Constant Factor Slides
Middle School Mathematics
Ratios & Proportions
Algebra Foundations
Writing Equations for
Proportional Relationships
Connecting tables, unit rates, and the master rule: \(y = kx\)
Constant Multiplier
Ratio Analysis
Equation Modeling
Core Mathematical Concept
What is a Proportional Relationship?
1. Constant Ratio (\(k\))
The ratio between dependent variable (\(y\)) and independent variable (\(x\)) never changes.
Constant of Proportionality Formula:
\(k = \frac{y}{x}\)
2. Starts at Zero \((0, 0)\)
When the input is zero, the output must be zero. There is no baseline setup charge or starting offset.
Origin Test on a Graph:
Straight Line through \((0, 0)\)
Key Rule: If every \(\frac{y}{x}\) produces the same value \(k\), the relationship is proportional!
Equation Architecture
Anatomy of \(y = kx\)
\(y\) = \(k\) \(x\)
\(y\) : Output
Dependent Variable
The quantity that depends on the input (e.g., total cost, total distance, total pay).
\(k\) : Multiplier
Constant Rate (\(\frac{y}{x}\))
Unit rate that never changes (e.g., $15 per hour, 60 miles per hour, $3 per ticket).
\(x\) : Input
Independent Variable
The quantity you choose or change (e.g., hours worked, hours driven, items bought).
Notice: There is never an added or subtracted number like \(+ 5\) or \(- 2\).
Step-by-Step Blueprint
How to Write the Equation
1
Identify \(x\) and \(y\)
Determine which value is independent (input \(x\)) and which is dependent (output \(y\)).
Input = \(x\) • Output = \(y\)
2
Calculate \(k = \frac{y}{x}\)
Divide \(y\) by \(x\) for every single row. Confirm that all rows yield the exact same quotient.
\(k = \text{unit rate}\)
3
Substitute \(k\)
Replace \(k\) in the standard formula \(y = kx\) with your calculated number. Keep \(y\) and \(x\) as variables!
Example: \(y = 4.5x\)
Common Mistake Alert: Always compute \(\frac{y}{x}\) (output over input), never \(\frac{x}{y}\)!
Worked Example
Bakery Cookie Batches
Flour Needed for Cookies
| Batches (\(x\)) | Cups Flour (\(y\)) | Ratio \(\frac{y}{x}\) |
|---|
| 2 | 6 | \(\frac{6}{2} = 3\) |
| 4 | 12 | \(\frac{12}{4} = 3\) |
| 7 | 21 | \(\frac{21}{7} = 3\) |
| 10 | 30 | \(\frac{30}{10} = 3\) |
Every ratio equals 3!
Find Constant Rate:
\(k = 3\) cups of flour per batch
Standard Form:
\(y = kx\)
Substitute \(k\):
\(y = 3x\)
Prediction power: To make 50 batches, \(y = 3(50) = 150\) cups of flour!
The equation translates word problems and tables into universal algebraic models.
Critical Discrimination
Proportional or Not?
Proportional
\(y = 12x\)
Direct multiple, no constant added.
\(y = 0.75x\)
Decimals and fractions can be \(k\)!
When \(x = 0\), \(y = 0\). Pure multiplication.
NOT Proportional
\(y = 12x + 5\)
Added \(\mathbf{+ 5}\) breaks constant ratio!
\(y = x^2\)
Exponent makes the rate change continuously.
When \(x = 0\), \(y = 5\). Does NOT pass through \((0,0)\).
Quick Test: If an equation has addition or subtraction (\(y = kx + b\)), it is NOT strictly proportional.
Active Class Practice
Your Turn: Write the Equation
Problem A: Wages
Marcus works at a summer camp. In 4 hours, he earns $68. In 7 hours, he earns $119.
1. Find \(k = \frac{68}{4} = \frac{119}{7} =\) ?
2. Write equation: \(y =\) _____ \(x\)
Answer: \(k = 17\) • Equation: \(y = 17x\)
Problem B: Travel Speed
A high-speed train travels 180 miles in 2 hours and 450 miles in 5 hours.
1. Find \(k = \frac{180}{2} = \frac{450}{5} =\) ?
2. Write equation: \(y =\) _____ \(x\)
Answer: \(k = 90\) • Equation: \(y = 90x\)
Key Takeaway: Always find the unit rate \(k\), then plug it directly into \(y = kx\)!
