🧠 Rudimentary Brain Freeze
Foundational math and reading from before 9th grade
All the math and reading elements from yesteryear that we definitely forgot and probably should have remembered.
∑ Math Foundations Before 9th Grade
Multiplication Table 12 x 12
Fractions HS Completion
A fraction has a numerator (top number) and a denominator (bottom number). The denominator tells you how many equal parts the whole is split into. The numerator tells you how many of those parts you have.
1
Same denominator: just add or subtract the top numbers. The bottom stays exactly the same. 2/7 + 3/7 = 5/7
2
Different denominators: find the Least Common Denominator (LCD), convert both fractions so they share that denominator, then add or subtract.
3
Multiplying fractions: multiply straight across. Top times top, bottom times bottom. Then simplify if possible.
4
Dividing fractions: flip the second fraction (its reciprocal) and then multiply. "Keep, Change, Flip."
5
Simplifying: find the greatest common factor of the top and bottom, then divide both by it. 6/9 = 2/3 (divided both by 3)
Worked Examples
Adding (same denom): 2/5 + 1/5 = 3/5
Adding (diff denom): 1/2 + 1/3 → LCD = 6 → 3/6 + 2/6 = 5/6
Subtracting: 3/4 - 1/6 → LCD = 12 → 9/12 - 2/12 = 7/12
Dividing: 2/3 ÷ 4/5 = 2/3 x 5/4 = 10/12 = 5/6 (flipped 4/5 to 5/4)
Adding (diff denom): 1/2 + 1/3 → LCD = 6 → 3/6 + 2/6 = 5/6
Subtracting: 3/4 - 1/6 → LCD = 12 → 9/12 - 2/12 = 7/12
Dividing: 2/3 ÷ 4/5 = 2/3 x 5/4 = 10/12 = 5/6 (flipped 4/5 to 5/4)
Multiplying Fractions: Deep Dive HS Completion
Multiplying fractions is actually the easiest operation. No common denominator needed. Just go straight across.
a/b x c/d = (a x c) / (b x d)
1
Multiply the two top numbers together. That is your new numerator.
2
Multiply the two bottom numbers together. That is your new denominator.
3
Cross-cancel first if you can. If the top of one fraction shares a factor with the bottom of the other, reduce before multiplying. It keeps the numbers smaller.
4
Simplify the final answer if it is not already in lowest terms.
Worked Examples
Basic: 2/3 x 3/4 = 6/12 = 1/2
Cross-cancel version: 2/3 x 3/4 → the 3 in 2/3 and the 3 in 3/4 cancel (3/3 = 1) → 2/1 x 1/4 = 2/4 = 1/2
Whole number times fraction: 4 x 3/5 → write 4 as 4/1 → 4/1 x 3/5 = 12/5 = 2 and 2/5
Mixed numbers: Convert to improper fractions first. 1 and 1/2 = 3/2 → 3/2 x 2/3 = 6/6 = 1
Cross-cancel version: 2/3 x 3/4 → the 3 in 2/3 and the 3 in 3/4 cancel (3/3 = 1) → 2/1 x 1/4 = 2/4 = 1/2
Whole number times fraction: 4 x 3/5 → write 4 as 4/1 → 4/1 x 3/5 = 12/5 = 2 and 2/5
Mixed numbers: Convert to improper fractions first. 1 and 1/2 = 3/2 → 3/2 x 2/3 = 6/6 = 1
Decimals HS Completion
3 4 7 . 8 9 2
hundreds tens ones . tenths hundredths thousandths
1
Adding / Subtracting: line up the decimal points so they stack perfectly, then compute normally. Add trailing zeros if needed so both numbers have the same length.
2
Multiplying: ignore the decimal points entirely and multiply as whole numbers. Then count the total decimal places across both original numbers and move the decimal that many places from the right in your answer.
3
Dividing: move the decimal in the divisor all the way to the right (making it a whole number), then move the decimal in the dividend the same number of places. Divide normally.
4
Decimal to percent: multiply by 100, which means slide the decimal point 2 places to the right. 0.75 = 75%
5
Percent to decimal: divide by 100, slide the decimal 2 places to the left. 32% = 0.32
Worked Examples
Adding: 3.25 + 1.4 → line up: 3.25 + 1.40 = 4.65
Multiplying: 2.3 x 1.2 → 23 x 12 = 276 → 2 total decimal places → 2.76
Dividing: 4.8 ÷ 0.6 → shift both: 48 ÷ 6 = 8
Multiplying: 2.3 x 1.2 → 23 x 12 = 276 → 2 total decimal places → 2.76
Dividing: 4.8 ÷ 0.6 → shift both: 48 ÷ 6 = 8
Angles and Degrees Geometry Prep
An angle is formed when two rays share a common endpoint (called the vertex). We measure angles in degrees (°).
Right angle = 90° | Straight angle = 180° | Full rotation = 360°
1
Acute angle: less than 90°. It looks like a sharp point.
