GRADES EIGHT THROUGH TWELVE MATHEMATICS

Glossary

absolute value. A number’s distance from zero
on the number line . The absolute value of -4 is
4; the absolute value of 4 is 4.

algorithm. An organized procedure for performing
a given type of calculation or solving a
given type of problem. An example is long
division.

arithmetic sequence . A sequence of elements,

. , such that the difference of
successive terms is a constant

for example, the sequence

where the common difference is 3.

asymptotes. Straight lines that have the property
of becoming and staying arbitrarily close to the
curve as the distance from the origin increases
to infinity. For example, the x-axis is the only
asymptote to the graph of sin(x)/x.

axiom. A basic assumption about a mathematical
system from which theorems can be deduced.
For example, the system could be the points
and lines in the plane. Then an axiom would be
that given any two distinct points in the plane,
there is a unique line through them.

binomial. In algebra, an expression consisting of
the sum or difference of two monomials (see
the definition of monomial), such as 4a-8b.

binomial distribution . In probability, a binomial
distribution gives the probabilities of k outcomes
A (or n-k outcomes B) in n independent
trials for a two-outcome experiment in which
the possible outcomes are denoted A and B.

dilation. In geometry, a transformation D of the
plane or space is a dilation at a point P if it
takes P to itself, preserves angles , multiplies
distances from P by a positive real number r,
and takes every ray through P onto itself. In
case P is the origin for a Cartesian coordinate
system in the plane, then the dilation D maps
the point (x, y) to the point (rx, ry).

dimensional analysis. A method of manipulating
unit measures algebraically to determine the
proper units for a quantity computed algebraically.
For example, velocity has units of the
form length over time (e.g., meters per second
[m/sec]), and acceleration has units of velocity
over time; so it follows that acceleration has
units (m/sec)/sec = m/(sec²).

expanded form . The expanded form of an algebraic
expression is the equivalent expression
without parentheses. For example, the expanded
form of (a + b)² is a² + 2ab + b².

exponent. The power to which a number or
variable is raised (the exponent may be any
real number).

exponential function. A function commonly
used to study growth and decay. It has the
form y = a^x with a positive.

factors. Any of two or more quantities that
are multiplied together. In the expression
3.712 × 11.315, the factors are 3.712 and 11.315.

function. A correspondence in which values of
one variable determine the values of another.

geometric sequence. A sequence in which there is
a common ratio between successive terms.
Each successive term of a geometric sequence
is found by multiplying the preceding term by
the common ratio. For example, in the sequence
the common ratio
is 3.

parallel. Given distinct lines in the plane that are
infinite in both directions, the lines are parallel
if they never meet. Two distinct lines in the
coordinate plane are parallel if and only if they
have the same slope.

permutation. A permutation of the set of numbers
is a reordering of these
numbers.

polar coordinates. The coordinate system for the
plane based on rθ, the distance from the origin
and θ, and the angle between the positive
x-axis and the ray from the origin to the point.

polar equation. Any relation between the polar
coordinates (r, θ) of a set of points (e.g., r =
2cosθ is the polar equation of a circle).

polynomial. In algebra, a sum of monomials; for
example, x² + 2xy + y².

prime. A natural number p greater than 1 is prime
if and only if the only positive integer factors
of p are 1 and p. The first seven primes are 2, 3,
5, 7, 11, 13, 17.

quadratic function. A function given by a polynomial
of degree 2.

random variable. A function on a probability
space.

range. In statistics, the difference between the
greatest and smallest values in a data set. In
mathematics , the image of a function.

ratio. A comparison expressed as a fraction. For
example, there is a ratio of three boys to two
girls in a class (3/2, 3:2).

rational numbers. Numbers that can be expressed
as the quotient of two integers; for
example, 7/3, 5/11, -5/13, 7 = 7/1.

real numbers. All rational and irrational numbers.

binomial theorem. In mathematics, a theorem
that specifies the complete expansion of a
binomial raised to any positive integer power.

box-and-whisker plot. A graphical method for
showing the median, quartiles, and extremes of
data. A box plot shows where the data are
spread out and where they are concentrated.

complex numbers . Numbers that have the form
a + bi where a and b are real numbers and i
satisfies the equation i² = -1. Multiplication is
denoted by (a+bi)(c+di) = (ac-bd) + (ad+bc)i,
and addition is denoted by (a+bi) + (c + di) =
(a+c) + (b+d)i.

congruent. Two shapes in the plane or in space
are congruent if there is a rigid motion that
identifies one with the other (see the definition
of rigid motion).

conjecture. An educated guess.

coordinate system. A rule of correspondence by
which two or more quantities locate points
unambiguously and which satisfies the further
property that points unambiguously determine
the quantities; for example, the usual Cartesian
coordinates x, y in the plane.

