While researching yesterday's topic, I discovered a short article or two on the skills needed for Calculus. It turns out, students need a good foundation in both Algebra and Trig!
The specific Algebraic skills needed include factoring, completing the square, know the basic functions and transformations, simplifying expressions, manipulating formulas, and knowing how to use a calculator for graphing.
For Trigonometry, students need to know the three basic trig ratios and their graphs, the values for the basic trig functions and understand what constitutes an exact answer.
If they do not have solid foundation in the above topics, it will make it much harder to do calculus. Since, I know these particular skills are the most important ones, I can focus on making sure my students learn these in Algebra I, Geometry, and Algebra II. In addition, there are certain algebraic mistakes students repeatedly make in the calculus class.
So what are the mistakes students regularly makes? Well -
1. Students try to apply an additive property to any function with an addition sign, not just the linear functions.
2. Students often cancel parts of an expression rather than the whole expression. I've seen students turn this - x^2 + 2/ x^2 -3 into 2/-3 because they've crossed out the x^2. They have trouble understanding they cannot do it.
3. Students often confuse negative exponents with fractional exponents. So sometimes they see 1/x^2 as x^1/2.
4. Students often have trouble with exponential rules and sometimes apply them incorrectly. For instance, you might see a student do this: 4(x - 1)^2 = (4x - 4)^2 which is not correct.
5. Incorrectly simplifying using a value that is not equivalent to one or they try to simplify an expression rather than an equation.
6. The order of operations and distributive properties when not followed correctly will create a problem.
These are really the six algebraic mistakes most students make in Calculus because their algebraic foundation is not as strong as it might be or the students do not quite understand the process.
If we know what the areas of Algebra and Geometry need to know well for Calculus and we know the most common mistakes they make, we can work on strengthening those areas so our students will be better prepared to do well in college.