Chapter 8

Test-Taking Strategies for Core Math

IN THIS CHAPTER

Bullet Saving time with shortcuts

Bullet Whittling down the answer choices

Bullet Coming up with the correct answer on your own

Mathematical knowledge and practice are very important for taking the Praxis Core exam’s math section, but the arsenal you take into battle doesn’t have to stop there. Test-taking strategies can make your weapons even more effective. What you know is extremely important, but so is what you do with what you know. This chapter is about making the best of the knowledge you have.

If you decide to use only one bit of advice on test-taking strategies, let it be this: Never stay on a question that looks like it will take an extreme amount of time to answer when you first get to it. You can save it for after you have answered the questions that do not take long to answer. This advice applies to any type of question. You can write down the number of the question in the hope of going back to it later. Some people get so caught up in a challenge to answer a time-consuming question that they use up time they could have used to answer four or five other questions.

All the Praxis Core exam math questions are worth the same number of points, so make sure you don’t treat any one of them as if it’s more important than the others. Whether you’re asked to solve a simple equation or find the volume of a remaining region of a cone, the question is worth as much as the other 55. Timing is part of the challenge, so use it well.

Using Helpful Shortcuts

Getting answers that are correct on the Praxis Core exam is really important. However, coming up with the right answers isn’t your only concern. You must consider something else of major relevance: The exam is timed. You have 90 minutes to answer 56 problems. Any methods you can use to reduce the time you take to answer questions without sacrificing accuracy will help you. The methods we outline in the following sections are real, and they work. We encourage you to give them a trial run as you take the practice tests in Part 5 and online.

Solving equations versus determining what must be solved

You will almost definitely have word problems that require setting up and solving algebraic equations when you take the Praxis Core exam. Solving these problems requires more than the ability to solve equations. You also need to determine what information the problem is asking you for, that is, what needs to be solved. You can use a variable to represent the unknown quantity that must be determined.

If the problem has more than one unknown, see whether you can represent the others in terms of the first. For example, if Bob’s score in bowling is in question and the problem says that Frank scored 10 points higher than Bob, you can use x to represent Bob’s score and to represent Frank’s score.

The next step is to put the algebraic expressions into an equation and solve the equation. The solution to the equation will be the value of the first unknown. Once you know it, you can find the values of any other unknowns by putting the value of the variable for which you just solved into the expressions used to represent the values you have not yet determined.

Example John is four years older than Adam. John was twice Adam’s age 33 years ago. What is John’s age now?

(A) 42

(B) 8

(D) 41

(E) 37

The correct answer is Choice (D). The question asks for John’s age, so you can make John’s age the first unknown and represent it with a variable, such as x. Because John is four years older than Adam, you can represent Adam’s age with the expression x – 4. The next step is to write an equation with the two expressions. Just translate the information in the word problem into mathematical language. John was twice Adam’s age 33 years ago. Their ages 33 years ago can be represented by subtracting 33 from their current ages:

John:
Adam:

“Twice Adam’s age 33 years ago” can be represented by 2 multiplied by 33 less than Adam’s current age.

Adam:

Then set up an equation that shows their ages 33 years ago with John’s being twice as much as Adam’s:

Notice the direct translation of worded information in that equation. It says exactly the same thing in a different way. The solution to the equation is 41. (For a review of how to solve algebraic equations, turn to Chapter 5.) John’s age is therefore 41.

Be extremely careful in making sure you know precisely what question you are being asked. It’s not always what you may assume it is. In the preceding example, for instance, you could have been asked Adam’s current age or even the sum of John’s and Adam’s ages. Don’t assume that the first unknown is the answer to the question. The need to use caution in identifying the exact question you’re being asked is often even greater on geometry questions.

Using estimation and approximation

Estimation is the act of using known information to get a number that is close to the answer to a question although it may not be the actual answer. This can be helpful on the Praxis Core exam because most of the test is multiple choice. When you estimate, you have a good chance of getting the answer right by choosing the one that’s closest to your estimate. Because the test is timed, estimation may be handy for a few last answers before time’s up.

Estimation can also help you to eliminate choices that are definitely wrong, and it can help you make sure the answer you get is not something beyond the fringe of reason or the result of a miscalculation. However, don’t spend too much time estimating for just that purpose. You’ll be performing a juggling act with time and caution.

Approximation is using numbers that are very close to numbers given in a problem for the purpose of making calculating easier. This, too, can save you time when time is running short, and it can also help you eliminate wrong answers. If you have to make a wild guess, your chances of guessing the correct choice increase when you can eliminate some of the choices.

When you multiply with irrational numbers in decimal form, you often have to approximate. An irrational number is a number that can’t be represented as a ratio of two integers. An irrational number in decimal form doesn’t terminate or repeat. In other words, the digits after the decimal don’t have a pattern and continue infinitely. Because you don’t have time to calculate with an infinite number of digits, approximation can often help.

