ISEE Mathematics Achievement:Content, Vocabulary and Samples

Mathematics Achievement: question counts, timing and content

This section gives the question counts and timing of the mathematics section at each level, and the distribution of questions across skill areas. The level is determined by the grade the student is applying to enter, not by the grade the student currently attends.

Level	Applying to grade	Mathematics section	Questions	Time	Other mathematics section at the same level
Primary 2	Grade 2	Mathematics	24	26 minutes	None
Primary 3	Grade 3	Mathematics	24	26 minutes	None
Primary 4	Grade 4	Mathematics	28	30 minutes	None
Lower	Grades 5–6	Mathematics Achievement	30	30 minutes	Quantitative Reasoning, 38 questions in 35 minutes
Middle	Grades 7–8	Mathematics Achievement	47	40 minutes	Quantitative Reasoning, 37 questions in 35 minutes
Upper	Grades 9–12	Mathematics Achievement	47	40 minutes	Quantitative Reasoning, 37 questions in 35 minutes

Source: official ERB ISEE materials.

Based on these question counts, the Lower Level allows an average of 1 minute per question, and the Middle and Upper Levels about 51 seconds per question. Online and paper tests have the same number of questions and the same timing. The official test instructions prohibit calculators in the testing room, except under an approved calculator accommodation. Scratch paper arrangements for each test format are as follows.

Test format	Scratch paper
Online test at a test site	The site provides pencils and scratch paper
Online test at home	Students may prepare four sheets of scratch paper
Paper test	Students work in the blank space of the test booklet

Questions by skill area (Lower / Middle / Upper)

The official “What to Expect on the ISEE” guides (the official guides) publish the approximate number of questions in each skill area by level. The table below reproduces the official figures; the Upper Level combines whole numbers with decimals, percents and fractions into a single Number Sense area.

Skill area	Lower (30 questions)	Middle (47 questions)	Upper (47 questions)
Whole Numbers	4–7	7–10	Part of Number Sense
Decimals, Percents, Fractions	4–7	7–10	Part of Number Sense
Number Sense	—	—	5–11
Algebraic Concepts	4–7	9–13	13–17
Geometry	2–5	4–6	5–8
Measurement	2–5	4–6	5–8
Data Analysis and Probability	4–7	5–9	8–13
Scored questions	25	42	42
Unscored trial questions	5	5	5

Source: official ERB ISEE materials.

Content assessed at each level

This section describes the content assessed at each level. It lists only what the official materials state and does not speculate about anything they do not publish.

The official score report explanation marks the mathematics section as applying to all levels. Its content follows the six strands of the National Council of Teachers of Mathematics (NCTM): number and operations, algebra, geometry, measurement, data analysis and probability, and problem solving. Unlike Quantitative Reasoning, Mathematics Achievement includes questions that require actual calculation, and some questions require knowledge of grade-appropriate mathematical vocabulary.

Primary 2 / 3 / 4: Mathematics

The official materials do not publish question counts by skill area for the Primary mathematics sections. Three points are certain:

1. The Primary levels have no separate Quantitative Reasoning section; Mathematics is the only scored mathematics section.
2. The Primary levels have no essay.
3. The test organizer provides one free online sample test each for Primary 2, 3 and 4.

EDUBUS recommends that Primary families gauge the difficulty from the official sample tests rather than applying the Lower Level skill-area table.

Lower / Middle / Upper: Mathematics Achievement

Level	Official difficulty reference
Lower	Calculation begins with simple addition and subtraction
Middle	Based on NCTM standards for grades 6 to 8; difficulty comparable to grades 6 and 7; skill areas similar to the Lower Level
Upper	Based on NCTM standards for grades 8 to 11; the most difficult questions reach second-year algebra (Algebra II)

Source: official ERB ISEE materials.

Official question-writing rules

The official guides state the following question-writing rules, which bear directly on preparation.

Question-writing rule	Implication for preparation
Wrong answer choices come from common errors in the solution process, such as arithmetic mistakes, the wrong operation or the wrong formula; the official materials state that there are no trick questions	Numbers reached partway through a solution often appear among the choices; confirm that the answer addresses what the question asks
Conversions between U.S. customary units (such as inches and feet) are given in the question	Students do not need to memorize U.S. customary conversions
Conversions within the metric system (such as centimeters and meters) are not given	Students must know metric conversions on their own
Each level is taken by applicants to two grades	Some questions may go beyond the student’s school curriculum; scores are compared only with those of other independent school applicants to the same grade

Common errors

This section lists the types of errors EDUBUS sees repeatedly in teaching and organizes the high-frequency mathematical English vocabulary associated with misreading.

