Current Digests: January 2025
Peering Into a Bleak, ‘Uninsurable Future’
Inside Climate News, January 17, 2025
At the beginning of this year, a series of wildfires in Southern California caused unprecedented destruction. More than 12,000 structures burned, contributing to a projected \$250 billion of damage. Insurance companies typically reimburse homeowners for catastrophic natural disasters. But as the frequency of disasters increases, driven by climate change, some insurers have stopped serving risky areas in Southern California. And according to reporter Kiley Price, the scale of the recent destruction may make it even harder for insurers to stay in business. This Inside Climate News article explains why the climate crisis changes the math of insurance policies. “Insurance premiums are getting higher, largely because they have started to factor climate risk exposure into their pricing models,” Price writes.
Classroom Activities: insurance, probability
- (Mid level) Complete these three math worksheets about insurance from Purdue University.
- Watch this short video for more about the value of an asset depreciating linearly over time.
- (Mid level) According to the article, insured losses from natural disasters reached \$140 billion in 2024. Answer the following based on what you have learned.
- What will the 2025 insured losses total be if losses increase by 15%?
- Based on your reading, list 3 controllable factors that would potentially increase insured losses, and 3 factors that would potentially decrease insured losses. (Hint: We are measuring only “insured” losses.)
—Max Levy
Rational or Not? This Basic Math Question Took Decades to Answer
Quanta Magazine, January 8, 2025
Rational numbers are any numbers—decimal, negative, or whole—that equal a ratio of two integers. Three mathematicians recently discovered a way of checking whether a number is rational or not. “It might seem surprising that mathematicians are still grappling with such a basic question about numbers,” Erica Klarreich writes in this Quanta Magazine article. “But even though rationality is an elementary concept, researchers have few tools for proving that a given number is irrational. And frequently, those tools fail.” This article tells the surprising history of the deceptive problem and describes how mathematicians solved it.
Classroom Activities: irrational numbers, power series, derivatives
- (Mid level) Which of the following are irrational numbers?
- 22/7
- 4.5
- $\pi$
- $e$
- $\sqrt{2}$
- Any prime divided by any other prime
- (High level) Read the article. Klarreich writes: “If you pick a point along the number line at random, it’s almost guaranteed to be irrational.” Explain in your own words why that is by following these guiding questions.
- Are all integers rational?
- 1.5 is an example of a rational number between 1 and 2, because it can be expressed as 3/2. What are two other examples between 1 and 1.5?
- Is $\pi$/2 rational? What about $\pi$/2.1 and $\pi$/2.0001?
- Does a number line from 0 to 100 contain more integers or non-integers?
- Does a number line from 0 to infinity contain more integers or non-integers? (Hint)
- (High level) Watch this introductory video to power series by Khan Academy.
- Explain in your own words what a power series is.
- How did the mathematicians in the Quanta article use a power series to solve their problem?
—Max Levy
The Helicone Numberscope: Mathematical Superpowers Hidden in a Simple Toy
Mathologer, January 6, 2025
In this YouTube video, the Mathologer (Monash University mathematician Burkard Polster) explores the mathematics behind a simple-looking toy called the helicone. By simulating much larger helicones, the Mathologer can discover which integers are good approximations to irrational numbers.
Classroom Activities: irrational numbers, approximation, recreational math
- (Mid level) Before watching the video, find:
- A rational number that is less than 0.01 away from $\pi$
- A rational number that is less than 0.005 away from $\frac{1 + \sqrt{5}}{2}$
- A rational number with a denominator of 8 that best approximates $\pi$. What is the error?
- A rational number with a denominator of 50 that best approximates $\pi$. What is the error?
- (All levels) Watch the intro section of the video. If possible, give students the opportunity to play with a real helicone.
- The video explores a helicone where every leaf is at a specified angle from the one below it: Either the golden angle (about 137.5$^{\circ}$) or 360$^{\circ}$ times the fractional part of $\pi$ (about 51$^{\circ}$). Watch the video from timestamp 17:22, until timestamp 37:24.
- (All levels) Research three examples of when approximating irrational numbers might be useful. You can search online, in books, or ask people you know. Share your examples with the class and discuss your reactions.
—Leila Sloman
Mathematician explains the trick of the “envelope system”, the formula for saving more than \$2,500 a year
Diario AS, January 30, 2025
“With a pack of envelopes, a pinch of discipline and some consistency, you can save over \$2,500 in a year,” writes William Allen in this article for Spanish sports publication Diario AS. The idea comes from Santi García Cremades, mathematician at Universidad Miguel Hernández de Elche. The technique: Using the envelopes, randomly select an even number between 2 and 100 each week. That’s the number of dollars you save that week. Over 50 weeks, you’ll save \$2 + \$4 + \$6 + \$8 + … + \$100 = \$2,550.
Classroom Activities: probability, series
- (Mid level) Read the article. Allen writes: “The mathematical key lies in the sum of these numbers. Each pair of extremes (2 and 100, 4 and 98, etc.) adds up to 102, and with 25 pairs, the total accumulated at the end of the year is exactly 2,550.” Write out this calculation in your own words, showing all your work.
- What is the sum of all numbers from 1 to 100?
- What is the sum of all multiples of 3 from 3 to 600?
- (High level) In general, if N is a natural number, what is the sum of all numbers from 1 to N?
- (High level, Probability) Imagine that you follow García Cremades’ savings technique. Each week, after you open your randomly chosen envelope, you throw that envelope away. Once you choose a specific number, you will never choose it again. This is called sampling without replacement. You could also try sampling with replacement. In this scenario, after you select your envelope and see what’s inside, you put the envelope back into the pile.
- If you sample with replacement, what is the maximum amount you could save over the course of 50 weeks? What is the minimum amount?
- What is the expected amount you’ll save?
- What is the standard deviation of how much you’ll save over 50 weeks?
—Leila Sloman
What Is Brutalist Architecture, and Why Is It So Controversial?
Art News, January 14, 2025
The new movie “The Brutalist” focuses on a fictional architect famous for his use of brutalist designs. Brutalism is a controversial style that conjures images of giant concrete buildings with square edges and minimal decorative flare. It’s both visually imposing and minimalistic—“two things that have proven divisive with critics and the public alike, who have often found this aesthetic tough to admire,” writes Alex Greenberger for Art News. This article explains the history of brutalism as an architectural trend that emphasizes utility over ornament.
Classroom Activities: geometry, architecture
- (Mid level) Assume that concrete has the same volume in liquid and solid form. Answer the following questions:
- Imagine building a cubic structure that measures 10 x 10 x 10 feet, with walls, floor and ceiling 12 inches thick. How many cubic feet of concrete would you need?
- Now imagine building a structure shaped like an equilateral triangle, as in this image. If your triangular building has the floor area as the above rectangular room, how many cubic feet of concrete do you need?
- Repeat for a dome (flat floor and hemispherical ceiling and walls).
- Which design uses more concrete per usable area? Why?
- Suppose that you can change the wall and roof thickness of the design that requires more concrete in order to use the same amount of concrete as the more efficient design. Calculate the new wall thickness.
- (High level) Explain with math why brutalist architecture may be considered more functional than other approaches to design. Hint: think about the volume, surface area, building materials and complexity.
Explore coverage of the recipients of the 2022 Fields Medals in Nature, Quanta Magazine, and The New York Times.
Read more recent digests of math in the media.
 |
 |
 |