Mathematics - Grade 6
Integers, Algebra, Volume, and Statistics
Table of Contents
1. Understanding Integers
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. The set of integers extends infinitely in both directions.
The Number Line:
... -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5 ...
← Negative integers | Zero | Positive integers →
Positive Integers (+)
- * Numbers greater than zero
- * Examples: +1, +5, +100
- * Real life: Profit, gain, above sea level
Negative Integers (-)
- * Numbers less than zero
- * Examples: -1, -5, -100
- * Real life: Loss, debt, below sea level
Real-Life Examples of Integers:
Temperature
-5°C (cold), +30°C (hot)
Bank Account
+₱500 (deposit), -₱200 (withdrawal)
Elevation
+2956m (Mt. Pulag), -10m (below sea level)
Sports
+3 (score increase), -2 (penalty)
Absolute Value:
The absolute value of an integer is its distance from zero on the number line (always positive or zero).
|+5| = 5 (5 steps from 0)
|-5| = 5 (also 5 steps from 0)
|0| = 0
2. Operations with Integers
Addition of Integers:
Same signs: Add the absolute values, keep the sign
(+5) + (+3) = +8 | (-5) + (-3) = -8
Different signs: Subtract absolute values, keep sign of larger
(+8) + (-3) = +5 | (-8) + (+3) = -5
Subtraction of Integers:
Rule: Change subtraction to addition of the opposite (add the additive inverse).
(+7) - (+3) = (+7) + (-3) = +4
(+7) - (-3) = (+7) + (+3) = +10
(-7) - (+3) = (-7) + (-3) = -10
(-7) - (-3) = (-7) + (+3) = -4
Multiplication of Integers:
Same signs = Positive
(+3) × (+4) = +12
(-3) × (-4) = +12
Different signs = Negative
(+3) × (-4) = -12
(-3) × (+4) = -12
Division of Integers:
Same rules as multiplication!
Same signs = Positive
(+12) ÷ (+4) = +3
(-12) ÷ (-4) = +3
Different signs = Negative
(+12) ÷ (-4) = -3
(-12) ÷ (+4) = -3
3. Order of Operations (PEMDAS/GEMDAS)
When solving expressions with multiple operations, follow the correct order to get the right answer. Use PEMDAS or GEMDAS to remember!
PEMDAS (GEMDAS):
Example 1:
Solve: 3 + 4 × 2
Step 1: Multiply first → 4 × 2 = 8
Step 2: Add → 3 + 8 = 11
Example 2:
Solve: (6 + 2) × 3 - 4
Step 1: Parentheses → (6 + 2) = 8
Step 2: Multiply → 8 × 3 = 24
Step 3: Subtract → 24 - 4 = 20
Example 3 (with exponents):
Solve: 2 + 3² × 4
Step 1: Exponents → 3² = 9
Step 2: Multiply → 9 × 4 = 36
Step 3: Add → 2 + 36 = 38
4. Introduction to Algebra
Algebra uses letters (variables) to represent unknown numbers. It helps us write and solve mathematical relationships in a general way.
Key Vocabulary:
Variable
A letter representing an unknown number (x, y, n)
Constant
A fixed number that doesn't change (5, -3, 100)
Coefficient
Number multiplied by a variable (3 in 3x)
Term
A number, variable, or their product (5x, 3, xy)
Translating Words to Algebra:
| Words | Operation | Algebraic |
|---|---|---|
| Sum of x and 5 | Addition | x + 5 |
| 7 less than n | Subtraction | n - 7 |
| Twice a number y | Multiplication | 2y |
| A number divided by 4 | Division | x ÷ 4 or x/4 |
| 3 more than twice n | Combined | 2n + 3 |
Remember: In algebra, when a number and variable are next to each other (like 3x), it means multiplication. We don't write 3 × x, we write 3x.
5. Algebraic Expressions
An algebraic expression is a combination of variables, numbers, and operations. We can evaluate expressions by substituting values for variables.
Evaluating Expressions:
To evaluate, replace the variable with the given value.