2
Right angle: exactly 90°. Always shown with a small square in the corner. Found in all rectangles and squares.
3
Obtuse angle: greater than 90° but less than 180°. It looks wide and open.
4
Straight angle: exactly 180°. It is a perfectly flat line.
5
Complementary angles: two angles that add up to exactly 90°. 30° + 60° = 90°
6
Supplementary angles: two angles that add up to exactly 180°. 110° + 70° = 180°
Worked Example
Find the complement of 35°:
Complementary angles add to 90° → 90 - 35 = 55°
Find the supplement of 112°:
Supplementary angles add to 180° → 180 - 112 = 68°
Complementary angles add to 90° → 90 - 35 = 55°
Find the supplement of 112°:
Supplementary angles add to 180° → 180 - 112 = 68°
Percentages HS Completion
Percent literally means "per hundred." So 45% means 45 out of every 100.
percent ÷ 100 = decimal | decimal x 100 = percent
1
Percent to decimal: divide by 100 (or slide the decimal 2 places left). 75% = 0.75
2
Decimal to percent: multiply by 100 (slide decimal 2 places right). 0.32 = 32%
3
Finding a percent of a number: convert the percent to a decimal, then multiply by the number.
4
Finding what percent one number is of another: divide the part by the whole, then multiply by 100.
Worked Examples
25% of 80:
25% = 0.25 → 0.25 x 80 = 20
What percent is 15 of 60?
15 ÷ 60 = 0.25 → 0.25 x 100 = 25%
What is 8% of 150?
8% = 0.08 → 0.08 x 150 = 12
25% = 0.25 → 0.25 x 80 = 20
What percent is 15 of 60?
15 ÷ 60 = 0.25 → 0.25 x 100 = 25%
What is 8% of 150?
8% = 0.08 → 0.08 x 150 = 12
Solving Basic Equations - Part 1 One-Step
The goal is always the same: get the variable alone on one side of the equals sign. The balance rule says whatever you do to one side, you must do to the other side.
Balance Rule: both sides of the = sign must always remain equal
1
Addition equation (x + a = b): subtract the same number from both sides. x + 5 = 12 → x = 12 - 5 = 7
2
Subtraction equation (x - a = b): add the same number to both sides. x - 3 = 7 → x = 7 + 3 = 10
3
Multiplication equation (ax = b): divide both sides by the coefficient. 3x = 15 → x = 15 ÷ 3 = 5
4
Division equation (x/a = b): multiply both sides by the divisor. x/4 = 6 → x = 6 x 4 = 24
All Four Types
x + 5 = 12 → subtract 5 → x = 7
x - 3 = 7 → add 3 → x = 10
3x = 15 → divide by 3 → x = 5
x/4 = 6 → multiply by 4 → x = 24
x - 3 = 7 → add 3 → x = 10
3x = 15 → divide by 3 → x = 5
x/4 = 6 → multiply by 4 → x = 24
Solving Basic Equations - Part 2 Two-Step
Two-step equations require two operations to isolate the variable. Work in reverse order of operations: deal with addition and subtraction first, then multiplication and division.
1
Identify what operations are being done to the variable. You will undo them in reverse order.
2
Step 1 - Undo add/subtract: add or subtract the constant from both sides.
3
Step 2 - Undo multiply/divide: divide or multiply both sides to isolate the variable.
4
Check your answer: plug it back into the original equation. Both sides should be equal.
Worked Examples
Example 1: 2x + 3 = 11
Step 1: subtract 3 from both sides → 2x = 8
Step 2: divide both sides by 2 → x = 4
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Example 2: 3x - 4 = 14
Step 1: add 4 to both sides → 3x = 18
Step 2: divide both sides by 3 → x = 6
Check: 3(6) - 4 = 18 - 4 = 14 ✓
Example 3: 5x + 7 = 32
Step 1: subtract 7 → 5x = 25
Step 2: divide by 5 → x = 5
Step 1: subtract 3 from both sides → 2x = 8
Step 2: divide both sides by 2 → x = 4
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Example 2: 3x - 4 = 14
Step 1: add 4 to both sides → 3x = 18
Step 2: divide both sides by 3 → x = 6
Check: 3(6) - 4 = 18 - 4 = 14 ✓
Example 3: 5x + 7 = 32
Step 1: subtract 7 → 5x = 25
Step 2: divide by 5 → x = 5
Video Resource
🎥 MathAntics
Find whatever you need here - this channel covers everything in this section and more.
✎ Reading Foundations HS Completion
Before we get into full essays, let us make sure the foundation is solid. These are the essay types you will see on the HiSET and in every English class.
Narrative Essay Story-Based
What it is: A story-based essay, real or imagined, that explores a theme or makes a point through experience. The story is the vehicle. The meaning is the destination.
