Ratios, Unit Rates & Percents

A ratio compares two quantities. It can be written as 3:5, 3 to 5, or 3/5.

• Unit rate: A ratio with a denominator of 1. "60 miles per hour" means 60 miles for every 1 hour.
• Equivalent ratios: 2:3 = 4:6 = 6:9 (multiply or divide both parts by the same number)
• Percent means "per hundred." 25% = 25/100 = 0.25

Ratio table tip: To complete a ratio table, multiply or divide both columns by the same factor.

Speed Math Tricks

• Multiply by 11: 35 × 11 = 385 (3_5, put 3+5=8 in the middle)
• Percent shortcut: 15% of 80 = 80% of 15 = 12 (swap the numbers!)
• Negative numbers: Subtracting a negative = adding. 5 - (-3) = 5 + 3 = 8
• Dividing fractions: Keep-Change-Flip. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8
• Finding 10%: Just move the decimal one place left. 10% of 250 = 25

Dividing Fractions, Negative Numbers & Absolute Value

Dividing fractions by fractions: Use "Keep, Change, Flip."

Negative numbers are numbers less than zero. They appear on the left side of a number line.

• Absolute value |x| = the distance from 0. |-7| = 7, |4| = 4
• Ordering: -5 < -2 < 0 < 3 < 7
• Coordinate plane: In 6th grade, you work with ALL four quadrants (negative x and y values)

Variables, Expressions & One-Step Equations

A variable is a letter that represents an unknown number (like x or n).

• Expression: 3x + 5 (no equal sign)
• Equation: 3x + 5 = 20 (has an equal sign)
• Evaluating: If x = 4, then 3x + 5 = 3(4) + 5 = 17
• Solving one-step equations: Use inverse operations to isolate the variable

Area, Surface Area & Volume

• Area of a triangle: A = ½ × base × height
• Area of a parallelogram: A = base × height
• Area of a trapezoid: A = ½(b1 + b2) × h
• Volume of rectangular prism: V = l × w × h
• Surface area: Sum of all face areas. For a rectangular prism: SA = 2(lw + lh + wh)

Mean, Median, Mode & Variability

• Mean (average): Add all values, divide by how many.
• Median: Middle value when sorted. If even count, average the two middle values.
• Mode: Most frequent value.
• Range: Max - Min. Measures spread.
• MAD (Mean Absolute Deviation): Average distance from the mean.

Cell Structure & Function

All living things are made of cells — the basic unit of life. There are two main types:

• Plant cells have a cell wall, chloroplasts (for photosynthesis), and a large central vacuole
• Animal cells have no cell wall or chloroplasts, but have smaller vacuoles and centrioles

Key organelles both share:

• Nucleus: The "brain" — contains DNA and controls cell activities
• Cell membrane: Controls what enters and exits the cell
• Mitochondria: The "powerhouse" — converts food into energy (ATP)
• Ribosomes: Make proteins
• Cytoplasm: Jelly-like substance filling the cell

Plate Tectonics, Rock Cycle & Erosion

Earth's outer layer (the lithosphere) is broken into tectonic plates that slowly move.

• Convergent boundary: Plates push together (mountains, volcanoes, trenches)
• Divergent boundary: Plates pull apart (mid-ocean ridges, rift valleys)
• Transform boundary: Plates slide past each other (earthquakes, like the San Andreas Fault)

The Rock Cycle: Rocks change form over time.

• Igneous: Formed from cooled magma/lava (granite, basalt)
• Sedimentary: Formed from compressed layers (sandstone, limestone)
• Metamorphic: Changed by heat & pressure (marble, slate)

Kinetic Energy, Potential Energy, Light & Sound

• Kinetic energy: Energy of motion. Faster objects have more KE.
• Potential energy: Stored energy. Higher objects have more gravitational PE.
• Energy transforms: PE → KE (falling ball), chemical → thermal (burning wood)
• Light waves: Travel in straight lines, can reflect and refract
• Sound waves: Need a medium (solid, liquid, gas) to travel. Sound is fastest in solids.

Greek & Latin Roots for 6th Grade

Many academic words come from Greek and Latin roots. Knowing these roots helps you decode unfamiliar words.

