Math Study Guide for the CBEST

General Information

No matter what your teaching specialization is, you will need to have some basic math skills to carry out the duties of your job. Math comes into play at some point in every area of teaching and a certain amount of knowledge in the subject is a must.

Here are the areas in which you will need to be competent in order to score well on this portion of the CBEST. Remember that most of the questions involve reading a word problem and that you are not allowed to use a calculator of any kind during the test.

Mathematics Estimation, Measurement, and Statistical Principles

Estimation and Measurement:

Standard Units

Measurement involves determining the size or dimension of a thing. You may measure length, surface area, volume or capacity, weight, temperature, and time. Measurement may be done using the metric system or the U.S. measurement system.

In the U.S. standard, length is expressed in inches, feet, yards, and miles. Liquid is measured in fluid ounces, cups, pints, quarts, and gallons; mass or weight is measured using ounces, pounds, and tons. Surface area is expressed in square inches, square feet, square yards, and acres. The units for time are seconds, minutes, hours, days, weeks, months, and years, while the unit for measuring temperature is degrees Fahrenheit.

Length and Perimeter

Length is measured when there is a need to determine how long or tall something is, how far apart two things are, or the distance between places or locations. The units inch or foot are used for shorter lengths, such as the length of paper or cloth. A standard ruler measures \(12\) inches or \(1\) foot. For longer lengths, such as distances between cities, a mile is the applicable unit.

Perimeter is the distance around a closed flat shape. To measure the perimeter of a square, simply find the sum of the lengths of its sides. A square with a side of \(5\) inches, for instance, has a perimeter of \(20\) inches \((5 + 5 + 5 + 5)\). The distance around a circle is called circumference \((C)\) and is computed using the formula:

\[C = 2πr\]

where \(r\) is the radius of the circle.

Time

CBEST will have questions involving addition or subtraction of time. The key is to add or subtract seconds, minutes, and hours separately, and to remember that:

\(1\) minute \(= \;60\) seconds \(1\) hour \(= \;60\) minutes

A question may go like this:

“You leave work at \(10\text:08\) a.m. after a \(5\)-and-a-half hour shift. What time did you arrive for work that day?”

This problem requires us to subtract \(5\text:30\) from \(10\text:08\). Do this by subtracting the minutes first, then the hours. However, subtracting \(30\) from \(8\) cannot be done directly, so we borrow \(1\) hour (or \(60\) minutes) from the hour side, and we now have \(68\) minutes instead of \(8\). Thus we have:

Minutes: \(68 – 30 = 38\)
Hours: \(9 – 5 = 4\)

The correct answer is \(4\text:38\) a.m.

Estimation

Estimating involves calculated guessing. If fully developed, it becomes a very useful skill, not only for this test but for everyday application. An intelligently estimated number or answer is close to an exact value and is often useful when you need an answer quickly and there is no calculator available.

Example:

Estimate \(497 \times 60.5\) by multiplying \(500\) by \(60\). That’s easily \(30,000\). The exact answer is \(30,068.5\)―our estimate is not that far from this amount.

This tip comes in handy when estimating. When you round one number up, round the other number down to minimize the discrepancy between the estimated value and the exact value.

Estimating answers is applicable whether you’re adding, subtracting, multiplying, or dividing. It is also useful to double-check answers obtained through computation.

For instance, you just calculated \(550,234 + 24,980\) and came out with an answer of \(552,732\). From a cursory estimate, adding the rounded numbers of \(550,000\) and \(25,000\), you know that the answer is something around \(575,000\). That’s significantly different from \(552,732\)! Checking again, you realize that you left out the \(0\) in \(24,980\) and got an erroneous answer.

This is the same technique used for estimating bills at a restaurant―you actually have a figure in your mind that shouldn’t be too far off from the actual bill. If the discrepancy is significant, it’s a clue that you need to check some items.

Statistical Principles

Operations

Statistical data can be described, summarized, presented, and interpreted using averages, range, minimum and maximum values, ratios, proportions, and percentile scores.

Mean, median, and mode are different ways of showing the average or central value of a set of data. Questions requiring to solve for the “average” mostly refer to the “mean” unless otherwise stated.

The mean of a set of data is computed by adding all the numbers in the set and dividing the resulting sum by how many numbers were added.

The median is the middle number of a set of data sorted from the lowest to the highest. When there are two middle numbers, simply add them and divide by \(2\).

The mode is the number that occurs the most number of times in the set. A set of data is said to be bimodal if it has two numbers occurring most often (equal number of times), and multimodal if there are more than two modes.

Probability

Probability measures the likelihood or the chance of an event occurring. To illustrate, imagine rolling a die. Let’s find out the probability of the die landing on the face with 5 dots:

A die has 6 faces. The possible outcomes when the die is thrown are: 1, 2, 3, 4, 5, and 6. Probability is computed as:

\[\text{Probability} = \frac{\text{number of ways an event can occur}}{\text{total number of outcomes}}\]

The probability that the die will land on the face with \(5\) dots is:

\[\text{Probability} = \frac{1}{6}\]

because there’s only \(1\) face with \(5\) dots, and there are \(6\) possible outcomes.

Test Score Interpretation

A student with a \(70\)th percentile score means that he or she scored as good as or better than \(70\%\) of the class. Percentile is also called percentile rank, the value that shows the percentage of scores at or below the score in this rank.

Stanine score comes from “standard nine” referring to the nine units used in measuring students’ performance in a test.

• Tests scores are scaled from \(1\) to \(9\), with \(5\) as the mean.

• Students’ scores are ranked from lowest to highest. The lowest \(4\%\) belong to stanine \(1\); the next \(7\%\) belong to stanine \(2\); the next \(12\%\) stanine \(3\); the next \(17\%\) stanine \(4\); the next \(20\%\) stanine \(5\); the next \(17\%\) stanine \(6\); the next \(12\%\) stanine \(7\); the next \(7\%\) stanine \(8\); the last \(4\%\) stanine \(9\).