EZ Arithmetic: Beyond the Basics

Arithmetic doesn't get a lot of respect these days. On the one hand, calculators have turned computation into something approaching a forgotten art. Learning traditional methods of computation can provide a foundation on which we can build more advanced mathematical skills and understanding--but such methods no longer have great importance for their own sake.

On the other hand, the more advanced use of arithmetic in problem solving has been largely neglected because of our desire to teach students algebra as early as possible. I once heard an ad for a tutoring service. A boy was struggling with algebra--and they discovered that his real problem was with percents. At first this seemed ridiculous to me. There aren't a lot of algebra problems that deal with percents. How could that be the root of the boy's problems?

I then realized that experience with percents could be of great value. Percentage problems can give students a background using reasoning that is similar to what is used in algebra. The way we transform problems working with percentages and the way we transform problems in algebra are very much alike. But percent problems give students a chance to learn such reasoning in a less abstract setting, and most students learn better when things are less abstract. Once students become familar with using such methods in a less abstract setting, it becomes much easier to understand what we are doing in the more abstract setting of algebra. Students will at least have a sense that they have seen something like this before--and that can make things a lot easier for them.

But the methods we use with percentage problems are only the last remaining vestige of the arithmetic methods that were once used to solve "algebra" problems. False position was widely used.

A number and a seventh the number add up to 17. What is the number? If the number were 7, adding a seventh would give us 8. In fact, we want our answer to be 17 which is 17/8 times as large. This means our number should be 17/8 times our first guess of 7. Our answer is 7 times 17/8 = 119/8 or 14 and 7/8. This problem and its solution were found in the Rhind Papyrus dating from about 1650 BC. Our first guess of 7 is the false position--a preliminary test solution which we will later correct to get our answer. No effort is made to make a realistic guess our test solution, but it is nice if we can pick a number that is easy to work with. In this case picking 7 makes it easy to add a seventh, and thus simplifies our arithmetic.

False position is a technique we can use with mental math. The method we have illustrated only works with problems that turn out to be proportions. But a modified version, double false position, will work with other linear problems.

Small cookies cost ten cents, and large ones cost thirty cents. If I paid 2 dollars for a dozen cookies, how many of each type did I buy? If we buy a dozen small cookies it costs $1.20 or 120 cents. In fact I spent 80 cents more.

Each large cookie costs 20 cents more than a small cookie. Since I paid an extra 80 cents, I must have bought 4 large cookies. 4 large cookies and 8 small cookies do cost 120 cents plus 80 cents or 200 cents as required.

Methods like false position extend what we can do with arithmetic. Flexible use of math at one level forms an important foundation for building math at the next level. Think in terms of gear shifts. When we shift gears, there is a range of speeds where we can make the shift. In a similar way there is a range of problem difficulty where we can make the shift from arithmetic to algebra. The best way to make the shift is for students to be able to see how a range of problems could be done using either approach. We start by learning the older methods using arithmetic.