Most Common Mistakes in Algebra?

Blackboard Lie AlgebrasImage by via Flickr
I've worked with tutoring students in Algebra for several years now, including 10 years as a tutor for Kaplan Test Prep. Over the years, I've noticed a handful of mistakes that always seem to rear their ugly heads. I thought I'd share them here, hoping to remind students to always be on the lookout as well as solicit tips from teachers on how to avoid them.

#3 Ranking Values - Least to Greatest

What if you saw the following numbers, and were asked to order them from least to greatest?

10^-1, 10^5, 1/2, 0, 10^-2, -10

After teaching science (specifically, Chemistry) for 10+ years, scientific notation was commonly one of the earliest units, and we would throw a problem like this on homework and a quiz. Students would often get confused with the negative numbers, the negative exponents, and/or fractions. Put them all together, and you've got triple trouble.  (-10, 0, 10^-2, 10^-1, 1/2, 10^5 = Answer)

#2 Dividing by a negative for inequalities

When working with inequalities, I often noticed that students would forget to flip the inequality sign. This combined with forgetting to check the "and" vs. "or" nature of the inequality, and you have nearly sure way to answer a problem incorrectly. A similar issue with inequalities involves graphing and then shading the graph to show the solution set. I often recommended using (0,0) as a test point whenever possible, and if it satisfied the inequality, shade towards the point.

-4X < 20 = X > -5

#1 Distribute the negative!

I'm willing to venture a guess that this is easily one of the most common mistakes made in algebraic problems. When there's a negative number outside a parentheses that calls for distributing that negative, students so often forget to multiply the negative number by all terms inside. Most often, the first term will get the negative and the second one will be forgotten. Here's a case where establishing the habit of changing the subtraction sign to a plus and a negative number may trigger a reminder to use the negative number when multiplying.

X - 4(X+2) = X + -4(X+2) = X -4X -8 = -3X - 8
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