Part One
Algebra 1
01
Quick Memory Points — Algebra
ISOLATE → get the variable alone on one side
BALANCE → whatever you do to one side, do to the other
DISTRIBUTE → multiply outside number to everything inside ( )
COMBINE → add/subtract like terms (same variable & exponent)
SUBSTITUTE → plug a value in to check your answer
BALANCE → whatever you do to one side, do to the other
DISTRIBUTE → multiply outside number to everything inside ( )
COMBINE → add/subtract like terms (same variable & exponent)
SUBSTITUTE → plug a value in to check your answer
Q1
Algebra
One-Step Equations
Jake has some baseball cards. After giving away 7 cards to his friend,
he has 15 cards left. How many cards did Jake start with?
Equation hint: Let \(x\) = cards at start. Write: \(x - 7 = 15\)
⚠️ Common mistake: Students subtract 7 again instead of adding 7 to both sides.
⚠️ Common mistake: Students subtract 7 again instead of adding 7 to both sides.
📘 Solution
①Set up equation: \(x - 7 = 15\)
②Add 7 to both sides: \(x - 7 + 7 = 15 + 7\)
③Result: \(x = 22\)
Memory word: ISOLATE — add 7 to cancel the −7 and isolate \(x\).
Q2
Algebra
Two-Step Equations
A taxi charges a base fee of $3 plus $2 per mile.
Maria paid $13 in total. How many miles did she travel?
Equation: \(2m + 3 = 13\)
⚠️ Common mistake: Forgetting to subtract the base fee first before dividing.
⚠️ Common mistake: Forgetting to subtract the base fee first before dividing.
📘 Solution
①Subtract 3 from both sides: \(2m = 10\)
②Divide both sides by 2: \(m = 5\)
Memory word: ISOLATE — undo addition first (−3), then undo multiplication (÷2).
Q3
Algebra
Distributive Property
Simplify: \(3(x + 4) - 2x\)
⚠️ Common mistake: Only multiplying 3 by \(x\) and forgetting to multiply 3 by 4.
Always distribute to every term inside the parentheses!
📘 Solution
①DISTRIBUTE: \(3 \cdot x + 3 \cdot 4 = 3x + 12\)
②COMBINE like terms: \(3x - 2x + 12 = x + 12\)
Answer: \(x + 12\)
Q4
Algebra
Solving Inequalities
Solve: \(-4x \leq 20\). Which answer shows the correct solution and graph direction?
⚠️ FLIP the inequality sign when you multiply or divide by a negative number!
This is the #1 mistake in inequality problems.
This is the #1 mistake in inequality problems.
📘 Solution
①Divide both sides by \(-4\): \(\frac{-4x}{-4} \leq \frac{20}{-4}\)
②FLIP the sign (dividing by negative!): \(x \geq -5\)
Memory word: FLIP — negative division → flip the inequality sign.
Q5
Algebra
Slope of a Line
A line passes through points \((2, 5)\) and \((6, 13)\). What is the slope?
Formula: \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) = "rise over run"
⚠️ Keep the same point on top and bottom — don't mix them up!
⚠️ Keep the same point on top and bottom — don't mix them up!
📘 Solution
①\(m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = 2\)
Memory word: RISE over RUN — change in y ÷ change in x.
Q6
Algebra
Slope-Intercept Form
Which equation represents a line with slope \(-3\) and \(y\)-intercept \(7\)?
Formula: \(y = mx + b\) where \(m\) = slope, \(b\) = \(y\)-intercept
⚠️ The \(y\)-intercept \(b\) is the number added, not the coefficient of \(x\).
⚠️ The \(y\)-intercept \(b\) is the number added, not the coefficient of \(x\).
📘 Solution
①\(m = -3\), \(b = 7\)
②Plug into \(y = mx + b\): \(y = -3x + 7\)
Memory: m is SLOPE, b is WHERE it BEGINS (y-intercept)
Q7
Algebra
Systems of Equations
Two friends buy lunch. Together they spend $14.
One friend spends $2 more than the other. How much does the cheaper meal cost?
System: \(x + y = 14\) and \(y = x + 2\)
⚠️ Use SUBSTITUTION: plug the second equation into the first.
⚠️ Use SUBSTITUTION: plug the second equation into the first.
