1. Fractions, Decimals, and Percentages

Introduce sets. Explain integers, fractions, and rational numbers. Integers and fractions are subsets of the rational numbers.

Explain how to convert a rational number into a decimal and a decimal into a fraction. Explain how the decimal system is base-10 and what that means.

Explain how to convert a percentage into a number in decimal notation and a number in decimal notation into a percentage. Find some percentages of numbers.

Explain how the reciprocal of a number is just 1 divided by a number. Review division of one rational number by another

Bonus: Continued fractions

2. Exponents and Roots

Exponent laws:

Addition: Same bases multiplied. Example: 3^6 * 3^10 = 3^16

Subtraction: Same bases divided. Example: 2^10 / 2^6 = 2^4

Multiplication: Power to a power. Example: (2^10)^3 = 2^30

Power of zero: Always equals 1. Example: 100^0 = 1

Negative powers: 1 divided by the number to the power. Example: 10 ^-2 = 1/(10^2) = 1/100

Explain how the xth root of a number A is a number B such that, when B is multiplied by itself x times, it equals A. For example,

the 5th root of 32 is 2 because 2^5 = 32.

A root can be written as a number raised to a fraction. For example, 3^(1/2) = sqrt(3). If the number on top is not 1, you first raise the number to that power and then find the root of this new number. For example, 2^(6/2) = sqrt(64) = 8 = 2^3.

Bonus: Continued fractions and square roots

3. Number Theory

Extend set theory. Review natural numbers, integers, rational numbers, and real numbers. Explain how there are numbers such as the square root of 2 that are irrational because they cannot be written as a rational number

Greatest common divisors

Euclidean algorithm to find greatest common divisors

Factorization into prime powers. Fundamental Theorem of Arithmetic: Every integer > 1 is either prime or can be written uniquely as the product of prime powers

Congruences

Bonus: Numbers in different bases. Provide general method for converting from decimal notation to another base and from another base to decimal notation

4. Equations and Inequalities

Start with equalities. Solve for simple values of x. For instance, 3x + 2 = 11

For inequalities, do the same. For instance, 3x + 8 >= 15

Introduce or review the Cartesian Coordinate System. Show how an equation can be graphed using the x and y coordinates

Explain intersections of a graph with the x and y coordinates (finding zeros)

Introduce interval notation: () and []

Slope formula and general slope intercept form. Distance formula

5. Geometry

Number line. Line segments. Parallel and perpendicular. Angles made at intersections.

Triangles: perimeter, area, and angles. Always 180 degrees. Different varieties of triangles (e.g., equilateral). Pythagorean theorem.

Pythagorean theorem as related to the distance formula

Squares and rectangles: areas and perimeters. Square roots in relation to squares.

Circles: area and circumference. Relation of the diameter to the radius. Introduction of pi. Explain how pi is irrational.

Volumes

Bonus: Arc length of a circle.

6. Counting

Introduce the notion of one-to-one correspondence with the set of natural numbers

Introduce the factorial function

Permutations of a set of numbers: Explain as set of rearrangements and provide formula for finding the total.

Combinations of a set of numbers: Explain as a set of unique arrangements and provide formula for finding the total. Difference between permutations and combinations should be noted.

Pascal’s Triangle

Binomial Theorem and Pascal’s Triangle

7. Data and Statistics

Mean, median, and mode

Different graphs: Line, bar, and circle

Plots

Frequency Tables

Bonus: Standard deviation