SAT MATH TOOLKIT (copy)

An integer is any number that can be expressed without a fraction, decimal, percentage sign, or symbol.

Only integers can be odd or even- A fraction or symbolic number is neither odd or even.

A number line is a simple diagram that arranges numbers from least to greatest.

A prime number is a number that has exactly 2 factors: 1 and itself.

All prime numbers are positive

The only even prime number is 2.

1 is not a prime number because it has only 1 factor(itself), while prime numbers must have exactly 2 factors

Composite numbers can be divide by more than 2 numbers

Ratios, proportions, and percentages all express a relationship between two numbers.

A ratio is often written as a pair of numbers with a colon between them.

A proportion is usually written as a fraction with a number in the numerator compared to the number in the denominator.

If the relationship between two quantities is the kind where increasing one quantity results in a consistent increase in the other quantity, then we say those two quantities vary directly or are directly proportional.

If two quantities are related so that increasing one consistently decreases the other, then we say those two vary indirectly or are inversely proportional.

To make a fraction into a decimal, divide the numerator by the denominator.

To make a fraction into a percentage, divide the numerator by the denominator, then multiply by 100.

To make a percentage into a fraction, give the original percentage a denominator of 100, and then simplify if necessary.

The distance formula is Distance= Rate x Time(d=rt).

A rate is a ratio that tells us how often one even happens in relation to another event happening. Rates are usually but not always expressed in terms of time.

To find the distance traveled by an object, we can multiply the rate at which the object travels by the time that the object spends traveling at that rate. d=rt

We can use the same formula to find out how long an object takes to cover a certain distance in a given time. We can also use the formula to find out the speed of an object that covers a certain distance in a given time. The same relationship can be used to discuss the rates at which other processes happen, not just the rate of movement of an object.

For our purposes, the complex conjugate of a complex number is another complex number whose real component is identical but whose imaginary component has the opposite sign but is otherwise identical. For example, if we’re given the imaginary number 5+2i, then its complex conjugate is 5-2i.

To do this, we take the fraction with the complex denominator and multiply it by a fraction whose numerator and denominator are both equal to the complex conjugate of the denominator of the first fraction.

In the context of SAT “MATH”, a set is a group of specific things- usually a group of numbers. In set notation, there are curly braces on either end of the set, and commas between the elements of the set. Example:The set of positive integers less than 4 is as follows: {1,2,3}

For our purposes, algebra is the process of using variables like x to stand for unknown numbers in mathematical expressions, and then manipulating those expressions to find the values of one or more of those unknown numbers.

On the SAT, an equation is a statement that involves an algebraic expressions and an equal sign.

 A system of equations contains two or more equations with the same variables.

A solution to a system of equations is a set of values that creates a valid statement when plugged into each equation in the system. It is also an intersection point between two equations on a graph.

On the SAT, inequalities are statements that show a particular amount is greater than or less than a second amount.

Inclusive means included in that range. The phrase “from 0 to 10, inclusive,” means that 0 and 10 are included in the range.

Exclusive means not included in that range. The phrase “from 0 to 10, exclusive,” means that 0 and 10 are not included in the range.

You solve an inequality the same way you solve an equation, with one difference: when you divide or multiply by a negative number to solve for a variable, you have to switch the direction of the inequality symbol.

When we graph an inequality, we end up with a shaded region on the side of the line that contains all the points that are solutions to the inequality.

We can check whether a particular point is a solution of an inequality by plugging that point into that inequality. If the result is a valid statement, then the point is a solution of the inequality.

When we graph a system of inequalities, the solution region will be the set of points in the coordinate plane that satisfy both inequalities.

INITIAL AMOUNT OF SUBSTANCE(rate of growth or decay)^amount of time. Or A(b)^x

Growth and Decay refer to the rate the function is increasing or decreasing. If as x approaches infinity/increase the rate increases that’s exponential Growth. If as x approaches infinity/increases the rate decreases it’s exponential decay. EXPONENTIAL GROWTH OR DECAY DOES NOT MEAN THAT THE VALUE OF THE FUNCTION/ Y ITSELF IS INCREASING OR DECREASING AS X INCREASES, IT ONLY TALKS ABOUT THE RATE.

Because the negative turn the fraction into its reciprocal which is an increasing rate/greater than 100%. A(0.80)^-x —> A(4/5)^-x —> A(5/4)^x, 5/4= 1.25 =125%.