📘 Solution
①Substitute \(y = x + 2\) into \(x + y = 14\): \(x + (x+2) = 14\)
②Simplify: \(2x + 2 = 14\) → \(2x = 12\) → \(x = 6\)
Cheaper meal = $6. More expensive = $8. Check: \(6 + 8 = 14\) ✓
Q8
Algebra
Exponent Rules
Simplify: \(x^3 \cdot x^5\)
⚠️ Common mistake: Students multiply the exponents. Remember — for multiplication, you add exponents!
📘 Solution
①Same base → ADD exponents: \(x^3 \cdot x^5 = x^{3+5} = x^8\)
Memory: MULTIPLY bases → ADD exponents | POWER of power → MULTIPLY exponents
Q9
Algebra
Word Problem — Proportions
A car travels 120 miles in 2 hours.
At the same speed, how many miles will it travel in 5 hours?
Set up proportion: \(\dfrac{120}{2} = \dfrac{x}{5}\)
⚠️ Cross-multiply correctly: \(120 \times 5 = 2 \times x\)
⚠️ Cross-multiply correctly: \(120 \times 5 = 2 \times x\)
📘 Solution
①Speed = 60 mph (120 ÷ 2)
②Distance in 5 hours = 60 × 5 = 300 miles
Or cross-multiply: \(120 \times 5 = 2x\) → \(x = 300\).
Q10
Algebra
Percent Problems
A shirt originally costs $40. It is on sale for 25% off.
What is the sale price?
⚠️ Common mistake: Students find 25% of $40 = $10 and stop there.
Remember to subtract the discount from the original price!
📘 Solution
①Find discount: \(40 \times 0.25 = \$10\)
②Sale price: \(40 - 10 = \$30\)
Or: \(40 \times 0.75 = \$30\) (multiply by what's LEFT after discount)
Part Two
Geometry
02
Quick Memory Points — Geometry
SUPPLEMENTARY → two angles add to 180°
COMPLEMENTARY → two angles add to 90°
PYTHAGOREAN → a² + b² = c² (right triangles only, c = hypotenuse)
AREA formulas → Triangle: ½bh | Rectangle: lw | Circle: πr²
PARALLEL lines → transversal creates equal alternate interior angles
COMPLEMENTARY → two angles add to 90°
PYTHAGOREAN → a² + b² = c² (right triangles only, c = hypotenuse)
AREA formulas → Triangle: ½bh | Rectangle: lw | Circle: πr²
PARALLEL lines → transversal creates equal alternate interior angles
Q11
Geometry
Supplementary Angles
Two angles are supplementary. One angle measures 65°.
What is the measure of the other angle?
SUPPLEMENTARY = sum is 180°
⚠️ Don't confuse with complementary (90°)! Trick: Supplementary → Straight line = 180°
⚠️ Don't confuse with complementary (90°)! Trick: Supplementary → Straight line = 180°
📘 Solution
①Supplementary: angles sum to 180°
②\(180° - 65° = 115°\)
Memory: S = Straight = 180° | C = Corner = 90°
Q12
Geometry
Pythagorean Theorem
A right triangle has legs of length 6 and 8.
What is the length of the hypotenuse?
Formula: \(a^2 + b^2 = c^2\) (\(c\) is always the longest side — hypotenuse)
⚠️ The hypotenuse is opposite the right angle, not the side next to it.
⚠️ The hypotenuse is opposite the right angle, not the side next to it.
📘 Solution
①\(6^2 + 8^2 = c^2\)
②\(36 + 64 = 100 = c^2\)
③\(c = \sqrt{100} = 10\)
Famous triple: 3-4-5 (×2 = 6-8-10) ← memorize this!
Q13
Geometry
Area of a Triangle
A triangle has a base of 10 cm and a height of 6 cm.
What is its area?
Formula: \(A = \dfrac{1}{2} \times b \times h\)
⚠️ Students often forget the \(\dfrac{1}{2}\). Think of it as "half a rectangle."
⚠️ Students often forget the \(\dfrac{1}{2}\). Think of it as "half a rectangle."
📘 Solution
①\(A = \frac{1}{2} \times 10 \times 6\)
②\(A = \frac{1}{2} \times 60 = 30 \text{ cm}^2\)
Memory: HALF of base times height
Q14
Geometry
Circle — Circumference
A circle has a radius of 5 cm. What is its circumference?
