Self-Study Worksheet · Grade 6–8

Pre-Algebra &
Geometry

Core problems, memory keys, and worked examples — designed for independent study.

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Pre-Algebra

10 Questions
1
Order of Operations  Easy
Evaluate:  3 + 4 × 2 − 1
⚡ Memory Key
"Please Excuse My Dear Aunt Sally"
PEMDAS Multiply BEFORE Add Left → Right
📌 Worked Example

Simplify 2 + 3 × 4

Step 1 — Multiply first: 3 × 4 = 12

Step 2 — Then add: 2 + 12 = 14

⚠️ Wrong shortcut: 2 + 3 = 5, then 5 × 4 = 20 ✗

2
Solving One-Step Equations  Easy
Solve for x:  x + 7 = 15
⚡ Memory Key
Undo the operation — do the OPPOSITE on both sides.
INVERSE OPERATION BALANCE ISOLATE x
📌 Worked Example

Solve x + 5 = 12

Subtract 5 from both sides: x = 12 − 5 = 7

Check: 7 + 5 = 12 ✓

3
Two-Step Equations  Medium
Solve:  2x − 5 = 11
⚡ Memory Key
Undo ADD/SUBTRACT first, then undo MULTIPLY/DIVIDE.
ADD/SUB FIRST THEN ÷
📌 Worked Example

Solve 3x + 2 = 14

Step 1 — Subtract 2: 3x = 12

Step 2 — Divide by 3: x = 4

4
Ratios & Proportions  Easy
A recipe uses 2 cups of flour for every 3 cups of milk. If you use 8 cups of flour, how many cups of milk do you need?
⚡ Memory Key
Cross-multiply to solve a proportion.
CROSS-MULTIPLY EQUAL RATIOS
📌 Worked Example

Set up: 23 = 8x

Cross-multiply: 2x = 24

Divide: x = 12

5
Percentages  Easy
A shirt originally costs $40. It is on sale for 25% off. What is the sale price?
⚡ Memory Key
"Percent" = Per hundred. Multiply by the decimal.
% ÷ 100 DISCOUNT = % × ORIGINAL SALE = ORIGINAL − DISCOUNT
📌 Worked Example

30% off $50 shirt

Discount: 0.30 × 50 = $15

Sale price: 50 − 15 = $35

6
Negative Numbers  Medium
The temperature was −3°C in the morning and rose by 9°C by noon. What was the temperature at noon?
⚡ Memory Key
Adding a positive = moving RIGHT on the number line.
NUMBER LINE NEGATIVE + POSITIVE
📌 Worked Example

−5 + 7 = ?

Start at −5, move 7 right → land on 2

7
Distributive Property  Medium
Expand and simplify:  3(2x + 4) − 6
⚡ Memory Key
Multiply the outside number by EVERY term inside the parentheses.
DISTRIBUTE a(b+c) = ab+ac
📌 Worked Example

Expand 4(x + 3) − 5

4·x + 4·3 − 5 = 4x + 12 − 5 = 4x + 7

8
Writing Algebraic Expressions  Easy
Jake has n apples. Maria has 5 more than twice as many apples as Jake. Which expression represents Maria's apples?
⚡ Memory Key
"More than" → add. "Times as many" → multiply. Order matters!
MORE THAN = + TIMES = × TRANSLATE CAREFULLY
9
Inequalities  Medium
Solve and graph on a number line:  2x + 3 > 11
Which value of x is a solution?
⚡ Memory Key
When you MULTIPLY or DIVIDE by a NEGATIVE, FLIP the inequality sign!
FLIP SIGN when ÷ NEGATIVE OPEN ○ = not included CLOSED ● = included
📌 Worked Example

Solve 3x − 1 > 8

Add 1: 3x > 9

Divide by 3: x > 3  → open circle at 3, shade right

10
Word Problem — Rate  Medium
A car travels at 60 miles per hour. How long will it take to travel 150 miles?
⚡ Memory Key
Distance = Rate × Time. Rearrange to find what you need.
D = R × T T = D ÷ R R = D ÷ T
📌 Worked Example

A bike rides at 15 mph for 45 miles.

