School in a Book: Algebra and Geometry

When it comes to algebra and geometry, most schools emphasize skills practice while spending almost no time helping students understand the ideas they are putting to use. Studying the definitions of commonly used higher-level math terms might help further your grasp of these subjects and allow you to converse about them more easily. Fluency in these ideas might also ease transitions between math teachers and curriculum and shorten your review time before exams.

Note that calculus and trigonometry terms are not included in this book, as these ideas require the kind of in-depth explanations that aren’t practical in this format. Also, this treatment of algebra and geometry focuses on the ideas and processes that are most useful for a general audience.

Basic Algebra

Algebra: An extension of arithmetic in which unknown numbers can be represented by letters

Variable: Any letter that stands for a number

Expression: Any string of numbers and symbols that makes sense when placed on one side of an equation; for example 5x + 4x

Term: Any part of an expression that is separated from the other parts by either a plus sign or a minus sign; for example, 3x and 5x in the expression 3x – 5x

Coefficient: The numerical part of a term; for example, the term 3x has a coefficient of 3

Constant: A number without a variable; for example, the number 2 in 6m + 2 = x

Like terms: Terms whose variables (with any exponents) are the same; for examples, 3x and 5x

Order of operations: The correct sequence of operations to use when solving an expression with multiple operations. Mathematical symbols are often used to indicate this sequence; for example, in (3x + 5x)/2, 3x and 5x are to be added before that number is divided by 2.

Theorum: A mathematical proposition that has been proven true, such as the Pythagorean Theorum

Rational number: A number that can be made by dividing two integers (an integer is a number with no fractional part)

Irrational number: A real number that can NOT be made by dividing two integers (an integer has no fractional part)

The Commutative Rule of Addition: The rule that states that when two terms are added, the order of addition does not matter

Commutative Rule of Multiplication: The rule that states that when two terms are multiplied, the order of multiplication does not matter

Associative Rule of Addition: The rule that states that when three or more terms are added, the order of addition does not matter

Distributive Rule of Multiplication: The rule that states that when a number is multiplied to an addition of two numbers, it results in the output which is same as the sum of their products with the number individually. The equation for the for this is: a × (b + c) = (a × b) + (a × c). For example, x2 × (2x + 1) = (x2 × 2x) + (x2× 1).

The inverse property of addition: The rule that states that for every number a, a + (-a) = 0 (zero)

The inverse property of multiplication: The rule that states that for every non-zero number a, a times (1/a) = 1

Factorization: The mathematical process of breaking a number down into smaller numbers that, multiplied together, equal the original number

Prime number: A positive number that has exactly two factors, 1 and itself

Square root: The number that, multiplied by itself once, equals the number of which it is a root. For example, the square root of 16 is 4 because 4 x = 16.

Root: The number that, multiplied by itself one or more times, equals the number of which it is a root. For example, the number 2 is a cube root of 8 because 2 x 2 x 2 equals 8.

Radical: The symbol √ that is used to indicate the square root or nth root of a number

Exponent: A number that indicates how many times to multiply its associated number. An exponent is written in a smaller font at the top right-hand corner of its associated number.

Exponential growth: The rapid numerical growth that occurs when numbers are multiplied, then multiplied again, with each iteration folding in the previous total and multiplying it by x number.

Second-degree term: A variable raised to the second power, like x2, or the product of exactly two variables, like x and y

Linear equation: An equation in which the highest power of the variable is always one. The standard form of a linear equation with one variable is: Ax + B = 0. These are some of the easier algebraic equations to solve, and are introduced early in the subject.

Linear model: A model that assumes a linear relationship between the input variables (x) and the single output variable (y)

Quadratic equation: An equation that has a second-degree term and no higher terms

Quadratic formula: A formula that provides a solution to the quadratic equation ax2 + bx + c = 0. The quadratic formula is obtained by solving the general quadratic equation.

Polynomial: A mathematical expression with one or more algebraic terms, each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx2)

Monomial: A polynomial with only one term

Binomial: A polynomial with only two terms

Trinomial: A polynomial with only three terms

Degree of a polynomial: The sum of the exponents of variables that occur in that term (if there is no exponent written on a variable, such as in 3x, the exponent is one). The degree of a polynomial is the greatest degree of any term in the polynomial (for instance, for the polynomial 4x2 + 7xyz, the degree is 3 because of the last term).

Function: An expression that states a relationship between one variable (the independent variable) and another variable. These expressions can be graphed on a coordinate plane.

Nonlinear function: A function whose graph is not a line or part of a line

Vector: A quantity that has both magnitude and direction but not position. Examples of such quantities are velocity and acceleration

Simple interest: Interest that is calculated on the principle amount only

Compound interest: Interest that is calculated on both the principal amount as well as the interest accumulated over the previous period

Amortization: A method for calculating interest payments wherein a much higher proportion of the total interest is charged first, and reduced at a regular rate over the life of a loan

Scientific notation: A way of writing very large or very small numbers in a shorter form, using symbols; for example, 650,000,000 can be written as 6.5 ✕ 10^8

Relation: A collection of ordered pairs containing one object from each set

Transformation: A general term for four specific ways to manipulate the shape and/or position of a point, line, or geometric figure

Simultaneous linear equation: The two linear equations in two or three variables solved together to find a common solution

Other Algebra Skills

  • Using algebraic symbols
  • Solving for variables
  • Solving and graphing inequalities
  • Calculating ratios, rates, percentages and proportions (as when finding taxes, discounts, markups, gratuities, commissions, simple interest, the percent rate of change, exponential growth and more)
  • Finding prime numbers and square roots
  • Solving quadratic equations
  • Working with radicals
  • Comparing functions

Basic Geometry

Plane geometry: The mathematics of flat, two-dimensional shapes like lines, circles and triangles

Solid geometry: The mathematics of three dimensional objects like cubes, prisms, cylinders and spheres

Point: A specific position on a line, plane, or in space. A point is a theoretical construct. It has no dimensions, only position.

