Current Digests: November 2024
Math reveals how skateboarders can ramp up their half-pipe power
Science News Explores, November 11, 2024.
In certain situations, a skateboarder can increase their speed just by bending and straightening their knees. This motion is called “pumping.” “As skateboarders roll along a U-shaped ramp called a half-pipe, they build speed and climb higher by pumping,” Kendra Redmond writes for Science News Explores. Her article covers a recent study that mathematically modeled how to maximize gains in speed with pumping.
Classroom Activities: vectors, physics
- (Mid level) The article includes an embedded video on the physics of pumping. Watch the video.
- The explanation relies on understanding vector-valued forces. Read this lesson from Physics Classroom on free body diagrams.
- Learn about decomposing vectors into perpendicular components with this lesson from Soft Schools. Work through the examples.
- (Mid level) Watch the video again.
- Explain, in your own words, how pumping works.
- Suppose you are skateboarding on a flat surface, and then go down a ramp, after which the surface becomes flat again. If you want to go as fast as possible, when should you pump? Why? (Include diagrams in your explanation.)
- (Mid level) Suppose you are on a ramp angled 30 degrees from the ground, and you pump, applying a force of 1 Newton in a direction perpendicular to the ramp.
- Draw a diagram showing the forces involved.
- How much of the pumping force is in the horizontal direction? How much is in the vertical direction?
- Repeat your calculations for a ramp angled 60 degrees from the ground.
- Is pumping more helpful or less helpful when the angle is greater? Explain.
—Leila Sloman
Which Airline Is the Most Reliable?
NerdWallet, November 6, 2024.
After a nightmarish flight delay, you may vow never to choose the same airline again. But what does the data say? This NerdWallet article analyzes data from major airlines about everything that goes wrong: late arrivals, cancellations, diversions, lost baggage, and denied boarding. “U.S. airlines have struggled to deliver on reliability over the past few years,” wrote JT Genter. Mathematical analyses like this allow us to compare performance more objectively than firsthand experiences.
Classroom Activities: data analysis, normalization
- (All levels) Which airlines were the most and least reliable? Explain why using the data.
- (Mid level) Enter all the data from the article into one spreadsheet table with rows for each airline and columns for each factor.
- List 3 examples of airlines performing well in one category and poorly in another.
- Is a higher score in each category good or bad?
- Based on the methodology described in the article, do you have enough information to recreate the “Reliability” scores (the first table)?
- (High level) Create your own reliability score.
- First, find a function to transform the data from each category into values ranging from 0 to 1. The worst performer should get a score of 0, and the best performer a score of 1.
- Now, make this reliability score personal to you by assigning a “scaling factor” of 2 to the two categories that matter most to you. These categories will now count double for reliability.
- Sum the new values up for each airline. Discuss the results of your analysis.
- Is a personal reliability score objective? Why or why not?
—Max Levy
Chimpanzees Could Never Randomly Type the Complete Works of Shakespeare, Study Finds
Smithsonian Magazine, November 8, 2024.
If a chimpanzee spent an infinite amount of time randomly typing, they would eventually type the works of Shakespeare in correct order, according to the “infinite monkey theorem.” This is because given infinite opportunities, any strange pattern will appear at some point. But true infinity doesn’t exist in practice. In this Smithsonian article, Sarah Kuta writes about a new study that determines whether the infinite monkey theorem could be realized before the end of time as we know it—the so-called heat death of the universe.
In the study, mathematicians assumed that a chimpanzee could type 1 key per second. Even with 200,000 chimpanzees working round the clock simultaneously, the probability of reproducing Shakespeare remains virtually zero. “Beyond that, a single chimpanzee has just a 5 percent chance of randomly typing the word “bananas” within its lifetime,” Kuta writes. “The odds of a chimpanzee typing a short phrase like ‘I chimp, therefore I am’ are 1 in 10 million billion billion.”
Classroom Activities: probability, expected value
- (Mid level) Calculate probability of randomly typing the following words or phrases on a 30-key keyboard, starting from your first keystroke (no extraneous letters). Assume every key has equal probability of being typed at each keystroke.
- “Bananas”
- Your first name
- Your full name
- (High level) Assuming the “heat death of the universe” will occur in 10100 years, calculate the following:
- How many seconds until the end of the universe?
- How many keys can one chimpanzee press in one year?
- How many seconds is it expected to take for one chimpanzee to type the first letter correctly? (Hint: $\sum_{k=1}^N k x^{k-1} = \frac{d}{dx} \sum_{k=0}^N x^k$. Approximate $x^{10^{100}}$ as 0 if $x$ is less than 1.)
- How many seconds is it expected to take for one chimpanzee to type the first two letters correctly?
- If there are 200,000 chimpanzees working simultaneously, how many do you expect to type the first two letters correctly in the first two seconds?
—Max Levy
Introduction to linear algebra
The Michigan Daily, November 17, 2024.
Algebra was introduced by Mohammed al-Khowarizmi in ninth-century Baghdad. The word “algebra” comes from the Arabic word “al-jabr” meaning restoration or completion. As the name implies, algebraic principles “restore” a sort of balance thrown off by unknowns. Algebra lets us solve problems in science and engineering. For instance, we can use it to determine the value of three unknown variables given three independent equations. In this article for The Michigan Daily, Madinabonu Nosirova writes about al-Khowarizmi’s contributions, as well as contributions from lesser-known Muslim researchers.
Classroom Activities: algebra, math history, systems of equations
- (Mid level) Solve the following algebra problems
- 4x + 6 = 28 | x = ?
- 2x + 3y = 33 ; y – x = 1 | x = ? y = ?
- (Mid level) Complete the “Systems of equations with elimination” activity from Khan Academy. (Hint: watch the accompanying video lesson.)
- (All levels) Read more about a mathematician from the Muslim world with this article from the University of St. Andrews Discuss what you’ve learned about their contributions in small groups.
—Max Levy
What’s the ‘coastline paradox’?
Live Science, November 11, 2024.
In this article in Live Science’s “Life’s Little Mysteries” series, Alice Sun writes about the “coastline paradox.” The coastline paradox observes that if you try to measure the length of a coastline, you can get two completely different numbers depending on how long a ruler you use. As the ruler gets smaller—allowing you to capture more bends and inlets in the coastline—the measurement will soar higher and higher.
Classroom Activities: measurement, fractals
- (All levels) Read the article. Answer:
- How could the Congressional Research Service and the National Oceanic and Atmospheric Administration get such different measurements for Alaska’s coastline?
- How much bigger is NOAA’s measurement than the Congressional Research Service’s measurement?
- Which agency used a “smaller ruler” in their measurement?
- (All levels) Students will try measuring the perimeter of the contiguous United States using different-sized rulers. Give each student a map of the US and split the class into two groups. Give each student in Group 1 a straightedge that is one inch long, and each student in Group 2 a straightedge that is two inches long. Using their straightedge and the map scale, each student will measure the length of the US perimeter.
- Students will answer the following questions:
- What result did you get?
- If you were in the other group, would your result have been larger? Smaller? Estimate what you think the difference might have been.
- Which parts of the map were easiest to measure? Which parts were hardest? Why?
- Write on the board all the answers from Group 1, and all the answers from Group 2. As a class, discuss reactions to the results.
- Students will answer the following questions:
- (Mid level) Calculate the mean and standard deviation for the results of Group 1, and the results of Group 2.
- In small groups, discuss how and why the results would differ if the US was (a) a perfect square or (b) a perfect circle.
—Leila Sloman
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.
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