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ISEE (Independent School Entrance Examination) Mathematics Achievement Syllabus

The Mathematics Achievement syllabus for ISEE (Independent School Entrance Examination) in full: 5 chapters, 21 topics and 30 sub-topics, every one listed below, plus 50 Mathematics Achievement flashcards with answers.

5Chapters
21Topics
30Sub-topics
50Flashcards

Mathematics Achievement syllabus: all 5 chapters

Expand any chapter to see its topics and sub-topics.

  1. Numbers and Operations

     5 topics
    • Whole Numbers and Integers
      • Place value and number lines
      • Operations with positive and negative integers
    • Fractions and Decimals
      • Equivalence, comparison, and ordering
      • Four operations with fractions and decimals
      • Converting among fractions, decimals, and percents
    • Ratios, Proportions, and Percents
      • Setting up and solving proportions
      • Percent increase, decrease, and of-a-number
    • Factors, Multiples, and Primes
      • GCF and LCM
      • Prime factorization
      • Divisibility rules
    • Exponents and Roots
      • Powers and exponent rules
      • Square roots and perfect squares
      • Scientific notation (Upper Level)

  2. Algebraic Concepts

     4 topics
    • Expressions and Variables
      • Evaluating and simplifying expressions
      • Combining like terms and the distributive property
    • Equations and Inequalities
      • Solving one- and two-step equations
      • Solving and graphing inequalities
    • Linear Relationships (Upper Level)
      • Slope and intercepts
      • Graphing lines and interpreting equations
    • Polynomials and Factoring (Upper Level)
      • Multiplying binomials
      • Basic factoring techniques

  3. Geometry

     4 topics
    • Plane Figures and Angles
      • Lines, rays, and angle relationships
      • Triangles, quadrilaterals, and polygons
    • Perimeter, Area, and Volume
      • Area and perimeter of 2D shapes
      • Surface area and volume of solids
      • Circumference and area of circles
    • Coordinate Geometry
      • Plotting points and quadrants
      • Distance and reflections on the grid
    • The Pythagorean Theorem (Upper Level)
      • Finding missing side lengths
      • Recognizing right-triangle applications

  4. Measurement

     4 topics
    • Customary and metric units
    • Unit conversion within and between systems
    • Time, money, temperature, and rate problems
    • Reading scales and measurement tools

  5. Data Analysis, Statistics, and Probability

     4 topics
    • Interpreting charts, graphs, and tables
    • Measures of central tendency and range
    • Simple and compound probability
    • Counting principles and basic combinations

ISEE (Independent School Entrance Examination) Mathematics Achievement flashcards — 50 questions with answers

  1. What is the order of operations (PEMDAS) for evaluating an arithmetic expression?

    Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right).

  2. What is an integer, and what numbers does the set of integers include?

    An integer is a whole number with no fractional part. The set includes all positive whole numbers, all negative whole numbers, and zero (..., -2, -1, 0, 1, 2, ...).

  3. What are the rules for the sign of a product or quotient of two nonzero numbers?

    Same signs give a positive result; different signs give a negative result. (e.g., negative times negative = positive; negative times positive = negative.)

  4. How do you add two integers with different signs?

    Subtract the smaller absolute value from the larger absolute value, then keep the sign of the number with the larger absolute value.

  5. What is the absolute value of a number?

    Its distance from zero on the number line, always nonnegative. For example, |-7| = 7 and |7| = 7.

  6. How do you add or subtract fractions that have different denominators?

    Find a common denominator (often the least common denominator), rewrite each fraction with that denominator, then add or subtract the numerators and keep the denominator.

  7. How do you multiply two fractions, and how do you divide one fraction by another?

    To multiply, multiply numerators together and denominators together. To divide, multiply the first fraction by the reciprocal of the second (flip the divisor).

  8. How do you convert a fraction to a decimal?

    Divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75.

  9. How do you convert a terminating decimal to a fraction?

    Write the decimal over a power of 10 matching the number of decimal places, then simplify. For example, 0.35 = 35/100 = 7/20.

