This ISEE Upper Level math review walks through every topic the test covers, from order of operations to quadratic functions, with the time limits, formulas, and strategies you need to score in the top stanines. The Upper Level math is built from two distinct sections—Quantitative Reasoning and Mathematics Achievement—and the smartest preppers tailor study time to the topics that matter most.
The ISEE Upper Level tests math in two separate sections that look almost like two different exams. Quantitative Reasoning comes early in the test and rewards logical thinking and shortcuts. Mathematics Achievement comes later and rewards content knowledge—knowing the formula, doing the calculation, and choosing the right answer fast.
| Section | Questions | Time | Per question | Calculator? |
|---|---|---|---|---|
| Quantitative Reasoning | 37 | 35 min | ~57 sec | No |
| Mathematics Achievement | 47 (42 scored + 5 experimental) | 40 min | ~51 sec | No |
Quantitative Reasoning is 37 questions in 35 minutes—about 57 seconds per question. The section blends two question types: word problems (roughly 18–21 per test) that ask you to set up and solve a scenario, and quantitative comparison questions (roughly 14–17 per test) that present two quantities labeled Column A and Column B and ask which is greater.
Average difficulty on QR is about 3.4 out of 5 based on analysis of official ISEE practice tests, and many students consider it the toughest section of the entire ISEE because the question style is unfamiliar and there is no calculator. The good news: the math content is mostly the same as Mathematics Achievement, so studying one section helps the other.
Mathematics Achievement is 47 questions in 40 minutes (51 seconds each), making it the longest section of the test. Of those 47, only 42 count toward your score—the other 5 are experimental questions E3n (formerly ERB, the Educational Records Bureau) uses to test new items. You will not know which is which, so answer every question.
Mathematics Achievement leans more on raw content fluency: identify what kind of problem it is, recall the formula or procedure, and run the calculation. Average difficulty here is about 3.5 out of 5—comparable to Quantitative Reasoning, but the path to the answer is usually more direct once you spot the topic.
Both sections are no-calculator, both are timed tightly, and both draw from the same five content areas: number sense, algebra, geometry, measurement, and data analysis & probability. Both also reward neatness—your scratch work happens in the test booklet, and a sloppy column of arithmetic is a fast way to misread your own answer.
Number sense shows up in 5–11 questions on Mathematics Achievement and quietly underlies almost every word problem in Quantitative Reasoning. If your fractions and exponents are slow, every other math topic feels harder than it is.
PEMDAS (parentheses, exponents, multiplication/division, addition/subtraction) is non-negotiable on the ISEE. The most common arithmetic mistake on this test is doing operations left to right, ignoring order. Watch especially for negative signs in front of squared terms—−3² is −9, not 9, because the exponent is applied before the negative.
Exponent rules to memorize: same-base multiplication adds exponents (x² · x³ = x⁵), division subtracts (x⁵ / x² = x³), and a power-of-a-power multiplies ((x²)³ = x⁶). A negative exponent flips a base into the denominator: x⁻² = 1/x².
Worked Example
Setup: Compute 24 ÷ 6 + 3 × (7 − 2)² without a calculator.
Be able to convert between all three forms in seconds. Memorize the common ones: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/8 = 0.125 = 12.5%, 1/3 ≈ 0.333 ≈ 33.3%. When a question mixes forms, convert to whichever is easiest to compute with—percents work well for "increase/decrease" problems, while fractions are usually faster for division.
A common ISEE trap is successive percent changes. Two 10% discounts do not add up to 20% off—the second discount is applied to the already-reduced price, so the total cut is closer to 19%. Always apply percent changes step by step.
A ratio is a relative comparison; a proportion is a statement that two ratios are equal. Solve proportions by cross-multiplying. ISEE problems often hide ratios inside word problems ("for every 3 girls, there are 2 boys")—translate the phrase into a fraction immediately.
Number-property questions test whether you can spot integers, primes, factors, and multiples quickly. Know that 1 is not prime, that 2 is the only even prime, and that the GCF (greatest common factor) and LCM (least common multiple) often unlock fraction problems.
Algebra is the single largest content bucket on Mathematics Achievement, with 13–17 questions per test out of 42 scored items—roughly one-third of your section score. It also drives most word problems on Quantitative Reasoning. If you can only deeply review one topic, make it algebra.
| Content area | Questions per test | Sample concept |
|---|---|---|
| Algebra | 13–17 | Solve 3x − 5 = 22 |
| Data analysis & probability | 8–13 | Find P(blue) from a 5-color spinner |
| Number sense | 5–11 | Order rational numbers on a line |
| Geometry | 5–8 | Find the area of a sector |
| Measurement | 5–8 | Convert 3.5 ft to inches |
A linear equation in one variable looks like 3x − 5 = 22. To solve, isolate the variable using inverse operations on both sides. Inequalities work the same way with one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign.
Systems of two linear equations can be solved by substitution (solve one equation for a variable, plug into the other) or elimination (add or subtract the equations to cancel a variable). On the ISEE, either method works—pick whichever leaves the cleanest arithmetic.
