πŸ“
Math Practice Notebook
Pre-Algebra & Geometry Β· Self-Study Edition Β· Interactive Quiz
✏️ 20 Problems 🧠 Memory Points πŸ“– Worked Examples 🎯 Multiple Choice πŸŽ‰ Instant Feedback
πŸ“˜ Pre-Algebra
πŸ“ Geometry
Pre-Algebra Score: 0 / 10
πŸ“˜ Section 1: Pre-Algebra
1
Order of Operations
⚑
PEMDAS Parentheses β†’ Exponents β†’ Multiplication/Division β†’ Addition/Subtraction
πŸ“Œ Worked Example Solve: \(3 + 2 \times 4\)
Step 1: Multiply first β†’ \(2 \times 4 = 8\)
Step 2: Add β†’ \(3 + 8 = \mathbf{11}\) βœ“
Evaluate: \(5 + 3^2 \times 2 - (4 + 1)\)
Start inside the ( ) first, then handle the exponent \(3^2\), then multiply, then add/subtract left to right.
πŸ“– Explanation
\(5 + 3^2 \times 2 - (4+1)\)
= \(5 + 9 \times 2 - 5\)  (Parentheses & Exponent)
= \(5 + 18 - 5\)  (Multiply)
= \(\mathbf{18}\) βœ“  (Left to right Add/Subtract)
2
Negative Numbers ⚠️ TRICKY
🧲
SAME signs β†’ ADD Β· DIFFERENT signs β†’ SUBTRACT Subtracting a negative = ADDING a positive:  \(a - (-b) = a + b\)
πŸ“Œ Worked Example \(-3 - (-7) = -3 + 7 = \mathbf{4}\)
Two negatives together flip to positive!
What is \(-8 - (-3) + (-5)\)?
πŸ“– Explanation
\(-8 - (-3) + (-5)\)
= \(-8 + 3 + (-5)\)  (flip the double negative)
= \(-8 + 3 - 5\)
= \(-5 - 5 = \mathbf{-10}\) βœ“
3
Fractions β€” Adding
πŸ•
LCD first! (Least Common Denominator) Same bottom β†’ add tops. Different bottoms β†’ find LCD, convert, THEN add.
πŸ“Œ Worked Example \(\dfrac{1}{3} + \dfrac{1}{4}\) β†’ LCD = 12
= \(\dfrac{4}{12} + \dfrac{3}{12} = \dfrac{7}{12}\) βœ“
Calculate: \(\dfrac{2}{3} + \dfrac{3}{4}\)
πŸ“– Explanation
LCD of 3 and 4 = 12
\(\dfrac{2}{3} = \dfrac{8}{12}\),   \(\dfrac{3}{4} = \dfrac{9}{12}\)
\(\dfrac{8}{12} + \dfrac{9}{12} = \mathbf{\dfrac{17}{12}}\) βœ“  (improper fraction β€” bigger than 1!)
4
Solving One-Step Equations ⚠️ TRICKY
βš–οΈ
BALANCE the scale Whatever you do to one side β†’ do the SAME to the other. Goal: isolate the variable!
πŸ“Œ Worked Example Solve: \(x + 5 = 12\)
Subtract 5 from both sides: \(x = 12 - 5 = \mathbf{7}\)
Solve for \(x\):   \(3x - 7 = 14\)
πŸ“– Explanation
\(3x - 7 = 14\)
Add 7 to both sides: \(3x = 21\)
Divide both sides by 3: \(x = \mathbf{7}\) βœ“
Check: \(3(7) - 7 = 21 - 7 = 14\) βœ“
5
Percent Problems
πŸ’―
IS / OF = % / 100 "Percent OF something" β†’ multiply by decimal.   \(p\% = \dfrac{p}{100}\)
πŸ“Œ Worked Example What is 30% of 80?
\(0.30 \times 80 = \mathbf{24}\) βœ“
A shirt costs $45. There is a 20% discount. What is the sale price?
πŸ“– Explanation
Discount = \(20\% \times \$45 = 0.20 \times 45 = \$9\)
Sale price = \(\$45 - \$9 = \mathbf{\$36}\) βœ“
Shortcut: 80% of $45 = \(0.80 \times 45 = \$36\)
6
Ratios & Proportions ⚠️ TRICKY
βœ–οΈ
CROSS-MULTIPLY to solve proportions \(\dfrac{a}{b} = \dfrac{c}{d}\) β†’ \(ad = bc\)
πŸ“Œ Worked Example \(\dfrac{3}{5} = \dfrac{x}{20}\) β†’ \(3 \times 20 = 5x\) β†’ \(60 = 5x\) β†’ \(x = \mathbf{12}\)
If \(\dfrac{4}{7} = \dfrac{x}{35}\), what is \(x\)?
