contribute, too. But in recent years, the magnitude of the advantage has been decreasing, statistical analyses show. Rule changes such as the addition of instant replays and coaches’ challenges are leveling the playing field. Plus, travel isn’t as hard on athlete’s bodies as it used to be, thanks to increased budgets and advances in sports medicine.

Classroom activities: statistics, sports

• (Introductory Statistics) Suppose that the Middletown Wanderers won 20 out of their 30 home games last season, and won 15 out of their 30 road games that same season.
• Create a list of possible reasons that these figures might not represent a true home-field advantage. (For example, think about where where the team faced their most highly ranked opponents, or how the team’s performance might have varied throughout the season.)
• Assuming that none of the factors you listed are at play, use a two-proportion Z test at at the 5% significance level to determine if there is statistical evidence that the Wanderers had a home-field advantage last season.
• (Introductory Statistics) Gather win-loss data from a recent season for all the teams in a local sports league. Calculate the percentage of games in which the home team won.
• Are there any confounding factors that might skew the data?
• If not, use a one-proportion Z test at the 5% significance level to determine if the league as a whole had a home-field advantage that season.

Related Mathematical Moments: Holding the Lead.

—Scott Hershberger

### Is There an Optimal Driving Speed that Saves Gas—and Money?

Wired, April 1, 2022

How fast should you drive to work? You can find a precise answer to that question, at least if your goal is to save money. In an article for Wired, physicist and science popularizer Rhett Allain explains how to calculate the most cost-effective driving speed given your car’s fuel efficiency, your hourly pay rate, and the price of gas. Allain says that the cheapest speed is not too slow (if you’re idling, you’re getting 0 mpg) and not too fast (because of factors like air resistance and friction), but at a sweet spot in the middle. As a bonus, Allain shares a snippet of code that does the math, which you can edit to calculate your own optimal speed.

Classroom activities: fractions, ratios, physics, optimization, coding, programming

• (All levels) Let’s develop an intuitive understanding of the equation for average velocity described in the article. Try to answer the following questions based on your experience. Then compare those with the answers you get from the equation $v = \Delta x / \Delta t$, where $v$ stands for average velocity, $\Delta x$ stands for the distance a car traveled, and $\Delta t$ stands for the amount of time the trip took. (Note: Speed is the absolute value of velocity, so in the following two examples you can treat “velocity” and “speed” interchangeably.)
• If Car A and Car B both drove for 1 hour, but Car A traveled 15 miles and Car B traveled 45 miles, which car had the higher average speed?
• If Car A and Car B both drove 40 miles, but Car A’s trip took 1 hour and Car B’s took 2 hours, which car had a higher average speed?
• (High level) In general, when you increase the absolute value of $\Delta x$, does average speed increase or decrease? What about when you increase the value of $\Delta t$? Does this match with your intuition about what it means for something to travel at a high or low speed?
• (High level) The article mostly discusses “speed” rather than “velocity.” Why?
• (High level) Experiment with the interactive coding tool in the article by clicking the pencil icon and changing the values of the variables basempg, G, R, and dx, then clicking the play/run icon. Try some of the following:
• Compare the cheapest speeds for a car that has an EPA listed mileage of 25 mpg versus a car that has a mileage of 50 mpg. For which car is the cheapest speed faster?
• Make a prediction about whether increasing the pay rate R will increase or decrease the cheapest speed. Then test your prediction using the code.
• Does the commuting distance dx have any effect on the optimal speed? (The answer to this question is in the article.)

—Tamar Lichter Blanks

### What Could You Stuff in a Post Office Mailer to Exceed the Weight Limit?

Popular Mechanics, April 25, 2022

In an April 20 tweet that attracted over 75,000 likes, attorney Paul Sherman claimed that “It is physically impossible to exceed the 70-pound domestic weight limit for a small flat rate box” shipped by the US Postal Service. A small flat rate box filled with pure osmium—the densest naturally occurring element—would weigh in at around 61.5 pounds, he wrote. In this article, writer Caroline Delbert compares osmium to other dense materials. While osmium may top the scales on Earth, trying to mail the material that neutron stars are made of would leave you on the hook for a hefty overweight charge.

