Proportional Relationships - Sixth Grade
Complete Notes & Formulas
1. What is a Proportional Relationship?
Definition
A proportional relationship exists when
two quantities maintain a CONSTANT RATIO
y = kx
Key Components
| Component | Symbol | Meaning |
|---|---|---|
| Dependent Variable | y | Output value |
| Independent Variable | x | Input value |
| Constant of Proportionality | k | The constant ratio y/x |
Key Point: In a proportional relationship, if x doubles, y doubles. If x triples, y triples!
2. Constant of Proportionality (k)
Formula
k = y/x
k = constant of proportionality
Also called: unit rate or slope
How to Find k
Step 1: Choose any point (x, y) from the relationship
Step 2: Divide y by x
Step 3: k = y ÷ x
Step 4: Check with other points (k should be the same!)
Example: Find k
Point (3, 12)
k = y/x = 12/3 = 4
The equation is: y = 4x
Answer: k = 4
3. Identifying Proportional Relationships from Tables
How to Check a Table
Calculate y/x for EACH pair
If all ratios are THE SAME → Proportional ✓
If ratios are DIFFERENT → Not Proportional ✗
Example 1: Proportional Table
| x | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| y | 6 | 12 | 18 | 24 |
| y/x | 6/2 = 3 | 12/4 = 3 | 18/6 = 3 | 24/8 = 3 |
✓ PROPORTIONAL! All ratios equal 3, so k = 3
Example 2: Non-Proportional Table
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 3 | 5 | 7 | 9 |
| y/x | 3/1 = 3 | 5/2 = 2.5 | 7/3 = 2.33 | 9/4 = 2.25 |
✗ NOT PROPORTIONAL! Ratios are different
4. Identifying Proportional Relationships from Graphs
Three Key Characteristics
1. It's a STRAIGHT LINE
2. It passes through the ORIGIN (0, 0)
3. It has a CONSTANT SLOPE
Visual Comparison
Quick Check: If the line doesn't pass through (0, 0), it's NOT proportional!
5. Graphing Proportional Relationships
Steps to Graph
Step 1: Find the constant of proportionality (k)
Step 2: Start at the origin (0, 0)
Step 3: Plot additional points using y = kx
Step 4: Draw a straight line through all points
Step 5: Extend the line in both directions
Example: Graph y = 2x
Step 1: k = 2
Step 2: Start at (0, 0)
Step 3: Make a table of values:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 0 | 2 | 4 | 6 |
Plot: (0,0), (1,2), (2,4), (3,6) and connect!
6. Interpreting Graphs of Proportional Relationships
What Can We Learn from the Graph?
1. The Slope (k): How steep the line is = constant of proportionality
2. The Rate: How fast y changes compared to x
3. Predictions: Use the line to find unknown values
4. The Equation: y = kx where k is the slope
Reading a Graph
To find k from a graph:
• Choose any point (x, y) on the line (except origin)
• Calculate k = y/x
• This gives you the equation: y = kx
Example: Real-World Interpretation
Situation: A graph shows the relationship between hours worked (x) and money earned (y). The line passes through (4, 60).
Find k: k = 60/4 = 15
Equation: y = 15x
Interpretation: The person earns $15 per hour
Prediction: After 8 hours: y = 15(8) = $120
The constant of proportionality (15) represents the hourly wage!
7. Common Mistakes to Avoid
Mistake 1: Forgetting the Origin
✗ Wrong: "The line is straight, so it's proportional"
✓ Correct: "The line must be straight AND pass through (0, 0)"
Mistake 2: Wrong Ratio Calculation
✗ Wrong: k = x/y
✓ Correct: k = y/x (output divided by input)
Mistake 3: Curved Lines
✗ Wrong: Thinking a curve can be proportional
✓ Correct: Proportional relationships are ALWAYS straight lines
8. Proportional Relationship Checklist
Is It Proportional? Check ALL These:
☑ From a table: All y/x ratios are the same
☑ From a graph: Straight line through origin
☑ From an equation: Form y = kx (no added constant)
☑ The constant k: Same for all pairs
☑ When x = 0: y must also equal 0
Quick Reference: Proportional Relationships
| Concept | Formula/Rule |
|---|---|
| Equation Form | y = kx |
| Constant of Proportionality | k = y/x |
| Graph Characteristics | Straight line through (0, 0) |
| Table Test | All y/x ratios equal |
| Slope | k (constant of proportionality) |
💡 Important Tips to Remember
✓ Equation form: y = kx (no added number!)
✓ Graph must pass through (0, 0) - the origin!
✓ All y/x ratios are the same in a table
✓ k = y/x - output divided by input
✓ Straight line only - no curves!
✓ k represents unit rate or slope
✓ When x doubles, y doubles (constant ratio)
✓ Check multiple points to confirm proportionality
✓ Real-world meaning: k is the rate per one unit
✓ No y-intercept except 0 in proportional relationships
🧠 Memory Tricks & Strategies
Proportional Graph:
"Straight line, origin spine - that's proportional, every time!"
Constant k:
"Y over X, that's the test - k stays constant, never rest!"
Table Check:
"Divide y by x in every row - same answer? Proportional show!"
Origin Test:
"No origin pass? Not the class! (Not proportional)"
The Equation:
"y equals k times x - that's the proportional mix!"
Real World:
"k is the rate per one - tells how fast or how it's done!"
Master Proportional Relationships! 📈 📊 🎯
Remember: Straight line + Origin = Proportional!