Algebra 1 · Geometry
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Algebra 1 & Geometry · Word Problems

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Algebra 1
Problems 1–10
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Super Memory Keys · Algebra
ISOLATE → move variables to one side, numbers to the other
DISTRIBUTE → multiply EVERYTHING inside the parentheses
OPPOSITE → undo operations with their opposite (+ ↔ −, × ↔ ÷)
BALANCE → whatever you do to one side, do to the other
FOIL → First·Outer·Inner·Last (for expanding two binomials)
01
Algebra
Linear Equations
TOPIC · SOLVING ONE-STEP EQUATIONS
Sarah has some apples. After giving away 7 apples to her friend, she has 15 left. How many apples did Sarah start with?
⚡ Common Trap: Students subtract instead of add. Remember: "giving away" means you need to find the original amount — add back what was taken.
02
Algebra
Linear Equations
TOPIC · TWO-STEP EQUATIONS
A taxi charges a flat fee of $3 plus $2 per mile. If Maria paid $11 total, how many miles did she travel?
⚡ Common Trap: Don't divide by 2 before subtracting 3. Always subtract the flat fee FIRST, then divide.
03
Algebra
Inequalities
TOPIC · SOLVING INEQUALITIES / SIGN FLIP
You need at least 60 points to pass. Your current score is 45. Each bonus question is worth 3 points. What is the minimum number of bonus questions you must answer correctly?
⚡ Common Trap: "At least" means ≥ not >. Also: when you divide or multiply an inequality by a NEGATIVE number, FLIP the sign!
04
Algebra
Systems of Equations
TOPIC · SYSTEMS OF LINEAR EQUATIONS
A store sells pens for $2 each and notebooks for $5 each. Jake buys a total of 8 items and spends $25. How many pens did Jake buy?
⚡ Common Trap: Set up TWO equations. Let p = pens, n = notebooks. Then: p + n = 8 AND 2p + 5n = 25. Solve by substitution!
05
Algebra
Proportions & Rates
TOPIC · PROPORTIONS
A recipe uses 3 cups of flour to make 24 cookies. How many cups of flour are needed to make 40 cookies?
⚡ Common Trap: Cross-multiply correctly: 3/24 = x/40 → 24x = 120 → x = 5. Don't flip the ratio!
06
Algebra
Exponents
TOPIC · EXPONENT RULES
A bacteria colony doubles every hour. If you start with 1 bacterium, how many bacteria will there be after 6 hours?

Hint: Use the formula \(N = 2^t\) where t = hours.
⚡ Common Trap: Students multiply 2 × 6 = 12 instead of computing \(2^6\). "Doubling" means EXPONENTIAL growth, not multiplication!
07
Algebra
Slope & Linear Functions
TOPIC · SLOPE / RATE OF CHANGE
A car rental company charges $50 on day 1 and $170 on day 5. Assuming the cost increases linearly, what is the daily charge after the first day?
⚡ Common Trap: Slope = (change in y) ÷ (change in x). Days go from 1 to 5, so Δx = 4 (not 5!). Count the INTERVALS, not the points.
08
Algebra
Factoring
TOPIC · FACTORING QUADRATICS
The area of a rectangular garden is \(x^2 + 7x + 12\) square feet. If the length is \((x + 4)\) feet, what is the width?
⚡ Common Trap: Factor: find two numbers that MULTIPLY to 12 and ADD to 7. That's 3 and 4. So \(x^2+7x+12 = (x+3)(x+4)\). Width = (x+3).
09
Algebra
Word Problems · Age
TOPIC · TRANSLATING WORDS INTO EQUATIONS
Tom is 3 times as old as his sister Emma. In 6 years, Tom will be twice as old as Emma. How old is Emma now?
⚡ Common Trap: "In 6 years" means ADD 6 to BOTH ages. Set up: Tom now = 3E. In 6 years: (3E + 6) = 2(E + 6). Solve for E.
10
Algebra
Percent & Mixture
TOPIC · PERCENT CHANGE
A jacket originally costs $80. It is on sale for 25% off. Then an additional 10% off is taken on the sale price. What is the final price?
⚡ Common Trap: 25% + 10% is NOT 35% off! Apply each discount SEPARATELY. First: $80 × 0.75 = $60. Then: $60 × 0.90 = $54. Discounts are NOT additive!
Geometry
Problems 11–20
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Super Memory Keys · Geometry
PYTHAGOREAN → a² + b² = c² (c is always the HYPOTENUSE — the longest side!)