Evaluate 3x + 5 when x = 4
Step 1: Replace x with 4 → 3(4) + 5
Step 2: Multiply → 12 + 5
Step 3: Add → 17
Like Terms:
Like terms have the same variable raised to the same power. Only like terms can be combined.
Like Terms (can combine)
3x and 5x → 8x
2y² and 7y² → 9y²
Unlike Terms (cannot combine)
3x and 5y (different variables)
2x and 2x² (different powers)
Simplifying Expressions:
Simplify: 4x + 3 + 2x - 1
Step 1: Group like terms → (4x + 2x) + (3 - 1)
Step 2: Combine → 6x + 2
6. Solving Simple Equations
An equation states that two expressions are equal. To solve an equation means finding the value of the variable that makes the equation true.
Golden Rule:
What you do to one side, you must do to the other side!
Addition/Subtraction Equations:
Solve: x + 7 = 15
Subtract 7 from both sides:
x + 7 - 7 = 15 - 7
x = 8
Multiplication/Division Equations:
Solve: 3x = 21
Divide both sides by 3:
3x ÷ 3 = 21 ÷ 3
x = 7
Two-Step Equations:
Solve: 2x + 5 = 13
Step 1: Subtract 5 → 2x + 5 - 5 = 13 - 5 → 2x = 8
Step 2: Divide by 2 → 2x ÷ 2 = 8 ÷ 2
x = 4
Checking Your Answer:
Substitute your answer back into the original equation to verify:
If x = 4: 2(4) + 5 = 8 + 5 = 13 ✓
7. Volume of Solid Figures
Volume measures the amount of space inside a 3D (three-dimensional) object. It is measured in cubic units (cm³, m³, etc.).
Cube:
📦
All sides are equal (s)
V = s × s × s = s³
Example: s = 4 cm
V = 4³ = 64 cm³
Rectangular Prism:
📦
Has length (l), width (w), height (h)
V = l × w × h
Example: l=5, w=3, h=2 cm
V = 5 × 3 × 2 = 30 cm³
Cylinder:
🥫
Has radius (r) and height (h)
V = π × r² × h
Example: r=3, h=7 cm (π ≈ 3.14)
V = 3.14 × 9 × 7 = 197.82 cm³
Cone:
🍦
1/3 the volume of a cylinder
V = (1/3) × π × r² × h
Example: r=3, h=6 cm
V = (1/3) × 3.14 × 9 × 6 = 56.52 cm³
Sphere:
🏀
Perfectly round, has radius (r)
V = (4/3) × π × r³
Example: r=3 cm
V = (4/3) × 3.14 × 27 = 113.04 cm³
8. Statistics and Data Analysis
Statistics is the study of collecting, organizing, analyzing, and interpreting data. We use measures of central tendency to describe data sets.
Sample Data Set:
Test Scores: 85, 90, 78, 92, 85, 88, 85
Mean (Average):
Add all values and divide by the count.
Mean = (85+90+78+92+85+88+85) ÷ 7
Mean = 603 ÷ 7 = 86.14
Median (Middle Value):
Arrange in order and find the middle number.
Ordered: 78, 85, 85, 85, 88, 90, 92
Median = 85 (4th value of 7)
For even count, average the two middle numbers.
Mode (Most Frequent):
The value that appears most often.
85 appears 3 times (most frequent)
Mode = 85
A data set can have no mode, one mode, or multiple modes.
Range:
The difference between the highest and lowest values.
Range = Highest - Lowest
Range = 92 - 78 = 14
When to Use Each:
- Mean: Best for data without extreme values (outliers)
- Median: Best when there are extreme values
- Mode: Best for categorical data or finding most common value
Key Takeaways
Integers
- * Positive, negative, or zero
- * Same signs: add, keep sign
- * Different signs: subtract, keep larger sign
- * Multiply/divide: same signs = +, diff = -
PEMDAS
- * Parentheses first
- * Then Exponents
- * Multiply/Divide (left to right)
- * Add/Subtract (left to right)
Algebra
- * Variables represent unknowns
- * Combine like terms only
- * Balance equations on both sides
Statistics
- * Mean = sum ÷ count
- * Median = middle value
- * Mode = most frequent
- * Range = max - min