Math Digests September 2024

JWST sees light invisible to the eye. These scientists decide how to color it

Astronomy, August 17, 2024.

When you see images of galaxies many lightyears away, the colors may not be exactly what you’d see if you flew past them on a spaceship. The advanced telescopes that astronomers use typically rely on wavelengths of light invisible to us, such as x-rays and infrared waves. “The process of deciphering data from non-optical telescopes is often called false colorization, but the word ‘false’ does it a disservice,” wrote Randall Hyman for Astronomy. This process is not a creative exercise. Scientists strive to reveal visible details of the cosmos by tinkering with the math of light waves.

Classroom Activities: color theory, hexadecimal numbers

  • (Mid level) Refer to this Teach Engineering resource and this UC Berkeley resource on the science of color for the following questions:
    • Describe in your own words the difference between how colored pigments vs colored light mix.
    • Rank these colors in ascending order of a) wavelength, and b) energy: Blue; violet; infrared; ultraviolet; microwave; yellow.
  • (High level) On a computer, colors are often displayed by a string of digits that represent a mixture of red, green, and blue light. The intensity of each color ranges from 0 (minimum) to 255 (maximum), such that (0,0,0) appears black and (255,0,0) appears bright red, (0,255,0) appears bright green, and (0,0,255) appears bright blue.
    • How many possible color combinations exist in this space?
    • Rather than using the integers 0–255, colors are usually programmed in a hexadecimal number system with two digits, each ranging from 0 to F. (000000:black; FFFFFF:white). Explain in your own words why the numbers for each digit go from 0 to F. (Hint: what does “hexadecimal” mean?)
    • Play the game Hexcodle or Hexcodle Mini to practice this representation of color in a fun game.

—Max Levy

Unveiling the math behind your calendar

The Daily, September 14, 2024.

What happens when several people try to find a meeting time by randomly filling out a when2meet poll? In a recent paper, three physicists created a simple model for this process, and calculated the probability that there is no time which works for all participants. Case Western Reserve University publicized the research with a press release in their newsletter, the Daily.

Classroom Activities: probability, data analysis

  • (Mid level) In the study, the authors use the symbol $\pi_0$ to represent the probability that there is no time which works for all participants. The simplest version of their model has \[ \pi_0 = (1-p^m)^{\ell} \] where $p$ is the probability that a participant is free for a given timeslot, $m$ is the number of participants, and $\ell$ is the number of timeslots. For $p=0.5$, $\ell = 40$, calculate $\pi_0$ for $m = 0, 1, 2, 3, 4, 5, 6$. Explain in your own words the meaning of your results.
    • The study found that, using a more complicated model, when scheduling a meeting with $p = 0.5$ and $\ell = 40$, there’s a 92.5% chance of finding a time that works for four people. But if you add just one more participant, that number shrinks down to 72%. Are these numbers consistent with your results?
  • (Mid level) Using the Desmos graphing calculator, plot $\pi_0$ as a function of the number of meeting participants $m$ for
      • $\ell = 40$ and $p = 0.1$, $0.3$, $0.5$, $0.7$, and $0.9$,
      • $p = 0.5$ and $\ell = 5$, $10$, $50$, and $100$.How do the graphs change as you vary $\ell$ and $p$? What do these results mean? Explain what you think is going on (both mathematically, and in the meeting-scheduling scenario.)
      • (Mid level) Create your own data using the below table. For each empty cell in the table, flip a coin to decide whether the participant is free at that time or not. Once you are done, write down whether there is a time at which
        • Participants 1 & 2 are both free,
        • Participants 1, 2 & 3 are all free,
        • Participants 1, 2, 3 & 4 are all free,
        • Participants 1, 2, 3, 4, & 5 are all free,
        • All six participants are free.

    Now, pool the entire class’s data. (The more students, the better the dataset will be.) Together, brainstorm how to use the data to calculate $\pi_0$ for meetings with 1, 2, 3, 4, 5, and 6 participants. Then, plot the results. How close are the results to the $\pi_0$ predicted by the model?

