Proportional Relationships, Unit Rates & Percents

A proportional relationship is one where two quantities always have the same ratio (constant of proportionality).

• Constant of proportionality (k): In y = kx, k is the unit rate. If 3 pounds of apples cost $6, then k = $2/lb.
• Identifying proportional relationships: In a table, check if y/x is always the same. On a graph, it must be a straight line through the origin.
• Complex percent problems: Markups, markdowns, tips, tax, simple interest, and percent change.
• Percent change: (New − Original) / Original × 100%

Speed Math Tricks

• Integer operations: Same signs → positive product. Different signs → negative product. −3 × −4 = 12, −3 × 4 = −12
• Percent of a number: 15% of 80 = 80% of 15 = 12 (swap trick!)
• Proportional thinking: If 3 items cost $12, then 1 costs $4 (divide), and 7 cost $28 (multiply).
• Fraction × whole number: 3/4 × 20 = 3 × 20/4 = 60/4 = 15
• Converting between forms: 1/8 = 0.125 = 12.5%,   3/8 = 0.375,   5/6 ≈ 0.833

Adding, Subtracting, Multiplying & Dividing Rational Numbers

In 7th grade, you work with all rational numbers — positive and negative fractions, decimals, and integers.

• Adding integers: Same sign → add and keep the sign. Different signs → subtract and take the sign of the larger absolute value.
• Subtracting: a − b = a + (−b). Change subtraction to adding the opposite.
• Multiplying/dividing: Same signs → positive result. Different signs → negative result.
• Converting fractions to decimals: Divide numerator by denominator. 3/8 = 0.375. 1/3 = 0.333... (repeating)

Two-Step Equations, Inequalities & Expressions

In 7th grade, you solve multi-step equations and inequalities.

• Two-step equation: 3x + 5 = 20. Step 1: subtract 5 → 3x = 15. Step 2: divide by 3 → x = 5.
• Distributive property: 2(x + 4) = 2x + 8
• Combining like terms: 5x + 3 − 2x + 7 = 3x + 10
• Inequalities: Solve like equations, but FLIP the inequality sign when multiplying or dividing by a negative.

Angle Relationships, Area, Circumference & Volume

• Supplementary angles: Two angles that add to 180°
• Complementary angles: Two angles that add to 90°
• Vertical angles: Opposite angles formed by two intersecting lines — always equal.
• Area of a circle: A = πr²
• Circumference: C = 2πr = πd
• Volume of a prism: V = Bh (base area × height)
• Cross-sections: Slicing a 3D shape reveals a 2D cross-section.

Random Sampling, Comparative Inferences & Probability Models

• Random sampling: A random sample fairly represents the population. Larger samples give better estimates.
• Comparing populations: Use measures of center (mean, median) and variability (MAD, IQR) to compare two data sets.
• Probability: P(event) = favorable outcomes / total outcomes. Always between 0 and 1.
• Experimental vs. theoretical: Experimental probability comes from actual trials. Theoretical is calculated.
• Compound events: P(A and B) = P(A) × P(B) for independent events.

Cell Processes & Human Body Systems

In 7th grade, you go deeper into how cells work and how body systems interact.

• Cell division (mitosis): One cell divides into two identical cells. Stages: Prophase → Metaphase → Anaphase → Telophase.
• Photosynthesis: CO₂ + H₂O + sunlight → glucose + O₂ (in chloroplasts)
• Cellular respiration: glucose + O₂ → CO₂ + H₂O + energy (ATP) (in mitochondria)
• Body systems: Circulatory (heart, blood), Respiratory (lungs), Digestive (stomach, intestines), Nervous (brain, nerves), Skeletal, Muscular
• Homeostasis: The body's ability to maintain stable internal conditions (temperature, blood sugar, etc.)

Ecosystems, Food Webs & Biodiversity

An ecosystem includes all living things (biotic) and nonliving things (abiotic) in an area.

• Producers: Make their own food (plants, algae) via photosynthesis
• Consumers: Eat other organisms. Primary (herbivores), Secondary (carnivores that eat herbivores), Tertiary (top predators)
• Decomposers: Break down dead organisms (fungi, bacteria), recycling nutrients
• Food web: Shows interconnected food chains in an ecosystem
• Energy pyramid: Only about 10% of energy transfers to the next trophic level
• Symbiosis: Mutualism (both benefit), Commensalism (one benefits, other unharmed), Parasitism (one benefits, other harmed)

Kinetic Energy, Potential Energy & Thermal Energy Transfer

• Kinetic energy: Energy of motion. KE = ½mv². Faster & heavier objects have more KE.
• Potential energy: Stored energy due to position or condition. Gravitational PE increases with height.
• Thermal energy transfer: Heat moves from hot to cold via conduction (contact), convection (fluid flow), radiation (waves).
• Conservation of energy: Energy cannot be created or destroyed, only transformed.
• Chemical energy: Stored in bonds of molecules (food, fuel, batteries)

Context Clues & Academic Vocabulary for 7th Grade

In 7th grade, you use context clues to determine word meaning and analyze how authors use language for effect.

• Definition clue: The word is defined directly. "The arid, or extremely dry, desert..."
• Example clue: Examples reveal meaning. "Nocturnal animals, such as owls and bats, are active at night."
• Contrast clue: Opposite meaning gives a hint. "Unlike the tranquil lake, the ocean was wild and chaotic."
• Connotation: The feeling a word carries beyond its definition. "Thrifty" (positive) vs. "stingy" (negative) — both mean careful with money.
• Denotation: The literal dictionary definition of a word.
• Word relationships: Analogies, synonyms, antonyms, and word families

Theme Development, Point of View & Literary Devices

• Theme development: In 7th grade, analyze how the theme develops over the course of the text. How do events, characters, and dialogue contribute?
• Comparing perspectives: How do different characters view the same events? How does the narrator's perspective shape the story?
• Dramatic irony: When the reader knows something the characters don't.
• Tone and mood: Tone is the author's attitude. Mood is how the reader feels.
• Analyzing poetry: How do form, structure, and word choice contribute to meaning?

