ACT Math Checklist

How to Divide and Multiply Fractions

To divide fractions, multiply by the reciprocal of the second fraction. To multiply, multiply the numerators and denominators.

How to Add and Subtract Fractions

Find a common denominator, then add or subtract the numerators while keeping the denominator the same.

How to divide whole number by fractions and vice versa

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

Graphing (y=mx+b)

In the equation y=mx+b, m represents the slope and b represents the y-intercept.

Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction.

Finding Slope Using Coordinates

The slope (m) can be calculated using the formula m = (y2 - y1)/(x2 - x1).

Geometry (SAS, ASA, SSS, AAS)

These are criteria for triangle congruence: SAS (Side-Angle-Side), ASA (Angle-Side-Angle), SSS (Side-Side-Side), AAS (Angle-Angle-Side).

Finding perimeters and areas of shapes

Use specific formulas; perimeter is the sum of all sides, and area varies based on the shape.

Matrixes/Matrices

A matrix is a rectangular array of numbers or variables arranged in rows and columns.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number.

Inequalities

An inequality compares quantities and shows the relationship using symbols like

Surface Area

Surface area is the total area that the surface of a three-dimensional object occupies.

TAN, SIN, COS

These are trigonometric functions: Tan (tangent), Sin (sine), Cos (cosine), used to relate angles to side lengths.

Pre-Algebra

The branch of mathematics that prepares students for algebra by introducing basic concepts.

Algebra 1

An introductory algebra course focusing on variables, equations, and basic functions.

Algebra 2

An advanced algebra course covering polynomial equations, functions, and complex numbers.

mean/median/mode

Mean is the average, median is the middle value in a data set, and mode is the most frequently occurring value.

Decimals (Adding, Subtracting, Dividing, Multiplying)

Operations with decimals follow the same rules as integers, aligning decimal points where necessary.

Circles (Surface Area, Perimeter)

The perimeter (circumference) of a circle is C=2πr, and the area is A=πr².

Geometry Formulas

Specific mathematical formulas used to calculate dimensions related to geometric figures.

Integers

Whole numbers that can be positive, negative, or zero, but not fractions or decimals.

Zero Product Property

If the product of two factors is zero, at least one of the factors must be zero.

Systems of Equations

A set of equations with the same variables that can be solved together to find the values of the variables.

FOIL

A method for multiplying two binomials: First, Outside, Inside, Last.

Slope Formula

m = (y2 - y1)/(x2 - x1); calculates the steepness of a line.

Slope-Intercept Form

A linear equation format: y = mx + b, where m is slope and b is y-intercept.

Midpoint Formula

The coordinates of the midpoint between two points (x1,y1) and (x2,y2) are ((x1 + x2)/2, (y1 + y2)/2).

Distance Formula

d = √((x2 - x1)² + (y2 - y1)²); calculates the distance between two points in a plane.

Cross Multiplication

A method used to solve proportions by multiplying across the equals sign.

Speed and Rates Distance Formula

d = rt; where d is distance, r is rate, and t is time.

Combined Work Formula

If multiple workers are involved, the formula combines their rates to find the total work done.

Quadratics and Polynomials

Quadratic equations are a type of polynomial of degree 2, often in the form ax² + bx + c.

Vertex Form

A way to express a quadratic function as y = a(x - h)² + k, where (h, k) is the vertex.

Factored Form

When a quadratic is expressed as y = a(x - r1)(x - r2), where r1 and r2 are the roots.

Standard Form

A quadratic expressed as y = ax² + bx + c.

Quadratic Formula

Used to find the roots of a quadratic equation, x = (-b ± √(b² - 4ac))/(2a).

Completing the Square

A method for solving quadratics by rewriting the equation in vertex form.

Finding Solutions using the Discriminant

The discriminant (b² - 4ac) indicates the nature of the roots of the quadratic equation.

Exponent Rules

Rules governing operations with exponents, including multiplying, dividing, and power of a power.

Radical Rules

Rules that apply when simplifying expressions involving square roots.

The Definition of a Logarithm

Logs are the inverse operations of exponentiation, expressed as log_b(a) = c if b^c = a.

Common Logarithms

Logs with base 10; often written simply as log(x) instead of log_10(x).

Natural Logarithms

Logs with base e, denoted as ln(x), where e is approximately 2.718.

Change of Base Formula

Used to convert logs of one base to another: logb(a) = logk(a)/log_k(b).

Properties of Logarithms

Include rules for multiplication, division, and exponentiation of logarithmic expressions.

Circle Equation

The general form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Ellipse Equation

In standard form: (x-h)²/a² + (y-k)²/b² = 1, where a and b are the semi-major and semi-minor axes.

Equation of a Hyperbola

Standard form: (x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/b² - (x-h)²/a² = 1.

Types of Graphs

Includes linear, quadratic, exponential, and logarithmic graphs, each with different shapes.

Horizontal Shift

Moving the graph left or right on the coordinate plane.

Vertical Shift

Moving the graph up or down on the coordinate plane.

End Behavior

Describes the behavior of a graph as x approaches positive or negative infinity.

Key Math Terms

Includes essential terminologies used in various math problems and concepts.

Percentages

A way of expressing a number as a fraction of 100.

Sequences

Ordered lists of numbers, typically following a specific rule or pattern.

Averages (Mean, Median, Mode, Range)

Mean is the average, median is the middle value, mode is the most common, and range is the difference between the max and min.

The Fundamental Counting Principle

If one event can occur in m ways and a second can occur in n ways, the two events can occur in m*n ways.

Factorial

The product of all positive integers up to a certain number n, denoted as n!.

General Permutations

The different arrangements of a subset of items where order matters.

Combinations

Selections from a larger set where the order does not matter.

Basic Probability Formula

P(E) = number of favorable outcomes / total number of outcomes.

Trigonometric Functions

Functions that relate angles of a triangle to the lengths of its sides.

Law of Sines

States that the ratio of the length of a side to the sine of its opposite angle is constant in a triangle.

Law of Cosines

Relates the lengths of the sides of a triangle to the cosine of one of its angles.

Straight Lines and Circles

Basic concepts in geometry addressing properties of lines and circles.

Triangles

Three-sided polygons, fundamental in geometry.

Triangle Similarity

When two triangles have the same shape but possibly different sizes.

Triangle Area

Calculated using base and height: Area = 1/2 * base * height.

Pythagorean Theorem

In a right triangle, a² + b² = c², where c is the hypotenuse.

Pythagorean Triples

Sets of three integers that satisfy the Pythagorean theorem, e.g., (3, 4, 5).

Trigonometry in Triangles

The study of the relationships between the angles and sides of triangles.

Polygons

Multi-sided shapes, with properties that vary depending on the number of sides.

Volume and Surface Area of a Cube

Surface Area = 6a²; Volume = a³, where a is the length of a side.

Volume and Surface Area of a Rectangular Prism

Surface Area = 2(lw + lh + wh); Volume = lwh.

Volume and Surface Area of a Sphere

Surface Area = 4πr²; Volume = (4/3)πr³, where r is the radius.

Volume and Surface Area of a Pyramid

Surface Area includes the base area plus the area of the triangular sides; Volume = (1/3) * base area * height.

Volume and Surface Area of a Right Circular Cone

Surface Area = πr(r + √(h² + r²)); Volume = (1/3)πr²h.

Surface Area of a Cylinder

Surface Area = 2πr(h + r); Volume = πr²h.

Volume of a Prism

Volume = base area * height.