Straight lines vs. curves

Yesterday we looked at some of the curves from Frank Farris’s book Creating Symmetry

Frank Farris’s Patterns

The idea for that project had caught my eye because of our two prior projects with Anna Weltman’s loop-de-loops:

Anna Weltman’s loop-de-loops

Anna Weltman’s loop-de-loops part 2

In a brief conversation over Twitter, Weltman asked about something that was already on my mind:

I’d definitely been wondering about the mathematical steps that take you from here:
  

to here:

IMG_0375

Honestly, I still don’t have a good feel for what the right progression is. I’m sure that Weltman’s activity is a great activity for kids to do, and I think the shapes in Farris’s book are great for kids to see and even play with on a computer even if the math required to understand those curves is too advanced for them to understand.

I’m confused as to why I’m so confused about the right way to connect the two ideas. It is pretty incredible to me that both ideas are so easy to share. Both ideas can lead to lots of great math-related conversations with kids.

As I’ve been thinking about the connection for the last two days, on other possible similar connection came to mind.

The latest math problem to go viral is this one from a math test in Scotland. Here’s the problem:

Problem

Christopher Long has a great solution here if you want to see something other than the usual calculus solution:

Solving a math puzzle using physics

This straight line problem has a generalization with curves, too. I remember the more general problem because it was very last (and a double star!) problem in my high school differential equations book. Since I don’t have that book around anymore, here’s the similar problem from the 1959 Putnam exam:

A sparrow, flying horizontally in a straight line, is 50 ft below an eagle and 100 ft directly above a hawk. Both the hawk and the eagle fly directly toward the sparrow reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far did each bird fly. At what rate does the eagle fly?

I need to spend a bit more time thinking about the jump from ideas with straight lines to ideas with curves. Showing the boys some of Farris’s work make me think that there is a lot of fun to be had talking about curves even if the math needs to stay in the background. I’m also a little surprised at why bringing the math out of the background seems so difficult.

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