OAE Middle Grade Math

Horizontal Stretch/Shrink (Transformation)

A number,a, multiplied inside the function causes a horizontal compression (shrink) and 1/a causes a stretch

Sample

A subset of the population. It has statistics, which are used to estimate the population parameters

Tessellation

A tiling of the plane by geometric figures with no gaps and no overlap. In order for this to happen, the sum of the measures of the angle around any vertex must be 360.

Population

All items of interest. It has parameters, descriptive measures (ie. mean and standard deviation)

Variance

Average of standard deviation from mean

Mean

Average! By adding all values of the members of a set and then dividing by the number of members

Quantitative Graphs

Histogram, Stem and Leaf Plot, and Box and Whisker Plot

Factoring Trinomials

If a trinomial has a leading coefficient 1, like ax^2 + bx + c, we look for numbers that multiply to c and add to b For example: x^2 + 8x + 7 where two values multiply to 7 and add to 8 = (x + 7)(x + 1)

The 2nd Derivative

The rate of change of the first derivative (how fast the slope is changing)

Three Regular Polygons that Tesselate in Euclidean Plane

Triangle (equilateral), Square, and Hexagon

Bar Graphs

Useful when comparing different quantities or when tracking change over time

Pie Charts

Useful when examining how parts make up a whole

>

greater than

Zero Correlation

The two measurements are not related to each other along a straight line

If two chords intersect in a circle, then what?

Then the product of the segments of the first chord is equal to the product of the segment of the second chord

Non-Mutually Exclusive Events

Those which can occur simultaneously! A and B are NOT disjoint. P(A or B) = P(A) + P(B) - P(A and B)

Mutually Exclusive Events

Those which cannot occur simultaneously! A and B are disjoint. P(A or B) = P(A) + P(B)

Independent Events

Two (or more) events are independent if the occurence of one event has no effect upon whether or not the other event occurs. P(A and B) = P(A)*P(B)

Voltage Formula

V = IR

Work Formula

W = Fd

Conditional Probability

What's the probability of event A happening given that event B happens P(A|B) = P(A and B) / P(B)

Permutation

With dependent choices, consider using factorials. The order DOES matter!

Percentage of Data (in a circle)

[degree of central angle / degree of circle] x 100

Rhombus (Kite)

2 sets of congruent adjacent sides

Parallelogram

2 sets of opposite parallel sides

Rectangle

4 right angles

Square

4 right angles and 4 congruent sides

Quadrilateral

4-sided and sum of interior angles = 360

"They have 5 more x than y"

5 + y = x

Intersection Venn Diagram

A (upside down U) B

Area of Rectangle

A = lw

Union Venn Diagram

A U B

Set (or Elements)

A collection of related terms such as the members of a team or the single digit even numbers

Sequence

A function that assigns a value to each term

Horizontal Shift (Transformation)

A number added or subtracted inside the function represents a horizontal shift in the opposite direction as what you might suspect

Vertical Shift (Transformation)

A number added or subtracted outside the function represents a vertical shift

Reflections (Transformation)

A number outside the function reflects across the x-axis, and a negative number inside the function reflects across the y-axis

Vertical Stretch/Shrink (Transformation)

A number, a, multiplied by the outside of the function causes a stretch and 1/a causes a shrink

Reflexive Property

A quantity is congruent to itself

Population Growth

A(t) = Ae^(kt) *Can be dealt with Half-Life*

Positive Correlation

As one measurement increases, the other measurement also increases

Negative Correlation

As one measurement increases, the other measurement decreases

Volume of a Prism (area of base)

Base x Height

Rational Numbers

Be represented as the ratio of two integers. That is, as a fraction where the denominator is always positive

Prime Number

Can be divided only by one and itself without leaving a reminder

Pascal's Triangle

Can be used to solve Combination problems

Subset

Contains from zero to all of the members of the set

Dot Plot

Create by putting a dot for each value or group of values. Similar to Stem and Leaf Plot but rotated 90 degrees

Density Formula

D = m/v

Box and Whisker Plot

Divides data into quartiles and plot extremes. Provide visual comparisons between data sets. 50% of data in Interquartile Range (IQR). Minimum Q1 = 25th percentile Q2 (Median) = 50th percentile Q3 = 75th percentile Maximum IQR = Q3 - Q1

Histogram

Dividing data into even categories

Combination

Each choice is independent, meaning what you chose in one group does not affect the options you may choose from in the next. Order does not matter!

30-60-90 Triangle

Equilateral Triangle

Simple Random Sampling

Every possible of n individuals is equally likely to be selected

Newton's 2nd Law

F = ma

Complement Rule

For any event A, the probability of A not happening P(not A) = 1 - P(A)

Identity Property of Addition

For every real number a, a + 0 = a

Identity Property of Multiplication

For every real number a, a x 1 = a

Arithmetic Sequence

Has a common difference (like: 2, 4, 6,....)

Geometric Sequence

Has a common ratio of successive terms, called r

Transitive Property

If a = b and b = c, then a = c

Subtraction Property

If a = b, then a - c = b - c

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angle opposite those sides are congruent

How can we tell Two Triangles are Similar?