cosine. Cos(θ) is the x-coordinate of the point on
the unit circle so that the ray connecting the
point with the origin makes an angle of θ with
the positive x-axis. When θ is an angle of a
right triangle, then cos(θ) is the ratio of the
adjacent side with the hypotenuse.

histogram. A vertical block graph with no spaces
between the blocks. It is used to represent
frequency data in statistics.

inequality. A relationship between two quantities
indicating that one is strictly less than or less
than or equal to the other.

integers. The set consisting of the positive and
negative whole numbers and zero; for example,
.

irrational number. A number that cannot be
represented as an exact ratio of two integers.
For example, the square root of 2 or π.

linear expression. An expression of the form
x is variable and a and b are
constants; or in more variables, an expression
of the form ax + by +c, ax + by + cz + d, etc.

linear equation. An equation containing linear
expressions.

logarithm. The inverse of exponentiation; for
example, .

mean. In statistics, the average obtained by
dividing the sum of two or more quantities by
the number of these quantities.

median. In statistics, the quantity designating the
middle value in a set of numbers.

mode. In statistics, the value that occurs most
frequently in a given series of numbers.

monomial. In the variables x, y, z, a monomial is
an expression of the form , in which m,
n, and k are nonnegative integers and a is a
constant (e.g., 5x², 3x²y or 7x³yz²).

nonstandard unit. Unit of measurement expressed
in terms of objects (such as paper clips,
sticks of gum, shoes, etc.).

reflection. The reflection through a line in the
plane or a plane in space is the transformation
that takes each point in the plane to its mirror
image with respect to the line or its mirror
image with respect to the plane in space. It
produces a mirror image of a geometric figure.

rigid motion. A transformation of the plane or
space, which preserves distance and angles.

root extraction. Finding a number that can be
used as a factor a given number of times to
produce the original number; for example, the
fifth root of 32 = 2 because 2 × 2 × 2 × 2 × 2 =
32).

rotation. A rotation in the plane through an angle
θ and about a point P is a rigid motion T fixing
P so that if Q is distinct from P, then the angle
between the lines PQ and PT(Q) is always θ.
A rotation through an angle θ in space is a
rigid motion T fixing the points of a line l so
that it is a rotation through θ in the plane
perpendicular to l through some point on l.

scalar matrix. A matrix whose diagonal elements
are all equal while the nondiagonal elements
are all 0. The identity matrix is an example.
scatterplot. A graph of the points representing a
collection of data.

scientific notation. A shorthand way of writing
very large or very small numbers. A number
expressed in scientific notation is expressed as
a decimal number between 1 and 10 multiplied
by a power of 10 (e.g., 7000 = 7 × 10³ or
0.0000019 = 1.9 × 10^-6).

similarity. In geometry, two shapes R and S are
similar if there is a dilation D (see the definition
of dilation) that takes S to a shape congruent
to R. It follows that R and S are similar if
they are congruent after one of them is expanded
or shrunk.

sine. Sin(θ) is the y-coordinate of the point on the
unit circle so that the ray connecting the point
with the origin makes an angle of θ with the
positive x-axis. When θ is an angle of a right
triangle, then sin(θ) is the ratio of the opposite
side with the hypotenuse.

square root. The square roots of n are all the
numbers m so that m² = n. The square roots
of 16 are 4 and -4. The square roots of -16 are
4 i and -4 i.

standard deviation. A statistic that measures the
dispersion of a sample.

symmetry. A symmetry of a shape S in the plane
or space is a rigid motion T that takes S onto
itself (T(S) = S). For example, reflection
through a diagonal and a rotation through a
right angle about the center are both symmetries
of the square.

system of linear equations. Set of equations of
the first degree (e.g., x + y = 7 and x - y = 1).
A solution of a set of linear equations is a set of
numbers

. so that when the variables

are replaced by the numbers all the equations
are satisfied. For example, in the equations
above, x = 4 and y = 3 is a solution.

translation. A rigid motion of the plane or space
of the form X goes to X + V for a fixed vector V.

transversal. In geometry, given two or more lines
in the plane a transversal is a line distinct from
the original lines and intersects each of the
given lines in a single point.

unit fraction. A fraction whose numerator is 1
(e.g., 1⁄π, 1⁄3, 1⁄x). Every nonzero number may be
written as a unit fraction since, for n not equal
to 0, n = 1/(1/n).

variable. A placeholder in algebraic expressions;
for example, in 3x + y = 23, x and y are variables.

vector. Quantity that has magnitude (length) and
direction. It may be represented as a directed
line segment.

zeros of a function. The points at which the value of a function is zero.