Self-Study Workbook · Grade 9

Algebra 1
& Geometry 1

Core problems with memory keys — practice at your own pace.

Algebra 1 — 10 Core Problems

Variables, equations, inequalities, functions, and systems — the most commonly missed topics.

ALGEBRA SCORE
0 / 10
01
Solving Equations
Solving One-Step Equations
Students forget to apply the same operation to BOTH sides.
⚡
Memory Key
BALANCE = whatever you do LEFT, do RIGHT
ISOLATE = get the variable alone
Example
Solve: \(x + 7 = 15\)
Subtract 7 from both sides: \(x + 7 - 7 = 15 - 7\)
Answer: \(x = 8\)
Your Turn
Solve for \(x\):
\(x - 4 = 11\)
✏️ Hint: Add 4 to both sides.
\(x - 4 + 4 = 11 + 4\) → \(x = 15\)
02
Solving Equations
Two-Step Equations
Common mistake: dividing before removing the constant.
⚡
Memory Key
ORDER: ADD/SUBTRACT first → then MULTIPLY/DIVIDE
Example
Solve: \(3x + 5 = 20\)
Step 1 — subtract 5: \(3x = 15\)
Step 2 — divide by 3: \(x = 5\)
Your Turn
Solve for \(x\):
\(2x - 6 = 10\)
✏️ Add 6 to both sides → \(2x = 16\) → divide by 2 → \(x = 8\)
03
Inequalities
Solving & Graphing Inequalities
🚨 Most missed: flipping the sign when multiplying/dividing by a negative.
⚡
Memory Key
FLIP the sign when you × or ÷ by a NEGATIVE number
"Negative → Reverse"
Example
Solve: \(-3x > 12\)
Divide both sides by \(-3\) and FLIP: \(x < -4\)
Graph: open circle at \(-4\), arrow pointing left.
Your Turn
Solve: \(-2x \leq 8\)
(Enter the simplified inequality direction: write "greater" or "less")
✏️ Divide by \(-2\) → sign FLIPS from \(\leq\) to \(\geq\) → \(x \geq -4\)
Answer: x is greater than or equal to \(-4\).
04
Linear Functions
Slope Formula
Students mix up which y and x is subtracted from which.
⚡
Memory Key
SLOPE = RISE / RUN = (y₂ - y₁) / (x₂ - x₁)
"y on top, x on bottom — always consistent order"
Example
Find the slope through \((1, 2)\) and \((3, 8)\):
\(m = \dfrac{8-2}{3-1} = \dfrac{6}{2} = 3\)
Your Turn
Find the slope of the line through \((2, 5)\) and \((6, 13)\).
✏️ \(m = \dfrac{13-5}{6-2} = \dfrac{8}{4} = 2\)
05
Linear Functions
Slope-Intercept Form
Confusing slope (m) with y-intercept (b) on a graph.
⚡
Memory Key
y = mx + b
m = Mountain (steepness/slope) b = Beginning (where line starts on y-axis)
Example
Line: \(y = 3x - 2\)
Slope \(m = 3\), y-intercept \(b = -2\) → starts at \((0, -2)\)