If two triangles of different sizes have all the same angle measurement, then these triangles are similar

Degree Measure of an Arc

In a circle, it is equal to the measure of the central angle that intercepts the arc

45-45-90 Triangle

Isosceles Right Triangle

Length of an Arc

It is a portion of the circumference. arc length / circumference = arc measure / 360 degrees

Kinetic Energy Formula

KE = 1/2 mv^2

Deductive Reasoning

Logic based on previously agreed upon assumptions, facts, or definitions. This is the goal in any type of proof. For example: Valentine's Day is February 14. Today is February 14, so I can conclude that today is Valentine's Day

Inductive Reasoning

Making an unproved statement based on extending observed pattern. For example: "I've always gotten A's in math classes before, so I should get an A this semester" Inductive reasoning is NOT a good way to justify arguments, but is used to extend patterns

Median

Middle Most Value! By putting the member of a set into sequential order and selecting the middle value

Probability

Number of ways to get desired outcomes / total possible outcomes

Momentum Formula

P = mv

Categorical Graphs

Pie Chart, Bar Graph, Dot Plot and Time Plot

Composite Numbers

Positive integers that can be divided by a number greater than one. Any non-prime positive integer greater than one is a composite. It can be factored into its prime factors - 105 = (3)(5)(7)

Projectile Motion

Quadratic function are commonly used to model projectile motion, using the function h(t) = gt^2 + vt + h where g is -16ft/sec or -4.9m/sec, v is the initial velocity, and h is the initial height. It is important to consider the vertex (maximum height), x-intercepts (time when the object hits the ground) and y-intercept, h

Types of Function that are Discontinous

Removable Discontinuity: a hole in the graph of the function. Nonremoveable Discontinuity: A jump or an asymptote in the graph (If the function factors and the bottom terms cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it. If the terms do not cancel, then the graph has a vertical asymptote)

Dilation (Transformation)

Retain shape but changes size. Resulting image is in proportion (or scale) to original (preimage). Center of dilation by a scale factor constant

Stratified Random Sampling

Separate population into mutually exclusive, homogeneous sets (strata) and randomly sample each stratum in population to the overall population

Domain

Set of input values (x-values)

Range

Set of output values (y-values)

Triangle Congruence Criteria

Side-Side-Side (SSS) Side-Angle-Side (SAS) Angle-Side-Angle (ASA) Side-Angle-Angle (SAA) Hypotuse-Leg (HL) (Two figures are congruent if there is a sequence of basic rigid motions (rotation, reflection, translation) that maps one figure onto the other)

The 1st Derivative

Slope (tangent) at a point. If the derivative is zero, then the slope is horizontal, at an optimum (min or max) point. Velocity

Standard Deviation

Square root of variance

Stem and Leaf Plot

Tens digit is the stem, unit digits are the leaf

Whole Numbers

The Non-negative integers, the set of Natural Numbers plus zero. - Exclude: fractions and decimals - An infinite set

Finite Arithmetic Series

The Sum of n terms with first term A1 and last term An

Rotation by 180 degree about the origin

The general rule: (A,B) --> (-A,-B)

Rotation by 90 degree about the origin

The general rule: (A,B) --> (-B,A)

Rotation by 270 degree about the origin

The general rule: (A,B) --> (B,-A)

Not Affected by Outliers

The mean and IQR

Affected by Outliers

The mean and the standard deviation are largely affected by outliers in the data

Mode

The member of a set that occurs the MOST

Integers, Z

The natural numbers, the negatives of the natural numbers and zero

Intersection

The set of all members contained in both sets For example: A = {1,3,5} B = {1,2,3} A and B = {1,3}

Union

The set of all members contained in either set For example: A = {1,3,5} B{1,2,3} A u B = {1,2,3,5} (no repetition)

Natural Numbers, N

The counting numbers, starting with one. - Exclude: negative numbers, zero, and fractions - Either prime or the multiple of primes

Rules of Changing the Direction of the Inequality Equations

- Multiplying (or dividing) both sides by a negative number - Swapping left and right hand sides

i^2

-1

How many inches are in a foot?

1 foot = 12 inches

How many seconds are in an hour?

1 hour = 3600 seconds

How many pounds are in a kilogram

1 kilograms = 2.2 pounds

How many feet are in a mile?

1 mile = 5280 ft

Trapezoid

1 set of opposite parallel side

How many inches are in a yard?

1 yard = 36 inches

Measure of an external angle of a circle

1/2 (large arc degree - small arc degree)

Additive Property "Grouping" of addition and multipication

(Multiplication) (3 x 2) x 5 = 3 x (2 x 5) (ab)c = a(bc)

Commutative Property of addition and multiplication

(Multiplication) 3 x 2 = 2 x 3 ab = ba

Graphing 2 Variable Linear Inequalities

- Graph as if an equation. - If < or >, change line to dash (note: points on this dashed line are NOT solutions). - Pick a point on either side of the line and test for inclusion. - If successful, that is the correct side. Otherwise, the other side is correct

<

less than

Standard Form of Parabola

x = - (b/2a) Will give you the x-coordinate of the vertex. Once you find the x-coordinate, plus it into the equation to find the y-coordinate.

"There is twice as many x as y"

x = 2y

Direct Variation

y = kx