**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**

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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 5

^{th}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**

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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**

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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

Bonus:. Quadratic Formula

**5. Geometry**

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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

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**7. Data and Statistics**

Mean, median, and mode

Different graphs: Line, bar, and circle

Plots

Frequency Tables

Bonus: Standard deviation