Classroom activities: density, mass, volume, media literacy

• (Pre-algebra, Algebra I) Let’s check Sherman’s calculations and do some of our own.
• Find the inside dimensions of a USPS small flat rate box. What is the volume of the box in cubic inches? What is the volume in cubic centimeters?
• The density of osmium is 22.59 g/cm3. What mass of osmium could you fit in the box? How does this compare to the USPS limit of 70 pounds?
• (Mid level) The density of a neutron star is on the order of 1014 g/cm3. What mass of neutron star material could fit in the box (in kilograms and in pounds)? What volume of neutron star material would weigh 70 pounds?
• (Upper level) Ask each student to pick a substance (water, rock, cotton candy, etc.) and look up its density. Calculate the weight of a small flat rate box filled with that substance. Then calculate the volume of 70 pounds of that substance.
• (High school) Ask students to find examples of claims on the Internet that cite simple calculations. Have them try to confirm or refute the claims, either by doing the calculations themselves or searching for other reputable sources online.

—Scott Hershberger

### Hate math? You’ll still love this cornucopia of simple-yet-seductive math games

Ars Technica, April 6, 2022

To Ben Orlin, a teacher and author, playing is essential to learning. That’s especially true in math. Orlin recently wrote a book called Math Games with Bad Drawings. It’s an illustrated collection of multiplayer games in which intriguing puzzles emerge from simple mathematical rules. You just need household items like pencil and paper, your hands, the internet, and Goldfish crackers, writes Jennifer Ouellette in Ars Technica. Ouellette interviewed Orlin about his new book, which includes more than 50 “math-y” games. According to Orlin, the book doesn’t try to create educational puzzles for math-lovers. He instead sees each game as a thought experiment enjoyable for anyone. “Games are constantly generating new puzzles,” he says. “With a puzzle, you solve it, you’re done. A game is like a fountain of puzzles that’s constantly pouring out new puzzles for you.”

Classroom Activities: math games

• (All levels) In partners, play Ultimate Tic-Tac-Toe. Discuss what is “math-y” about it and how your strategy is different from your strategy in standard Tic-Tac-Toe. (You can find more info on Wikipedia.)
• (All levels) In partners, play Sequencium, one of Orlin’s own games.
• Show your final board to another pair of students and compare the strategies that you used.
• If both players are able to move until the board fills up, what is the maximum possible score? What is the minimum possible score?
• (All levels) Read about the game of Set via examples from pop culture on Orlin’s webpage, and watch a PBS video about the geometry of Set.
• (Advanced) Try to solve the challenge problem at the end of the PBS video (then see the solution).

—Max Levy

### Scientists Teach Fish to Do Basic Math

CNET, April 1, 2022

Math is a language, and humans aren’t the only creatures who understand it. Nature is full of animals that seem to perform simple mathematical operations. Animals as big as lions tally their competitors; animals as small as bees distinguish small numbers from large ones. The list is always growing as scientists put more animals to the test. In an article for CNET, Monisha Ravisetti writes about new research involving fish. Researchers from Germany taught cichlids and stingrays to add and subtract 1 from numbers up to 5. Their experiment worked like this: Fish can’t read our numerals, so the team showed the fish some number of blue or yellow shapes. Blue meant “add 1,” and yellow meant “subtract 1.” The fish earned treats for choosing correct answers and eventually, they learned. “The fish passed with flying colors,” writes Ravisetti.

Classroom activities: counting, symbolic math, numerals, box plots, p-values

• (All levels) Let’s count like fish. Read the article to learn how the scientists tested the fish. (More info in the full study here.) In groups of two (or a class) challenge each other to create a prompt and pick the correct answer. For example, what would be the correct response to a symbol with a blue square, a blue circle, and a blue triangle?
• (Mid level) Each student can create their own language of symbolic arithmetic different from the colors and shapes here. This math language should communicate addition, subtraction, multiplication, and division. They can use shapes and colors or something else entirely, like emojis as digits. Get creative! Then, students can form small groups, teach each other, and quiz each other. Discuss the advantages and disadvantages of each system as compared to our standard system.
• (Upper level, Statistics) Discuss the significance of Figures 5 and 7 in the scientific paper and whether they provide compelling evidence for their claim that these fish can “add and subtract 1.”

—Max Levy