Math Digests May 2024

Can mathematicians help to solve social-justice problems?

Nature News, May 22, 2024.

In this article for Nature, Rachel Crowell discusses the work of several mathematicians who have brought their quantitative skills to bear on social justice issues — from developing a tool that helps residents of Rhode Island’s Woonasquatucket River Watershed find healthcare resources to exploring the dependencies among Sustainable Development Goals set by the United Nations. “Mathematicians can experience first-hand the messiness and complexity — and satisfaction — of applying maths to problems that affect people and their communities,” writes Crowell.

Classroom Activities: statistics, data science

  • (All levels) Read the article. Choose one of the featured projects and describe how mathematics helps the relevant social justice cause.
    • (Mid level) Identify a social justice cause you care about. How might mathematics be helpful in studying it?
  • (Mid level) Navigate to the CDC’s interactive CovidVaxView page for adults. View Figure 3A. This figure shows the percent of a population over time that has received the latest Covid-19 booster. Under “Jurisdiction,” select “National.” For each of the following “Demographics” selections, plot the 95% confidence intervals of the most recent figures. Then, interpret the results. Which of the demographic differences are statistically significant?
    • “Disability Status”
    • “Gender Identity”
    • “Health Insurance”
    • “Poverty Status”
    • “Race/Ethnicity”
  • (Mid level) Try these statistics activities on the relationship between earnings, education, and gender from the US Census Bureau’s Statistics in Schools program: One on plotting and analyzing scatterplots for 8th graders, and one interpreting box plots for 9th graders.

—Leila Sloman

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

60 Minutes, May 5, 2024.

Last year, two teenagers from New Orleans announced a rare trigonometric proof of the Pythagorean theorem, which states that a right triangle’s hypotenuse squared equals the sum of its shorter sides squared ($a^2 + b^2 = c^2$). Many trigonometric principles depend on the Pythagorean theorem, so some mathematicians once thought using these principles was self-referential. In this 60 Minutes segment, the two students describe their work and achievement in discovering up to five new proofs.

Classroom Activities: trigonometry, algebra

  • (All levels) Read more about the teens’ proof presented last year in this Scientific American article. (Note: The Scientific American article was written by the editor of this column.)
    • Describe in your own words the flaw in using the equation $\sin^2(\theta) + \cos^2(\theta) = 1$ to prove the Pythagorean theorem.
    • Complete the AMS activities based on the initial news from last year.
  • (High level) Follow along with the students’ “waffle cone” proof in the 60 Minutes video and go deeper with this step-by-step video from Polymathematic.
    • Why does the proof require reflecting the initial right triangle? (Hint: Can you create the “waffle cone” without doing so?)
    • What is the law of sines, and how does it enable this proof?
    • Explain why using “similar” right triangles allows Johnson to draw a general conclusion for her proof. (Hint: think of convergent infinite series.)

—Max Levy

Call a mathematician, we’ve got a Monty Hall problem

RNZ, May 23, 2024.

In the famous Monty Hall Problem, you are a contestant on a fictitious game show. In front of you are three closed doors, labeled 1, 2, and 3. One of the doors — you must guess which — conceals a valuable prize. To start, you choose a door. The game show host responds by opening one of the other two doors to reveal there is no prize behind it. You now get a chance to make your final guess: Stay with your first guess, or switch? On May 23, mathematician Chris Tuffley joined Radio New Zealand to explain what the right choice is, and why.

Classroom Activities: probability

  • (All levels) Before learning the solution to the Monty Hall Problem, answer the following questions individually. Then, discuss answers as a class. After a brief discussion, vote on the right answers.
    • Should you switch your choice? Why or why not?
    • At the beginning of the game, what is the probability that your first choice is the right answer?
    • Does this change after the host makes the reveal? Why or why not?
  • (All levels) Split into pairs and play 6 rounds of the Monty Hall Problem with your partner, using a quarter hidden underneath one of three plastic cups. Take turns playing the host. Jot down anything you notice while playing. (You can also use this lesson plan, in which students play the game on a simulator such as this one and track the results.)
    • Revisit the questions from the previous exercise and vote again if it seems like the class’s perspective has changed.
  • (Mid level) Read Keith Ellis’s explanation of the solution. Write out the reasoning in your own words. Read this article about how mathematicians and scientists reacted to this counterintuitive problem in 1990.
  • (Mid level) Consider a variant of the Monty Hall Problem in which the placement of the prize is not totally random. How does the optimal strategy change if everything stays the same, but at the beginning of the game:
    • There is a 50% chance the prize is behind Door 1, a 25% chance it’s behind Door 2, and a 25% chance it’s behind Door 3
    • There is a 40% chance the prize is behind Door 1, a 30% chance it’s behind Door 2, and a 30% chance it’s behind Door 3
    • There is a 20% chance the prize is behind Door 1, a 30% chance it’s behind Door 2, and a 50% chance it’s behind Door 3
    • There is a 20% chance the prize is behind Door 1, a 40% chance it’s behind Door 2, and a 40% chance it’s behind Door 3

—Leila Sloman 

An amazing thing about 276

Numberphile, May 1, 2024.

Is any single number more special than the rest? It depends on what question you ask. For mathematician Ben Sparks, the number 276 stands out. It’s not because 276 is an even number or a “triangular” number. It’s because of a special application of number theory called an aliquot sequence, in which you sum a number’s divisors (excluding itself), then sum that number’s divisors, and so on. Under this procedure, Spark’s number packs a strange surprise. In this Numberphile video, Sparks explains the aliquot sequence process and plots the results to reveal surprising mathematical behaviors.

Classroom Activities: aliquot sequences

  • (All levels) Using only pen and paper, write the aliquot sequences for the following numbers: 10, 33, 15, 100.
    • Which number has the longest sequence?
    • Was this surprising? Describe in your own words what factor(s) allow some numbers to have longer or shorter sequences than others.
  • (Mid level) Use this GeoGebra program to generate and plot the aliquot sequences for the numbers below.

—Max Levy

A Rosetta Stone for Mathematics

Quanta Magazine, May 6, 2024.

More than 80 years ago, French mathematician André Weil first wrote about a surprising connection between two disparate fields of mathematics: geometry and number theory. Weil’s “Rosetta stone” connected the two topics with a third, the study of finite fields: “Finite fields are a place where number theory and geometry begin to blend,” writes Kevin Hartnett. In this Quanta Magazine article, Hartnett tells the story of Weil’s discovery and how it laid the groundwork for the Langlands program, a “grand project to unify disparate fields of mathematics.”

Classroom Activities: number theory, geometry, finite fields

  • (Mid level) Based on the reading and your own online searching, describe each of these areas of math in your own words, and write a simple example problem related to each.
    • Geometry
    • Number theory
    • Finite fields
  • (Mid level) Write three different polynomial expressions that can be written in a finite field with the elements 0 and 1.
    • Write the binary form of each polynomial, as described in the article. What whole number does each binary form correspond to?
    • What polynomial expression in this field would correspond to the whole number 121?
    • Are the polynomials you listed irreducible or reducible? Explain.

—Max Levy

Some more of this month’s math headlines: