Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)

FMMs (FREQUENTLY MADE MISTAKES)

This section of the book is one like no other. I bet you will not find one like this in a traditional textbook (at least I haven’t yet). This section is strictly dedicated to highlighting the most common and “frequently made mistakes” by students. By honing in on these common mistakes, I hope you will be able to quickly recognize and avoid them.

This section is also different than the other sections in this book in the way it is set up. If it’s a topic that hasn’t been covered in the book yet, I will give it its own new section. If it’s a mistake I’ve already explained in a previous section, I will list it with a brief introduction and provide the hyperlink to direct you back to that respective section of the book.

The Two Meanings of “Cancelling Out”

“Cancelling out” can mean two different things:

1.  Cancelling out to zero, or

2. Cancelling out to one.

Students and instructors often just say “cancelling out,” which is a bit ambiguous and can cause confusing. You must be able to properly differentiate which context is being used and when each is happening.

Terms are cancelled to zero when opposite terms are added (meaning adding and subtracting the same term. This is often seen when you are adding or subtracting the same term to each side of the equation in order to move a term to the opposite side. It is also seen during “combine like-terms,” when terms happen to be opposites of each other.

Terms cancel out to 1 when:

·   A number or term is divided by itself, or

·   A fraction is multiplied by its reciprocal.

This is often seen during:

·   Reducing fractions;

·   Multiplying fractions;

·   The final step of solving a simple algebraic equation of one variable, where you divide both sides by the coefficient in front of the variable you’re solving for; and

·   Factoring a GCF out of a series of terms.

Students usually don’t have problems remembering that adding opposites cancels them to zero. But sometimes the mistake is when students think canceling always results to zero. You must not forget that when a term is divided by itself, it equals “1,” as shown in Property Crises of Zeros, Ones and Negatives. This typically involves fractions (either during reducing individual fractions or multiplication or division of fractions).

  But the most common time it is forgotten is when you find a GCF in a series of terms and then factor that GCF out (by dividing each term by it), leaving the GCF out front, multiplied by the parentheses containing the remaining factors of each term.

For example, in the following expression:

18x³ – 6x² + 3x

the GCF is 3x. To simplify this, you would factor 3x out of each term. The intermediate step (which you wouldn’t always show) shows each term divided by 3x:

Students often incorrectly answer this as:  

.

When asked about it, they will respond that “three x over three x cancels out,” which is true, but it cancels to “1,” not zero, so the 1 must be shown, as in the correct answer shown here:

While on the subject of “cancelling out,” this plays a role in multiplication and division of fractions by means of “cross cancelling.” Another common mistake or area of confusion is when students mix up cross cancelling with cross multiplication. This is explained in: Cross Multiplying vs. Cross Cancelling. Follow this link to see the proper way to cross multiply.

Checking Your Answers

The last step of any problem-solving procedure is to check your answer. Specifically, that means taking the value of the unknown you determined, substituting it back into the original equation, simplifying, and reviewing the outcome. If, after simplifying, the left side equals the right side, this affirms that your answer is correct. However, if it doesn’t, there are three possible reasons why.

1.  You made an error doing the problem.

2. You’ve identified an extraneous solution.

3. You made an error in your math in the check step.

One clear reality is that students either forget, or just hate doing the check step. If I had to guess, it’s because it costs extra time and workspace to do, and students just want to be done with a problem, especially problems that are long to begin with. But to attain complete answers, you’re expected to check your answers, and there are times when checking will help you discover an error or anomaly. This could save you valuable points on a test.

First, it will simply draw your attention to an error you made during problem solving. If you can’t find your mistake by reviewing your work, consider starting it over without looking at the last way you did it. Consequently, you can also make an error during the check steps which may lead you to think you made an error in the original steps, but didn’t. Either way, your answers should check out.

There’s also another major reason an answer might not check out, and that is due to an extraneous solution. Extraneous solutions often occur when a variable is in the denominator in an equation (or inside a radical).

Miscellaneous Mistakes

· When multiplying factors of a common base with exponents, sometimes students mistakenly multiply the exponents. When factors of a common base are multiplied, their exponents are added.

· Sometimes students are required to distribute an exponent through a term of multiple (variable) bases with exponents. This is taking the power of each base to the power being distributed. There is often a coefficient attached to the variables, and when there is, students often forget to apply the power (from outside the parentheses) to the coefficient. The reason might be because students are used to taking the power of the power of each variable base, and they just forget about the coefficient. When distributing an exponent through a group of bases with a coefficient, don’t forget to apply the exponent to the coefficient.

· When given an equation with a trinomial, or a quadratic equation, sometimes students will successfully factor it, but then forget to do the last step, which is to solve.

· Students commonly make the mistake of using “zero” and “no slope” or “undefined” interchangeably, but they have completely different meanings.

· Click the link for common mistakes students make during the Substitution Method.

· Click the link for common mistakes students make during the Addition/Elimination Method for solving a system of two linear equations.

· Students often make a mistake when a negative sign is in front of a fraction by not properly distributing the negative sign through, changing the sign of each term in the series.