EDUBUS teaching experience shows that a considerable share of the points lost in Mathematics Achievement by students from Chinese-speaking families comes from the English question stems rather than from calculation ability. Many Irvine families use Chinese-language textbooks to study mathematics ahead of school outside class; students master the methods but not the English expressions for the same concepts. The official materials state that Mathematics Achievement requires knowledge of grade-appropriate mathematical vocabulary, and calculation ability cannot compensate for this.

Types of errors

No.	Error type	Description
1	Misreading the question word	The question asks for greatest, least, NOT or remainder, and the student gives an intermediate result; all three distractors in Sample 2 belong to this type
2	Reversing the direction of subtraction or comparison	“5 less than x” is translated word by word as 5 − x, giving a result with the opposite sign
3	Including or excluding equality incorrectly	At least, at most and no more than all include equality; exceed does not
4	Unfamiliarity with metric conversions	The official materials state that metric conversions such as centimeters to meters are not given in the question; students must know them
5	Choosing an intermediate result	The official materials state that wrong answer choices come from common errors in the solution process; seeing a familiar number among the choices does not mean the solution is complete
6	Weak mental and written calculation	No calculator is allowed at any point, so decimal division and common denominators must be worked quickly on scratch paper
7	Poor time allocation	The Middle and Upper Levels allow about 51 seconds per question on average; lingering on one question takes time from later questions

The vocabulary tables below are organized by how often each term appears in questions and how likely it is to be misread. Students should master both the meaning and the common error for each term.

Mathematical English: operations and properties of numbers

English	Meaning	Common error
sum	The result of addition	Confused with some
difference	The result of subtraction	Read as a way in which things are unlike
product	The result of multiplication	Read as an item for sale
quotient	The result of division	Confused with the remainder
remainder	The amount left over after division	Giving the quotient when the remainder is asked for
factor	A number that divides another number evenly	Confused with multiple, the opposite relationship
multiple	The product of a number and a whole number	Confused with factor, the opposite relationship
prime number	A number whose only factors are 1 and itself	Treating 1 as a prime number
integer	A whole number, including negatives and 0	Used interchangeably with whole number
whole number	A non-negative integer (0, 1, 2 and so on)	Assumed to include negative numbers
consecutive	Following one after another	Consecutive even integers differ by 2
digit	A single numeral in a place value position	The tens digit is the digit in the tens place, not ten digits
reciprocal	1 divided by the number	Confused with the opposite of a number

Mathematical English: comparisons and constraints

English	Meaning	Common error
at least	No less than, ≥	Omitting equality
at most	No greater than, ≤	Omitting equality
no more than	Does not exceed, ≤	Treated as strictly less than, omitting equality
no fewer than	Not less than, ≥	Reversing the direction because of the negative word
exceed	To be strictly greater than	Assumed to include equality
5 less than x	x − 5	Written as 5 − x by following the word order
x subtracted from y	y − x	Written as x − y by following the word order
twice as many as	Two times the amount of	Reversing the two quantities in the multiple relationship
per	For each	Overlooked in unit rate questions
which of the following is NOT	Which option does not hold	Overlooking NOT
must be / could be	Is always true / may be true	Treating may be true as always true

Mathematical English: geometry, statistics and algebra

English	Meaning	Common error
perimeter	The distance around a figure	Confused with area
circumference	The distance around a circle	Mixed up with the area formula
radius / diameter	Half the width of a circle / the full width of a circle	Substituting the diameter into a radius formula
parallel / perpendicular	Never meeting / meeting at a right angle	Confusing the two words
isosceles / equilateral	Two equal sides / all sides equal	Confusing the two words
mean (average)	The sum divided by the number of values	Confused with the median
median	The middle value	Forgetting to put the values in order first
mode	The most frequent value	Confused with the median
range	The greatest value minus the least value	Read as a span and written as an interval
probability	The likelihood of an event	Using the wrong denominator
expression / equation	A combination of terms / a statement that two expressions are equal	Solving an equation when the question asks for simplification
evaluate	To find the value by substitution	Read as to judge the quality of
round to the nearest tenth	Round to the tenths place	Confused with the tens place

Original sample questions

The following three questions were written by the EDUBUS teaching team, one each for the Lower, Middle and Upper Levels, and all focus on understanding the question stem. Each explanation first restates the stem, then explains the solution and the distractors.

Sample 1 (Lower Level · division and remainders)

A number is divided by 7. The quotient is 5 and the remainder is 3. What is the number? (A) 15 (B) 32 (C) 35 (D) 38

Explanation

Restating the stem: a number divided by 7 gives a quotient of 5 and a remainder of 3. The question asks for the number.

The answer is D. The dividend equals the divisor times the quotient plus the remainder: 7 × 5 + 3 = 38. Check: 38 ÷ 7 = 5 remainder 3.

