Just speedran the pre-algebra course before I dive into heavier stuff.

A true return to high school.

Quick Reference - Key Formulas

Percentages

• Percent to decimal: $\text{percent} \div 100$
• Find percentage: $\frac{\text{part}}{\text{whole}} \times 100$
• Find part: $\text{percent} \times \text{whole}$

Rates and Proportions

• Unit rate: $\frac{\text{quantity}}{\text{time or unit}}$
• Cross multiplication: $\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$

Exponent Rules

• $a^m \times a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^{-n} = \frac{1}{a^n}$

Linear Equations

• Slope-intercept form: $y = mx + b$
• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Point-slope form: $y - y_1 = m(x - x_1)$

Scientific Notation

• Standard form: $a \times 10^n$ where $1 \leq a < 10$

Distance and Midpoint

• Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Factors and Multiples

Prime factorization breaks any number into its building blocks:

$72 = 2^3 \times 3^2 = 8 \times 9$

GCD and LCM become straightforward once you see the prime factorization.

Patterns

Number sequences follow predictable rules:

• Arithmetic: $2, 5, 8, 11, \ldots$ (add 3 each time)
• Geometric: $3, 6, 12, 24, \ldots$ (multiply by 2 each time)

Writing expressions for patterns: the $n$th term of $2, 5, 8, 11$ is $3n - 1$.

Ratios and Rates

Ratios compare quantities. $3:4$ means 3 parts to 4 parts.

Rates include units: 60 miles per hour, $12 per pizza.

Unit rates make comparisons easy:

$\frac{240 \text{ miles}}{4 \text{ hours}} = 60 \text{ mph}$

Percentages

Percentages are fractions with denominator 100:

$25\% = \frac{25}{100} = 0.25$

Finding percentages:

• 15% of 80: $0.15 \times 80 = 12$
• What percent is 12 of 80? $\frac{12}{80} = 0.15 = 15\%$

Exponents and Order of Operations

Exponents are repeated multiplication: $2^4 = 2 \times 2 \times 2 \times 2 = 16$

Order matters: PEMDAS

$3 + 2^3 \times 4 = 3 + 8 \times 4 = 3 + 32 = 35$

Variables and Expressions

Variables represent unknown numbers. Substitute and evaluate:

$3x + 5 \text{ when } x = 4$

$= 3(4) + 5 = 12 + 5 = 17$

Combine like terms: $2x + 3x + 7 = 5x + 7$

Distributive property: $3(x + 4) = 3x + 12$

Equations and Inequalities

Solve by isolating the variable:

$2x + 7 = 15$ $2x = 8$ $x = 4$

Inequalities work the same, but flip the sign when multiplying or dividing by negatives:

$-2x > 6$ $x < -3$

Proportional Relationships

When two quantities are proportional: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$

Cross multiply to solve:

$\frac{3}{4} = \frac{x}{12}$ $3 \times 12 = 4 \times x$ $36 = 4x$ $x = 9$

Constant of proportionality: in $y = kx$, $k$ is how much $y$ changes per unit of $x$.

Multi-Step Equations

Work backwards through order of operations:

$2(x - 3) + 5 = 17$ $2(x - 3) = 12$ $x - 3 = 6$ $x = 9$

Variables on both sides: collect terms on one side:

$3x + 7 = x + 15$ $2x = 8$ $x = 4$

Roots, Exponents, and Scientific Notation

Square roots undo squares: $\sqrt{25} = 5$ because $5^2 = 25$

Exponent rules:

• $a^m \times a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^{-n} = \frac{1}{a^n}$

Scientific notation for large numbers:

$6{,}000{,}000 = 6 \times 10^6$ $0.00034 = 3.4 \times 10^{-4}$

Linear Equations in Two Variables

Linear equations graph as straight lines.

Slope-intercept form: $y = mx + b$

• $m$ is the slope (rise over run)
• $b$ is where the line crosses the y-axis

Finding slope from two points:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For points $(2, 5)$ and $(4, 11)$:

$m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$

Functions

Functions take an input and give exactly one output:

$f(x) = 2x + 3$ $f(5) = 2(5) + 3 = 13$

Linear functions graph as straight lines. Non-linear functions curve.

Systems of Equations

Solve multiple equations simultaneously.

Substitution method:

$y = 2x + 1$ $3x + y = 11$

Substitute the first into the second:

$3x + (2x + 1) = 11$ $5x + 1 = 11$ $x = 2, \quad y = 5$

Graphing method: find where the lines intersect.

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Personal insight: I enjoy math, I didn’t in high school. I think the key difference is that in high-school it felt like there was little “purpose” to what I was learning. Also the teachers sucked. Now I have a clear goal (comp-sci) and my teacher (khan) has a lot more passion.