  10. What is a ratio, and what does the proportion a/b = c/d allow you to do?

    A ratio is a comparison of two quantities by division. A proportion states two ratios are equal; you can solve it by cross-multiplying: a·d = b·c.

  11. How do you convert a percent to a decimal and a decimal to a percent?

    To convert a percent to a decimal, divide by 100 (move the decimal two places left). To convert a decimal to a percent, multiply by 100 (move two places right).

  12. How do you find a given percent of a number?

    Convert the percent to a decimal and multiply by the number. For example, 20% of 50 = 0.20 × 50 = 10.

  13. How do you calculate percent increase or percent decrease?

    Percent change = (change in amount ÷ original amount) × 100. Use the difference between new and original values as the change.

  14. What is the difference between a factor and a multiple of a number?

    A factor divides the number evenly (e.g., factors of 12: 1,2,3,4,6,12). A multiple is the number times an integer (e.g., multiples of 12: 12,24,36,...).

  15. What is a prime number, and what is the only even prime?

    A prime number has exactly two distinct factors: 1 and itself. The only even prime number is 2.

  16. What is the prime factorization of a number?

    Expressing the number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

  17. What is the difference between the GCF and the LCM of two numbers?

    The GCF (greatest common factor) is the largest number that divides both. The LCM (least common multiple) is the smallest number that both divide into.

  18. What does a positive exponent mean, as in a^n?

    It means multiply the base a by itself n times. For example, 2^4 = 2 × 2 × 2 × 2 = 16.

  19. What are the product and quotient rules for exponents with the same base?

    Product rule: a^m × a^n = a^(m+n). Quotient rule: a^m ÷ a^n = a^(m−n).

  20. What is the value of any nonzero number raised to the power of 0, and what does a negative exponent mean?

    Any nonzero number to the 0 power equals 1. A negative exponent means the reciprocal: a^(−n) = 1/a^n.

Show the other 30 Mathematics Achievement flashcards

  1. What is a square root, and what does √a represent?

    A square root of a number is a value that, when multiplied by itself, gives that number. √a denotes the nonnegative (principal) square root, e.g., √25 = 5.

  2. What are the first several perfect squares from 1 to 12?

    1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (the squares of 1 through 12).

  3. In algebra, what is the difference between a variable, a coefficient, and a constant?

    A variable is a letter standing for an unknown value; a coefficient is the number multiplying a variable; a constant is a fixed number with no variable.

  4. What are like terms, and why do they matter?

    Like terms have the same variable raised to the same power (e.g., 3x and 5x). Only like terms can be combined by adding or subtracting their coefficients.

  5. How do you evaluate an algebraic expression for a given value of the variable?

    Substitute the given number in place of the variable, then simplify using the order of operations.

  6. How do you solve a one-step equation such as x + 7 = 12?

    Perform the inverse operation on both sides. For x + 7 = 12, subtract 7 from both sides to get x = 5.

  7. What is the key rule when solving an inequality that differs from solving an equation?

    When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

  8. How do you solve a two-step equation such as 3x − 4 = 11?

    Undo addition/subtraction first, then multiplication/division. Add 4 to both sides (3x = 15), then divide by 3 to get x = 5.

  9. What is the slope-intercept form of a linear equation, and what do its parts represent?

    y = mx + b, where m is the slope (rate of change) and b is the y-intercept (the y-value where the line crosses the y-axis).

  10. How is the slope of a line calculated from two points (x₁, y₁) and (x₂, y₂)?

    Slope m = (y₂ − y₁) / (x₂ − x₁), the change in y divided by the change in x (rise over run).

  11. How do you multiply two binomials such as (x + 2)(x + 3) using FOIL?

    Multiply First, Outer, Inner, Last terms and combine: x² + 3x + 2x + 6 = x² + 5x + 6.

  12. How do you factor a quadratic of the form x² + bx + c?

    Find two numbers that multiply to c and add to b; write them as (x + p)(x + q). For x² + 5x + 6, the numbers are 2 and 3, giving (x + 2)(x + 3).