Worked Example
Setup: Solve x² − 5x + 6 = 0 by factoring.
FOIL (First, Outer, Inner, Last) multiplies two binomials: (x + 2)(x + 3) = x² + 5x + 6. Going the other direction—factoring a trinomial back into two binomials—shows up constantly. Memorize the two factoring patterns: difference of squares (a² − b² = (a + b)(a − b)) and perfect-square trinomials (a² + 2ab + b² = (a + b)²).
Pull out the greatest common factor first whenever possible. 2x² + 4x = 2x(x + 2)—a common ISEE setup is to ask for an equivalent expression, and the factored form is usually what shows up in the answer choices.
Quadratic equations take the form ax² + bx + c = 0. Solve them by factoring when possible, otherwise use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac tells you how many real solutions exist (positive = 2, zero = 1, negative = 0).
Function notation f(x) is just a fancy way to name a formula. To evaluate f(x) = 2x + 3 at x = 4, plug in 4: f(4) = 2(4) + 3 = 11. ISEE function questions sometimes ask for f(g(x))—just work from the inside out.
Perimeter, area, and angle questions are among the most frequently tested concepts on Quantitative Reasoning, and geometry adds another 5–8 Mathematics Achievement questions on top. Because no formula sheet is provided, the single highest-ROI study habit is to memorize a short formula list and review it daily.
| Topic | Formula | Used for |
|---|---|---|
| Pythagorean theorem | a² + b² = c² | Right-triangle side lengths |
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a | Solving ax² + bx + c = 0 |
| Slope | m = (y₂ − y₁) / (x₂ − x₁) | Line steepness from two points |
| Distance | d = √((x₂ − x₁)² + (y₂ − y₁)²) | Distance between two points |
| Area of a circle | A = πr² | Circles and sectors |
| Circumference | C = 2πr | Perimeter of a circle |
| Area of a triangle | A = ½ × base × height | All triangles |
| Volume of a cylinder | V = πr²h | 3-D solids and capacity problems |
| Volume of a cone | V = ⅓πr²h | 3-D solids problems |
| Probability | P(event) = favorable / total | All probability problems |
Every triangle's interior angles add up to 180°. The Pythagorean theorem (a² + b² = c²) applies to right triangles only. Recognize the common Pythagorean triples on sight: 3-4-5, 5-12-13, 8-15-17, and any multiple of them (6-8-10, 9-12-15, etc.).
Special right triangles let you skip the Pythagorean theorem. A 45-45-90 triangle has legs in ratio 1 : 1 : √2. A 30-60-90 triangle has sides in ratio 1 : √3 : 2. Memorizing these saves seconds on every triangle problem.
Worked Example
Setup: A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
For a circle with radius r: circumference is C = 2πr and area is A = πr². A sector is a slice of a circle, so its area is the central-angle fraction times the whole-circle area: A = (θ/360°) × πr². Arc length follows the same logic: (θ/360°) × 2πr.
The interior angles of any n-sided polygon sum to (n − 2) × 180°—so a triangle is 180°, a quadrilateral is 360°, a pentagon is 540°, and a hexagon is 720°. Each interior angle of a regular polygon is the total divided by n.
Volume formulas you must know: rectangular prism (length × width × height), cylinder (πr²h), cone (⅓πr²h), and sphere ((4/3)πr³). Surface area for a cylinder is 2πr² + 2πrh—the two circles plus the rectangle that wraps around.
Coordinate geometry questions usually need just three formulas. The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The slope is (y₂ − y₁)/(x₂ − x₁)—rise over run.
Data analysis is the single most frequently tested topic on Mathematics Achievement, with 8–13 questions per test across data, statistics, and probability. These are usually fast points—the math is light, but you have to read the prompt and the chart carefully.
Mean is the arithmetic average: add the values and divide by how many there are. Median is the middle value of an ordered list (or the average of the two middle values if the list has an even count). Mode is the most frequent value—lists can have one mode, multiple modes, or no mode at all. Range is the largest minus the smallest.
A common ISEE trick: changing one value in a data set affects the mean immediately but may leave the median unchanged. Be ready for "if the largest value increases by 10, what happens to the mean and median?" style questions.
Basic probability is favorable outcomes divided by total outcomes: P(event) = favorable / total. For two independent events both happening, multiply the probabilities. For "this OR that" with mutually exclusive events, add them. For events that can overlap, use P(A or B) = P(A) + P(B) − P(A and B).
Permutations care about order; combinations do not. If 5 students line up for a photo and order matters, you have 5! = 120 arrangements. If you pick 2 of those 5 to form a team and order does not matter, you have C(5, 2) = 10 ways.
Bar charts, line graphs, pie charts, and scatter plots all show up. Always read the axis labels and units before computing anything—many wrong answers are correct numerically but for the wrong column. Pie charts test percentages: a wedge labeled "30%" of a 200-student total represents 60 students.
On Quantitative Reasoning, expect at least one or two questions where you compare totals, ratios, or differences pulled from a chart. The fastest path is usually to compute one specific value, not to convert the whole chart.
