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.
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 $$\frac{2x^2+1}{3x-5}.$$ 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 $\frac{2}{x+2}+\frac{2}{(x+2)(x-1)},$ I instruct my students to find a common denominator and contribute the missing factors to the necessary fractions, resulting in $$\frac{2(x-1)}{(x+2)(x-1)}+\frac{2}{(x+2)(x-1)}=\frac{2x}{(x+2)(x-1)}.$$ 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 $\frac{2}{x+2}+\frac{2}{(x+2)(x-1)}=0,$ I instruct my students to multiply the entire equation by the common denominator and cancel factors as necessary: $$(x+2)(x-1)\big(\frac{2}{x+2}+\frac{2}{(x+2)(x-1)} \big)=2(x-1)+2=0.$$ 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.
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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 Hanoi, Vietnam. Your comments are always welcome. Archives
May 2021
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