My students arrived to class a few minutes before I did today, and this is what they drew on the board for me before I got there. The bow tie this character is wearing might give you a hint of who the picture is of, although I don't think my head is that big. Aside from the flattering depiction, I was most excited to see them adopting the phrase "Independent Problem Solver." I've been telling them all year that they are working to become Independent Problem Solvers, and it was great to see them using it themselves. The "PoW" stands for "Problem of the Week," and if you're unsure about the "What are those?" quote, you should watch this video.
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The first four regular polygons have whole number angle measurements, but a heptagon has a decimal 128.57 angle measurement. Why did the whole numbers go away? My students are learning to become Independent Problem Solvers and one approach they use when solving a problem is Trying Multiple Examples. Students calculated the interior angle measurements for several more regular polygons and made observations. (I've reworded them slightly.) $n$ is the number of sides of a regular polygon.
Some of these observations are incorrect, and none of them exactly capture when the angle is a whole number. However, they all contain the germ of an idea. I was impressed to see my students willing to make a hypothesis, even if it was incomplete. This is the heart of being an Independent Problem Solver.
I recently covered a chapter on rational expressions and equations in my Algebra 2 class. A rational expression is *basically* when you divide by a variable as in $$\frac{2x^2+1}{3x5}.$$ There are two situations that regularly arise in the study of these expressions: simplifying rational expressions and solving rational equations. Both use the concept of a common denominator, which is a unifying idea. Simplifying Rational Expressions When adding fractions as in $\frac{2}{x+2}+\frac{2}{(x+2)(x1)},$ I instruct my students to find a common denominator and contribute the missing factors to the necessary fractions, resulting in $$\frac{2(x1)}{(x+2)(x1)}+\frac{2}{(x+2)(x1)}=\frac{2x}{(x+2)(x1)}.$$ Making these common denominators requires multiplying a numerator and denominator of a fraction by the same factor. Solving Rational Equations When solving equations such as $\frac{2}{x+2}+\frac{2}{(x+2)(x1)}=0,$ I instruct my students to multiply the entire equation by the common denominator and cancel factors as necessary: $$(x+2)(x1)\big(\frac{2}{x+2}+\frac{2}{(x+2)(x1)} \big)=2(x1)+2=0.$$ Students solve the resulting equation normally. Rather than multiplying numerator and denominator of a fraction by the common denominator, we multiply both sides of the equation. I've noticed that in my explanation students get these two situations confused. Students will attempt to multiply an entire expression by a common denominator, which is incorrect. We can only multiply by a common denominator in an equation, because the multiplication is balanced on the other side of the equation.
Perhaps by teaching these two types of problems in different ways I have obscured much of the similarity between them and prevented students from seeing some of the consistent patterns. Perhaps it would be clearer to teach solving rational equations by finding a common denominator as we did when simplifying expressions. 
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
