Math Study Guide for the PSAT/NMSQT Exam

Specific Skills to Practice

Before taking this test, you should practice such skills as graphing, simplifying, combining like terms, and solving equations.

Graphing Linear Equations

Linear equations can always be graphed by plotting two points and connecting them with a straight line that passes through both. However, depending on the form in which the linear equation is given, it may be easier to note the information given and use that to construct the line.

For example, if the slope-intercept form is found, the \(y\)-intercept can immediately be graphed, and the slope can be used to quickly find another point:

For \(y=3x+2\), the line passes through the point \((0,2)\) and has a slope of \(3\), meaning the next point is \((0+1,2+3)=(1,5)\).

Manipulating Units of Measurement

Recall that dimensional analysis enables the conversion of one unit of measurement to another by multiplying the original value by a ratio called the conversion factor.

For example, consider the conversion of \(15\) meters per second to kilometers per minute.

\(1\) kilometer = \(1\text{,}000\) meters or \(\frac{\text {1 } km}{1000 \; m}\)

\(1\) minute = \(60\) seconds or \(\frac{1 \text { min}}{60 \text { s}}\) which is the same as \(\frac{\text {60 }s}{1 \text { min}}\)

Using these conversion units, we multiply them by the quantity we want to convert:

\[15\frac{m}{s} \cdot \frac{\text {1 } km}{1000 \text{ m}} \cdot \frac{\text {60 }s}{1 \text { min}} = \frac{15 \cdot 60 \text { km}}{1000 \text { min}} = 0.9 \frac{km}{min}\]

Notice that, in the multiplication, the seconds cancel out from the numerator and denominator leaving minutes, and that meters cancel from the numerator and denominator leaving kilometers. Always verify that the resulting units match those required by the question.

Solving Linear Equations

To completely solve a linear equation, a system of equations must be provided. In the case when only one linear equation is provided, the equation can only be solved in terms of one of the variables. Solving a linear equation for y enables the equation to be graphed immediately because both the slope and y-intercept are known.

Simplifying Expressions

Expressions containing solely numbers and operations can be simplified by following the order of operations. The order of operations proceeds as follows:

Operations inside of parentheses are performed first.
Exponents are evaluated next.
Multiplication and division from left to right are performed next.
Addition and subtraction from left to right are performed last.

If you take the bolded letters above in order, it spells “PEMDAS”. And easy way to remember this is to use the phrase, “Please excuse my dear Aunt Sally.”

Expressions involving variables can be simplified by combining like terms, and factoring where possible. In order for terms to be combined, they must be of the same type. For example, a term with a single \(x\) cannot be combined with a term with an \(x^2\).

Working with Fractions

Recall that fractions represent the ratio of two values. The relationship between a part to a part, or a part to a whole can be represented with a fraction.

It is important to remember the rules governing the combination of fractions. The addition and subtraction of fractions require a common denominator. Multiplication of fractions entails the product of the numerators over the product of the denominators. Division of fraction is the same as multiplication by the reciprocal.

It is imperative that all of these rules are understood and can be comfortably used.

Extra Tips

Know How to “Grid In” an Answer

About eight of the questions in the two math sections combined will ask you to “grid in” your answer. Be sure you understand how this is done by referring to this College Board SAT publication about the Math Test.

Consider especially how to: grid mixed numbers and repeating decimals, what to do if an answer does not fit on the grid, and what to do about fractions on the grid.

The grid only has 4 spaces, so if your answer requires more space, it is incorrect and you need to refigure the problem. Also, you cannot put a 0 in the leftmost space on the grid, so something like 0.47 will need to be changed to .47(.) Reducing fractions is not necessary unless they won’t fit on the grid.

Know Your Calculator and Use It Appropriately

Although a calculator is allowed on certain tests, use of the calculator is not always necessary. In some cases, relying upon a calculator can require too much time. Rather than relying upon a calculator, it is highly advised that a strong number sense is developed so that estimations and calculations can be quickly performed mentally, and if necessary, the calculator can provide verification.

Of course, when performing tedious calculations, as in the multiplication or division of decimals, a calculator is an excellent tool to use. Additionally, as time passes, technology and the ability to use it appropriately are becoming increasingly important. And while it is not always the most fun thing to do, learning how to use a graphing calculator by reading the instruction manual (or some other method), can prove incredibly worthwhile, not only as a practical skill but also as a marketable attribute.

Finally, be sure you take, and are comfortable using, one of the approved calculators for this test. See more information here.

Quickly Write Out Each Step

Much like a calculator should not be relied upon until the foundational skills are established (so as not to develop a reliance upon a crutch), the foundational skills of problem-solving should be developed before attempting to skip or combine steps.

Problem solving begins with reading comprehension. Understanding what information is provided, what relevant equations should be used, and what information the problem is seeking, are all crucial steps in solving a problem. These steps can be further complemented with diagrams or drawings that help to visualize the problem and clarify the steps that should be taken to arrive at the solution.

Check Answers to Make Sure They Make Sense

There are two elements to this strategy. Due to the nature of algebra, we are often solving an equation for a particular value or values. Once a value is calculated, its correctness can be verified by plugging the solved value or values back into the original equations. If upon substitution, the value does not yield a true statement, then it is clear that a mistake has been made, most likely in the steps taken to arrive at the answer.

The validity of an answer can also be tested by way of the size and units of the answer found. A speed of 10 million is obviously incorrect, just as a speed of 6 meters is incorrect. Always assess a final answer to decide if it is in line with the nature of the problem.