The ISEE Quantitative Reasoning section is often considered the most difficult part of the test.
Those who excel on the ISEE are set up for success on their private school applications, so taking the time to prepare for each section in advance is worth the effort!
The ISEE (Independent School Entrance Exam, generally pronounced “i-see”) is a peer-normed, standardized test most commonly taken by students applying to middle and high school. To evaluate students’ reading and math skills, it consists of two ELA sections and two math sections:
| ISEE Section | Time | Number of Questions |
|---|---|---|
| Verbal Reasoning | 20 Minutes | 40 Questions (34 LL) |
| Quantitative Reasoning | 35 Minutes | 37 Questions (38 LL) |
| Break | 5-10 Minutes | |
| Reading Comprehension | 35 Minutes (25 LL) | 36 Questions (25 LL) |
| Mathematics Achievement | 40 Minutes (30 LL) | 47 Questions (30 LL) |
| Break | 5-10 Minutes | |
| Essay | 30 Minutes | |
| Total | ~2 Hours 40 Minutes (2:20 LL) | 160 + Essay (127 + Essay LL) |
Quantitative Reasoning is the second section of the ISEE exam at all levels (lower, middle, and upper), preceding the first 5-10 minute break. Test-takers are given 35 minutes to answer 37 questions (so, just under a minute allotted per question).
In this deep-dive into the ISEE Quantitative Reasoning test, we explain the types of questions you’ll encounter on the test, strategies for working through those math problems, and the best ways to study for this section of this exam.
ISEE Math Topics
Both the Quantitative Reasoning and Mathematics Achievement sections of the ISEE test students’ math skills. The ERB (which develops the ISEE) follows standards set forth by the NCTM (National Council of Teachers of Mathematics) to determine how the ISEE tests students’ math abilities.
Each level of the ISEE tests students on a corresponding level of math knowledge; naturally, Upper Level ISEE-takers will be tested on more advanced ideas in mathematics than those taking the Middle Level test. Across both math sections for all levels, the ISEE will test students on the following general concepts:
- Numbers and Operations. Understanding numbers, fractions and decimals, operations, comparing values, and relationships between numbers.
- Algebra. Understanding patterns, variables, functions, symbolic equations, and graphs.
- Geometry. Identifying, classifying, and analyzing 2-d and 3-d geometric objects, as well as using a coordinate plane and understanding congruency and symmetry.
- Measurement. Using formulas to make measurements of objects, converting between units, and identifying tools associated with length, weight, capacity, temperature, or time.
- Data Analysis & Probability. Analyzing and making inferences about sets of data, calculating mean, mode, range, and median, and finding probability about a scenario.
Learn more about the specific topics tested on each ISEE level on our review pages:
Types of Quantitative Reasoning Questions
There are two types of questions you will be asked on the ISEE Quantitative Reasoning section: word problems and quantitative comparisons. (On the ISEE Lower Level, students are only asked word problems.)
Approaching ISEE Math Questions
Having a systematic approach to answer each Quantitative Reasoning word problem will help you confront and work through each question efficiently.
| ISEE Math Problems: 5-Step Approach | ||||
|---|---|---|---|---|
| Step 1 | Do I UNDERSTAND the question? | Circle or underline important information, and write it out next to the problem. | If you get stuck, SKIP IT! Circle the question in the test booklet and answer it later. | |
| Step 2 | Do I know which FORMULAS I should use? | Quickly write out the necessary formulas to the side of the problem. | ||
| Step 3 | Will the question take a LONG TIME to complete? | Skip it, and come back when you have time. | ||
| Step 4 | Can I INPUT VALUES to solve the question? | Use simple values such as -1, 0, and 1 to help you evaluate expressions and equations. Keep in mind all properties of algebraic and geometric concepts. | ||
| Step 5 | Can I make an EDUCATED GUESS with what I know? | Eliminate incorrect answer choices, then input the remaining answers to check your work. | ||
| If you’re still unsure, DO NOT leave it blank. Choose the answer that makes the most sense. | ||||
Word Problems
Each ISEE word problem is followed by four answer choices. Several of these problems may be solved without actually doing any math on paper—these questions test your ability to decipher math problems from real-world scenarios and solve them.
Here’s an examples of how to work through an ISEE Upper Level word problem.
Let’s break it down further by working through this problem:
1. Where does the function f (x)= x² +7x + 10 cross the x-axis?
A. x = –5, –2
B. x = 5, 2
C. x = –5, 2
D. x = –2, –5
| UNDERSTAND |
First and foremost, decide if you understand the question. If you do: write out all important information to the side of the question. If you don’t: SKIP IT and return later. |
“I know this one! I need to figure out what my values for x are when f(x), or the output, is equal to zero. |
| CONTEXT |
Determine if there are any properties, rules, or formulas you know that can help. |
"I know that my output needs to be zero— I’ll replace f(x) with 0. To solve quadratics, I need to factor out the equation… so I’ll start with two parentheses."
0 = x² + 7x + 10 0 = ( ) ( ) |
| LONG TIME |
Quickly determine if this question will take you a very long time to complete.
If it will: SKIP IT and return later. |
(LONG TIME) “I’ve done problems like these before, but I’m really rusty on them. I’ll come back to it later."
(SHORT TIME) “We went over this recently in class, I know exactly what I need to do!”
|
| ESTIMATION |
Before you start solving the problem, see if estimation can help.