Your Turn
In the equation \(y = -4x + 7\), what is the y-intercept?
✏️ In \(y = mx + b\), the number without \(x\) is \(b\). Here \(b = 7\).
Systems & Polynomials
06
Systems of Equations
Substitution Method
Forgetting to substitute back to find BOTH variables.
⚡
Memory Key
PLUG IN → SOLVE → PLUG BACK → CHECK
"Always find both x AND y"
Example
\(y = 2x\) and \(x + y = 9\)
Substitute: \(x + 2x = 9\) → \(3x = 9\) → \(x = 3\)
Then: \(y = 2(3) = 6\). Answer: \((3, 6)\)
Your Turn
Solve the system: \(y = x + 1\) and \(2x + y = 7\)
What is the value of \(x\)?
✏️ Sub \(y = x+1\) into \(2x + (x+1) = 7\) → \(3x + 1 = 7\) → \(3x = 6\) → \(x = 2\)
07
Polynomials
Distributing with Negatives
🚨 Top mistake: forgetting to distribute the negative to ALL terms.
⚡
Memory Key
−(a + b) = −a − b ← the minus hits EVERY term
"Negative spreads like a virus — infects all terms inside"
Example
Simplify: \(5 - (2x - 3)\)
Distribute the negative: \(5 - 2x + 3\)
Combine: \(8 - 2x\)
Your Turn
Simplify: \(10 - (3x - 4)\)
What is the constant term in the simplified expression?
✏️ \(10 - 3x + 4 = 14 - 3x\). The constant is \(14\).
08
Exponent Rules
Product & Power Rules
Confusing adding exponents (×) with multiplying exponents (power of power).
⚡
Memory Key
xᵃ · xᵇ = xᵃ⁺ᵇ ← same base × → ADD exponents
(xᵃ)ᵇ = xᵃˣᵇ ← power of power → MULTIPLY exponents
Example
\(x^3 \cdot x^4 = x^{3+4} = x^7\)
\((x^3)^4 = x^{3 \times 4} = x^{12}\)
Your Turn
Simplify: \(x^2 \cdot x^5\)
What is the exponent in the answer?
✏️ Same base → ADD: \(2 + 5 = 7\) → \(x^7\)
09
Factoring
Factoring Out GCF
Missing the greatest common factor when it's a variable.
⚡
Memory Key
GCF = Greatest Common Factor
FIND → DIVIDE → WRITE as product: GCF(remaining)
Example
Factor: \(6x^2 + 9x\)
GCF = \(3x\)
Answer: \(3x(2x + 3)\)
Your Turn
Factor: \(4x^2 + 8x\)
What is the GCF?
✏️ Both terms share \(4\) and \(x\) → GCF = \(4x\) → \(4x(x + 2)\)
10
Functions
Evaluating Functions
Students forget to substitute the entire input expression.
⚡
Memory Key
f(a) = substitute EVERY x with the value a
"f(2) → replace ALL x's with 2"
Example
\(f(x) = 2x + 3\), find \(f(4)\):
\(f(4) = 2(4) + 3 = 8 + 3 = 11\)
Your Turn
If \(f(x) = 3x - 1\), find \(f(5)\).
✏️ \(f(5) = 3(5) - 1 = 15 - 1 = 14\)