• bene = good (benefit, benevolent)
• mal = bad (malfunction, malicious)
• chron = time (chronological, synchronize)
• auto = self (autobiography, autonomy)
• spec/spect = look (spectator, inspect, perspective)
• cred = believe (credible, incredible)
• duc/duct = lead (conductor, deduce, produce)
• scrib/script = write (describe, manuscript, prescription)

Theme, Character Development & Figurative Language

• Theme: The universal message or life lesson. Support your theme with evidence from the text.
• Character development: How characters change over the story. Look for moments of conflict, decision, and growth.
• Plot structure: Exposition → Rising Action → Climax → Falling Action → Resolution
• Figurative language in 6th grade:
  • Allusion: Reference to a well-known person, place, or event ("He had the Midas touch")
  • Irony: When the opposite of what's expected happens
  • Symbolism: An object represents something deeper (a dove = peace)

Central Idea, Argument Structure & Author's Purpose

When reading nonfiction in 6th grade, you need to:

• Identify the central idea and explain how it is supported by key details
• Trace the argument: What claims does the author make? What evidence supports each claim?
• Evaluate reasoning: Is the evidence relevant? Sufficient? From a credible source?
• Determine author's purpose: To inform, persuade, entertain, or explain
• Analyze structure: How does the author organize ideas? (cause/effect, compare/contrast, problem/solution)

Complex Sentences, Semicolons & Active/Passive Voice

• Complex sentence: Has an independent clause + a dependent clause.
  "Although it was raining, we continued the experiment."
• Semicolons join two related independent clauses WITHOUT a conjunction.
  "The experiment failed; we tried a different approach."
• Active voice: Subject performs the action. "The scientist conducted the experiment." (preferred)
• Passive voice: Subject receives the action. "The experiment was conducted by the scientist."

Mesopotamia, Egypt, Greece & Rome

The first civilizations arose near rivers that provided water for farming.

• Mesopotamia ("land between rivers" — Tigris & Euphrates): Invented writing (cuneiform), the wheel, and the first code of laws (Code of Hammurabi)
• Ancient Egypt (Nile River): Built pyramids, developed hieroglyphics, mummification, and a 365-day calendar
• Ancient Greece: Birthplace of democracy, philosophy (Socrates, Plato, Aristotle), the Olympics, and theater
• Ancient Rome: Built roads and aqueducts, developed the republic system (Senate), Roman law influenced modern legal systems

Democracy, Republic & Rights

• Democracy: Government by the people. Citizens vote on laws directly (direct democracy) or elect representatives (representative democracy).
• Republic: Citizens elect representatives to make decisions on their behalf. The United States is a democratic republic.
• Monarchy: Ruled by a king or queen. May be absolute or constitutional.
• Rights: Freedoms guaranteed to citizens (speech, religion, press, assembly)
• Responsibilities: Duties of citizens (voting, jury duty, obeying laws, paying taxes)

Supply, Demand, Scarcity & Trade

• Scarcity: Resources are limited, but wants are unlimited. This forces choices.
• Supply: How much of something producers are willing to sell at a given price.
• Demand: How much of something consumers want to buy at a given price.
• Equilibrium: Where supply meets demand — the "right" price.
• Trade: Countries trade to get goods they can't produce efficiently. This benefits everyone through specialization.

Words You'll Need

• SENATE — Governing body in Rome and modern governments (6 letters)
• PHARAOH — Ancient Egyptian ruler (7 letters)
• DEMOCRACY — Government by the people (9 letters)
• SCARCITY — Limited resources vs unlimited wants (8 letters)
• TRADE — Exchange of goods between people or nations (5 letters)
• PYRAMID — Ancient Egyptian tomb structure (7 letters)
• REPUBLIC — Government where citizens elect representatives (8 letters)

Knights & Knaves, Truth Tables

In Knights & Knaves puzzles, knights ALWAYS tell the truth and knaves ALWAYS lie.

• Strategy: Assume each person is a knight, check if their statement is consistent.
• Truth tables: Organize all possibilities to test which ones work.
• Contrapositive: "If A then B" is logically the same as "If not B, then not A."

Primes, GCF, LCM & Divisibility Rules

• Divisibility rules:
  • By 2: last digit is even
  • By 3: digit sum divisible by 3
  • By 4: last two digits divisible by 4
  • By 6: divisible by both 2 AND 3
  • By 9: digit sum divisible by 9
• GCF via prime factorization: GCF(48, 36) = 2² × 3 = 12
• LCM via prime factorization: LCM(48, 36) = 2⁴ × 3² = 144

Permutations, Combinations & Expected Value

Probability = favorable outcomes ÷ total outcomes

• Permutations (order matters): P(n,r) = n!/(n-r)!. How many ways to arrange 3 books from 5? P(5,3) = 60
• Combinations (order doesn't matter): C(n,r) = n!/[r!(n-r)!]. Choose 3 from 5? C(5,3) = 10
• Expected value: Average outcome = sum of (value × probability) for each outcome

Patterns, Function Machines & Pattern-to-Equation

• Function machine: Input → Rule → Output. If the rule is "×3 + 1": input 4 → 4×3+1 = 13
• Finding the rule: Look at how inputs relate to outputs. Test with multiple values.
• Pattern to equation: Position 1=3, 2=5, 3=7, 4=9... The rule is y = 2n + 1