Grading a recent Geometry test on triangle centers, I came across this poem on a student's test. Clearly this student had other things on her mind when taking the test. She should have been thinking about altitudes and orthocenters. Baked treats are fine, but they are no substitute for a triangle's medians and centroid. Also, I was surprised to hear about this student's children. That said, I hope she teaches them about the angle bisectors and incenters she's being tested on. It is a sign of a good upbringing when a child can explain the intricacies of a perpendicular bisector and circumcenter. ...Oh, wait a minute! Instead of chiding the student for being distracted, I should be praising her ingenuity. The poem was actually cleverly designed to help the student on her test. Can you see how? All the clues are in the story.
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Some functions, like $\ln (x)$, grow to infinity very slowly. Others, like $e^x$, grow very quickly. Most functions fall somewhere in between. Armed with L'Hôpital's Rule and precalculus algebra skills, students were able to create a precise ordering of functions according to their rates of growth. Only afterwards, did I create this speed chart to summarize their results. (The animal additions were inspired by some student doodling.) In a second, twoday investigation, students were asked to evaluate improper integrals based on the functions whose speeds we had ranked. For each function $f$ in our ranking, they considered the integral $$\int_2 ^\infty \frac{1}{f(x)}dx,$$ trying to decide if it converges or diverges. On the first day of the investigation, students were not permitted a calculator and were able to evaluate some of the simpler integrals. On the second day, students could use a graphing calculator. By comparing integrals of unknown convergence to previously evaluated integrals, students were able to determine the convergence/divergence of all the remaining integrals. My students are so smart. They articulated their findings, "If the function is faster than $x$, that integral converges. If the function is $x$ or slower, that integral diverges." (That summary is so close to correct!) There is a relationship between a function's rate of growth and whether the corresponding integral converges!

About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome. Archives
September 2017
