Testinar Grade 6 math

Proportional vs. Non-Proportional Relationships

Students will build Grade 6 mastery of proportional vs. non-proportional relationships using models, equations, and test-style reasoning.

CH01-12 Ratios, Rates, and Percents

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Score: ____ / 8

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Which table shows a proportional relationship between $x$ and $y$?
1. Option A
2. Option B
3. Both A and B
4. Neither
A recipe calls for 2 cups of flour for every 3 cups of sugar. Which statement correctly describes the table?
1. Proportional; $s/f = 2/3$ throughout
2. Not proportional; the ratio changes
3. Proportional; $f/s = 2/3$ throughout
4. Cannot determine from the table
Which graph represents a proportional relationship between $x$ and $y$?
1. Graph A only
2. Graph B only
3. Both graphs
4. Neither graph
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?
1. $c = 2m$
2. $c = 2m + 1$
3. $m = 2c$
4. $c = m + 2$
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?
1. Unit rate: $12 per hour; equation: $p = 12h$
2. Unit rate: $24 per hour; equation: $p = 24h$
3. Unit rate: $12 per hour; equation: $h = 12p$
4. The relationship is not proportional
Which of the following does NOT represent a proportional relationship?
1. Option A
2. Option B
3. Both A and B
4. Both are proportional
The graph below shows distance traveled ($d$, in miles) over time ($t$, in hours). Does the relationship show a constant speed?
1. No; the line passes through the origin
2. No; the line is not straight
3. Yes; the graph is a straight line through the origin
4. Cannot determine from the graph
Which table represents a non-proportional relationship?
1. Option A; the ratio is 7:1
2. Option B; the ratio is not constant
3. Option A; each $y$ is $x$ squared
4. Neither; both are proportional

Teacher and parent reference

Answer key and explanations

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.
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.
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. $c = 2m$
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. 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.
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 $10/6 \approx 1.67$.
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.
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.