Proportional Relationships and Slope: Grade 8 Math Test Answer Key – PDF Download

Question 1: Unit rate from a table (2 points)

Unit rate: \(\frac{13.50}{3}=4.50\), \(\frac{22.50}{5}=4.50\), \(\frac{36}{8}=4.50\). Equation: \(y=4.5x\).

For 11 pounds: \(4.5(11)=49.5\), so the cost is $49.50.

Scoring: 1 point for the unit rate and equation; 1 point for the correct cost with the dollar sign.

Question 2: Graph a proportional equation (3 points)

1. For \(x=2\), \(y=3\); for \(x=4\), \(y=6\). Points \((2,3)\) and \((4,6)\) (also \((0,0)\)).
2. Draw a straight line through the plotted points and the origin.
3. The slope is \(\frac{3}{2}\).

Scoring: 1 point for two correct points, 1 point for a straight line through the origin, 1 point for slope 3/2 (accept 1.5).

Question 3: Slope and equation from a graph (2 points)

The line passes through the origin and \((5,4)\): slope \(=\frac{4}{5}\).

Equation: \(y=\frac{4}{5}x\) (or \(y=0.8x\)).

Scoring: 1 point for slope 4/5; 1 point for the equation.

Question 4: Similar triangles and slope (3 points)

1. The run grows from 4 to 12, which is 3 times larger, so the rise is \(3\times 3=9\). Rise = 9.
2. Both triangles have a right angle and share the angle at the origin, so they are similar. In similar triangles the ratios of corresponding sides are equal, so \(\frac{3}{4}=\frac{9}{12}\). The slope is \(\frac{3}{4}\) in both.

Scoring: 1 point for rise 9; 1 point for stating the triangles are similar; 1 point for concluding that rise over run is equal.

Question 5: Compare two speeds (3 points)

Car A: \(\frac{150}{3}=50\) miles per hour. Car B: slope \(58\) miles per hour.

Since \(58>50\), Car B is faster, by 8 miles per hour.

Scoring: 1 point for Car A’s rate 50; 1 point for Car B’s rate 58 and the comparison; 1 point for the difference of 8.

Question 6: Slope and intercept (2 points)

Slope: \(-\frac{3}{4}\). \(y\)-intercept: 6.

It is not proportional, because the \(y\)-intercept is 6, not 0, so the line does not pass through the origin.

Scoring: 1 point for the slope and intercept; 1 point for the answer not proportional with a reason.

Question 7: Points to an equation (2 points)

Slope \(=\frac{-3-5}{1-(-3)}=\frac{-8}{4}=-2\).

Use \((1,-3)\): \(-3=-2(1)+b\), so \(b=-1\). Equation: \(y=-2x-1\). Check with \((-3,5)\): \(-2(-3)-1=5\).

Scoring: 1 point for slope -2; 1 point for the equation with b = -1.

Question 8: Streaming plan (3 points)

1. \(y=3x+5\).
2. \(3(9)+5=32\). The cost is $32.
3. \(3x+5=41\), so \(3x=36\) and \(x=12\). She rented 12 movies.

Scoring: 1 point per part.

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