Self-Study Worksheet · Interactive

Pre-Algebra &
Geometry

20 carefully crafted problems targeting the most commonly missed concepts. Work at your own pace.

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Part A — Pre-Algebra
Variables · Equations · Ratios · Proportions · Integers
01
Order of Operations
Evaluate: 3 + 4 × 2² − (8 ÷ 4)
★ Common mistake: students add 3+4 first!
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Memory key → PEMDAS  Parentheses · Exponents · Multiply/Divide · Add/Subtract (left→right)
02
Solving Equations
Solve for x:  2x + 5 = 17
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Memory key → UNDO  Do the opposite operation in reverse order. Subtract first, then divide.
03
Ratios & Proportions
A map uses a scale of 1 inch = 25 miles. Two cities are 3.5 inches apart on the map. What is the actual distance?
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Memory key → CROSS-MULTIPLY  Set up a/b = c/d, then multiply diagonals: a×d = b×c
04
Integers
What is (−3) × (−4) + (−6)?
★ Trick spot: sign rules for multiplication vs. addition!
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Memory key → SAME=PLUS / DIFF=MINUS  (−)×(−) = (+)  |  (−)×(+) = (−)
05
Percentages
A shirt originally costs $40. It goes on sale for 25% off. What is the sale price?
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Memory key → MULTIPLY BY (1 − rate)  25% off → multiply by 0.75. Faster than finding discount first!
06
Combining Like Terms
Simplify: 5x + 3y − 2x + 7y − x
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Memory key → MATCH LETTERS  Only combine terms with the exact same variable. x-terms with x, y-terms with y.
07
Inequalities
Solve: −3x > 12
★ The #1 most forgotten rule in algebra!
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Memory key → FLIP SIGN  Whenever you multiply or divide both sides by a NEGATIVE number, the inequality sign flips!
08
Exponents
Simplify: 2³ × 2⁴
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Memory key → SAME BASE = ADD EXPONENTS  aᵐ × aⁿ = aᵐ⁺ⁿ  |  Never multiply the base!
09
Word Problem – Rate
Jake drives at 60 mph for 2.5 hours, then at 40 mph for 1 hour. What is his total distance?
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Memory key → D = R × T  Distance = Rate × Time. Calculate each leg separately, then add totals.
10
Distributive Property
Expand: −2(3x − 5)
★ Watch out for the negative sign outside!
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Memory key → DISTRIBUTE EVERYTHING  Multiply the outside term by EACH term inside, keeping all signs. −2 × (−5) = +10!
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Part B — Geometry
Angles · Triangles · Area · Perimeter · Pythagorean Theorem · Circles
11
Angles
Two angles are supplementary. One angle measures 73°. What is the other angle?
★ Don't confuse supplementary (180°) with complementary (90°)!
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Memory key → S=Straight=180 / C=Corner=90  Supplementary → Straight line = 180° | Complementary → Corner = 90°
12
Triangle Angles
A triangle has angles 55° and 82°. What is the third angle?
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Memory key → TRIANGLE SUM = 180  All three angles of any triangle always add up to exactly 180°.
13
Pythagorean Theorem
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
a² + b² = c²  →  6² + 8² = c²
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HYPOTENUSE = LONGEST SIDE  Memorize 3-4-5 and 6-8-10 as "Pythagorean triples" — they appear constantly on tests!
14
Area of Triangle
A triangle has a base of 12 cm and a height of 7 cm. What is its area?
★ Students often forget the ÷ 2!
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Memory key → HALF-BASE-HEIGHT  A = ½ × b × h. A triangle is exactly half of a rectangle with the same base and height.
15
Circle — Circumference
A circle has a diameter of 10 cm. What is its circumference? (Use π ≈ 3.14)
C = π × d   OR   C = 2πr
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Memory key → DIAMETER vs RADIUS  d = 2r. If given diameter, use C = πd directly. Don't divide then multiply again!
16
Circle — Area
A circle has a radius of 7 cm. What is its area? (Use π ≈ 3.14)
A = π × r²
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Memory key → π r SQUARED  "Pie Are Squared!" The r is squared, not just multiplied. 7² = 49, not 14!
17
Perimeter
A rectangle has a length of 15 m and a width of 8 m. What is its perimeter?
★ Common error: forgetting to double both sides!
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Memory key → WALK AROUND  P = 2l + 2w = 2(l + w). You walk around all 4 sides — two lengths AND two widths.
18
Volume of Rectangular Prism
A box is 4 cm long, 3 cm wide, and 5 cm tall. What is its volume?
V = l × w × h
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Memory key → LAYER × LAYERS  Think of it as area of base (l × w) times the height. How many base layers stack up?
19
Parallel Lines & Transversal
Two parallel lines are cut by a transversal. One angle formed is 65°. What is its alternate interior angle?
★ Classic geometry trap — know your angle pair names!
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Memory key → Z-ANGLE = EQUAL  Alternate interior angles form a "Z" shape between the parallel lines → they are always EQUAL.
20
Coordinate Geometry
What is the distance between points A(1, 2) and B(4, 6)?
d = √[(x₂−x₁)² + (y₂−y₁)²]
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Memory key → PYTHAGOREAN ON GRID  The distance formula IS the Pythagorean theorem! Δx and Δy are the two legs.