Unit 1 · Integers & Absolute Value
Integers on the Number Line
Integers include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …
The absolute value |x| is the distance from 0, always ≥ 0.
|5| = 5 |−5| = 5 |0| = 0
Adding opposites: a + (−a) = 0
Subtracting: a − b = a + (−b)
Example
Evaluate: −3 + (−8)
= −11 (same sign → add magnitudes, keep sign)
|neg| = pos
+ × − = −
− × − = +
a−b = a+(−b)
Unit 2 · Fractions, Decimals & Percents
Rational Number Conversions
A fraction a/b converts to a decimal by division. A percent means "per hundred": p% = p/100.
Fraction → Decimal: divide numerator ÷ denominator
Decimal → Percent: multiply by 100
Percent → Decimal: divide by 100
% of a number: Part = (Percent/100) × Whole
Example
What is 35% of 80?
= (35/100) × 80 = 0.35 × 80 = 28
3/4 = 0.75 = 75%
1/3 ≈ 0.333
Part=R×W
Unit 3 · Ratios, Rates & Proportions
Proportional Reasoning
A ratio compares two quantities. A rate is a ratio with different units. A proportion states two ratios are equal.
Cross-multiplication: a/b = c/d → ad = bc
Unit rate: divide first quantity by second
Scale factor: New/Original = Scale ratio
Example
Solve: x/5 = 12/20
20x = 60 → x = 3
Cross-multiply
ad = bc
Unit Rate
Unit 4 · Exponents & Scientific Notation
Powers & Exponent Rules
An exponent shows repeated multiplication. Scientific notation writes numbers as a × 10ⁿ where 1 ≤ a < 10.
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1/aⁿ
Sci. Notation: 3,400 = 3.4 × 10³
Example
Simplify: 2³ × 2⁴
= 2⁷ = 128
aᵐ·aⁿ=aᵐ⁺ⁿ
a⁰=1
a⁻ⁿ=1/aⁿ
Unit 5 · Variables, Expressions & Equations
Algebra Basics
A variable is a letter representing an unknown. An expression has no = sign; an equation does. Solving means isolating the variable using inverse operations.
Like terms: combine coefficients of same variable
Distributive: a(b + c) = ab + ac
Solve: perform same operation on both sides
Example
Solve: 3x − 7 = 11
3x = 18 → x = 6
Inverse ops
Like terms
Distribute
Unit 6 · Inequalities
Solving & Graphing Inequalities
Solve inequalities like equations, BUT flip the sign when multiplying or dividing by a negative number.
x + 3 > 7 → x > 4
−2x ≤ 8 → x ≥ −4 (flip! divided by −2)
Compound: a < x < b (two boundaries)
Example
Solve: −3x + 6 > 0
−3x > −6 → x < 2 (flip sign!)
Flip on ÷/×neg
Open circle ≠ closed
Unit 7 · Graphing on the Coordinate Plane
Coordinate Geometry
Points are written as (x, y). The x-axis is horizontal, y-axis is vertical. Slope measures steepness: rise over run.
Slope m = (y₂ − y₁)/(x₂ − x₁)
Slope-intercept: y = mx + b
Quadrants: I(+,+) II(−,+) III(−,−) IV(+,−)
Example
Slope through (1, 2) and (3, 8)?
m = (8−2)/(3−1) = 6/2 = 3
m=rise/run
y=mx+b
4 Quadrants
Unit 8 · Geometry: Area, Perimeter & Volume
Essential Geometry Formulas
Rectangle: A = lw P = 2(l+w)
Triangle: A = ½bh
Circle: A = πr² C = 2πr
Rectangular Prism: V = lwh
Cylinder: V = πr²h
Pythagorean: a² + b² = c²
Example
Area of triangle: base = 10, height = 6
A = ½ × 10 × 6 = 30 sq units
A=½bh
A=πr²
a²+b²=c²
Unit 9 · Statistics & Probability
Data Analysis Basics
Mean = sum ÷ count. Median = middle value. Mode = most frequent. Range = max − min.
Mean = (Σx) / n
Probability = favorable outcomes / total outcomes
P(A and B) = P(A) × P(B) [independent]
P(A or B) = P(A) + P(B) − P(A∩B)
Example
Mean of 4, 7, 10, 3, 6?
= (4+7+10+3+6)/5 = 30/5 = 6
Mean=sum/n
Median=middle
P=fav/total
Unit 10 · Functions & Patterns
Introduction to Functions
A function assigns exactly one output to each input. Written as f(x). The set of inputs is the domain; outputs are the range.
Function notation: f(x) = expression
Linear function: f(x) = mx + b
Vertical Line Test: each x has one y
Arithmetic sequence: aₙ = a₁ + (n−1)d
Example
If f(x) = 2x + 3, find f(5)
= 2(5) + 3 = 13
One output/input
f(x)=mx+b
VLT