πŸ“– Explanation
Cross-multiply: \(4 \times 35 = 7 \times x\)
\(140 = 7x\)
\(x = \dfrac{140}{7} = \mathbf{20}\) βœ“
7
Exponents & Powers
πŸ”‹
POWER RULES \(x^0 = 1\) (anything!)  Β·  \(x^m \cdot x^n = x^{m+n}\)  Β·  \((x^m)^n = x^{mn}\)
πŸ“Œ Worked Example \(2^3 \times 2^2 = 2^{3+2} = 2^5 = 32\) βœ“
Simplify: \(3^4 \div 3^2\)
πŸ“– Explanation
\(\dfrac{3^4}{3^2} = 3^{4-2} = 3^2 = \mathbf{9}\) βœ“
(When dividing same base β†’ subtract exponents!)
8
Combining Like Terms ⚠️ TRICKY
🍎
LIKE TERMS = same variable, same exponent You can only add apples to apples!   \(3x + 2y \neq 5xy\)
πŸ“Œ Worked Example \(4x + 3y - 2x + y\)
= \((4x - 2x) + (3y + y) = 2x + 4y\) βœ“
Simplify: \(5x^2 - 3x + 2x^2 + 7x - 4\)
πŸ“– Explanation
Group like terms:
\(x^2\) terms: \(5x^2 + 2x^2 = 7x^2\)
\(x\) terms: \(-3x + 7x = 4x\)
Constants: \(-4\)
Answer: \(\mathbf{7x^2 + 4x - 4}\) βœ“
9
Inequalities
πŸ”„
FLIP the sign when multiplying/dividing by NEGATIVE! \(-2x > 6\) β†’ divide by \(-2\) β†’ \(x < -3\) (sign flips!)
πŸ“Œ Worked Example Solve: \(-3x \leq 9\)
Divide by \(-3\) and FLIP: \(x \geq -3\) βœ“
Solve: \(-2x + 5 > 11\)
πŸ“– Explanation
\(-2x + 5 > 11\)
Subtract 5: \(-2x > 6\)
Divide by \(-2\) and FLIP the sign: \(x < -3\) βœ“
⚠️ Most common mistake: forgetting to flip the inequality sign!
10
Distributive Property ⚠️ TRICKY
πŸ“¦
DISTRIBUTE = multiply EVERYTHING inside the parentheses \(a(b + c) = ab + ac\)  Β·  Watch the sign when \(a\) is negative!
πŸ“Œ Worked Example \(-3(x - 4) = -3 \cdot x - (-3)(4) = -3x + 12\) βœ“
Negative Γ— Negative = Positive!
Expand and simplify: \(2(3x - 5) - 4(x + 1)\)
πŸ“– Explanation
\(2(3x-5) - 4(x+1)\)
= \(6x - 10 - 4x - 4\)  (distribute each)
= \((6x-4x) + (-10-4)\)
= \(\mathbf{2x - 14}\) βœ“
⚠️ Don't forget: \(-4 \times 1 = -4\) (not +4!)
Geometry Score: 0 / 10
πŸ“ Section 2: Geometry
1
Angle Types
πŸ“
ANGLE NAMES by degrees Acute < 90Β°  Β·  Right = 90Β°  Β·  Obtuse 90°–180Β°  Β·  Straight = 180Β°
πŸ“Œ Worked Example Two angles are supplementary. One is 65Β°. Find the other.
Supplementary = sum is 180Β° β†’ \(180Β° - 65Β° = \mathbf{115Β°}\) βœ“
Two angles are complementary. One angle is 38°. What is the other? ⚠️ TRICKY
Complementary angles add up to 90Β°. (Supplementary = 180Β°. Don't mix them up!)
πŸ“– Explanation
Complementary β†’ sum = 90Β°
Other angle = \(90Β° - 38Β° = \mathbf{52Β°}\) βœ“
Memory trick: Complementary β†’ Corner (right angle = 90Β°)
2
Area of a Triangle
πŸ”Ί
Triangle Area = HALF of rectangle \(A = \dfrac{1}{2} \times base \times height\)  Β·  Height must be PERPENDICULAR to base!
h b ←————→
A triangle has a base of 12 cm and a height of 9 cm. What is its area?
πŸ“– Explanation
\(A = \dfrac{1}{2} \times 12 \times 9 = \dfrac{1}{2} \times 108 = \mathbf{54 \text{ cm}^2}\) βœ“
3
Pythagorean Theorem ⚠️ TRICKY
🐍
aΒ² + bΒ² = cΒ²  (c = hypotenuse, always the LONGEST side!) Only works for RIGHT triangles. The hypotenuse is ALWAYS opposite the 90Β° angle.
πŸ“Œ Worked Example Legs = 3 and 4. Find hypotenuse.
\(3^2 + 4^2 = 9 + 16 = 25\) β†’ \(c = \sqrt{25} = \mathbf{5}\) βœ“  (3-4-5 triangle!)
In a right triangle, legs are 5 cm and 12 cm. What is the hypotenuse?
πŸ“– Explanation
\(a^2 + b^2 = c^2\)
\(5^2 + 12^2 = 25 + 144 = 169\)
\(c = \sqrt{169} = \mathbf{13 \text{ cm}}\) βœ“  (5-12-13 is a classic Pythagorean triple!)
