Math Mastery — Pre-Algebra & Geometry

Part 1

Pre-Algebra

Integers, fractions, variables, equations, and inequalities — the foundation of all higher math.

0 / 10 answered

Pre-Algebra ⚠ Tricky

🧠 Memory Key NEGATIVE × NEGATIVE = POSITIVE | think: "double negative = positive"

Sarah has a bank account. She owes her friend $15 and writes it as −$15. If she "undoes" (negates) that debt 3 times, what is the result?

Calculate: (−15) × (−3)

💡 Quick Example

(−4) × (−5) = +20 ← two negatives cancel out, result is always positive

📖 Explanation

Two negatives multiply to a positive. (−15) × (−3) = +45. A common mistake is forgetting to flip the sign. Rule: negative × negative = positive; negative × positive = negative.

Pre-Algebra⚠ Tricky

🧠Memory Key ORDER OF OPS: PEMDAS — Parentheses · Exponents · ×÷ · +−

What is 3 + 4 × 2 − 1?

💡 Quick Example

2 + 3 × 4 → do × first: 2 + 12 = 14 (NOT 20!)

📖 Explanation

PEMDAS: multiplication before addition/subtraction. So: 3 + (4×2) − 1 = 3 + 8 − 1 = 10. Students who go left-to-right get 13 — the most common mistake here!

Pre-Algebra

🧠Memory Key FRACTION DIVISION: "Keep · Change · Flip" (KCF)

A recipe needs 3/4 cup of sugar per batch. You have 3 cups of sugar. How many complete batches can you make?

Calculate: 3 ÷ 3/4

💡 Quick Example

2 ÷ 1/3: Keep 2, Change ÷ to ×, Flip 1/3 to 3/1 → 2 × 3 = 6

📖 Explanation

KCF: 3 ÷ (3/4) = 3 × (4/3) = 12/3 = 4 batches. Many students multiply instead of flipping — always flip the second fraction when dividing!

Pre-Algebra⚠ Tricky

🧠Memory Key INEQUALITY FLIP: multiply / divide by NEGATIVE → flip the sign (< becomes >)

Solve the inequality: −2x + 5 > 11

Which value of x makes this true?

💡 Quick Example

−3x > 9 → divide by −3 AND flip: x < −3

📖 Explanation

Step 1: −2x + 5 > 11 → −2x > 6. Step 2: Divide by −2 and FLIP the inequality → x < −3. Forgetting to flip is the #1 mistake in inequality problems!

Pre-Algebra

🧠Memory Key PERCENT: Part ÷ Whole × 100 | "IS over OF = % over 100"

A jacket originally costs $80. It is on sale for 30% off. What is the sale price?

💡 Quick Example

20% off $50: discount = 0.20 × 50 = $10, sale price = $50 − $10 = $40
Or use multiplier: $50 × 0.80 = $40

📖 Explanation

Discount = 30% of $80 = 0.30 × 80 = $24. Sale price = $80 − $24 = $56. Common error: students choose $24 (the discount) instead of the sale price!

Pre-Algebra⚠ Tricky

🧠Memory Key LIKE TERMS: same variable + same exponent → can combine | "Match the dress code!"

Simplify: 3x² + 5x − 2x² + x − 4

💡 Quick Example

4y² + 2y − y² + 3y → (4−1)y² + (2+3)y = 3y² + 5y

📖 Explanation

Group like terms: x² terms: 3x² − 2x² = x². x terms: 5x + x = 6x. Constants: −4. Answer: x² + 6x − 4. Don't mix x² with x — they are different terms!

Pre-Algebra

🧠Memory Key RATIO → PROPORTION: cross-multiply | a/b = c/d → ad = bc

A car travels 150 miles in 3 hours. At the same speed, how many miles does it travel in 5 hours?

💡 Quick Example

60 miles / 2 hrs = x / 5 hrs → cross-multiply: 60 × 5 = 2x → x = 150 miles

📖 Explanation

Speed = 150 ÷ 3 = 50 mph. Distance = 50 × 5 = 250 miles. Or use proportion: 150/3 = x/5 → 3x = 750 → x = 250.

