How to Solve SAT® Math Problems
Tricks and Strategies
In the xy-plane, the graph of which of the following equations is a line that is perpendicular to the line shown in the graph above?
|A. y = ─ 3 / x + 6|
|B. y = ─ 2 / x + 1 /|
|C. y = 2 / x + 3|
|D. y = 3 / x + 6|
Hint: Perpendicular lines have slopes that are negative reciprocals of each other.
y ≥ x + 4
4x + 5y ≤ 20
Which of the following has a shaded region that represents the solution set in the xy-plane to the system of inequalities above?
Hint: The graph of a linear inequality of the form y > mx + b or y ≥ mx + b has shading above the boundary line.
The graph of a linear inequality of the form y < mx + b or y ≤ mx + b has shading below the boundary line.
A research study conducted on a group of adult males determined that the formula H = 0.98A + 44.09 can be used to approximate the height H, in millimeters, of an adult male in this group, based on his arm span A, in millimeters. What is the meaning of 0.98 in this context?
|A. The approximate increase in a man's height, in millimeters, for each one-millimeter increase in arm span.|
|B. The approximate increase in a man's height, in millimeters, for each increase of 44.09 millimeters in arm span.|
|C. The approximate increase in a man's arm span, in millimeters, for each one-millimeter increase in height.|
|D. The approximate increase in a man's arm span, in millimeters, for each increase of 44.09 millimeters in height.|
Hint: The linear model is in slope-intercept form y = mx + b, where m is the rate of change in quantity y per quantity x.
Problem Solving and Data Analysis
While all SAT Math problems require problem-solving, this category refers to solving problems that rely on ratios, percentages, and the collection and use of one or two variables. Recognize these questions by their reference to statistical terms like mean, median, and mode; their use of scatter plots and lines of best fit; their description of experiments and data collection or sampling methods; and their inclusion of percent or ratio relationships. Also, though only present on some exams, box-and-whiskers plot questions on the SAT belong to this section.
Note that the Digital SAT continues the pen-and-paper SAT's use of this category of questions in name and relevant content.
This domain accounts for just under a third of all math questions on the pen-and-paper SAT on any given exam. This category accounts for much less on the Digital SAT—only about a sixth of all math questions.
Tricks and strategies to ace SAT Problem Solving and Data Analysis questions
As you prepare for this section, keep track of definitions for technical terms and, when they refer to values or sets of values, how to calculate them. SAT math questions frequently include wrong answers derived from common errors or misconceptions (e.g., if a question asks about a dataset's mean, you can be sure the answer choices will include the median). Once you've mastered the relevant vocabulary, you'll approach this domain much more confidently and answer the question with less distraction.
Another common trap in this section is that the data points in these questions appear randomly rather than in numerical order. When deriving statistical values like the median, numerical order is critical, so reordering any values is always worth taking a few seconds.
Problem Solving and Data Analysis examples
Mr. Staine is one of the marine biologists at Coastal University, which has 1,816 students. He selected 50 of his marine biology students at random and asked each whether they enjoy swimming. Of the 50 students surveyed, 39 responded that they enjoy swimming. Based on the design of the study, which of the following is the largest group of students to whom the results of Mr. Staine's survey can be generalized?
|A. All of Mr. Staine's marine biology students at Coastal University|
|B. All of the marine biology students at Coastal University|
|C. All of the students at Coastal University|
|D. All of the marine biology students in the country|
Hint: The results of a study can be generalized only to a population from which the participants were randomly selected.
Hint: A box plot is a graph of a data distribution and is based on five numbers: the minimum, 1st quartile (Q1), median, 3rd quartile (Q3), and maximum.
Two units of volume used to measure dry goods in the United States are bushels and pecks. A grocer purchased 160 bushels of apples. If 1 peck is equivalent to 0.25 bushels, which of the following best approximates the volume of the apples in pecks?
Hint: To convert from one unit to another, multiply by a conversion factor that cancels the old unit and replaces it with the new unit.
h(x) = (x + 1)(x + 9)
The function h is defined above and the graph of h in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of h ?
|A. −10 < x < −9|
|B. −9 < x < 1|
|C. 1 < x < 9|
|D. 9 < x < 10|
Hint: The vertex of a parabola is the point at which the minimum or maximum of a quadratic function occurs.
Daisy modeled the growth of monarch butterflies during the caterpillar stage by estimating the length of a monarch on each day of the caterpillar stage. She estimated that a monarch measures 1.77 millimeters on the first day of the caterpillar stage, with a 28% daily increase in length until it reaches the pupa stage. Which of the following functions models L(t), the length of a monarch t days after the first day of the caterpillar stage?
|A. L(t) = 1.77t|
|B. L(t) = 1.771.28t|
|C. L(t) = 1.77(0.72)t|
|D. L(t) = 1.77(1.28)t|
Hint: Model the length of a monarch butterfly over time with an exponential equation of the form y = a(b)x, where a is the initial value and b is the growth (b > 1) or decay (b < 1) factor.
For the function h in the table above, values of h(x) are shown in terms of constants a, b, c, and d. If a < b < c < d, which of the following could NOT be the graph of y = h(x) in the xy-plane?
Hint: The table gives values of h(x) for increasing values of x. Identify the corresponding points and use the given relationship between a, b, c, and d to determine which could NOT be the graph of h(x).