Distractor analysis: (A) 15 multiplies the quotient by the remainder (5 × 3), showing no understanding of the relationship between quotient and remainder. (B) 32 subtracts the remainder from 35, reversing the operation. (C) 35 multiplies only the divisor by the quotient and omits the remainder.

Skill tested: the terms quotient and remainder, and the relationship dividend = divisor × quotient + remainder.

Sample 2 (Middle Level · consecutive even integers)

The sum of three consecutive even integers is 78. What is the greatest of the three integers? (A) 24 (B) 26 (C) 27 (D) 28

Explanation

Restating the stem: the sum of three consecutive even numbers is 78. The question asks for the greatest of the three.

The answer is D. Let the least even number be n, so the three numbers are n, n + 2 and n + 4. From 3n + 6 = 78, n = 24; the three numbers are 24, 26 and 28, and the greatest is 28. Alternatively, find the mean first: 78 ÷ 3 = 26, which is the middle number.

Distractor analysis: (A) 24 is the least number and does not answer the question asked. (B) 26 is the middle number; the student found the mean and stopped. (C) 27 is the greatest of three ordinary consecutive integers (25, 26, 27), from reading consecutive even integers as consecutive integers.

Skill tested: the meaning of consecutive even integers and the question word greatest. Three of the four answer choices in this question result from misreading.

Sample 3 (Upper Level · applying inequalities)

A phone plan charges a monthly fee of $15 plus $0.15 for each text message. Maya wants her monthly bill to be no more than $45. What is the greatest number of text messages she can send in one month? (A) 100 (B) 199 (C) 200 (D) 300

Explanation

Restating the stem: a phone plan charges a $15 monthly fee plus $0.15 per text message. Maya wants her monthly bill to be no more than $45. The question asks for the greatest number of text messages she can send in one month.

The answer is C. Let t be the number of text messages. The inequality 15 + 0.15t ≤ 45 gives 0.15t ≤ 30, so t ≤ 200. No more than includes equality, so the maximum is 200 messages.

Distractor analysis: (A) 100 divides the monthly fee of 15 by 0.15, using the wrong data. (B) 199 treats no more than as strictly less than, omitting equality. (D) 300 divides 45 directly by 0.15 without subtracting the monthly fee.

Skill tested: matching no more than to ≤, and setting up and solving a linear inequality. Calculators are not permitted, so 30 ÷ 0.15 should first be rewritten as 3000 ÷ 15 and then calculated mentally.

Practice methods

This section describes how to organize weekly practice. The steps below should be followed in order, and the first two continue throughout the preparation period.

1. Build a mathematics vocabulary notebook. Start from the three vocabulary tables on this page, pair each term with a short English sentence the student writes, and review the notebook once a week.
2. Read each question in three steps. First circle the question sentence (usually the last sentence), then underline every numerical condition, and finally restate the question in the home language. Begin working only after the question can be restated clearly.
3. Record errors in three categories. Label each incorrect answer as a misreading, a concept error or a calculation error. Whichever words the misreadings cluster around should be added to the vocabulary notebook.
4. Practice calculation without a calculator. Set aside a fixed time each day for mental and written calculation with decimals, fractions and percents, aiming first for accuracy and then for speed.
5. Follow official timing. Once practice by skill area is stable, complete full timed sets: 30 questions in 30 minutes at the Lower Level, and 47 questions in 40 minutes at the Middle and Upper Levels.
6. Use the free official resources. The test organizer provides a free official guide for each of the Lower, Middle and Upper Levels, each containing one paper practice test, and one free online sample test for each of the six levels. Students should complete one partway through preparation to become familiar with the question types and the testing interface.

For Primary families, preparation centers on reading alongside the student. Applicants to grades 2 through 4 are still developing English reading skills, so parents may read the questions with the student, but the student should state what the question asks, and parents should not translate it into the home language for the student.

EDUBUS mathematics courses

The EDUBUS assessment: preparation for Mathematics Achievement should give mathematical English the same weight as calculation and concepts, rather than relying on practice questions alone. The course arrangements are designed on this basis.

Level	Formats offered by EDUBUS	Mathematics content
Primary 2 / 3 / 4	One-on-one private tutoring	Follows the student’s grade-level progress, combined with question-reading practice
Lower / Middle / Upper	Fall class + one-on-one private tutoring	The Quantitative Reasoning with Math module of the fall class, available as a single-module enrollment
All levels	The ISEE online learning system built by EDUBUS	Practice by module, realistic practice tests, error analysis and daily vocabulary check-ins

Classes meet at the EDUBUS Irvine teaching center. Quantitative Reasoning and mathematics are taught in the same module because the two sections use the same mathematical concepts and the same mathematical English, and practicing them separately is less efficient. Class hours and prices for each course are listed on the ISEE courses and assessment page.