Pre-Algebra⚠ Tricky

🧠Memory Key EXPONENT RULES: x^a × x^b = x^(a+b) | "Same base → ADD the powers"

Simplify: x³ × x⁴ ÷ x²

💡 Quick Example

y² × y³ = y⁵ (add) y⁶ ÷ y² = y⁴ (subtract)

📖 Explanation

x³ × x⁴ = x^(3+4) = x⁷. Then x⁷ ÷ x² = x^(7−2) = x⁵. Students often multiply the exponents instead of adding — only do that with the Power Rule: (x³)⁴ = x¹².

Pre-Algebra

🧠Memory Key SLOPE: rise over run | m = (y₂−y₁)/(x₂−x₁) | "rise/run, never run/rise"

Find the slope of the line passing through (2, 5) and (6, 13).

💡 Quick Example

Points (1,2) and (3,8): slope = (8−2)/(3−1) = 6/2 = 3

📖 Explanation

m = (13−5)/(6−2) = 8/4 = 2. Remember: always subtract y-values first (top), then x-values (bottom). Reversing gives you the reciprocal!

Pre-Algebra⚠ Tricky

🧠Memory Key DISTRIBUTIVE: a(b+c) = ab+ac | "Multiply EVERYONE inside the parentheses"

Solve for x: 2(3x − 4) = 5x + 6

💡 Quick Example

2(x + 3) = 10 → distribute: 2x + 6 = 10 → 2x = 4 → x = 2

📖 Explanation

Distribute: 6x − 8 = 5x + 6. Subtract 5x: x − 8 = 6. Add 8: x = 14. Students often forget to distribute the 2 to the −4, writing 6x − 4 instead of 6x − 8.

Part 2

Geometry

Angles, triangles, circles, area, perimeter, and the Pythagorean theorem — spatial reasoning at its core.

0 / 10 answered

Geometry⚠ Tricky

🧠Memory Key PYTHAGOREAN: a² + b² = c² | c = HYPOTENUSE (longest side, opposite 90°)

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

💡 Quick Example

Legs 3 and 4: 3² + 4² = 9 + 16 = 25 → c = √25 = 5
Memorize this triple: 3-4-5 (and multiples: 6-8-10, 9-12-15…)

📖 Explanation

a² + b² = c² → 6² + 8² = 36 + 64 = 100 → c = √100 = 10. This is the famous 3-4-5 triple scaled by 2: 6-8-10. Recognizing these triples saves time!

Geometry

🧠Memory Key SUPPLEMENTARY = 180° | COMPLEMENTARY = 90° | "S comes before C, 180 > 90"

Two angles are supplementary. One angle measures 117°. What is the measure of the other angle?

💡 Quick Example

Supplementary: 65° + ? = 180° → ? = 115°

📖 Explanation

Supplementary angles sum to 180°. So: 180° − 117° = 63°. Don't confuse with complementary (90°)! The trick: Supplementary starts with 'S' like 'Straight' (a straight line = 180°).

Geometry⚠ Tricky

🧠Memory Key CIRCLE AREA: A = πr² | CIRCUMFERENCE: C = 2πr = πd | "Area needs r-SQUARED"

A circular pizza has a diameter of 14 inches. What is its area? (Use π ≈ 3.14)

💡 Quick Example

Diameter = 10 → radius = 5 → Area = π × 5² = 3.14 × 25 = 78.5 in²

📖 Explanation

Diameter = 14, so radius = 7. Area = πr² = 3.14 × 7² = 3.14 × 49 = 153.86 in². COMMON ERROR: using diameter (14) instead of radius (7) gives 615.44 — the #1 circle mistake!

Geometry

🧠Memory Key TRIANGLE ANGLES: always sum to 180° | "Three corners, one-eighty total"

A triangle has angles of 48° and 75°. What is the measure of the third angle?