ISEE Upper Level Quantitative Reasoning rewards smart shortcuts more than any other section of the test. Use the strategies below to convert content knowledge into actual points on test day.
Plugging in turns abstract algebra into arithmetic. If a problem says, "If n is a positive even integer, which of the following is always odd?" pick n = 2, test every answer, then double-check with n = 4 to be safe. Avoid 0 and 1—they have weird properties that can mislead you.
Backsolving is the cousin trick. When the question gives numeric answer choices and asks for a value, run each answer through the equation. Because ISEE answer choices are usually listed in order, start with the middle option—if it is too small, eliminate it and the smaller ones; if too big, eliminate it and the larger ones.
Every quantitative comparison question on the ISEE uses the same four answer choices. Memorize them once and you have a permanent edge.
| Choice | Meaning |
|---|---|
| A | The quantity in Column A is greater |
| B | The quantity in Column B is greater |
| C | The two quantities are equal |
| D | Cannot be determined from the information given |
The hardest answer to spot is D, "cannot be determined". It comes up when a variable could be positive or negative, or when a geometry figure is "not drawn to scale." If you cannot pin down the relationship after testing a couple of values, D is the right call.
You have about 57 seconds per Quantitative Reasoning question and 51 seconds per Mathematics Achievement question. Spending 90 seconds on one question to get it right means two other questions get rushed. The fix: plan to answer the questions you find easy first, mark the rest with a star, and come back with whatever time is left.
The ISEE has no wrong-answer penalty, so before time is called fill in every blank with your best guess. A blind guess gives you a 25% shot; even a single eliminated answer choice raises that to 33%.
Enter how many questions you have left and how many minutes remain to see the seconds-per-question target.
Most students benefit from one to three months of focused review for ISEE Upper Level math. The plan below assumes about an hour of practice per day, six days a week, and adjusts depending on how recently you have seen Algebra II and geometry in school.
Take a full-length, timed practice test in week one—do not study first. The diagnostic shows you which content areas are already strong and which need work. After scoring, sort your missed questions by topic. If you missed five algebra problems and one geometry problem, your plan is obvious.
Spend the rest of weeks one and two on number sense, exponent rules, fraction-decimal-percent fluency, and order of operations. These skills are prerequisites for everything that follows, and they are usually quick to refresh.
Spend at least 60% of your study time on algebra and data analysis—together they account for about 21–30 of the 42 scored Mathematics Achievement questions. Move on to geometry in week four, prioritizing the formula list. Use the topic-weight lookup below to plan each session deliberately.
Pick a Mathematics Achievement content area to see how many questions it accounts for and how to prioritize study time.
Build a personal formula sheet during these weeks, even though you cannot bring it into the test. The act of writing the formulas down each session, then quizzing yourself on them, is the single most efficient way to memorize them.
In the final weeks, take at least three full-length timed math sections under realistic conditions—no calculator, no breaks mid-section, scratch paper only. Score them, then spend twice as much time on the review as you spent on the test itself. Every missed question becomes a topic to re-drill.
Use the readiness checklist below to make sure you have not skipped anything. Every box checked equals one less worry on test day.
The ISEE Upper Level math sections cover pre-algebra, algebra (through Algebra II concepts), geometry, measurement, data analysis, probability, and basic statistics. Algebra is the largest content area on Mathematics Achievement with 13–17 questions per test, while data analysis is the most frequently tested concept overall. Some advanced topics like matrices, sequences, and standard deviation also appear.
ISEE Upper Level math is challenging because it covers content through Algebra II in a no-calculator format with strict time limits. Average question difficulty is 3.4 out of 5 for Quantitative Reasoning and 3.5 out of 5 for Mathematics Achievement, making them the toughest sections for many students. Difficulty can also feel higher because younger applicants are tested at the same level as eleventh graders.
No, calculators are not allowed on either the Quantitative Reasoning or Mathematics Achievement section of the ISEE Upper Level. Students must rely on mental math, paper-and-pencil computation, and estimation. Practicing without a calculator is essential during prep so that arithmetic does not become a bottleneck on test day, especially on multi-step problems.
No, the ISEE does not provide a formula sheet. Students must memorize geometry formulas like the area of circles and triangles, volume of cylinders and cones, the Pythagorean theorem, the quadratic formula, and the slope and distance formulas. Building a personal formula sheet during practice and reviewing it daily is the most effective way to retain them.
You have 35 minutes for 37 Quantitative Reasoning questions (about 57 seconds each) and 40 minutes for 47 Mathematics Achievement questions (about 51 seconds each). The two math sections are not back-to-back—Reading Comprehension separates them—but both demand quick, efficient work. Pacing practice with a timer is essential during prep.
Most students benefit from 1 to 3 months of focused review for ISEE Upper Level math. Strong students who only need topic refreshers can succeed with 4–6 weeks; students who are weaker in algebra or geometry should plan 2–3 months of practice. Starting early is critical because the test cannot be retaken in the same six-month window.