ISEE (Independent School Entrance Examination) Quantitative Reasoning Syllabus

What does 'logic without computation' mean on the ISEE?

Using reasoning about size, sign, or relationships to reach an answer without performing the full arithmetic.

Without computing, is the product of two numbers each greater than 1 larger or smaller than either factor?

Larger than either factor.

Without calculating, what happens to a positive number when you multiply it by a fraction between 0 and 1?

The result gets smaller than the original number.

Without calculating, what happens to a number when you divide it by a fraction between 0 and 1?

The result gets larger than the original number.

What two quantities are compared in the ISEE Quantitative Comparison format?

Column A and Column B; you decide which is greater, if they are equal, or if it cannot be determined.

What are the four answer choices in an ISEE Quantitative Comparison question?

(A) Column A is greater, (B) Column B is greater, (C) the two columns are equal, (D) the relationship cannot be determined.

In Quantitative Comparison, what is a smart strategy when both columns share the same term or operation?

Cancel or ignore the identical part and compare only what differs between the columns.

In Quantitative Comparison, why should you test multiple kinds of numbers?

Plugging in positives, negatives, zero, and fractions can reveal that the relationship changes, pointing to 'cannot be determined.'

In Quantitative Comparison, what should you do before choosing an answer if a variable has no stated restrictions?

Test extreme and special values (0, 1, negatives, fractions) to see if the comparison stays consistent.

When is the correct Quantitative Comparison answer 'cannot be determined'?

When different valid values for the unknown produce different relationships (e.g., sometimes A is bigger, sometimes B is).

If Column A = x² and Column B = x with no restriction on x, why is the answer 'cannot be determined'?

For x = 2, x² is larger; for x = 1/2, x is larger; for x = 1 they are equal—so no single relationship holds.

What is a key sign that a Quantitative Comparison answer is NOT 'cannot be determined'?

If both columns are made only of specific numbers (no variables), there is always a definite answer—never D.

On a bar graph, what does the height (or length) of each bar represent?

The value or quantity of the category that bar stands for, read against the labeled axis.

What does a line graph best show?

How a quantity changes over time or shows a trend, with the slope indicating increase or decrease.

On a line graph, what does a steeper line segment indicate?

A faster rate of change (greater increase or decrease) over that interval.

What does a pie chart represent, and what must its slices total?

It shows parts of a whole as proportions; all slices together total 100% (a full circle of 360°).

If a pie chart slice is 25% of the whole, how many degrees is its central angle?

90° (because 25% of 360° = 90°).

How do you find the actual amount represented by a pie chart slice?

Multiply the slice's percentage (as a decimal) by the total value the whole circle represents.

In a pictograph, how do you read a row's value?

Multiply the number of symbols by the value each symbol represents (given in the key), counting partial symbols proportionally.

What is a frequency table?

A table showing how many times (the frequency) each value or category occurs in a data set.

How do you find the mean (average) of a data set?

Add all the values, then divide by the number of values.

How do you find the median of a data set?

Order the values from least to greatest; the median is the middle value (or the average of the two middle values if the count is even).

How do you find the mode of a data set?

The mode is the value that appears most often; a set can have no mode or more than one mode.

How do you find the range of a data set?

Subtract the smallest value from the largest value.

Find the mean, median, mode, and range of 4, 8, 8, 10, 5.

Mean = 35/5 = 7; median = 8 (ordered: 4,5,8,8,10); mode = 8; range = 10 - 4 = 6.

If a set has an even number of values, how is the median calculated?

Average the two middle values after ordering the data.

How do you find a missing value when you know the mean and the other values?

Multiply the mean by the total count to get the sum, then subtract the known values to find the missing one.

What is the basic formula for the probability of an event?

Probability = number of favorable outcomes divided by total number of possible outcomes.

What is the probability of rolling a 3 on one standard six-sided die?

1/6 (one favorable outcome out of six equally likely outcomes).

What range of values can a probability take, and what do 0 and 1 mean?

Probability ranges from 0 to 1; 0 means impossible and 1 means certain.

How do you find the probability that an event does NOT happen?

Subtract the probability that it does happen from 1 (the complement rule).

A bag has 3 red and 5 blue marbles. What is the probability of drawing a red marble?

3/8 (3 red out of 8 total marbles).

When drawing conclusions from a data set, why is it important to read titles, labels, and units first?

They tell you what the data measures and in what units, preventing misinterpretation of the numbers.

What is the danger of drawing a conclusion that goes beyond what the data shows?

Data only supports claims it directly measures; assuming causes or future results not shown leads to invalid conclusions.

What is a quick way to round a number to a given place value?

Look at the digit just to the right of that place: round up if it is 5 or more, keep the same if it is 4 or less.

Round 3,847 to the nearest hundred.

3,800 (the tens digit 4 is less than 5, so the hundreds digit stays).

Why use estimation/approximation before computing an exact answer?

It gives a quick ballpark to check that the exact answer is reasonable and to eliminate impossible answer choices.

To estimate 48 x 21 quickly, what should you do?

Round to 50 x 20 = 1,000, giving a close estimate of the product.

What does it mean to check the 'reasonableness' of an answer?

Judging whether the answer makes sense in size and context—e.g., a person's age shouldn't be 200, or a discount price shouldn't exceed the original.

If a question asks for a discounted price and your answer is higher than the original, what does that tell you?

The answer is unreasonable; you likely added instead of subtracting and should recheck.

What is a mental-math trick for multiplying a number by 10, 100, or 1000?

Move the decimal point right by 1, 2, or 3 places (or append that many zeros to a whole number).

What is a fast mental-math way to find 10% of a number?

Move the decimal point one place to the left.

How can you quickly find 5% of a number using mental math?

Find 10% (move decimal left one place) and then take half of that.

What mental-math strategy helps multiply 25 x 16?

Use 25 = 100/4: 100 x 16 = 1600, then divide by 4 = 400.

How can rearranging addends speed up adding 17 + 25 + 3?

Group friendly pairs: 17 + 3 = 20, then 20 + 25 = 45.

What is the difference between the mean and the median as 'averages'?

The mean is the arithmetic average of all values; the median is the middle value. The median is less affected by extreme outliers.

Why can an outlier make the mean misleading?

A very large or very small value pulls the mean toward it, so the median may better represent a typical value.

On a double bar graph, how do you compare two groups for one category?

Compare the heights of the two side-by-side bars within that category against the value axis.

How do you translate 'the quotient of a number and 4 increased by 9' into an expression?

n/4 + 9.

What should you do when two pie chart percentages are given but you need a count?

Convert each percentage to a decimal and multiply by the total, then compare or combine the resulting counts.

In a quantitative comparison where Column A = the average of 10 and 20 and Column B = 15, what is the answer?

They are equal (C); the average of 10 and 20 is 15.