💡 Quick Example

Angles 40° and 90°: third = 180° − 40° − 90° = 50°

📖 Explanation

180° − 48° − 75° = 57°. Always remember: all three angles in ANY triangle must add up to exactly 180°. This is the Triangle Angle Sum Theorem.

Geometry⚠ Tricky

🧠Memory Key VOLUME of CYLINDER: V = πr²h | "Circle base × height"

A cylindrical water tank has a radius of 4 ft and a height of 10 ft. What is its volume? (Use π ≈ 3.14)

💡 Quick Example

r=3, h=5: V = 3.14 × 3² × 5 = 3.14 × 9 × 5 = 141.3 ft³

📖 Explanation

V = πr²h = 3.14 × 4² × 10 = 3.14 × 16 × 10 = 502.4 ft³. Mistake alert: using diameter 4 instead of radius 4 (it's already the radius here!), or forgetting to square the radius.

Geometry

🧠Memory Key SIMILAR TRIANGLES: corresponding sides are PROPORTIONAL | "Same shape, different size"

Two similar triangles have corresponding sides in a ratio of 3 : 5. The smaller triangle has a perimeter of 24 cm. What is the perimeter of the larger triangle?

💡 Quick Example

Ratio 2:3, small perimeter=10 → 10/2 × 3 = 15 (scale up)

📖 Explanation

Scale factor = 5/3. Larger perimeter = 24 × (5/3) = 40 cm. Perimeters scale by the same ratio as the sides. Note: areas scale by the ratio SQUARED, but perimeters scale linearly.

Geometry⚠ Tricky

🧠Memory Key PARALLEL LINES + TRANSVERSAL: alternate interior angles are EQUAL | "Z-angles are equal"

Two parallel lines are cut by a transversal. One of the alternate interior angles is 68°. What is the measure of its alternate interior angle partner?

💡 Quick Example

The two angles that form a "Z" shape between parallel lines are always equal.

📖 Explanation

Alternate interior angles formed by parallel lines cut by a transversal are congruent (equal). So the answer is 68°. Students often confuse these with co-interior (same-side) angles, which are supplementary (sum to 180°).

Geometry

🧠Memory Key AREA of TRAPEZOID: A = ½(b₁ + b₂) × h | "Average the two bases, then × height"

A trapezoid has parallel bases of 8 cm and 14 cm, and a height of 6 cm. What is its area?

💡 Quick Example

Bases 4 and 6, height 5: A = ½(4+6) × 5 = ½ × 10 × 5 = 25 cm²

📖 Explanation

A = ½(8 + 14) × 6 = ½ × 22 × 6 = 11 × 6 = 66 cm². Forgetting the ½ gives 132 — the most common error. Think of the trapezoid as a "squished" parallelogram cut in half!

Geometry⚠ Tricky

🧠Memory Key EXTERIOR ANGLE THEOREM: exterior angle = sum of TWO non-adjacent interior angles

In a triangle, two interior angles are 42° and 58°. What is the measure of the exterior angle at the third vertex?

💡 Quick Example

Interior angles 30° and 70°: exterior angle = 30° + 70° = 100°

📖 Explanation

Exterior Angle Theorem: exterior = 42° + 58° = 100°. The third interior angle = 180° − 42° − 58° = 80°, and the exterior angle = 180° − 80° = 100°. Both methods work! Students often just subtract from 180° (find interior) and forget to find the exterior.

Geometry⚠ Tricky

🧠Memory Key SURFACE AREA of RECTANGULAR PRISM: SA = 2(lw + lh + wh) | "2 of each face pair"

A gift box is 5 cm long, 3 cm wide, and 4 cm tall. How much wrapping paper (surface area) is needed to cover it completely?

💡 Quick Example

l=2, w=3, h=4: SA = 2(2×3 + 2×4 + 3×4) = 2(6+8+12) = 2(26) = 52 cm²

📖 Explanation

SA = 2(lw + lh + wh) = 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 2 × 47 = 94 cm². Forgetting to multiply by 2 gives 47 — because each face has a matching opposite face!

Math MasteryWorksheet

Math Mastery
Worksheet