  • (All levels) In their paper, the authors make simplifying assumptions. For instance, they assume that the probability of a participant being free during a given time block doesn’t depend on the time of day or who the participant is. What do you think of these assumptions? Do you think they accurately reflect the process of meeting scheduling? Why do you think the authors made them?

—Leila Sloman

Weymouth mathematician unravels Shakespeare-era card trick

Dorset Echo, September 26, 2024.

Modular arithmetic—the set of rules that, among other things, describes the mathematics of timekeeping—was developed around 1800. But about two centuries before that, the London theater crowd was enjoying a card trick whose success depends on modular arithmetic principles. Mathematician Colin Beveridge explored those principles in a September 23 blog post. “Even knowing the maths behind it, I think this is still a pretty impressive trick,” Beveridge wrote. Dorset Echo reporter Andy Jones interviewed Beveridge about the blog post a few days later.

Classroom Activities: modular arithmetic

  • (All levels) Read the first two sections of Beveridge’s blog post. Break into pairs, and try the card trick yourself, according to his instructions. Take turns acting as the “volunteer” and the “trickster.” Does the trick work?
    • Beveridge’s son asked if the trick would still work if the volunteer counts to a number other than 15. Try again, using the numbers 13, 18, and 22. Does it still work?
  • (Mid level) Read the rest of the blog post, including Beveridge’s explanation of the mathematical principles that make the trick work. To learn more about modular arithmetic, do this lesson from Khan Academy.
    • In your own words, write out a derivation of why the trick works. Prove that it still works if you use a number other than 15.
  • (High level) Can you modify the trick to work with 10 cards in a circle, instead of 13?

—Leila Sloman 

Math tricks for everyday life

NPR Life Kit, September 5, 2024.

In math classes, students’ work is often nothing more than mimicry, according to Ben Orlin, a mathematician and author. We learn concepts carefully compressed into formulas and notations, vetted over thousands of years, but don’t always learn to make sense of them. “It’s this strange game with no obvious connection to their lives,” Orlin said. In a recent episode of NPR Life Kit, Orlin gives tips on how to better learn math by connecting it to language and everyday life.

Classroom Activities: language, arithmetic

  • (All levels) Around 5:00, Orlin talks about the concept of “numbers as nouns.” Listen and explain in your words how naming works in mathematics and why it’s important for the study of mathematics. (Hint: think about how the names/labels are used in math.)
  • (All levels) Around 7:40, Orlin talks about the “verbs” of math — what we do to numbers — with an example of measuring the depth of the ocean. Write your own example of how calculations allow us to convert from “numbers we have” to “numbers we want.” What are the verbs and nouns in your example?
  • (All levels) Orlin recommends doing mental math faster by rounding consistently. Practice by adding the following numbers in your head: 11 + 18 + 92 + 75.
    • Check your work against a calculator.
    • What percent error does rounding give you? Discuss when that kind of error is acceptable versus unacceptable.
  • (Mid level) Around 12:00, Orlin talks about negative numbers, with examples of negative temperatures and debt. How else could you explain to someone the concept of negative numbers? Come up with 3 examples.

—Max Levy

The mystic and the mathematician: What the towering 20th-century thinkers Simone and André Weil can teach today’s math educators

The Conversation, August 20, 2024.

Siblings Simone and André Weil had strong opinions about math education. As a French philosopher and math teacher in the early 1900s, Simone wanted her students to recognize math as a subject rich with culture and history, rather than just an ensemble of facts and rules. As a mathematician, André believed teachers “must motivate students by providing them meaningful problems and provocative examples,” wrote Scott Taylor in The Conversation.

Classroom Activities: philosophy of math

  • (All levels) Taylor wrote “As a math teacher, I frequently see students grit their teeth and furrow their brow, developing only a headache and resentment. According to Simone, however, true attention arises from joy and desire.” In small groups, discuss with your classmates what math topics have most held your attention. What makes these topics different to you than others? Which topics are the hardest to pay attention to? Why?
  • (High level) In the small groups, discuss which parts of your life you expect mathematics to be most meaningful for. (For example, think about your hobbies or potential career.) Write a math problem for one of your classmates to solve based on this area of interest.

—Max Levy

Some more of this month’s math headlines: