A focused Grade 6 learning session for Proportional vs. Non-Proportional Relationships using visual models, worked examples, strategy guidance, and a short mastery check.
Student goal: I can solve problems involving proportional vs. non-proportional relationships.
Proportional vs. Non-Proportional Relationships means choosing a model, naming what each number means, and explaining the strategy.
Question: Which table shows a proportional relationship between \(x\) and \(y\)?
| x | \(1\) | \(2\) | \(3\) |
|---|---|---|---|
| y (option A) | \(5\) | \(10\) | \(15\) |
| y (option B) | \(3\) | \(5\) | \(8\) |
Why it works: A relationship is proportional if \(y/x\) is constant. Option A has \(5/1=5\), \(10/2=5\), and \(15/3=5\), so it is proportional. Option B has \(3/1=3\) and \(5/2=2.5\), so the ratio is not constant.
Answer: Option A
Question: Which graph represents a proportional relationship between \(x\) and \(y\)?
Answer: Graph A only
Question: A table shows the number of hours worked (\(h\)) and the amount of pay earned (\(p\), in dollars). Which option gives both the correct unit rate and the correct equation for the relationship?
| Hours (\(h\)) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| Pay (\(p\)) | \(12\) | \(24\) | \(36\) | \(48\) |
Answer: Unit rate: $12 per hour; equation: \(p = 12h\)
Question: A taxi charges $2 per mile traveled. If \(m\) is the number of miles and \(c\) is the total cost in dollars, which equation represents this proportional relationship?
Why it works: Since the taxi charges $2 per mile with no additional fee, the total cost is directly proportional to miles: \(c = 2m\). This passes through the origin when \(m = 0, c = 0\). Options B, C, and D either add a constant or reverse the relationship.
Answer: \(c = 2m\)
Proportional vs. Non-Proportional Relationships becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
Understand the model before choosing the operation.
Use scratch paper. The explanation matters more than simply seeing whether the choice is correct.
Which table shows a proportional relationship between \(x\) and \(y\)?
Answer A: Option A
A relationship is proportional if \(y/x\) is constant. Option A has \(5/1=5\), \(10/2=5\), and \(15/3=5\), so it is proportional. Option B has \(3/1=3\) and \(5/2=2.5\), so the ratio is not constant.
A recipe calls for 2 cups of flour for every 3 cups of sugar. Which statement correctly describes the table?
Answer C: Proportional; \(f/s = 2/3\) throughout
The ratio of flour to sugar is \(f/s = 2/3\), \(4/6 = 2/3\), \(6/9 = 2/3\), \(8/12 = 2/3\). The ratio remains constant, so the relationship is proportional.
Which graph represents a proportional relationship between \(x\) and \(y\)?
Answer A: Graph A only
A proportional relationship must pass through the origin \((0, 0)\). Graph A passes through the origin; Graph B starts at \((0, 1)\), so it does not pass through the origin.
A taxi charges $2 per mile traveled. If \(m\) is the number of miles and \(c\) is the total cost in dollars, which equation represents this proportional relationship?
Answer A: \(c = 2m\)
Since the taxi charges $2 per mile with no additional fee, the total cost is directly proportional to miles: \(c = 2m\). This passes through the origin when \(m = 0, c = 0\). Options B, C, and D either add a constant or reverse the relationship.
A table shows the number of hours worked (\(h\)) and the amount of pay earned (\(p\), in dollars). Which option gives both the correct unit rate and the correct equation for the relationship?
Answer A: Unit rate: $12 per hour; equation: \(p = 12h\)
The ratio \(p/h\) is constant: \(12/1 = 24/2 = 36/3 = 48/4 = 12\). That means the unit rate is $12 per hour, and the equation \(p = 12h\) represents the proportional relationship.
Which of the following does NOT represent a proportional relationship?
Answer B: Option B
Option A has a constant ratio: \(6/2 = 3\), \(12/4 = 3\), and \(18/6 = 3\). Option B does not: \(4/2 = 2\), \(7/4 = 1.75\), and \(10/6 \approx 1.67\).
The graph below shows distance traveled (\(d\), in miles) over time (\(t\), in hours). Does the relationship show a constant speed?
Answer C: Yes; the graph is a straight line through the origin
A proportional distance-time graph is a straight line through the origin. Here, the line passes through \((0,0)\), \((1,3)\), and \((2,6)\), so the speed stays constant at \(3\) miles per hour.
A store charges $5 per notebook. An employee has a discount and gets the first notebook free, then pays $5 for each additional one. For \(n\) notebooks, the total cost is \(c = 5(n - 1)\). Is this a proportional relationship?
Answer D: No; the ratio of cost to notebooks is not constant
This is not proportional because the cost per notebook is not constant. For example, \(1\) notebook costs \($0\), but \(2\) notebooks cost \($5\), so the ratio of cost to notebooks changes.
Which table represents a non-proportional relationship?
Answer B: Option B; the ratio is not constant
Option A has a constant ratio: \(7/1 = 7\), \(21/3 = 7\), and \(35/5 = 7\). Option B does not: \(2/1 = 2\), \(4/3 \approx 1.33\), and \(6/5 = 1.2\), so Option B is non-proportional.
A cyclist travels at 15 miles per hour. Which equation represents the distance \(d\) (in miles) after \(t\) hours?
Answer A: \(d = 15t\)
The proportional relationship between distance and time is \(d = 15t\). When \(t = 0\), \(d = 0\), indicating the line passes through the origin. Options B and D include constants (non-proportional), and option C reverses the variables.
Look at the data in the table. Is the relationship between hours worked and pay proportional?
Answer A: No; the first data point is not at the origin
For a proportional relationship, when hours \(= 0\), pay must \(= 0\). Here, when hours \(= 0\), pay \(= $5\) (a fixed fee), so the relationship is not proportional. The equation is \(p = 5h + 5\), which includes a non-zero constant term.
