The Power of One in Math

The number 1 is such a simple number:

·         It’s the first number when you count positive integers.

·         It has no effect when multiplying or dividing by 1.

·         It also has no effect as an exponent.

·         Adding or subtracting 1 simply yields the neighboring integer.

Yet the number 1 can be very powerful in math.

One way to convert units is to multiply by 1. For example, to convert 4 ft. to inches, you could multiply by 1, writing 1 as a fraction: 12 in. / 1 ft. equals 1. Multiplying 4 ft. by 12 in. / 1 ft. yields 48 in. (Observe that the feet cancel out.)

Fractions can be added by finding a common denominator, and the denominators are made common by multiplying by 1. For example, to add 1/4 to 1/3, the least common denominator is 12. Multiply 1/4 by 3/3 (which is 1) to make 3/12 and multiply 1/3 by 4/4 (also 1) to make 4/12. Now add these together to get 7/12.

The number 1 is used to turn numbers into their reciprocals. For example, the reciprocal of 5 is 1/5. The reciprocal of x is 1/x. Any number times its reciprocal equals 1. For example, 2 times 1/2 makes 1.

We use this principle all the time in algebra. For example, consider 4 x = 12. In this case, isolate the unknown, x, by multiplying by the reciprocal of its coefficient. The coefficient of x is 4; the reciprocal of 4 is 1/4. Multiply both sides by 1/4 to make x = 12/4 = 3. (Note that dividing both sides by 4 is equivalent to multiplying both sides by 1/4.)

The equation x + y⁰ = 2 wouldn’t have a unique solution if not for the fact that y⁰ = 1. (This follows from the rule that y^a/y^b = y^a–b. Let b = a. Then y^a/y^a = 1 = y^a–a = y⁰.) Knowing this, the original equation becomes x + 1 = 2, or x = 1.

In trigonometry, the number 1 often comes in handy through trig identities. For example, the equation 1 – sin²x = 0.5 cos x would be difficult to solve because it includes two different trig functions, except that 1 can be written as sin²x + cos²x, which reduces this equation to cos²x = 0.5 cos x. We can then factor it as cos x (cos x – 0.5) = 0, which means that cos x = 0 or cos x = 0.5, for which the solutions are 90, 60, or 300 degrees.

In probability, the number 1 represents 100%. If you reach into a bag with 3 red marbles and 5 blue marbles, you have a 3/8 probability of selecting red and 5/8 probability of selecting blue. If you add these together, you get 3/8 + 5/8 = 8/8 = 1. That’s because there is 100% that you’ll select a marble of some color.

There are many derivations and proofs in higher levels of math that become eloquent when a term is multiplied by 1 and the 1 is written in terms of an identity (such as the summation expression for the completeness theorem in group theory – which itself is another example of the power of one, in establishing unity for a complete set).

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

Issues with Multiple Choice Tests in Algebra

Suppose that an algebra problem asks a student to solve an equation for an unknown. The goal of the problem is to test whether or not the student knows how to apply a particular algebraic technique.

However, if the question is multiple choice, the student doesn’t actually need to know how to solve the equation in order to determine the correct answer. The student could simply plug each answer into the equation to see which one works.

Example: 3x + 20 = 12x + 2.

(A) x = 1 (B) x = 2 (C) x = 3 (D) x = 4 (E) x = 5

A student could simply try each answer. Plugging in x = 1, it’s easy to see that 23 doesn’t equal 14. Trying x = 2, 26 = 26. The student already has the answer, but hasn’t done any algebra.

If the correct answer is (E), or if the student makes a mistake in the calculation, it will take longer, but in principle, a student can get the correct answer without doing any algebra.

If there are fractions in the answers, that makes the calculation a little more cumbersome. Many students prefer to avoid fractions, so this may deter students from avoiding the algebra.

Suppose that the choices had been:

(A) x = 1/2 (B) x = 18/15 (C) x = 22/15 (D) x = 2 (E) x = 22/9

Some problems can be solved faster by actually doing the algebra. But more time-consuming problems, like using the quadratic or solving a system of equations, might be solved faster by just plugging the answers into the original equations. When algebra yields the fastest solution, this provides an incentive to solve the problem as intended.

If a student relies on plug and chug for every problem, the student risks not finishing the test. But students probably won’t try to solve every problem this way. Just the hard ones.

Fortunately, there are many problems that tests can ask that can’t be solved this way. For example, many problems ask students to simplify an expression, and there are also conceptual, strategic, and logical questions. This helps to ensure that students must grasp some algebra concepts in order to pass the class.

The main goal of an algebra course is for students to become fluent in solving for unknowns in a variety of types of equations. Multiple choice is convenient, especially in large classes. It’s worth considering whether or not students might find ways to succeed in the course without actually mastering the techniques.

One way is for the instructor to become acquainted with the students, perhaps by checking written solutions of homework or quizzes (or adding a few written problems to exams, where feasible), helping students solve problems on the board, or interacting with students during office hours. If students who ordinarily struggle solving equations are better able to figure out the right answers on multiple choice problems, this might be a signal (of course, this could also be the result of studying, tutoring, and improvement). If instead experience with their written solutions corresponds well with their ability to solve similar problems on multiple choice exams, then there may not be any reason to worry that the students who most need to improve their fluency might be finding an easy way out (which may very well be the case for many of the students – it might be the clever problem-solvers who are most likely to think of this).

Chris McMullen, author of the Improve Your Math Fluency series of workbooks