NES Essential Academic Skills Subtest III: Mathematics key terms

The value a digit has because of its position in a numeral; in 4,703 the 7 means 700 while the 4 means 4,000. Regrouping and expanded form both depend on this idea.

A numeral written as the sum of the values of its digits, such as 4,703 = 4,000 + 700 + 0 + 3. It makes place value visible and supports mental computation.

Replacing a number with a nearby value of a chosen place, decided by whether the next digit is 5 or more. Used to estimate sums, products, and quotients quickly.

Finding an answer close enough for a purpose without exact computation, often by rounding or using compatible numbers. Useful for checking whether an exact answer is reasonable.

Numbers adjusted to friendlier values that divide or multiply easily, such as treating 4,180 ÷ 62 as 4,200 ÷ 60. They make mental estimation faster.

Factors are whole numbers multiplied to give a product; multiples are the products of a number with whole numbers. The factors of 12 are 1, 2, 3, 4, 6, 12; its multiples include 12, 24, 36.

A whole number greater than 1 whose only positive factors are 1 and itself, such as 2, 3, 5, 7, and 11. Every composite number can be written uniquely as a product of primes.

A whole number greater than 1 that has at least one positive factor besides 1 and itself, such as 9 or 14. It contrasts with prime numbers and with 1, which is neither.

Writing a composite number as a product of primes, for example 84 = 2 × 2 × 3 × 7. It underlies finding greatest common factors and least common multiples.

The largest whole number that divides two or more given numbers without a remainder; the GCF of 18 and 24 is 6. Used to simplify fractions and factor expressions.

The smallest positive number that is a multiple of two or more given numbers; the LCM of 6 and 8 is 24. It gives a common denominator when adding fractions.

Fractions naming the same value, produced by multiplying or dividing numerator and denominator by the same nonzero number; 2/3 equals 8/12. They allow fractions to be compared and combined.

Rewriting a fraction with the smallest possible whole-number numerator and denominator by dividing both by their greatest common factor; 18/24 becomes 3/4. The value is unchanged.

A shared multiple of the denominators of two or more fractions, letting them be added or subtracted directly. The least common denominator is the least common multiple of the denominators.

Any number expressible as a fraction of two integers with a nonzero denominator, including integers and terminating or repeating decimals. Its decimal form repeats or stops.

A real number that cannot be written as a fraction of integers; its decimal neither terminates nor repeats. Familiar examples are pi and the square root of 2.

Dividing the numerator by the denominator to express a fraction in decimal form; 3/8 becomes 0.375 and 1/3 becomes 0.333... Reversing the process uses place value over powers of ten.

A ratio out of 100, so 45% means 45/100 or 0.45. Percent problems are solved with the relationship part = percent × whole, often using a proportion or an equation.

The agreed sequence for evaluating expressions: grouping symbols, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right. It makes an expression have one value.

Rules such as commutative, associative, distributive, and identity properties that let expressions be rewritten without changing value. The distributive property links multiplication with addition over a sum.

A mathematical phrase combining numbers, variables, and operation signs but containing no equals sign, such as 3x + 5. It can be evaluated or rewritten but not solved.

A statement that two expressions are equal, such as 4x − 7 = 21. Solving means finding the value or values of the variable that make the statement true.

Operations that undo each other, namely addition with subtraction and multiplication with division. Solving an equation isolates the variable by applying them equally to both sides.

An equation whose variable appears only to the first power, so its graph is a straight line; 2x + 5 = 13 is one. It has at most one solution in one variable.

A statement comparing expressions with symbols such as <, >, ≤, or ≥. Its solution is a range of values, and multiplying or dividing by a negative reverses the symbol.

The steepness of a line, computed as rise over run or as the change in y divided by the change in x between two points. Parallel lines share a slope; perpendicular slopes multiply to −1.

The grid formed by a horizontal x-axis and vertical y-axis meeting at the origin, with points named by ordered pairs (x, y). It displays relationships between two variables.