T = D ÷ R = 45 ÷ 15 = 3 hours

Geometry

10 Questions
11
Area of a Rectangle  Easy
A rectangle has a length of 9 cm and a width of 5 cm. What is its area?
⚡ Memory Key
Area = Length × Width. Always write units SQUARED.
A = l × w cm² m² ft²
12
Perimeter vs. Area Trap  Easy
A square has a perimeter of 24 cm. What is its area?
⚡ Memory Key
Perimeter = around the outside (1D). Area = inside space (2D). Don't mix them up!
PERIMETER = 4s AREA = s² FIND s FIRST
📌 Worked Example

Square with perimeter = 20 cm

Side s = 20 ÷ 4 = 5 cm

Area = 5² = 25 cm²

13
Area of a Triangle  Easy
A triangle has a base of 10 cm and a height of 6 cm. What is its area?
⚡ Memory Key
A triangle is HALF of a rectangle with the same base and height.
A = ½ × b × h HALF of b×h
14
Circumference of a Circle  Easy
A circle has a diameter of 10 cm. What is its circumference? (Use π ≈ 3.14)
⚡ Memory Key
"Cherry Pie Delicious" → C = πd  |  "Apple Pie R-Squared" → A = πr²
C = πd C = 2πr d = 2r
📌 Worked Example

Circle with diameter = 7 cm

C = π × 7 = 3.14 × 7 = 21.98 cm

15
Pythagorean Theorem  Medium
A right triangle has legs of 3 cm and 4 cm. What is the length of the hypotenuse?
⚡ Memory Key
The hypotenuse is ALWAYS the longest side, opposite the right angle.
a² + b² = c² 3-4-5 TRIPLE c = HYPOTENUSE
📌 Worked Example

Legs: 5 and 12. Find hypotenuse.

5² + 12² = 25 + 144 = 169

c = √169 = 13 cm

16
Angles in a Triangle  Easy
A triangle has two angles measuring 45° and 85°. What is the third angle?
⚡ Memory Key
Triangle angles always add up to exactly 180°. Always.
∠A + ∠B + ∠C = 180° TRIANGLE SUM
17
Volume of a Rectangular Prism  Easy
A box is 5 cm long, 4 cm wide, and 3 cm tall. What is its volume?
⚡ Memory Key
Volume = three dimensions multiplied. Units are CUBED.
V = l × w × h cm³ m³ ft³
18
Supplementary & Complementary Angles  Medium
Two angles are supplementary. One angle measures 112°. What is the other angle?
⚡ Memory Key
"C" comes before "S" in alphabet → Complementary (90°) is smaller than Supplementary (180°).
COMPLEMENTARY = 90° SUPPLEMENTARY = 180° "S for Straight line"
19
Area of a Circle  Medium
A circle has a radius of 5 cm. What is its area? (Use π ≈ 3.14)
⚡ Memory Key
"Apple Pie R-Squared" → A = πr². Don't confuse radius with diameter!
A = πr² RADIUS = d ÷ 2 SQUARE the RADIUS
📌 Worked Example

Circle with radius = 3 cm

A = π × 3² = 3.14 × 9 = 28.26 cm²

⚠️ Common mistake: 3.14 × 3 = 9.42 ✗ (forgot to square!)

20
Similar Figures & Scale Factor  Medium
Two similar triangles have sides in the ratio 1 : 3. The smaller triangle has an area of 4 cm². What is the area of the larger triangle?
⚡ Memory Key
If the sides scale by k, the AREA scales by k². This trips up everyone!
AREA RATIO = k² VOLUME RATIO = k³ SIDE RATIO = k
📌 Worked Example

Sides ratio = 1 : 2, small area = 5 cm²

Area ratio = 1² : 2² = 1 : 4

Large area = 5 × 4 = 20 cm²