Additional Topics in Math
“Additional Topics in Math” suggests something of a catch-all, but the category focuses heavily on geometry, with the remaining questions in the section covering complex numbers. Recognize these questions by their references to, and diagrams or equations of, shapes and angles, as well as by using the imaginary constant i.
The Digital SAT renames this category to “Geometry and Trigonometry, and the content follows suit, retaining the geometric questions but not those related to complex numbers.
On the pen-and-paper SAT, this domain accounts for a tenth of all math questions on any given exam. This domain accounts for a little more on the Digital SAT—about a sixth of all math questions.
Tricks and strategies to ace SAT Additional Topics in Math questions
As Geometry makes up the core of this domain, familiarity with geometric rules and theorems is vital. In particular, be sure you can recognize how angle values relate to each other within sets of parallel lines or triangles, noting which angles must be congruent (equivalent) or supplementary (summing to 180 degrees). Circle theorems comprise a small proportion of this section, so they do not appear on every exam. Still, many students need to familiarize themselves with how to arrive at the size and angles of a given circle, so once you're confident in the other categories, take time to memorize these applications.
Separately, for those taking the pen-and-paper SAT, the difficulty of complex number questions depends almost entirely on your comfort with the constant i. If you study the exponential powers of i and practice FOILing (multiplying two binomials) expressions that use them, you're likely to master these questions.
Additional topics in Math examples
In the figure above, line m is parallel to line n. If y = 75 and z = 50, what is the value of x ?
Hint: When two parallel lines are intersected by a third line (called a transversal), they create pairs of corresponding angles that are equal in measure.
In the figure above, AB = BC. What is the value of x ?
Hint: First label the figure with the given information that AB = BC, which means that triangle ABC is isosceles.
The base angles of an isosceles triangle are equal in measure, so ∠BAC and ∠BCA have equal measures.
Which of the following complex numbers is equivalent to (1 − i) − (4 − 3i) ? (Note: i = √-1 )
|A. −3 − 4i|
|B. −3 + 2i|
|C. 5 − 4i|
|D. 5 + 2i|
Hint: To add or subtract complex numbers, add or subtract their real and imaginary parts separately.
How To Fill the Grid-Ins on SAT Math
Most SAT Math test questions come in the form of multiple-choice questions. However, some questions do not provide answer choices and require you to come up with an answer on your own and fill it in on the answer sheet. These are called Grid-in questions. This type of question provides a grid where you write the answer and fill in bubbles for the corresponding digits or symbols (like decimals or fractions). Here are some points to remember:
- Make sure that when you write an answer, you also bubble it in. Do not bubble more than one circle per column. Providing a written answer or a bubble will not be considered for scoring.
- Remember that decimal answers cannot start from zero. They must be written and bubbled in starting from the decimal itself, such as .77, .56, .08, etc.
- If your decimal answer is larger than four digits, you may round up or fill in an abbreviated version. For example, .777992 may be bubbled in as .778 or .777.
- If a question has more than one correct answer, be sure to only bubble in one answer per question.
- You can provide your answer as a decimal or fraction for some questions, but be sure to convert any mixed numbers to improper fractions. However, you do not need to reduce fractions.
- Grid-in questions will not provide you with a negative sign, so negative answers are not possible for these questions.
Studying for the SAT Math Section can feel overwhelming due to the variety of question types. However, you can tackle these questions with less intimidation by dividing them into the exam's four domains. As you practice SAT math questions, group them into these categories and, when possible, study one domain at a time until you’re comfortable and rarely miss a related question. By arranging your study plan this way, you can gradually master the subject one step at a time as the topics within a domain progressively build upon each other. This approach prevents you from overwhelming yourself by tackling the subject simultaneously.
If you are unfamiliar with the domains’ content, uncomfortable with classifying the questions on your own, or would simply prefer a more organized way to study the Section, UWorld’s SAT Math QBank does this grouping for you and allows you to easily build a test of questions and comprehensive explanations for each domain, one subsection at a time.
Frequently Asked Questions (FAQs)
As previously mentioned, the Heart of Algebra domain constitutes most of the SAT Math Section. Consequently, linear functions, equations, and inequalities are the most prevalent SAT math question types on the exam.
Most SAT Math mistakes, including those deemed “silly mistakes,” often occur due to stress or unfamiliarity with the content. The most effective approach is to practice with similar questions to prevent these common errors. Take the time to identify potential traps in the questions and write out your step-by-step process for each problem. As you become more familiar with a particular question type, you will naturally avoid miscalculating or misreading parts of the question.
Remember that each question carries equal weight in your score, so focus on answering the questions you feel more confident about first. Due to the limited time allocated for each SAT math question (approximately a minute and a half, on average), don’t be afraid to skip questions you don’t know where to begin or feel you’d only be able to answer through a lengthy process. Once you finish the questions you’re more comfortable with, you can return to the challenging questions with less pressure.
If you encounter a challenging question while practicing, be sure to study its explanation and note it for later review (if you’re practicing with UWorld) or discuss it with peers, a tutor, or your math teacher (if you’re studying on your own). When possible, follow up on that study by answering similar questions; in doing so, you’ll apply the lessons you learned from the first explanation and cement those methods into your memory.