TRIANGLE SUM → all 3 angles ALWAYS add up to 180°
STRAIGHT LINE → supplementary angles = 180° · Vertical angles = EQUAL
AREA vs PERIMETER → Area = INSIDE (units²) · Perimeter = AROUND (units)
CIRCLE → Area = πr² · Circumference = 2πr (r = radius, not diameter!)
11
Geometry
Triangles
TOPIC · TRIANGLE ANGLE SUM
In a triangle, two angles measure 55° and 72°. What is the measure of the third angle?
⚡ Common Trap: All triangles have angles that sum to exactly 180°. Subtract both known angles: 180 − 55 − 72. Don't forget to subtract BOTH!
12
Geometry
Pythagorean Theorem
TOPIC · PYTHAGOREAN THEOREM
A ladder leans against a wall. The base of the ladder is 6 feet from the wall, and the ladder reaches 8 feet up the wall. How long is the ladder?
⚡ Common Trap: The ladder IS the hypotenuse (c). Use \(a^2 + b^2 = c^2\): \(6^2 + 8^2 = 36 + 64 = 100\). So \(c = \sqrt{100} = 10\). Don't add 6 + 8!
13
Geometry
Area · Circles
TOPIC · CIRCLE AREA
A circular pizza has a diameter of 14 inches. What is the area of the pizza? (Use \(\pi \approx 3.14\))
⚡ Common Trap: The formula uses RADIUS, not diameter! Radius = diameter ÷ 2 = 7. Then Area = π × 7² = π × 49 ≈ 153.86. Half the diameter first!
14
Geometry
Angles · Parallel Lines
TOPIC · PARALLEL LINES CUT BY TRANSVERSAL
Two parallel lines are cut by a transversal. One of the angles formed is 65°. What is the measure of the angle that is co-interior (same-side interior / consecutive interior) to it?
⚡ Common Trap: Co-interior angles are SUPPLEMENTARY (add to 180°), NOT equal! Alternate interior angles are equal. Know which pair is which!
15
Geometry
Perimeter · Composite Shapes
TOPIC · PERIMETER OF COMPOSITE SHAPES
An L-shaped room has the following measurements: the long side is 10 m, the short bottom is 6 m, the step-up height is 4 m, the inner top is 4 m, the inner drop is 6 m, and the left side is 10 m. What is the perimeter?
⚡ Common Trap: Count ALL sides, including the inner "step" sides. For an L-shape, there are 6 sides total. Add all 6: 10 + 6 + 4 + 4 + 6 + 10 = 40 m.
16
Geometry
Similar Triangles
TOPIC · SIMILAR TRIANGLES / SHADOW PROBLEMS
A 6-foot tall person casts a 4-foot shadow. At the same time, a nearby tree casts a 14-foot shadow. How tall is the tree?
⚡ Common Trap: Set up the proportion carefully: person height / person shadow = tree height / tree shadow. → 6/4 = x/14. Cross-multiply: 4x = 84, x = 21.
17
Geometry
Volume · Rectangular Prism
TOPIC · VOLUME OF 3D SHAPES
A fish tank is 40 cm long, 25 cm wide, and 30 cm tall. How many liters of water does it hold when full? (1 liter = 1,000 cm³)
⚡ Common Trap: Volume = length × width × height. Then divide by 1,000 to convert cm³ → liters. Don't forget the unit conversion!
18
Geometry
Coordinate Geometry
TOPIC · DISTANCE FORMULA / MIDPOINT
Point A is at coordinates (2, 3) and Point B is at (8, 11). What is the midpoint of segment AB?
⚡ Common Trap: Midpoint formula: \(\left(\dfrac{x_1+x_2}{2},\, \dfrac{y_1+y_2}{2}\right)\). Add x-coordinates and divide by 2. Same for y. Don't subtract — ADD!
19
Geometry
Area · Trapezoid
TOPIC · AREA OF TRAPEZOID
A trapezoidal garden has parallel sides of 8 m and 12 m, with a height of 5 m. What is the area of the garden?
⚡ Common Trap: Area of trapezoid = \(\dfrac{1}{2}(b_1 + b_2) \times h\). Students forget to multiply by ½! = ½ × (8+12) × 5 = ½ × 20 × 5 = 50 m²
20
Geometry
Angles · Polygons
TOPIC · INTERIOR ANGLES OF POLYGONS
What is the sum of all interior angles of a hexagon (6-sided polygon)?
⚡ Common Trap: Formula: Sum = (n − 2) × 180°. For a hexagon: n = 6, so (6 − 2) × 180 = 4 × 180 = 720°. Students often use n directly, not (n−2)!
Final Score
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