1. Where does the function f(x)= x² + 7x + 10 cross the x-axis?
A. x = –5, –2
B. x = 5, 2
C. x = –5, 2
D. x = –2, –5 |
“I don’t think estimation will help me too much here, but I know +10 is the y-intercept. My teacher also said that factors of the y-intercept help determine x-intercepts… my guess is that it’s the factors 5 and 2, and it has to be their negatives that are the answers…"
0 = x² + 7x + 10
0 = ( )( )
0 = (x + 5)(x + 2)
x + 5 = 0
x + 2 = 0 |
There is no penalty for wrong answers on the ISEE, so be sure you answer every question by making educated guesses and prioritizing problems you know.
Some word problems can take a significant amount of time to read through and solve. ISEE-takers are allotted only 57 seconds per question on this section, so conserve your time by prioritizing questions that are short, easy, or you’re most confident in answering. Once you work through the section once, go back and attempt the harder, longer questions.
Quantitative Comparisons
The ISEE Quantitative Reasoning section contains quantitative comparison questions, in addition to word problems. Each quantitative comparison question contains two values that you must compare. Some questions will include extra information to help you answer the question.
Each answer choice designates a particular relationship between the two quantities:
A. The quantity in Column A is greater.
B. The quantity in Column B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
Use the systematic approach above to work through quantitative comparison problems. In short:
- Determine if you understand the problem.
- Identify which formulas to use.
- Decide if the question will take a long time to complete.
- Input values* to help you determine (or get closer to) the answer. (More on this below.)
- Make an educated guess.
*How to Input Values: Some quantitative comparisons use variables to ask you to compare operations. Use simple values such as -1, 0, and 1 in place of an unknown variable. If your answer changes with each of those values, choose D like in the example above.
Here’s an example of the process of working through an ISEE Middle Level quantitative comparison.
1.
| Column A | Column B |
|
64 |
62 + 62 |
A. The quantity in Column A is greater.
B. The quantity in Column B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
| UNDERSTAND |
First and foremost, decide if you understand the question. If you do: write out all important information to the side of the question. If you don’t: SKIP IT and return later. |
I know this one. I just need to multiply 6 by itself 4 times to find the value of column A, then multiply 6 by itself twice and add the sums for the value of column B.” |
| CONTEXT |
Determine if there are any properties, rules, or formulas you know that can help. |
"According to the order of operations, I need to apply the exponents ahead of addition for column B." |
| LONG TIME |
Quickly determine if this question will take you a very long time to complete.
If it will: SKIP IT and return later. |
(LONG TIME) “My mind is blanking on exponents right now. The question is short, so if I remember later I should have time to work it out."
(SHORT TIME) “We went over this recently in class, I know exactly what I need to do!”
|
| ESTIMATION |
Before you start solving the problem, see if estimation can help.
64 vs. 62+62 |
“I know that 62 is 6 x 6, which is equal to 36. 64 would be 6x6x6x6… I don’t know it off the top of my head, but I know that the total would be the same as 36x36, way more than 36+36!”
64 vs. 62 + 62
6 x 6 x 6 x 6 vs. 6 x 6
64 > 62 + 62
|
In the above case, there’s a clear answer to which column has a greater value. However, some quantitative comparison questions will present you with unsolvable equations with the correct answer being D, that the relationship cannot be determined. This is the hardest aspect of QC questions—you must differentiate between unsolvable questions and solvable ones that you don’t understand. Don’t automatically assume something is unsolvable!
Here’s one more example of working through a quantitative comparison, but for the ISEE Upper Level test. See how it can be solved by simply inputting values:
2.
| Column A | Column B |
|
15k – 2 |
15(k – 2) |
A. The quantity in Column A is greater.
B. The quantity in Column B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
| INPUT VALUES |
“I’m not given a value for k, but it’s in both columns. Maybe I should pick D, but I’ll try putting in my own values first. I’ll try 1, 0, and -1…” |
For 1: A. 15(1) – 2 = 13 B. 15(1) – 15(2) = -15 For 0: A. 15(0) – 2 = -2 B. 15(0) – 15(2) = -30 For -1: A. 15(-1) – 2 = -17 B. 15(-1) – 15(2) = -45 Column A’s total is larger for every value—this means the answer is that column A is greater than B! |
How to Prepare for Quantitative Reasoning Questions
The most effective way to study for the ISEE Quantitative Reasoning is by practicing the math topics that are tested, especially through practice questions. The ISEE test prep process is a cycle of working through practice questions and relearning topics between benchmark ISEE exams. Based on your practice test results, you can determine which topics you have a good understanding of versus the ones you should review more.
Make sure to use practice questions and tests designed for the ISEE exam. By practicing with questions that resemble what you’ll encounter on the ISEE Quantitative Reasoning section, you’ll become comfortable with solving those types of problems. Use the strategies we’ve described above as you practice, and you’ll be familiar with what to do come test day.
In addition to regularly practicing with test-specific questions, make sure to take at least one benchmark for your ISEE ahead of test day. Not only will this help you gauge your current skills and see how much you need to improve, but taking multiple practice exams is proven to improve test-taking performance.
Personalized Online ISEE Prep
Use these ISEE Quantitative Reasoning strategies as you prepare for this section of the test! Remember that taking practice tests before your ISEE test day (and retaking it once you learn from your first attempt) will grant you the most opportunities to learn from your mistakes and give a strong performance. If you’re looking for ISEE math practice to help you utilize the strategies we talked about above, consider Piqosity’s ISEE courses!
Along with our full-length ELA and Math online courses for grades 6-11, we offer market-leading ISEE prep courses for all levels of the exam. Each course includes detailed topic lessons, hundreds of questions, and 12 full-length practice tests, together with personalized practice software that identifies your weaknesses and helps you study effectively. Sign up for an account and you can take a No-Cost ISEE Mini Diagnostic exam, plus 2 free full-length ISEE practice tests—no credit card required!
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