Why this matters for FAST: Proportional relationships are heavily tested on FAST. Students must identify proportional relationships in tables, graphs, and equations, determine the constant of proportionality, and write equations in the form y = kx.
Students think any straight-line graph shows a proportional relationship. This is WRONG! Proportional relationships MUST pass through the origin (0, 0).
"A proportional relationship is a SPECIAL type of linear relationship. It must pass through (0, 0). Why? Because if you have 0 of something, the other quantity must also be 0. No hours worked = $0 earned!"
Students confuse the constant of proportionality (k) with a point in the table or a y-value on the graph.
"The constant of proportionality k is the RATIO y/x. It tells us how much y changes for every 1 unit of x. In a table, divide EVERY y by its x - if you get the same number every time, that's k!"
Students think proportional means "add the same amount each time" rather than "multiply by the same amount."
"In proportional relationships, we MULTIPLY x by k to get y. If k = 3, then when x = 2, y = 6 (not 5). The equation is y = kx, which is multiplication!"
Review ratios and equivalent ratios. "If a car travels 60 miles in 1 hour, how far in 2 hours? 3 hours?" Show this relationship builds from equivalent ratios.
"A proportional relationship exists when two quantities always have the same ratio. We call this constant ratio 'k' - the constant of proportionality. The equation is always y = kx."
Three Ways to Identify Proportional Relationships
Table: y/x = same value (k) for ALL rows
Graph: Straight line through the ORIGIN (0, 0)
Equation: y = kx (no added or subtracted number)
Example: Hourly Wage
| Hours (x) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Earnings (y) | $12 | $24 | $36 | $48 |
| y/x | 12 | 12 | 12 | 12 |
k = 12 (This is the unit rate: $12 per hour)
Equation: y = 12x
"When we graph a proportional relationship, it's ALWAYS a straight line through the origin. The constant k is also the SLOPE of the line - it tells us how steep it is!"
Key Features of Proportional Graphs
1. Passes through (0, 0) - the origin
2. Is a straight line
3. The point (1, k) is always on the line
Work through these together:
"Which equation represents a proportional relationship?"
A) y = 5x + 2 B) y = 7x C) y = x - 3 D) y = 4
Correct answer: B) y = 7x. This is the only equation in the form y = kx with no constant added. The others have a number added, subtracted, or no x at all.
For struggling students: Focus on the table method first. Have students calculate y/x for every row and check if they get the same value. Use simple whole numbers before introducing decimals.
For advanced students: Challenge them with problems where they must determine if a real-world relationship is proportional (e.g., taxi fares with a base fee are NOT proportional). Have them explain why using multiple representations.
For home: Send Parent Activity sheet. Families can find proportional relationships in gas mileage, recipes, and unit prices at the store.