Geometry 1 — 10 Core Problems

Angles, triangles, area, perimeter, and congruence — visualize before you calculate.

GEOMETRY SCORE
0 / 10
G1
Angles
Complementary vs. Supplementary Angles
Always mixed up — which pair adds to 90° and which to 180°?
⚡
Memory Key
C comes before S in alphabet → Complementary (90°) before Supplementary (180°)
"Corner = 90°, Straight = 180°"
Example
Angle A = 35°. Find its complement.
\(90° - 35° = 55°\). Complement = 55°
Your Turn
Angle B = 52°. What is its supplement?
✏️ Supplement adds to 180°: \(180° - 52° = 128°\)
G2
Triangles
Triangle Angle Sum
Forgetting the sum rule when one angle is missing.
⚡
Memory Key
ALL 3 angles of a triangle = 180°
"Triangle has 3 sides, angles sum to 180"
Example
Triangle with angles 60°, 80°, and \(x\):
\(60 + 80 + x = 180\) → \(x = 40°\)
Your Turn
A triangle has angles 55° and 75°. Find the third angle.
✏️ \(55 + 75 + x = 180\) → \(130 + x = 180\) → \(x = 50°\)
G3
Triangles
The Pythagorean Theorem
Using the wrong side as the hypotenuse (always the longest!).
⚡
Memory Key
a² + b² = c² (c = HYPOTENUSE = longest side = opposite right angle)
"c² is always the BIG one"
Example
Legs: 3 and 4. Find the hypotenuse.
\(3^2 + 4^2 = c^2\) → \(9 + 16 = 25\) → \(c = 5\)
Your Turn
A right triangle has legs of length 5 and 12. Find the hypotenuse.
✏️ \(5^2 + 12^2 = 25 + 144 = 169\) → \(c = \sqrt{169} = 13\)
G4
Area & Perimeter
Area of a Triangle
Forgetting to divide by 2 — the most common area mistake.
⚡
Memory Key
A = ½ × base × height
"Triangle is HALF a rectangle"
Example
Base = 8, Height = 5:
\(A = \dfrac{1}{2} \times 8 \times 5 = 20\) square units
Your Turn
A triangle has base = 10 and height = 6. Find the area.
✏️ \(A = \frac{1}{2} \times 10 \times 6 = 30\)
G5
Circles
Circumference of a Circle
Mixing up circumference and area formulas — both use π differently.
⚡
Memory Key
C = 2πr (circumference = "around") A = πr² (area = "inside")
"C has no square, A has r-squared"
Example
Circle with radius 7:
\(C = 2\pi(7) = 14\pi \approx 43.98\)
Your Turn
Find the circumference of a circle with radius 5.
(Leave answer in terms of π, e.g. "10π")
✏️ \(C = 2\pi r = 2\pi(5) = 10\pi\)
Parallel Lines & Congruence
G6
Parallel Lines
Alternate Interior Angles
Confusing alternate interior with co-interior (same-side interior) angles.
⚡
Memory Key
ALTERNATE interior angles = EQUAL (they're on opposite sides — like mirror twins)
CO-INTERIOR (same side) = ADD to 180°
Example
Two parallel lines cut by a transversal.
Alternate interior angle = 65° → the other alternate interior angle = 65°.
Your Turn
Two parallel lines are cut by a transversal. One alternate interior angle is 73°. What is the measure of the other alternate interior angle?
✏️ Alternate interior angles are EQUAL when lines are parallel. Answer: 73°
G7
Polygons
Sum of Interior Angles of a Polygon
Using the wrong formula — many students just add all sides instead.
⚡
Memory Key
Sum = (n − 2) × 180° where n = number of sides
"Subtract 2, then × 180"
Example
Pentagon (5 sides):
\((5-2) \times 180 = 3 \times 180 = 540°\)
Your Turn
What is the sum of the interior angles of a hexagon (6 sides)?
✏️ \((6-2) \times 180 = 4 \times 180 = 720°\)
G8
Area
Area of a Trapezoid
Forgetting to add BOTH parallel bases before halving.
⚡
Memory Key
A = ½(b₁ + b₂) × h
"Average the two bases, then multiply by height"
Example
Trapezoid: \(b_1 = 6,\ b_2 = 10,\ h = 4\)
\(A = \dfrac{1}{2}(6 + 10) \times 4 = \dfrac{1}{2}(16)(4) = 32\)
Your Turn
A trapezoid has bases 5 and 9, and height 6. Find the area.
✏️ \(A = \frac{1}{2}(5+9)(6) = \frac{1}{2}(14)(6) = 42\)
G9
Congruence
Congruence Shortcuts (SSS, SAS, ASA)
Students try to use SSA — which is NOT a valid congruence rule!
⚡
Memory Key
Valid: SSS · SAS · ASA · AAS · HL
INVALID: SSA ("donkey rule") and AAA (no size info)
Example
Two triangles share two sides and the included angle.
→ Use SAS (Side-Angle-Side) → triangles are congruent.
Your Turn
Two triangles have all three pairs of sides equal. Which congruence shortcut applies?
(Type: SSS, SAS, ASA, AAS, or HL)
✏️ Three pairs of equal sides → SSS (Side-Side-Side)
G10
Volume
Volume of a Rectangular Prism
Using area formula (length × width) instead of volume (add the height!).
⚡
Memory Key
V = l × w × h (3 dimensions = 3 measurements multiplied)
"Area is flat (2D), Volume is deep (3D) — add height"
Example
Box: length 4, width 3, height 5:
\(V = 4 \times 3 \times 5 = 60\) cubic units
Your Turn
Find the volume of a box with length 6, width 4, and height 3.
✏️ \(V = 6 \times 4 \times 3 = 72\) cubic units