Many Geometry textbooks instruct the teacher to distinguish between a line segment $\overline{DE}$ and its length $DE$ by writing a line over the letters and to distinguish between an angle $\angle BCD$ and its measure (in degrees) $m \angle BCD$ by including an $m$ for measure. However, I have gone the entire year not writing lines over my letters or extra $m$'s in front of my angle symbols, and not one student has asked me to clarify. It's the job of the teacher to sort through needed and needless distinctions and present the material predigested to the students. In our class, I have found this segment vs. length distinction is not needed.
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If you've been through a high school Algebra 2 course, you may remember polynomial long division. It's as much fun a regular long division, but now with algebra! Usually, after a few days of practicing long division, the teacher will introduce a faster way of completing polynomial division problems, called synthetic division. Essentially, this method allows the student to write less by keeping track of the variables in columns. Often students are resistant to this new method, simply because it is new. However, after practice, they usually concede that it is easier. Unfortunately, synthetic division only works when dividing by a monic, linear polynomial such as $x3$. Once we start dividing by quadratic polynomials such as $x^23x+5$ most textbooks do not provide any such shortcut... Today, I offered my students two options. Students could practice dividing by quadratic polynomials using long division, or, if they felt confident in that skill, they could opt to learn a synthetic division technique for quadratic polynomials that was not in the book. Four students had strong enough math skill and had sufficient interest to take me up on the offer to see if the synthetic shortcut for linear polynomials could be extended to quadratics. Below is the same problem worked out using both long division (left) and synthetic division (right). If you're familiar with synthetic division from high school, you might try to figure out how the procedure works. Overheard at the end of class, "This synthetic division is so much easier!"
Along the landscape of English vernacular, there are some words that we say because those are the words we say. Have you ever encountered an abode that was not “humble” or a “petard” that was doing something other than hoisting?
Mathematics is often criticized for having more than its fair share of these disconnected words. “Conjugate” and “discriminant” come to mind. The only connection we have to these words is from their usage in a very limited context. Why not change our terminology to allow our experience to influence our understanding of the mathematics? As a starting point, I nominate “imposter solution” to replace the outdated “extraneous solution”. 
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