Two numbers written in a fixed order, (x, y), locating a point on the coordinate plane; the first number is horizontal and the second vertical. Order matters: (3, 5) is not (5, 3).

An equation stating that two ratios are equal, such as 3/4 = x/12. Cross-multiplying solves it, and proportions model scaling, rates, and percent problems.

A comparison of two quantities by division, written as 3:5, 3 to 5, or 3/5. It expresses relative size rather than absolute amount and can be scaled up or down.

A rate expressed per one unit of the second quantity, found by dividing; 150 miles in 3 hours gives 50 miles per hour. Unit rates allow direct comparison of different offers or speeds.

The total distance around a closed plane figure, found by adding side lengths or applying formulas such as 2(l + w) for a rectangle. It is measured in linear units.

The number of square units needed to cover a plane figure; a rectangle's area is length times width. Formulas for triangles, circles, and composite figures build on this idea.

The number of cubic units a solid occupies, found for a rectangular prism by multiplying length, width, and height. It is measured in cubic units, unlike surface area.

The total area of all faces or surfaces of a three-dimensional figure, measured in square units. For a rectangular prism it is the sum of the areas of six rectangles.

The distance around a circle, equal to pi times the diameter or 2π times the radius. It plays the role for circles that perimeter plays for polygons.

The radius is the distance from a circle's center to its edge; the diameter is twice the radius, spanning the circle through the center. Both feed circumference and area formulas.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². It finds a missing side and tests whether a triangle is right.

Figures with the same shape whose corresponding angles are equal and corresponding side lengths are proportional. Their areas compare as the square of the scale factor.

The ratio relating corresponding lengths of a figure and its image under a dilation or similarity. Lengths scale by it, areas by its square, and volumes by its cube.

Changing a measurement to a different unit by multiplying by a conversion factor equal to 1, such as 12 inches per foot. Units cancel to confirm the setup is correct.

Solving measurement problems by tracking units and canceling them through multiplication by conversion ratios. It catches setup errors because the final unit must match what is asked.

The arithmetic average, found by adding all data values and dividing by how many there are. It is pulled toward outliers more than the median is.

The middle value of a data set arranged in order, or the average of the two middle values when the count is even. It resists distortion by extreme values.

The value or values that occur most often in a data set; a set may have one mode, several, or none. It applies to categorical data as well as numerical data.

The difference between the greatest and least values in a data set, a simple measure of spread. It uses only two values and ignores how the rest are distributed.

A data value far from the rest of the set, often from error or a genuinely unusual case. It can shift the mean substantially while barely affecting the median.

A number from 0 to 1 describing how likely an event is, computed for equally likely outcomes as favorable outcomes divided by total outcomes. It may also come from data or simulation.

The set of all possible outcomes of an experiment, such as the 36 ordered pairs from rolling two dice. Listing it systematically makes counting favorable outcomes reliable.

Events whose outcomes do not affect each other's probabilities, so the chance both occur is the product of their individual probabilities. Coin flips are independent; drawing without replacement is not.

The event that a given event does not happen; its probability is 1 minus the probability of the event. It often simplifies problems that ask for at least one occurrence.

If one choice can be made in m ways and a second in n ways, the pair can be made in m × n ways. It extends to longer sequences and underlies many probability counts.

Reading values, trends, and comparisons from bar graphs, line graphs, circle graphs, or scatter plots, including attention to scales and axis labels. Misleading scales are a common trap.

Judging whether an answer fits the situation and the numbers involved, using estimation, units, and context. It is the final check after any multi-step computation.

Breaking a word problem into sequential steps, identifying what is known and what is asked, then choosing operations. Working backward from the goal often reveals the needed order.

Regular arrangements of numbers or figures governed by a rule, such as adding 4 each time or doubling. Finding the rule allows prediction of later terms.

A line with evenly spaced marks representing numbers, used to compare values, show operations, and graph inequalities. Distance from zero gives absolute value.