· Equations and expressions are intended to be simplified completely. Often times, students do most of the problem correctly, but make one of two vital mistakes that could make or break an answer (especially when instructors don’t give partial credit). Sometimes students get near the end, but simply forget to simplify the answer. Or, sometimes students attempt to simplify, but do it wrong. Learn to avoid: The Wrong Way to Simplify a Rational Expression.

· When students use the Quadratic Formula, they often forget to simplify the last step. This is explained in: The Part Everyone Forgets: The Last Step of the Quadratic Equation

· When applying the “special case” shortcut method to multiplying out a binomial squared, students often make the common mistake of using the shortcut method for multiplying conjugate pair binomials. This mistake results in the missing “bx” term.

· When a negative sign is in front of a rational expression (a fraction with a polynomial in the numerator) students very often forget to distribute that negative sign through all terms in the numerator. This then incorrectly associates the negative to only the first term in the numerator, leaving the terms to follow with opposite signs than what they should be.

· Radicals Are Not Long Division. There’s not really much to say about this other than that the symbols and set up of radicals and long division are similar looking, but they are completely different operations. Anytime I’ve ever encountered a student attempting to apply long division to a radical may have been their desperate attempt to do something when they had no idea how to approach radicals (most likely due to lack of preparation). Long division is a process to find out how many times the divisor goes into the dividend, and the answer is the quotient. But radicals are used to answer: What number, which when taken to the power shown as the root, equals the radicand? The radicand won’t always be a perfect power number, and in that case, assuming you don’t use a calculator, you will break it down and simplify it using the rule of multiplication of roots, as briefly shown in Common Radical Fingerprints.

Scientific Notation on Your Calculator

Scientific notation is a standardized way of reporting numbers that are either very big or very small, with many zeros and/or decimal places. It is a way to express numbers into a manageable format, and is often used in science and statistics. Scientific notation is the alternate way of writing a number from its expanded form.

  Although I do not cover scientific notation in this book, I want to address the mistake students frequently make when putting scientific notation into a calculator. The mistake is some variation of not knowing how to properly put it into the calculator.

Since there are generally two types of calculators (scientific and graphing) with the scientific notation function, each type and brand varies in what buttons they have to accomplish this function, so it’s a good idea to be prepared for each possibility. There is also a completely wrong way to input scientific notation, which results in the number being off by an order of magnitude (a factor of 10, or in other words, off by one zero).

  Typically, on all calculators, you start off the same, by typing in the base number. Next, you must hit the exponent button, but not the same exponent button you would use for normal exponents. The button you want may look like any of the following:

[EE]  

[exp]  

[EXP]  

[x 10]  (meaning “times ten to the …”)

[10^x] (meaning “meaning base ten to the power of”)

[anti LOG] (often a 2^nd function to [LOG])

[e] (not to be confused with [e^x], which stands for “the number e to the x,” also known as “inverse LN,” which is “inverse natural log”).

If one of the functions shown above is a 2^nd function, meaning the symbol is shown in another color, above a primary button, you must hit or hold a button such as:

“2^nd,” “Shift,” or “Alt,”

often located at the top left corner of the calculator, then hit the button as it is shown (from the choices listed above). You might have to look around for it; it doesn’t always jump out at you at first.

To reiterate, you would first type your base number, then scientific notation button (shown above), then the exponent (of the 10).

Here is the place students often make a mistake… by manually typing out:

[the base #] [x] [10] [EXP] [the exponent]... In words, that would say, “base number times ten, times ten to the power of some number.” In other words, this causes a redundant multiple of 10, which will result in your number being off by a factor of ten. To prevent this, you must use either one or the other (either the [EXP, then the exponent], or [x 10 ^ the exponent]).

Consider the example of converting 9,400,000 to scientific notation, which would be

9.4 x 10⁶. You would type

Either: [9.4][x][10][^][6]

In words: Nine point four times ten to the sixth, using the exponent feature, not the scientific notation feature.

Or: [9.4][EXP][6]

This is the preferred way to input numbers in scientific notation. In words, this reads the same as above (“Nine point four times ten to the sixth”), but the buttons are clearly different. In this version, the scientific notation button is used, not manually typing the ten, the carrot, and the exponent six. I recommend getting use to the [EXP] button.

What Does “Error” on a Calculator Mean?

Often times, students will put an operation into the calculator and get the response: “Error.” Some misinterpret what that means. Sometimes students interpret that as “the student made an error,” but this many not be the case. When the output on the calculator is “Error,” it could mean one of the following things:

1.  “Error” is the correct and expected response. What we might call “Undefined” or “No Solution” to an arithmetic operation, the calculator will report as “Error.” For such examples involving division and radicals, see: Property Crises of Zeros, Ones and Negatives).

2. Sometimes, however, “Error” means you made a mistake in-putting your intended operation. In that case, you should check what you typed and look for an error in that respect. For instance, you may have accidentally typed two decimals in a number. If you check, and don’t find an input mistake, then there is a good chance “error” is the correct response for a reason.