For example, the number 7.32960584965105… is an irrational number. You may come across a number like that when you use your calculator. Rounding such a number to four decimal places should be sufficient to get a correct result in almost all cases. Rounding also saves time, and time is precious on the Praxis Core exam. In some cases, you can just leave a number in your calculator and do the necessary operation with it.

Using the calculator

An on-screen calculator is provided for you on the math section of the Praxis Core exam. That makes computation easier and saves time over doing calculations on paper. The calculator’s keypad includes keys for all the digits, the four basic operations (adding, subtracting, multiplying, and dividing), square roots, making numbers negative, a decimal, parentheses, memory, clearing, transfer display (which can transfer your answers to the answer box on constructed response questions), and, of course, the highly important equal sign. The calculator is also programmed to follow the order of operations. However, you still need to know the order of operations for solving variable equations.

Identifying calculations you can make in your head

We highly encourage caution when making calculations for the Praxis Core exam. That involves writing out work on scratch paper and using the calculator, but only when you actually have time to do those things. You should write and use the calculator even for the calculations you’re sure you can do in your head when you’re keeping a good pace, though we don’t recommend using the calculator for calculations that can come from pure memory.

However, when time becomes a major issue, you need to know what more complex calculations you can do in your head relatively safely. The first category to consider is problems for which writing and using the calculator don’t actually help you figure out the answer. For these calculations, the paper and calculator will only help you avoid careless errors. Suppose you have to calculate 2 + 2. We recommend using paper even for that when you’re good on time. Using paper allows you to write out every step of the calculations, better ensuring that you avoid careless errors. When you’re way behind on time, you should speed up by making such a calculation only in your head. Timing is important, but so is caution.

Make sure you can add and subtract all combinations of two single-digit numbers purely by memory. This will help you add numbers with more than one digit in your head when necessary. For example, if you know that , you can quickly conclude that . Also, make very sure that you know all the multiplication tables for 2 through 12.

While we believe in writing every step involved in answering a math question, we don’t think the calculator should be used for every single calculation. The more you have in your brain’s memory, the faster your pace will be.

The art of guessing as a last resort

If time really wears thin on the exam and you only have time to make wild guesses, go ahead and make nothing but wild guesses. A point may come where you don’t even have time to estimate, approximate, or do quick calculations in your head. That’s when you need to just start hitting answers as fast as you can.

If you make all of your choices randomly, you have some probability of getting all of them right. However, the flip side to that coin is that you have equal probability of getting all of them wrong. Still, you’ll most likely get some right if you answer enough of them that way.

We recommend choosing the same option every time. The Praxis Core exam tends to distribute answer choices close to evenly. Because of this, if you pick the third choice every time, you’ll almost surely get an answer right if you pick it enough times.

In the examples used in this book, the choices are lettered. That’s only for the purpose of being able to easily identify the choices when discussing them. On the actual Praxis exam, the choices are not lettered. They have elliptical figures beside them. With that being the case, you should pick a certain number in the order of choices and stick with it when you’re in wild guess mode. You could pick the first choice, second choice, third choice, and so on and stick with it every time. However, if you get a quick enough peek to have any reason at all to think any choice is even slightly better than the others, you should choose that one.

A small number of questions tell you that more than one answer choice may apply. If you notice one of those during a pure guessing phase, you should mark at least two answers of the choices presented.

A few of the questions don’t have choices presented with them. You should answer those last if you’re in a guessing phase. When you get around to guessing on them, see how much time you have left. Quickly look at the problem and give the best guess you can give. If you still have a few more seconds to actually think about the answer, do all of the thinking you have time to do and then give the best answer you can.

Working backward

When you’re in doubt about how to answer a math problem by using the conventional calculation methods, you can increase the probability of getting the correct answer by working backward. This approach can be used in various situations and in different ways. For example, if an end result is given and you’re asked about what was involved in getting the result, you can start at the end and work toward the beginning. This technique can be used for age problems, problems concerning final amounts of money, final locations, and other types of problems. The technique is sometimes the only one that will work.

Example Robert went for a walk on a trail that runs north and south after getting a ride on a four-wheeler. At the end of his walk, he was 14 miles north of his campsite. He had previously walked 10 miles south, and before that he walked 20 miles north. How far north of his campsite was Robert when he began his walk?

(A) 24 miles

(B) 0 miles

(D) 44 miles

(E) 6 miles

The correct answer is Choice (C). You can answer this problem by using positive and negative numbers and adding them, with north distances being positive and south distances being negative. However, you can also work backward from the final distance and trace Robert’s travel back to the beginning of the hike. If Robert was 14 miles north and had previously walked 10 miles south, he was 24 miles from his campsite before the final part of the walk. If he got to that point by walking 20 miles north, he was 20 miles back, which is 4 miles from the campsite. Choice (A) is the distance Robert was from his campsite before changing directions, but not the initial distance. Choice (D) results from adding 20 to 24 instead of subtracting it.