(Use \(\pi \approx 3.14\))
(Use \(\pi \approx 3.14\))
Formula: \(C = 2\pi r\) or \(C = \pi d\)
⚠️ Circumference uses radius, not radius squared. Area uses \(r^2\).
⚠️ Circumference uses radius, not radius squared. Area uses \(r^2\).
📘 Solution
①\(C = 2\pi r = 2 \times 3.14 \times 5\)
②\(C = 31.4 \text{ cm}\)
Memory: Circumference = 2πr (no square!) | Area = πr² (squared!)
Q15
Geometry
Interior Angles of a Triangle
A triangle has angles of 50° and 70°.
What is the measure of the third angle?
Rule: All three interior angles of any triangle always sum to 180°.
⚠️ This applies to ALL triangles — right, acute, obtuse. No exceptions!
⚠️ This applies to ALL triangles — right, acute, obtuse. No exceptions!
📘 Solution
①\(50° + 70° + x = 180°\)
②\(120° + x = 180°\) → \(x = 60°\)
Memory: Triangle = 180° (three sides, one-eighty)
Q16
Geometry
Perimeter of a Rectangle
A rectangular garden is 12 m long and 5 m wide.
How much fencing is needed to go all the way around it?
Formula: \(P = 2l + 2w\) or \(P = 2(l + w)\)
⚠️ "Fencing around" = perimeter (not area). Perimeter is the outside distance.
⚠️ "Fencing around" = perimeter (not area). Perimeter is the outside distance.
📘 Solution
①\(P = 2(12 + 5) = 2 \times 17 = 34 \text{ m}\)
Memory: PERIMETER = "PARAMETER" = outline = add ALL sides
Q17
Geometry
Vertical Angles
Two lines intersect. One of the angles formed is 42°.
What is the measure of the angle directly across from it (vertical angle)?
Vertical angles are always equal.
⚠️ Adjacent angles (next to each other) are supplementary = 180°, not equal!
⚠️ Adjacent angles (next to each other) are supplementary = 180°, not equal!
📘 Solution
①Vertical angles (across from each other) are always equal.
②Vertical angle = 42°
Memory: VERTICAL = EQUAL (V = same value)
Q18
Geometry
Area of a Circle
A circular pizza has a diameter of 14 inches. What is its area?
(Use \(\pi \approx 3.14\))
(Use \(\pi \approx 3.14\))
Formula: \(A = \pi r^2\) — note: radius, not diameter!
⚠️ The problem gives diameter (14 in). Radius = diameter ÷ 2 = 7 in. Don't forget to halve it!
⚠️ The problem gives diameter (14 in). Radius = diameter ÷ 2 = 7 in. Don't forget to halve it!
📘 Solution
①Radius = 14 ÷ 2 = 7 in
②\(A = \pi r^2 = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \text{ in}^2\)
Memory: DIAMETER ÷ 2 = RADIUS — always check which one is given!
Q19
Geometry
Exterior Angle Theorem
In a triangle, two interior angles measure 40° and 65°.
What is the measure of the exterior angle at the third vertex?
Exterior Angle Theorem: An exterior angle equals the sum of the two non-adjacent interior angles.
⚠️ Students add all three angles including the third interior one — but you only add the two remote interior angles.
⚠️ Students add all three angles including the third interior one — but you only add the two remote interior angles.
📘 Solution
①Exterior angle = sum of the two remote interior angles
②\(40° + 65° = 105°\)
Check: Third interior angle = \(180° - 105° = 75°\). Exterior + interior = 180° ✓
Q20
Geometry
Volume of a Rectangular Prism
A box is 4 cm long, 3 cm wide, and 5 cm tall.
What is its volume?
Formula: \(V = l \times w \times h\)
⚠️ Volume is 3D — multiply three dimensions, not two! (Two dimensions = area)
⚠️ Volume is 3D — multiply three dimensions, not two! (Two dimensions = area)
📘 Solution
①\(V = 4 \times 3 \times 5\)
②\(V = 60 \text{ cm}^3\)
Memory: VOLUME = 3 sides multiplied → answer in cubic units (³)