Math Digests January 2022

Equations built giants like Google. Who’ll find the next billion-dollar bit of maths?

The Guardian, January 24, 2022

Time and time again, mathematical ideas developed decades or even centuries ago find unexpected—and profitable—uses in industry. In an opinion piece for The Guardian, mathematician David Sumpter gives a quick tour of how mathematics has transformed the modern world and speculates about which branch of mathematics will lead to money-making applications next. Fractal geometry, chaos theory, and random walks are all possibilities, he writes. “You don’t need to be a mathematical genius yourself in order to put the subject to good use. You just need to have a feeling for what equations are, and what they can and can’t do.”

Classroom activities: fractals, probability, matrices

  • (All levels) Introduce students to fractals with K-12 activities from the Fractal Foundation. Topics include fractal triangles, coastlines, exponents, and more.
    • Ask students to find examples of fractals in their own home or neighborhood and share them with the class.
  • (Middle school) Introduce students to random walks using coin flips with this lesson plan from the National Museum of Mathematics.
  • (Linear Algebra) When teaching eigenvalues and eigenvectors, use Google’s PageRank algorithm as an example (see these Cornell notes). Discuss: what are some other situations where a similar approach might be useful? What are the limitations of PageRank?
    • (Advanced) Have students complete the exercises at the bottom of the Cornell notes.

—Scott Hershberger

The Math Behind False Positives and False Negatives

Slate, January 14, 2022

During this winter’s Omicron wave, you may have taken rapid COVID-19 tests. But interpreting the results is not always straightforward. In an article for Slate, mathematician Gary Cornell explains two statistical terms and how they relate to COVID tests. A high specificity means that a test gives very few false positives. A high sensitivity means that a test gives very few false negatives. Rapid COVID tests have a high specificity, but a not-so-high sensitivity, Cornell writes—so a positive test result means you are almost certainly infected, but a negative result cannot give total confidence that you are in the clear.

Classroom activities: statistics, specificity and sensitivity

  • (High school) Introduce students to sensitivity and specificity with this lesson plan from Penn State. The lesson also discusses positive predictive value and negative predictive value, which answer an important question: What is the chance that a person who tests positive is infected, or that a person who tests negative is not infected?
  • (High school) A test’s predictive value depends in large part on how common the condition is in the population. (For example, a recent New York Times investigation found that prenatal tests for rare disorders give far more false positives than true positives.) For each of the following hypothetical diseases, suppose you take a test with 99.9% specificity and 80% sensitivity. If you test positive, what is the probability that you actually have the disease? If you test negative, what is the probability that you do not have the disease? (Hint: create charts like those in the lesson plan.)
    • Disease W: prevalence is 10 out of every 100,000 people
    • Disease X: prevalence is 10 out of every 1,000 people
    • Disease Y: prevalence is 10 out of every 100 people
    • Disease Z: prevalence is 40 out of every 100 people

    Discuss how these results could be relevant during the different stages of a pandemic.

—Scott Hershberger

Mathematicians Clear Hurdle in Quest to Decode Prime Numbers

Quanta Magazine, January 13, 2022

Despite all we know about math, the field is still full of mysteries. Some may seem hopelessly abstract, but others have to do with fundamental concepts we all recognize, like primes. “Prime numbers are the most fundamental — and most fundamentally mysterious — objects in mathematics,” writes Kevin Hartnett. A prime number (like 3, 5, 23, or 419) is only divisible by 1 and itself. The mystery of primes is that they seem to follow no discernible pattern. Yet a 160-year-old idea called the Riemann hypothesis suggests that there is a pattern to be found, and mathematicians are hard at work trying to crack it. There is even a million-dollar prize on the line. In this article, Hartnett describes a groundbreaking new step toward solving this stubborn mystery.

Classroom activities: prime numbers, finding patterns, sequences

  • (All levels) Learn more about the Riemann hypothesis with these videos from Quanta Magazine and Numberphile.
  • (Mid level) Write out the prime numbers under 100, one below the other. There should be 25 of them. One column to the right, in the space between each consecutive prime, write the result of subtracting the smaller number from the larger number. Do you see a pattern? Discuss why or why not. Compare this to the pattern you get when subtracting consecutive Fibonacci numbers.
  • (High level) Compute the values of the function $f(n) = n^2 + n + 41$ for $n = 1, 2, 3, \text{and } 4$. Referencing your earlier work, what appears to be happening with this function? Can you find a counterexample to your conjecture?
  • (High level) Have each student come up with their own simple rule to create a sequence of numbers. The rule should involve just addition, subtraction, multiplication, or division and should involve either one, two, or three consecutive terms of the sequence. Ask students to swap sequences in partners and see if they can figure out each other’s patterns.

—Max Levy

Learn how to make a sonobe unit in origami – and unlock a world of mathematical wonder

The Conversation, January 4, 2022

With a few folds, a piece of paper can become a piece of art—and maybe more. In an article for The Conversation, mathematician Julia Collins writes about how origami can inspire mathematical discovery. Collins starts with a small, parallelogram-shaped bit of origami called the sonobe unit. With six of those units, you can build a cube. With more of them, you can create other mathematical shapes such as Platonic solids, Archimedean solids, and Johnson solids. You can also explore principles in the mathematical field of graph theory, like the Four-Color Theorem, by building a shape out of sonobe units of different colors. Origami may even be useful for technology like unfolding solar panels in space.

Classroom activities: origami, geometry, technology

—Tamar Lichter Blanks

Omicron upends mathematical models tracking COVID-19

CTV News, January 13, 2022

Throughout the COVID-19 pandemic, mathematical models have helped policymakers estimate infection risk based on factors like vaccination status, indoor versus outdoor setting, and crowd density. But one of the most important factors for determining transmission risk is also potentially misleading: positivity rate, or the percentage of tests that come back positive. The hyper-transmissible Omicron variant is straining test supplies, so a larger slice of positive cases is going unreported—skewing positivity rates. Mathematicians modeling the spread of COVID-19 are struggling to keep up. “We’re still adapting to flying blind in terms of reported cases,” one mathematician told CTV News reporter Sarah Smellie. In this article, Smellie explains how mathematicians need to adapt their models to keep up with the constantly changing pandemic.

Classroom activities: exponential growth, logarithms, data analysis

  • (Algebra II) The doubling times (how long it takes for the number of infections to double) for Omicron are “some of the fastest we’ve seen in the pandemic”—between 1.5 and 3 days in some regions. Imagine a city of 10 million people where two people are sick. If nobody is vaccinated or takes any precautions to prevent the spread, how many days would it take for 10% of all inhabitants to catch the disease if:
    • The cases doubled every 2 days
    • The cases doubled every 3 days
    • Discuss the implications for public health interventions.
  • (High level) Collecting enough data points is an important part of having a reliable model. To see why, gather two different colors of marbles (or pieces of paper or other item)—one will represent negative cases and the other positive. Place 10 marbles in each of four identical bags or boxes according to the following ratios of positive:negative: 1:9, 2:8, 3:7, 5:5. Now, scramble the bags and remove one marble from each bag. Write down a guess of which bag corresponds to which ratio. Repeat this until no marbles remain. How many rounds did it take until you were correct about all of the bags? Discuss how this relates to the challenge of determining COVID infection rates by sampling from different areas of the country.
    • Remote-friendly version: Use Wheel of Names with 4 different ratios of names “positive” or “negative” instead of marbles and bags.

Related Mathematical Moments: Resisting the Spread of Disease.

—Max Levy

Some more of this month’s math headlines:

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