A constant is a number that has a set, defined value that can’t change.

The Degree of a variable in a polynomial refers to the highest exponent to which that variable is raised in the expression.

Three types of special angle relationships that appear often on the SAT are vertical, supplementary, and complementary angles.

Vertical angles are the pairs of angles that lie across from each other when two lines intersect. In a pair of vertical angles, two angles are equal to each other.

Supplementary angles are pairs of angles whose measurements add up to 180 degrees. When supplementary angles are next to each other, they form a straight line.

Complementary angles describe angles whose measures add up to 90 degrees.

A transversal is the result when a line crosses two parallel lines.

The sum of the measures of the angles in any triangle is 180 degrees

(n-2)x180, where n is the number of sides.

Biggest angle.

Smallest angle.

60 degrees.

Two of the 3 sides. Two of the 3 angles.

A ninety degree angle as one of its 3 angles.

A Pythagorean triple is a set of numbers that can be all the lengths of the sides of the same right triangle. 

When we multiply each number in a pythagorean triple by the same number, we get another pythagorean triple.

Two triangles are similar triangles if they have all the same angle measurements.

Between two similar triangles, the relationship between any of your corresponding sides is the same as between any other two corresponding sides. 

Between any two points on a line, there is a midpoint that is halfway between the two points.

A set of points that fall on the same line.

We can use the pythagorean theorem or the distance formula.

Pythagorean theorem is A^2+B^2=C^2 Distance formula is -B+- sqrt B^2 -4ac /2a

Equal to each other.

360

90 degrees.

The perimeter of a two dimensional object that is not a circle is the sum of the lengths of its sides

The perimeter for a circle is the distance around the circle.

To find the perimeter of a non-circle, just add up the lengths of its sides.

The perimeter of a circle is the circumference which is 2πr

To find the area of a polygon besides a triangle, parallelogram, or circle, just divide the polygon into smaller triangles, polygons, and/or circles and find the area of these pieces.

A radius is a line segment drawn from the center point of a circle to the edge of the circle.

Length. Same distance.

A diameter is a line segment drawn from one edge of a circle, through the center of the circle and all the way to the opposite edge. 

A diameter always has a length equal to twice the radius of the circle, because a diameter can be broken into two opposite radii.

Formula for area of a circle is A= πr^2

A tangent is a line that intersects a circle at only 1 point. A tangent is perpendicular to the radius ending at the shared point.

An arc is a portion of a circle that is measured in degrees, like an angle. We can measure an arc by drawing radii to the endpoints of the arc, and then measuring the angles formed by the radii at the center of the circle.

Two points on the circumference of a circle can define two different arcs. One arc is the shorter distance between the two points, while the other arc is the longer distance between the two points. When context isn’t sufficient we can use the term minor and major arc.

A central angle is an angle whose vertex is the center of a circle and whose sides are radii of that circle.

A chord is a line segment whose endpoints are points on the circumference of a circle. Note that the diameter is a chord that passes through the center of the circle.

An inscribed angle is an angle created by two cords that meet on one end to form a vertex.

A sector is a portion of a circle defined by the center of the circle and two points on the circle. 

We can find the area of a sector by finding the area of the entire circle, and then multiplying by the fraction of the circle represented by the sector. 

The mean or arithmetic mean of a set of numbers is the result you get when you add all the numbers together and then divide by the number of things you added.

The median of a set of numbers is the number that appears in the middle of the set when all the numbers in the set are arranged from least to greatest . If there are 2 numbers left in the middle combine the 2 then divide it by 2.

The mode of a set of numbers is the number that appears most frequently in the set.

The range of a set of numbers is the difference between the highest number in the set and the lowest number in the set.

An outlier is a data point that doesn’t follow the trend established by the other data points in the set.

The probability of an event is a fraction from 0 to 1 that describes how likely the event is to happen. If the fraction is closer to 1, the event is more likely to happen; if the fraction is closer to 0, the event is less likely to happen.

To determine the probability, you first calculate the total number of possible outcomes and place this number in the denominator of the fraction; then, you determine the number of outcomes that satisfy the event’s requirements, and place this number in the numerator of the fraction.

To find the probability of two or more events happening in a sequence, we just find the probability of each event by itself, and then multiply them by each other. 

A sample should include as many different parts of the whole as possible, to maximize the chance that it will accurately reflect the whole