Please Teach the Vertical Line Test

I was re-writing an introductory lesson on functions this morning and was reminded of something from years ago.

An influential teacher was telling their followers not to teach the vertical line test because it was confusing. I strongly disagreed. The vertical line test is a great way to meaningfully connect several fundamental ideas: the definition of a function, the definition of a graph, domain and range. The influencer was unmoved, but conceded slightly, saying that the topic should be handled with caution. I responded, “Yes, any time a teacher teaches something they don’t fully understand themselves, they should be cautious!”

It’s sad to know that there are teachers out there not teaching the vertical line test because someone told them it’s too confusing.

Originally posted on Mastodon.

By MrHonner, ago

Charter Schools and Cafeterias

I visited a school recently and was surprised to learn they had no cafeteria. The school requires that students bring their lunches from home which they eat in their classrooms. The school can do this because it’s a charter school and is not bound by the same laws that require public schools to ensure that all students have access to meals.

On the one hand, this is precisely the kind of freedom charter school advocates would say can drive innovation. Rather than wasting a lot of money on building and operating a cafeteria, let’s spend that money elsewhere. On the other hand, requiring that students bring their lunch every day subtly, but effectively, screens out students with unstable home environments.

Originally posted on Mastodon.

By MrHonner, ago

How to Optimize Traffic

Yesterday while stuck in traffic I noticed that the dynamically priced express lanes were nearly empty *and* the price to use them was quite high. This seemed odd, as I thought that if the lanes weren’t being used, the price would come down to encourage drivers to use them. It seems like a straightforward optimization problem.

The twist is that the express lanes are privatized, so the operators price the lanes to maximize profit, not, as I had assumed, to optimize traffic flow. This different objective lead to different behavior in the system.

This is a nice, simple example of how every applied math problem depends on many different assumptions, some of which might be hidden from view. It’s also a good example of how math can be used to better understand the different approaches to managing public resources.

Note: I originally posted this on Mastodon. Jeremy Kun posted some nice resources in response, including this paper surveying the research on the topic and a link to the Transportation Research Board’s Managed Lanes Committee.

By MrHonner, ago

Math That Lets You Think Locally but Act Globally — Quanta Magazine

My latest column for Quanta Magazine explores some recent results in graph theory that use local information to draw global conclusions, a powerful tool in math! It begins with a puzzle.

In math, as in life, small choices can have big consequences. This is especially true in graph theory, a field that studies networks of objects and the connections between them. Here’s a little puzzle to help you see why.

Given six dots, your goal is to connect them to each other with line segments so that there’s always a path between any pair of dots, with no path exceeding two line segments in length.

You can see the solution to the puzzle and learn how it connects to new results in graph theory by reading the full article here for free.

By MrHonner, ago

The Symmetry That Makes Solving Math Equations Easy — Quanta Magazine

My latest column for Quanta Magazine is about one of the most dreaded mathematical objects in high school math: the quadratic formula!

x=\frac{-b \pm \sqrt{b^2-4ac}} {2a}

As complicated as the quadratic formula is, the cubic formula is much worse, but a simple geometric idea connects the two.

As intimidating as this looks, hiding inside is a simple secret that makes solving every quadratic equation easy: symmetry. Let’s look at how symmetry makes the quadratic formula work and how a lack of symmetry makes solving cubic equations much, much harder. So much harder, in fact, that a few mathematicians in the 1500s spent their lives embroiled in bitter public feuds competing to do for cubics what was so easily done for quadratics.

You can read the full article here.

By MrHonner, ago
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