Claims, Evidence, Reasoning & Counterarguments

In 7th grade, you write arguments with clear claims supported by relevant evidence.

• Claim: Your main argument or position on an issue.
• Evidence: Facts, data, quotes, or examples that support your claim.
• Reasoning: Explains WHY the evidence supports your claim (the logical connection).
• Counterargument: Acknowledge the opposing view, then explain why your position is stronger.
• Transitions: Use words like "furthermore," "in contrast," "consequently" to connect ideas.

Phrases, Clauses, Modifiers & Concise Writing

• Phrases vs. clauses: A phrase lacks a subject-verb pair ("running quickly"). A clause has one ("she ran quickly").
• Dangling modifiers: "Walking to school, the rain started." (Who was walking? Not the rain!) Fix: "Walking to school, I got caught in the rain."
• Misplaced modifiers: Put modifiers next to the word they describe. "She almost ate the entire pizza" vs. "She ate almost the entire pizza."
• Concise writing: Eliminate wordiness. "Due to the fact that" → "Because." "At this point in time" → "Now."
• Commas with coordinate adjectives: Use commas between adjectives that independently modify a noun: "a tall, elegant building."

Fall of Rome, Byzantine Empire, Islam & Medieval Europe

After the fall of the Western Roman Empire (476 CE), Europe entered the Middle Ages (roughly 500–1500 CE).

• Byzantine Empire: The Eastern Roman Empire survived for another 1,000 years. Capital: Constantinople. Preserved Greek and Roman knowledge.
• Rise of Islam: Founded by Prophet Muhammad in the 7th century. The Islamic Golden Age (800–1300 CE) advanced math, science, medicine, and astronomy.
• Feudalism: A political system where kings granted land (fiefs) to lords, who provided military service. Peasants (serfs) worked the land.
• The Crusades: Religious wars (1096–1291) between Christians and Muslims over the Holy Land. Led to increased trade and cultural exchange.
• Magna Carta (1215): Limited the king's power and established the idea that even rulers must follow the law.

Renaissance, Reformation & Age of Exploration

• Renaissance (1300–1600): "Rebirth" of art, learning, and culture. Started in Italy. Key figures: Leonardo da Vinci, Michelangelo, Shakespeare.
• Reformation (1517): Martin Luther challenged the Catholic Church's practices (95 Theses). Led to Protestantism and religious wars.
• Age of Exploration (1400s–1600s): European explorers sought new trade routes. Columbus (1492), Magellan (circumnavigation), and others. Led to colonization and the Columbian Exchange.
• Columbian Exchange: Transfer of plants, animals, diseases, and ideas between the Americas and the Old World. Potatoes and corn went to Europe; horses and wheat came to the Americas.

Mercantilism, Trade Networks & Economic Systems

• Mercantilism: Economic policy where a nation's strength comes from accumulating gold/silver through exports exceeding imports.
• Silk Road: Ancient trade route connecting China to the Mediterranean. Carried silk, spices, ideas, and diseases.
• Triangular Trade: Trade routes between Europe, Africa, and the Americas. Included the horrific transatlantic slave trade.
• Capitalism: Economic system based on private ownership and free markets. Emerged during the early modern period.
• Supply and demand: Still the fundamental driver of prices in market economies.

Words You'll Need

• FEUDAL — System where lords granted land to vassals in exchange for service (6 letters)
• CRUSADE — Religious military expedition to the Holy Land (7 letters)
• RENAISSANCE — Period of cultural rebirth in Europe (11 letters)
• REFORM — To change or improve (6 letters)
• TRADE — Exchange of goods between people or nations (5 letters)
• CHARTER — Document granting rights or privileges (7 letters)
• DYNASTY — Series of rulers from the same family (7 letters)

Knights & Knaves, Logical Deductions

In deductive reasoning, you start with general rules and draw specific conclusions.

• Syllogism: "All mammals are warm-blooded. A dog is a mammal. Therefore, a dog is warm-blooded."
• Contrapositive: "If A then B" is logically the same as "If not B, then not A."
• Knights & Knaves: Knights ALWAYS tell the truth. Knaves ALWAYS lie. Test each possibility.
• Process of elimination: Use given clues to rule out impossibilities until only one answer remains.

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
• Perfect squares and cubes: 144 = 12², 125 = 5³

Theoretical vs. Experimental Probability, Compound Events

Probability = favorable outcomes ÷ total outcomes

• Theoretical probability: What SHOULD happen based on math. P(heads) = 1/2
• Experimental probability: What actually DID happen in trials. If you got 47 heads in 100 flips, P(heads) = 47/100
• Compound events: P(A and B) = P(A) × P(B) for independent events
• Sample space: List ALL possible outcomes. For 2 dice: 6 × 6 = 36 outcomes
• Simulations: Use random number generators to model real situations

Patterns, Equations from Patterns & Function Tables

• Function table: Input → Rule → Output. If the rule is "×3 + 1": input 4 → 4×3+1 = 13
• Finding the rule from a pattern: Look at common differences (linear) or common ratios (exponential).
• Writing equations: Position 1=5, 2=8, 3=11, 4=14... Common difference = 3. Rule: y = 3n + 2
• Graphing proportional relationships: y = kx is a line through the origin with slope k.