Narrowing Down Answer Choices

You improve your chances of choosing the right answer — and choosing it quickly — by eliminating choices that you know are wrong. This narrows the set of choices to consider and helps lead you to the right answer. We discuss this earlier in this chapter to make a case for increasing the probability of getting the right answer, but another advantage is that it helps lead you to better considerations.

How to eliminate obviously wrong answer choices

If a choice is so outrageous that it couldn’t possibly be the correct answer, you should eliminate it from consideration. For example, the length of a rectangle can never be greater than the rectangle’s perimeter. A person’s age years ago can’t be greater than his or her current age. The mean of a set of data can never be greater than the highest number. These are just some examples of impossibilities you can readily notice.

Example If math , what is the value of j?

(A) 270

(B) 30

(D) 120

(E) 3

The correct answer is Choice (B). You can find the answer by solving the equation.

To help ensure you get the correct answer, you can eliminate Choices (A) and (D) because both of those choices are greater than 90. You have to multiply 3 by a positive number to get 90, and no positive number is greater than 3 times itself. Eliminating those choices leaves you with three choices to consider instead of five. Choices (C) and (E) are randomly incorrect.

Geometry of a circle with a radius of 12 meters to find the circumference of the circle.

Example The radius of the preceding circle is 12 meters. What is the circumference of the circle?

(A) 24π meters

(B) 12π meters

(D) 144π meters

(E) 6π meters

The correct answer is Choice (A). The formula for the circumference of a circle is C = 2πr. Substituting 12 m for r, you get 2π(12 m) = 24π m. You can eliminate Choice (C) right away because neither π nor a decimal number, which would result from multiplying 12 by a decimal approximation of π, is in the answer. Another reason Choice (C) can be eliminated is because the radius of a circle can’t possibly equal the circle’s circumference. Multiplying by 2π can never be the same as multiplying by 1; 2π is not equal to 1, and it never will be, as far as we know. Choice (B) can be eliminated because multiplying by π is not the same as multiplying by 2π. You can also eliminate Choice (E) because the number that precedes π in a representation of a circumference can’t be less than its radius unless different units are used. Multiplying a positive number by 2π has to result in an increase. The three eliminations leave you with Choices (A) and (D), and that puts you closer to the right answer. Choice (D) is incorrect because it is the area of the circle, not the circumference, except with m instead of m². If the unit were m², you could eliminate it right away because circumference is not properly expressed in square units.

Avoiding the most common wrong answers

Other choices to watch out for and be ready to eliminate are the ones that result from the most common calculation mistakes. Memorizing the rules for all the math topics you’ll face on the Praxis is important for avoiding these common mistakes.

For geometry questions, some of the most common mistakes involve getting formulas confused, such as area and circumference. If you have a question about the area or circumference of a circle, you may have an answer choice that gives you the number involved in a measure about which you’re not being asked. Be ready to avoid such answers.

Also remember the difference between the interior angle sums of triangles and quadrilaterals. Choices may be the result of using the wrong number. Confusion between supplementary and complementary angles is common, and so is confusion between formulas for surface area and volume. Forgetting the formulas for the volume of pyramids and cones also happens often. (Take a spin through Chapter 6 if you need a refresher on any of these geometry topics.)

As for algebra and number and quantity questions, losing track of the fact that a number is negative is one of the most common mistakes. The distributive property is very frequently used improperly. Remember that the term right before the parentheses is supposed to be multiplied by every term in the parentheses, not just the first one. math is equal to math , not math .

The rules for switching between decimals and percents can be easily confused. Keep in mind that moving a decimal two places to the right and writing it as a percent is equivalent to multiplying by 100. Taking a percent and moving the decimal two places to the left is the same as dividing by 100 because a percent is “out of one hundred.”

For statistics and probability questions, be careful about confusing mean, median, and mode. People often forget that the median of a set of data can be found only when the numbers are in order. When using scientific notation, be careful about the direction in which you move decimals. Numbers resulting from wrong decimal directions may be in the choices.

Example Which of the following has the same value as 35.937 percent?

(A) 3,593.7

(B) 3.5937

(D) 35,937

(E) 0.035937

The correct answer is Choice (C). To convert a percent to a decimal number, drop the percent symbol, which can be % or the word “percent,” and move the decimal two places to the left. Choice (A) results from dropping the percent symbol and moving the decimal two places to the right. The other choices result from moving the decimal points something other than two places to the right or left. The main lesson here is that Choice (A) can be reached through a very common mistake in converting percents to decimals. You want to avoid such wrong choices. They’re lurking, so be ready.