Math Digests July 2024

Record heat engulfs both U.S. coasts

Yale Climate Connections, July 9, 2024.

On July 7, a heat wave raised temperatures in Las Vegas, Nevada to 117 degrees Fahrenheit, the all-time high since records began 87 years ago. That blistering summer day smashed the previous high by 3 degrees, and two other days in the week topped the old record as well. “Unusually warm temperatures on both coasts are getting a boost from climate change,” wrote Bob Henson and Jeff Masters for Yale Climate Connections. Their story explains that a summer of heat waves across North America brought about the second-warmest June on record, and at least 10 record temperatures.

Classroom Activities: data analysis, climate, weather, records

  • (All levels) Read this Climate Math resource from Let’s Talk Science.
    • Answer questions 1–7.
  • (Mid level) Create a graph with an $x$-axis of months (January to December) and a $y$-axis of temperature (in degrees Fahrenheit).
    • Based on your own best estimation, plot a line of the average monthly temperatures in: your hometown; a hot place in the northern hemisphere; a cold place in the southern hemisphere.
    • Plot the average annual and average June temperatures from the past 50 years in Death Valley, based on this public data. Note any observations or trends.

—Max Levy

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Scientific American, July 1, 2024.

In 2000, the Clay Mathematics Institute announced a set of one-million-dollar prizes, awarded to anyone who solved one of seven unsolved problems in mathematics. One of these so-called Millennium Prize Problems is the Riemann Hypothesis, which attempts to make sense of where prime numbers appear in the infinite expanse of all numbers. “The Riemann hypothesis is the most important open question in number theory—if not all of mathematics,” wrote Manon Bischoff in Scientific American. Although the problem remains unresolved, this article describes a new “remarkable breakthrough” that brings mathematicians closer than ever.

Classroom Activities: Riemann hypothesis, prime numbers

  • (All levels) List all prime numbers between 0 and 100.
    • How many do you count?
    • According to the article, how many are “predicted?” Explain in your own words why there is a discrepancy.
    • Based on the prime number theorem, how many prime numbers should there be between 0 and 1000? Between 0 and 1,000,000?
    • Write down any patterns that you notice among the first 10 primes. Do any of these patterns continue after the first 10 primes?
  • (High level) Answer the following questions about the Riemann Hypothesis. (Refer to this video for a deeper explanation.)
    • Based on the three sections of the video, explain in your words why the “smooth graph” (0:00 – 3:30) suggests that there may be a general rule or equation for finding primes.
    • (8:15 – 9:30) What is a complex number?
    • (6:30 – 7:30; 9:30 – 10:30) Explain in your own words the relationship between the zeta function and prime numbers.

—Max Levy

Why expanding access to algebra is a matter of civil rights

The Conversation, June 19, 2024.

Math education expert Liza Bondurant believes that the United States has an algebra problem. According to her research, today’s poor algebra literacy demands more algebra exposure. In this article, Bondurant makes a case for expanding access to algebra classes and compares policies such as “opt-out” programs and automatic enrollment. “Math literacy is a civil right,” she writes, “a requirement to earning a living wage in modern society.”

Classroom Activities: algebra, statistics, data analysis

  • (Mid level) Navigate to the NAEP Report Card Mathematics Assessment. Scroll down to “Fourth-grade mathematics scores declined across all regions of the country and in 43 states/jurisdictions.” Select “Display as table,” then answer the following questions.
    • Which state(s) increased by the greatest percentage between 2019 and 2022?
    • Which state(s) decreased by the greatest percentage between 2019 and 2022?
    • What is the mean change for the states listed?
    • What is the standard deviation of scores in 2022?
  • (Mid level) Scroll down to the figure “Changes in eighth-grade NAEP mathematics scores between 2019 and 2022, by selected racial/ethnic groups” in the NAEP Report Card. Answer the following questions.
    • What conclusion can you draw about the 25th-percentile groups when comparing 2019 to 2022?
    • What can you observe about the 25th-percentile scores in 2022 between racial/ethnic groups? Might Bondurant’s article about algebra scores contribute to explaining this observation about math scores in general? Why or why not?
    • Use the data in the table and spreadsheet software to recreate the graphs.

—Max Levy

This intricate maze connects the dots on quasicrystal surfaces

Science News, July 29, 2024.

In an Ammann-Beenker tiling, squares and diamond shapes completely cover the $xy$-plane. The corners and edges of these shapes form an infinite network called a graph. In this article for Science News, Skyler Ware covers new work that finds a Hamiltonian cycle—a path in the graph that visits every single corner.

Classroom Activities: graph theory, Hamiltonian cycle, trigonometry

  • (Mid level) The Ammann-Beenker tiling is made up of two tiles, or shapes: a square and a rhombus. Suppose the squares have side length 1. Examine the graphic in the article and calculate:
    • The side lengths of the rhombus.
    • The angles of the rhombus corners.
  • (All levels) In an Ammann-Beenker tiling, the precise arrangement of the shapes is slightly different everywhere you look. And yet, at a quick glance, some patterns in the article’s graphic are evident. What patterns do you notice in the tiling shown in the article?
  • (Mid level) Read the article to learn what a Hamiltonian cycle is. Then examine the four graphs on this page. For each one, determine whether it has a Hamiltonian cycle. If you think there is one, draw it. If you think there is not one, explain why. (Teachers: Give students several minutes to work on this problem, and do not grade for correctness—finding Hamiltonian cycles can be extremely difficult.)

—Leila Sloman

Can you solve it? Can you outwit the wizards of Oz?

The Guardian, July 22, 2024.

When I was little, my great-grandfather made me a simple toy: a wooden board, with nails hammered all over it in a grid, painted bright blue. Along with the board, he gave me a small metal box of rubber bands. The idea was to create shapes on the board with the rubber bands. Apparently, the creators of the Australian math magazine Parabola were familiar with this type of contraption. One of their geometry problems, based on just this setup, was featured in The Guardian in July.

Classroom Activities: geometry, algebra

  • (Mid level) Read and solve the problem “Peg squares,” in The Guardian.
    • What restrictions are there on the kinds of shapes you can make with rubber bands?
  • (Mid level) Answer the following questions.
    • Find two different ways to construct a triangle of area 2 on the pegboard. What are the side lengths?
    • What is the area of the smallest octagon you can make?
    • Create your own shape and challenge a partner to calculate the area and side lengths.
  • (High level, Calculus) Suppose rather than being on a flat board, the pegs are on the surface of a sphere of radius 1. Imagine that they are placed on the grid of latitude and longitude lines shown here, at the points where multiples of 10 degrees meet (so, at points like $(30^{\circ}S, 60^{\circ}W)$ or $(0^{\circ}N, 40^{\circ}E)$). What would be the new area of the shapes shown in the “Peg squares” problem, if the bottom row of pegs shown is along the equator?

—Leila Sloman

Some more of this month’s math headlines