Middle Grades Math GACE

Natural Numbers

Counting numbers, beginning with 1 (1, 2, 3...)

Whole Numbers

Zero and the natural numbers (0, 1, 2, 3...)

Integers

The positive and negative values of the whole numbers (-1, 0, 1, 2...)

Rational Number

Any number that can be expressed as a fraction with an integer as the numerator, and a non-zero integer as the denominator

Irrational Number

Any non-terminating, non-repeating number that CANNOT be expressed as a fraction.

Real Numbers

The set of all numbers that fall into any of the above categories

Kinds of Numbers

Real, Natural, Whole, Rational, & Irrational

Complex Numbers

Any number that contains the imaginary number i

Additive Identity

any number plus zero equals that number

Additive Inverse

The opposite of a number

Closure

In addition, a + b is a real number. In multiplication, ab is a real number

Multiplicative Inverse

The reciprocal of a number

Whole numbers

the counting numbers along with zero.

Ex: 0, 1, 2, 3, 4, ...

Decimals

portions of ten (deci = part of ten).

Factors

Whole numbers that can be multiplied together to get another whole number.

Prime numbers

whole numbers greater than 1 that have only two factors - 1 and itself.

Composite numbers

whole numbers that have factors other than 1 and the number itself

Greatest common factor (GCF)

greatest number that is a factor of all the numbers given in a series. Can be no greater than the smallest number given in the series.

Least common multiple (LCM)

the smallest number in a group of numbers into which all of the given numbers will divide evenly. Will always be the largest of the given numbers or a multiple of the largest number.

Manipulatives

materials that students can physically handle and move.

Rational numbers

can be expressed as the ratio of two integers, a/b where b can not be equal to 0, for example 2/3, -4/5, 5=5/1

include integers, fractions, and mixed numbers, and terminating and repeating decimals.

Irrational numbers

real numbers that cannot be written as the ratio of two integers. Infinite non-repeating and non-terminating decimals. pi, 2.236...

Properties

rules that apply for addition, subtraction, multiplication, or division of real numbers.

Associative property

terms can be regrouped without changing the result

addition: a+(b+c) = (a+b) +c

multiplication: a(bc) = (ab)(c)

Identity property

the number is unchanged by the operation

additive: sum of any # and 0 is itself a+0=a

multiplicative: product of any # and 1 is itself

a x 1=a

Inverse property

operation results in 0 or 1

additive: a + (-a) = 0

multiplicative: a x (1/a) = 1

Perimeter

sum of the length of all sides of a polygon

Area

number of square units covered bu the figure or the space that a figure occupies

Circumference

distance around a circle

Cylinder

solid with 2 congruent bases that are parallel

Sphere

solid with all points the same distance from center

Cone

solid with circular base and single vertex

Square pyramid

solid with square base and 4 triangle-shaped faces.

Tetrahedron

solid with 4 triangle faces

Prism

solid with 2 congruent, parallel bases that are polygons

Net

2D figure that can be cut out and folded up to make a 3D solid

Point

indicated a place or position in space

Line

considered a straight set of points that does not end.

Plane

set of points composing a flat surface.

Line segment

straight set of points with 2 end points

Ray

has exactly one endpoint and extends indefinitely in one direction

Perpendicular lines

forms a 90 degree angle to each other.

Parallel lines

do not intersect

Skew lines

do not intersect because they do not lie on the same plane

Angle

formed by the intersection of 2 rays

Right angle

measures 90 degrees

Acute angle

measures more than 0 and less than 90 degrees

Obtuse angle

measures more than 90 degrees and less than 180 degrees

Straight angle

measures 180 degrees

Reflexive angle

measures more than 180 degrees and less than 360 degrees

Adjacent angles

have common vertex and one common side but no interior points in common

Complementary angles

add up to 90 degrees

Supplementary angles

add up to 180 degrees

Vertical angles

have sides that form 2 pairs of opposite rays

Corresponding angles

in the same corresponding position on 2 parallel lines cut by a transversal

Parallel lines postulate

If 2 lines are parallel and are cut by a transversal, corresponding angles have the same measure

Alternate Interior Angles Theorem

If 2 parallel lines are cut by a transversal, the alternate interior lines are congruent.

Congruent

same size and shape

Polygon

simple, closed, 2D figure composed of line segments

triangle

polygon with 3 sides

Quadrilateral

polygon with 4 sides. Sum of measure of angles = 360

Trapezoid

quadrilateral with 1 pair of parallel sides

Isosceles trapezoid

non-parallel sides are congruent

Parallelogram

quadrilateral with 2 pairs of parallel sides.

Rectangle

parallelogram with 1 right angle

Rhombus

parallelogram with all sides of equal length

Square

rectangle with all equal sides

Coordinate plane

consists of 2 perpendicular lines or axes that intersect at a point known as the origin.

Positive and negative directions for each axis.

x-axis = horizontal

y-axis = vertical

Divided into 4 quadrants

Coordinates

unique ordered pair of numbers that identify the location of a point on a coordinate plane. (x,y)

Slope

the slant of a line

(y-y)/(x-x)

Parallel lines have same

With perpendicular lines they are negative reciprocals of each other

Transformation

change in position, shape, or size of a geometric figure

the image of an object

if original object was labeled with ABCD, the image may be labeled with the same letters followed bu a prime symbol, A'B'C'D'.

Transformational geometry

the study of manipulating objects by flipping, twisting, turning, and scaling.

Symmetry

reflected across a line

Line of symmetry

line where a reflection occurs

Translation

transformation that keeps original size and shape but slides another a fixed distance in on direction

Reflection

same shape and size, but face in opposite directions across line

Rotation

transformation that turns a figure about a fixed point

Dilation

transformation that reduces or enlarges a figure by a scale factor

Arithmetic sequence

set of terms with a common difference between the terms.

Formula: an = a1 + ( n - 1 )d

an = the nth term

a1 = the first term

n = the number of the term sequence

d = the common difference

Geometric sequence

series of numbers in which a common ration can be multiplied by a term to yield the next term.

Formula: r = (an+1) / (an)

where r = common ratio

an = the nth term

r is then plugged into the formula: an = (a1)r^n-1

Relation

any set of ordered pairs

Domain

the set made of all the first coordinates of the ordered pairs (the x values)

Range (in functions)

the set made of all the second coordinates of the ordered pairs (the y values)

Function

a relation in which each value for the domain has a unique value for the range. X values are not repeated

Vertical line test

used to look for a function or to prove if a function or not.

If any vertical line intersects the graph of a relation in more than one point, it is not a function

Mapping

a diagram with arrows drawn from each element of the domain to the corresponding elements of the range. If 2 arrows are drawn from the same element of the domain it is not a function

Ratio

comparison of two numbers

Proportion

an equation in which a ratio is set equal to another in order to find an unknown quantity in one of the ratios

Slope intercept form:

y = mb + b

Mean

average

sum of numbers given divided by # of numbers

Median

the middle number (after being placed in numerical order)

Mode

the number that occurs the greatest frequency

Range (data)

difference between the highest and the lowest value in the data set

Tells the scope of the data.

Variance

the sum of the squares of the deviation of each data item x from the mean x- divided by the number of items N.

Looks as how far each value in the data set is from the mean

Standard Deviation

square root of the variance

defines how wide the spread of the data is.

Outlier

a number in a set of data that is much larger or smaller than most of the other numbers in the set

Counting Principle

n * m

to find the number of possibilities in a sequence of 2 distinct events, 1 not effecting the other

Addition Principle of Counting

If A & B are events, n(AorB) = n(A) + n(B) - n(AnB)

Permutation formula

order matters

Combinations formula

order does not matter