  13. What is the sum of the interior angles of any triangle, and of any quadrilateral?

    The interior angles of a triangle sum to 180°, and the interior angles of a quadrilateral sum to 360°.

  14. What are complementary angles and supplementary angles?

    Complementary angles add up to 90°; supplementary angles add up to 180°.

  15. How are acute, right, and obtuse angles defined?

    An acute angle is less than 90°, a right angle is exactly 90°, and an obtuse angle is greater than 90° but less than 180°.

  16. What is the difference between an equilateral, isosceles, and scalene triangle?

    Equilateral has all three sides (and angles) equal; isosceles has exactly two equal sides; scalene has no equal sides.

  17. What is the formula for the area of a triangle?

    Area = ½ × base × height, where the height is perpendicular to the chosen base.

  18. What are the formulas for the area and perimeter of a rectangle?

    Area = length × width; Perimeter = 2 × (length + width).

  19. What are the formulas for the circumference and area of a circle?

    Circumference = 2πr (or πd); Area = πr², where r is the radius and d is the diameter.

  20. What is the formula for the volume of a rectangular prism (box)?

    Volume = length × width × height.

  21. On the coordinate plane, what does the ordered pair (x, y) tell you?

    x is the horizontal position (right if positive, left if negative) and y is the vertical position (up if positive, down if negative), measured from the origin (0, 0).

  22. How are the four quadrants of the coordinate plane numbered, and what are the sign patterns?

    Quadrant I (+,+) top right, II (−,+) top left, III (−,−) bottom left, IV (+,−) bottom right, numbered counterclockwise.

  23. What is the Pythagorean Theorem and when does it apply?

    For a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse (the side opposite the right angle).

  24. In the U.S. customary system, how many inches are in a foot, feet in a yard, and feet in a mile?

    12 inches = 1 foot; 3 feet = 1 yard; 5,280 feet = 1 mile.

  25. What are the basic metric prefixes for kilo-, centi-, and milli-?

    Kilo- means 1,000; centi- means 1/100 (0.01); milli- means 1/1,000 (0.001). So 1 km = 1,000 m, 1 cm = 0.01 m, 1 mm = 0.001 m.

  26. How many ounces are in a pound, and how many pounds are in a ton (U.S. customary weight)?

    16 ounces = 1 pound; 2,000 pounds = 1 ton.

  27. What is the formula relating distance, rate, and time?

    Distance = rate × time (d = r × t). Equivalently, rate = distance ÷ time and time = distance ÷ rate.

  28. What is the difference between the mean, median, and mode of a data set?

    Mean is the average (sum ÷ count); median is the middle value when data is ordered; mode is the value that occurs most often.

  29. How do you find the range of a data set, and how is the probability of a single event calculated?

    Range = largest value − smallest value. Probability of an event = number of favorable outcomes ÷ total number of equally likely outcomes.

  30. How do you find the probability of two independent events both occurring, and how does the counting (multiplication) principle work?

    Multiply the probabilities of the individual events: P(A and B) = P(A) × P(B). Likewise, if one choice has m options and another has n options, there are m × n total combined outcomes.

ISEE (Independent School Entrance Examination) Mathematics Achievement FAQ

What is in the ISEE (Independent School Entrance Examination) Mathematics Achievement syllabus?

ISEE (Independent School Entrance Examination) Mathematics Achievement is split into 5 chapters: Numbers and Operations, Algebraic Concepts, Geometry, Measurement and Data Analysis, Statistics, and Probability. Together they contain 21 topics and 30 sub-topics.

How many chapters are in ISEE (Independent School Entrance Examination) Mathematics Achievement?

5 chapters, containing 21 topics and 30 sub-topics. The largest chapter is Numbers and Operations, with 5 topics.

Are there flashcards for ISEE (Independent School Entrance Examination) Mathematics Achievement?

Yes — a 50-card Mathematics Achievement deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.