Line: A one-dimensional figure that features length but no depth or height. A line is a theoretical construct.

Plane: A flat two-dimensional surface. A plane is a theoretical construct with no depth whose height and width are infinite or indefinite

Solid: A three-dimensional shape

Polygon: Any two-dimensional (plane) shape with straight sides, such as triangles, rectangles and pentagons

Quadrilateral: A polygon with four sides

Pentagon: A polygon with five sides

Hexagon: A polygon with six sides

Septagon/Heptagon: A polygon with seven sides

Octagon: A polygon with eight sides

Rhombus: A quadrilateral with parallel and equally-sized opposite sides; a diamond

Parallelogram: A quadrilateral with parallel but unequally-sized opposite sides

Trapezoid: A quadrilateral with two parallel and two nonparallel sides

Isosceles triangle: A triangle with two sides that are of equal length

Equilateral triangle: A triangle with equal sides and angles

Scalene triangle: A triangle with unequal sides and angles

Right triangle: A triangle with one internal 90-degree angle

Cube: A three-dimensional square

Cone: A three-dimensional triangle with a round base

Cylinder: A tube-shaped object

Sphere: A ball-shaped object

Pyramid: A three-dimensional figure on which the faces are triangular and converge to a single point at the top

Prism: A three-dimensional figure with identical ends of any shape. For example, a rectangular prism has identical rectangles at each end. Note that a cube is a prism.

Angle: Two lines that meet to form a corner

Parallel lines: Lines that do not intersect

Perpendicular lines: Lines that intersect at a 90-degree angle

Vertex: A corner point

Right angle: A 90-degree angle

Acute angle: An angle less than 90 degrees but greater than 0 degrees

Obtuse angle: An angle greater than 90 degrees but less than 180 degrees

Diameter: A straight line that passes through the center of a circle or sphere and ends at the circle or sphere’s outer edges

Radius: A straight line that extends from the center of a circle or sphere to the outer edge; half of a diameter

Chord: The line segment between two points on a curve

Face: A surface plane of a three-dimensional shape

Edge: The meeting place of two faces on a three-dimensional shape

Slope: The steepness and direction of a line as read from left to right

Transversal line: A straight line that intersects two other straight lines

Coordinate: Two numbers (or a letter and a number) that signify a specific point on a coordinate plane

Coordinate plane: A grid with a horizontal x-axis and a vertical y-axis that meet at a center point, with the center point value being 0 and each line on the grid representing whole numbers as they increase or decrease along each axis. The plane has four quadrants: quadrant I, with a positive x value and a positive y value; quadrant II, with a negative x value and a positive y value; quadrant III, with a negative x value and a negative y value; and quadrant IV, with a positive x value and a negative y value. A coordinate plane is used to graph points, lines and other objects.

X-axis: The horizontal axis in a coordinate plane

Y-axis: The vertical axis in a coordinate plane

Congruent: The same shape and size (though not necessarily positioned the same way)

Similar: The same shape, with the same angle degrees (though not necessarily the same size)

Pythagorean theorem: The rule of mathematics that states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. Written as a formula, this is: a2 + b2 = c2 (for a right-angled triangle).

Proof: Statements that prove that a mathematical concept is true

Formula for calculating the area of squares and rectangles: Multiply height by width: hxw. Note that some areas can be divided into multiple squares and rectangles and the results can be added together to find the total area.

Formula for calculating square footage: Use the same formula as for finding the area of a square, using feet as the measurement: hxw

Formula for calculating the area of a triangle: Multiple the height by the width, then divide by two: (h x w)/2

Formula for calculating diameter: Multiply pi by radius, then square this number: πR2

Formula for calculating perimeter: Add length and width, then multiply this by two: 2(length + width)

Formula for calculating the volume of a cube or rectangle-based shape: Multiply width, length and height: l x w x h

Formula for calculating the volume of a sphere: Cube the radius, then use this formula: 4/3 × π × R3

Formula for calculating the volume of a prism or cylinder: Find the area of the end shape, then multiply by its depth

Formula for calculating the volume of a cone or pyramid: Calculate the volume of the base as if the base were a square, then divide by 3.

Formula for measuring an angle: (n – 2) * 180

Trigonometry: The branch of mathematics that applies algebra and geometry skills to circular and periodic functions. It includes the use of sine, cosine and tangent.

Calculus: The branch of mathematics that works with series and sequences; probability and statistics; and limits and derivatives.

Other Geometry Skills

  • Measuring angles
  • Calculating scale
  • Calculating arc length
  • Graphing lines and slopes
  • Working with coordinate planes
  • Proving simple geometric theorems
  • Making geometric constructions based on a given set of numbers
  • Working with the Pythagorean theorem
  • Solving linear equations
  • Working with functions

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