A friend of mine teaches chemistry at a community college in South Carolina and is pulling out his remaining hair. It appears that in his introductory chemistry and remedial classes the vast majority of students (recent high school graduates) is unable to do simple calculations involving percentages. Forget about dimensional analysis.
Something must have gone awfully wrong with these students' math instruction in high school and probably even earlier in elementary school. I am on the case investigating.
I teach math to eighth graders and know that teaching math successfully need not be rocket science. Most of my students can now convert fractions to decimals and percents (and vice-versa) in their sleep. They can also solve the three different types of percent word problems (unkown rate, whole and part) in their sleep.
The following is an aspect of how I teach percentages after teaching the concept that a percentage is an equivalent fraction with a denominator of one hundred.
I have my students create a table with three columns. Each column is labeled fraction, decimal, percent. This visualizes the operations and helps reduce confusion. They can go from left to right to start with fractions and end up with percents, or from right to left to start with percents and end up with fractions.
Converting percents to fractions generally poses no problems when the percent is a whole number, e.g. 47% --> .47 --> 47/100. A special problem arises when the percent is a fraction like 5-1/4 %. This is where the students need to realize that an additional step is required. Many students want to enter 5.25 in the decimal column. The task of the teacher is to focus on this problem and to show that the students must first convert to 5.25%, then to the decimal .0525 and on to the fraction.
A good way to drive home this point is to contrast the conversion of 5-1/4 (without %) and 5-1/4% to decimals and writing 5-1/4% and 5.25% in the percent column. Only now can 5.25% leave the percent column and move to the decimal column to become .0525. Bingo!
I also have my students solve percent problems as proportions and equations. I drill them in identifying the rate (percent), part and whole in any percent problem. For equations I have them use the rb=a formula. r=rate (percent), b=base (whole) and a=amount (part). This is also a nice algebra exercise since they need to solve for a, b or r as the case may be.
I once again have them make a table with three columns and several rows. The columns are labeled r, b and a.
Then I have them look at various percent problems and ask them to identify what's known and unknown. If the percent is unknown, a question mark goes into the box of the r-column and what's known (whole and part) goes into the respective boxes of the b-column and a-column.
If a (the part) or b (the whole) is unknown, the procedure is repeated accordingly. This works amazingly well and the visualization once again helps to minimize confusion.
An added advantage is that the students learn how to solve for the various variables and how to manipulate the rb=a formula, e.g. r=a/b, b=a/r.
It is amazing, though, how many of these 8th graders find this manipulation of variables difficult. Adults might consider this manipulation child's play. But it really represents a huge jump to this age group to go from the concreteness of numbers to the abstractness of variables (letters).
This concludes the lesson on how to teach percents for understanding. Instructivist will give other lessons on how to teach math successfully to middle graders as he sees fit.