Structure:
1. Setting the scene
2. Rising action
3. Climax (the turning point)
4. Falling action
5. Resolution with reflection
Key features: First person is acceptable. Use vivid details and sensory language. There should be a clear theme or lesson. The story exists to make a point, not just to entertain.
Tone: Personal, descriptive, and engaging.
Structure:
1. Setting the scene
2. Rising action
3. Climax (the turning point)
4. Falling action
5. Resolution with reflection
Key features: First person is acceptable. Use vivid details and sensory language. There should be a clear theme or lesson. The story exists to make a point, not just to entertain.
Tone: Personal, descriptive, and engaging.
Argumentative Essay 5 Paragraphs
What it is: A persuasive essay that takes a clear position and defends it using evidence and logical reasoning. You are not just sharing an opinion. You are making a case.
Structure:
1. Introduction with a clear thesis
2. Body paragraph 1 (first argument plus evidence)
3. Body paragraph 2 (second argument plus evidence)
4. Body paragraph 3 (counterargument acknowledged and refuted)
5. Conclusion
Key features: You must acknowledge and refute the opposing viewpoint. No emotional language. Use facts, statistics, and logic only.
Tone: Formal, confident, direct.
Structure:
1. Introduction with a clear thesis
2. Body paragraph 1 (first argument plus evidence)
3. Body paragraph 2 (second argument plus evidence)
4. Body paragraph 3 (counterargument acknowledged and refuted)
5. Conclusion
Key features: You must acknowledge and refute the opposing viewpoint. No emotional language. Use facts, statistics, and logic only.
Tone: Formal, confident, direct.
Descriptive Essay Sensory Detail
What it is: An essay that paints a vivid picture of a person, place, object, or experience using sensory detail. The goal is to make the reader feel present.
Structure:
1. Introduction (establishing the subject)
2. Body paragraphs (each focusing on one sensory aspect or detail)
3. Conclusion (reflecting on significance)
Key features: Engages all five senses. Uses figurative language (simile, metaphor, personification). The reader should feel like they are there.
Tone: Evocative, immersive, sensory-rich.
Structure:
1. Introduction (establishing the subject)
2. Body paragraphs (each focusing on one sensory aspect or detail)
3. Conclusion (reflecting on significance)
Key features: Engages all five senses. Uses figurative language (simile, metaphor, personification). The reader should feel like they are there.
Tone: Evocative, immersive, sensory-rich.
Expository Essay Inform Only
What it is: An essay that explains or informs. Not to argue. Not to tell a story. Just to explain clearly and objectively. Think of it as a well-organized explanation.
Structure:
1. Introduction with thesis (what you will explain)
2. Body paragraph 1 (first aspect)
3. Body paragraph 2 (second aspect)
4. Body paragraph 3 (third aspect)
5. Conclusion
Key features: Completely neutral. No personal opinion. Present all sides fairly. Use facts, examples, definitions, and data.
Tone: Neutral, objective, clear.
Structure:
1. Introduction with thesis (what you will explain)
2. Body paragraph 1 (first aspect)
3. Body paragraph 2 (second aspect)
4. Body paragraph 3 (third aspect)
5. Conclusion
Key features: Completely neutral. No personal opinion. Present all sides fairly. Use facts, examples, definitions, and data.
Tone: Neutral, objective, clear.
Compare and Contrast Essay Two Structures
What it is: Examines the similarities and differences between two subjects. The comparison must serve a purpose.
Two structures:
Block method: cover Subject A fully in the first half, then Subject B fully in the second half.
Point-by-point method: compare both subjects on one specific point per paragraph.
Key feature: The conclusion must explain WHY the comparison matters. What does it reveal? What insight does it provide?
Tone: Analytical and balanced.
Two structures:
Block method: cover Subject A fully in the first half, then Subject B fully in the second half.
Point-by-point method: compare both subjects on one specific point per paragraph.
Key feature: The conclusion must explain WHY the comparison matters. What does it reveal? What insight does it provide?
Tone: Analytical and balanced.
Cause and Effect Essay Logical Chain
What it is: Explains why something happened (causes) and what resulted from it (effects). You are tracing a relationship between events or conditions.
Structure options:
Focus on causes: one effect, multiple paragraphs each exploring a different cause.
Focus on effects: one cause, multiple paragraphs each exploring a different effect.
Chain method: A causes B, which causes C, which causes D. Each event leads to the next.
Key feature: Strong transition words are essential. Use: therefore, as a result, consequently, because of this, which led to, this in turn caused.
Tone: Logical and analytical.
Structure options:
Focus on causes: one effect, multiple paragraphs each exploring a different cause.
Focus on effects: one cause, multiple paragraphs each exploring a different effect.
Chain method: A causes B, which causes C, which causes D. Each event leads to the next.
Key feature: Strong transition words are essential. Use: therefore, as a result, consequently, because of this, which led to, this in turn caused.
Tone: Logical and analytical.