4
Perimeter vs Area
🏠
Perimeter = FENCE (distance around) Β· Area = CARPET (space inside) Rectangle: \(P = 2l + 2w\)  Β·  \(A = l \times w\)
A rectangle has length 8 m and width 5 m. What is the difference between its area and its perimeter?
πŸ“– Explanation
Area = \(8 \times 5 = 40 \text{ m}^2\)
Perimeter = \(2(8) + 2(5) = 16 + 10 = 26 \text{ m}\)
Difference = \(40 - 26 = \mathbf{14}\) βœ“
⚠️ Notice area has unitsΒ² but perimeter has units β€” they are different types!
5
Circles β€” Area & Circumference ⚠️ TRICKY
β­•
C = 2Ο€r (Cherry Pie) Β· A = Ο€rΒ² (Apple Pie) Remember: d = 2r (diameter = 2 Γ— radius). Don't mix up r and d!
πŸ“Œ Worked Example Circle with radius 5:   \(A = \pi \times 5^2 = 25\pi \approx 78.5\)
A circle has a diameter of 10 cm. What is its area? (Use \(\pi \approx 3.14\))
πŸ“– Explanation
Diameter = 10 β†’ Radius = \(10 \div 2 = 5\) cm
\(A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = \mathbf{78.5 \text{ cm}^2}\) βœ“
⚠️ Most common mistake: using diameter instead of radius in the formula!
6
Triangle Angle Sum
πŸ”Ί
ALL triangles: angle sum = 180Β°. ALWAYS. Exterior angle = sum of the two NON-adjacent interior angles.
A triangle has angles of 47Β° and 85Β°. What is the third angle?
πŸ“– Explanation
Sum of all angles = 180Β°
Third angle = \(180Β° - 47Β° - 85Β° = \mathbf{48Β°}\) βœ“
7
Volume of Rectangular Prism
πŸ“¦
Volume = length Γ— width Γ— height (l Γ— w Γ— h) Units are CUBED: cmΒ³, mΒ³, etc. Volume = how many unit cubes fit inside!
A box is 6 cm long, 4 cm wide, and 3 cm tall. What is its volume?
πŸ“– Explanation
\(V = l \times w \times h = 6 \times 4 \times 3 = \mathbf{72 \text{ cm}^3}\) βœ“
8
Similar Triangles ⚠️ TRICKY
πŸ”
SIMILAR = same shape, different size. Sides are PROPORTIONAL. Corresponding sides β†’ same ratio (scale factor). Set up a proportion and cross-multiply!
πŸ“Œ Worked Example β–³ABC ~ β–³DEF. AB=4, DE=6, BC=5. Find EF.
\(\dfrac{4}{6} = \dfrac{5}{EF}\) β†’ \(EF = \dfrac{5 \times 6}{4} = 7.5\) βœ“
Two similar triangles have sides in ratio 3 : 5. The smaller triangle has a side of 9 cm. What is the corresponding side of the larger triangle?
πŸ“– Explanation
Set up the proportion: \(\dfrac{3}{5} = \dfrac{9}{x}\)
Cross-multiply: \(3x = 45\)
\(x = \mathbf{15 \text{ cm}}\) βœ“
9
Parallel Lines & Transversals
πŸ›€οΈ
Parallel lines cut by transversal: Alternate interior = EQUAL  Β·  Corresponding = EQUAL  Β·  Co-interior (same-side) = 180Β°
l₁ lβ‚‚ aΒ° aΒ° alt. int.
Two parallel lines are cut by a transversal. One alternate interior angle is \((3x + 10)°\) and the other is \((5x - 20)°\). Find \(x\). ⚠️ TRICKY
πŸ“– Explanation
Alternate interior angles are EQUAL:
\(3x + 10 = 5x - 20\)
\(10 + 20 = 5x - 3x\)
\(30 = 2x\)
\(x = \mathbf{15}\) βœ“
Check: \(3(15)+10 = 55Β°\), \(5(15)-20 = 55Β°\) βœ“
10
Coordinate Geometry β€” Distance & Midpoint ⚠️ TRICKY
πŸ“
Midpoint = AVERAGE the x's, AVERAGE the y's \(M = \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)\)
πŸ“Œ Worked Example Midpoint of \((2, 4)\) and \((8, 10)\):
\(M = \left(\dfrac{2+8}{2}, \dfrac{4+10}{2}\right) = \left(5, 7\right)\) βœ“
Point \(A = (1, -3)\) and point \(B = (7, 5)\). What is the midpoint of \(\overline{AB}\)?
πŸ“– Explanation
\(M = \left(\dfrac{1+7}{2},\ \dfrac{-3+5}{2}\right) = \left(\dfrac{8}{2},\ \dfrac{2}{2}\right) = \mathbf{(4,\ 1)}\) βœ“
⚠️ Careful with negatives: \(-3 + 5 = +2\), not \(-8\)!