Mathematics
Number Sense, Operations, Fractions, Decimals, Geometry, and Ratio & Proportion
Table of Contents
1. Place Value and Number Sense
Key Concept
Place value tells us the value of each digit based on its position in the number.
Place Value Chart (Up to Millions)
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Example: 3,456,789 = Three million, four hundred fifty-six thousand, seven hundred eighty-nine
Rounding Numbers
Rule: Look at the digit to the right of the place you're rounding to:
- If it's 5 or more, round UP
- If it's 4 or less, round DOWN
Example: Round 4,567 to nearest hundred
Look at tens digit (6). Since 6 β₯ 5, round up β 4,600
Comparing Numbers
>
Greater than
5 > 3
<
Less than
3 < 5
=
Equal to
5 = 5
2. Whole Number Operations (GEMDAS)
Order of Operations: GEMDAS
Follow this order when solving equations with multiple operations.
Grouping Symbols
Parentheses ( ), Brackets [ ]
Exponents
Powers like 2Β², 3Β³
Multiplication and Division
Left to right
Addition and Subtraction
Left to right
Example Problem
Solve: 8 + 2 Γ (5 - 3)Β²
Step 1 (G): 8 + 2 Γ (2)Β² = Solve parentheses first
Step 2 (E): 8 + 2 Γ 4 = Solve exponent
Step 3 (M): 8 + 8 = Multiply
Step 4 (A): 16 = Add
Multiplication Table Tips
Γ 9 Trick
9 Γ 7: Hold up 7 fingers. Fold down 7th finger. Left side = 6, Right side = 3 β 63
Γ 11 Trick
11 Γ 25: Add the digits (2+5=7), put in middle β 275
3. Fractions
Parts of a Fraction
Β³ββ
3 = Numerator (top) | 4 = Denominator (bottom)
Types of Fractions
Proper Fraction
Β²ββ
Numerator < Denominator
Improper Fraction
β·ββ
Numerator > Denominator
Mixed Number
1Β³ββ
Whole + Fraction
Operations with Fractions
Adding/Subtracting (Same Denominator)
Β²ββ + ΒΉββ = Β³ββ
Just add/subtract numerators, keep denominator
Adding/Subtracting (Different Denominators)
ΒΉββ + ΒΉββ = Β³ββ + Β²ββ = β΅ββ
Find LCD (Least Common Denominator) first
Multiplying Fractions
Β²ββ Γ Β³ββ = βΆβββ = ΒΉββ
Multiply numerators, multiply denominators, simplify
Dividing Fractions
Β²ββ Γ· ΒΉββ = Β²ββ Γ Β²ββ = β΄ββ = 1ΒΉββ
Multiply by reciprocal (flip the second fraction)
4. Decimals and Percent
Decimal Place Values
| Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|
| 1 | . | 0.1 | 0.01 | 0.001 |
| 3 | . | 4 | 5 | 6 |
3.456 = 3 and 456 thousandths
Converting Between Fractions, Decimals, and Percent
| Fraction | Decimal | Percent |
|---|---|---|
| ΒΉββ | 0.5 | 50% |
| ΒΉββ | 0.25 | 25% |
| Β³ββ | 0.75 | 75% |
| ΒΉββ | 0.2 | 20% |
| ΒΉββ | 0.333... | 33.33% |
Quick Conversions
- Fraction β Decimal: Divide numerator by denominator (ΒΉββ = 1Γ·4 = 0.25)
- Decimal β Percent: Multiply by 100 (0.25 Γ 100 = 25%)
- Percent β Decimal: Divide by 100 (25% Γ· 100 = 0.25)
5. Geometry
Important Formulas
| Shape | Perimeter | Area |
|---|---|---|
| Square | P = 4s | A = sΒ² |
| Rectangle | P = 2l + 2w | A = l Γ w |
| Triangle | P = a + b + c | A = Β½ Γ b Γ h |
| Circle | C = 2Οr or Οd | A = ΟrΒ² |
| Parallelogram | P = 2(a + b) | A = b Γ h |
| Trapezoid | P = a + b + c + d | A = Β½(bβ+bβ) Γ h |
Remember: Ο (Pi) β 3.14 or Β²Β²ββ
Use 3.14 for most calculations unless told otherwise
Types of Angles
Acute
< 90Β°
Right
= 90Β°
Obtuse
> 90Β° and < 180Β°
Straight
= 180Β°
Types of Triangles
By Sides
- Equilateral: 3 equal sides
- Isosceles: 2 equal sides
- Scalene: 0 equal sides
By Angles
- Acute: All angles < 90Β°
- Right: One 90Β° angle
- Obtuse: One angle > 90Β°
Key Fact
Sum of all angles in a triangle = 180Β°
6. Ratio and Proportion
What is a Ratio?
A comparison of two quantities. Can be written as 3:4, 3 to 4, or Β³ββ
Example: Ratio
In a class of 30 students, there are 18 boys and 12 girls.
- Ratio of boys to girls: 18:12 = 3:2 (simplify by dividing by 6)
- Ratio of girls to total: 12:30 = 2:5
Proportion
Proportion = Two equal ratios
a : b = c : d
Cross multiply: a Γ d = b Γ c
Solving Proportion Problems
Problem: If 3 pencils cost β±15, how much do 7 pencils cost?
Set up: 3 : 15 = 7 : x
Cross multiply: 3 Γ x = 15 Γ 7
3x = 105
x = 105 Γ· 3
x = β±35
Direct vs Inverse Proportion
Direct Proportion
When one increases, the other increases
Example: More items = More cost
Inverse Proportion
When one increases, the other decreases
Example: More workers = Less time
Key Takeaways
- βGEMDAS: Grouping, Exponents, Multiplication/Division, Addition/Subtraction
- βRound up if digit is 5 or more, down if 4 or less
- βTo divide fractions: multiply by reciprocal
- βDecimal β Percent: multiply by 100
- βArea of triangle = Β½ Γ base Γ height
- βArea of circle = ΟrΒ² (Ο β 3.14)
- βSum of angles in triangle = 180Β°
- βSolve proportion by cross multiplication