Slope with similar triangles in 8th grade

The correct choice is the one that says Mia added 3 to the x-value instead of the y-value. The shift is vertical (up 3), not horizontal.

1. 1Line 1 is correct. The mistake starts in Line 2, where Mia writes (x + 3, 2x) instead of (x, 2x + 3). Line 3 follows from that same error.
2. 2When x = 0, the point on y = 2x is (0, 0). On y = 2x + 3 it is (0, 3). The x-value stays 0; only y increases by 3.
3. 3Adding 3 shifts every point up 3 units, not right 3 units. Mia added 3 to the x-value when it should have been added to the y-value.

Common mistake. A common mistake is thinking +3 changes the x-input because the equation looks like 'something plus 3.' In y = 2x + 3, the +3 is added to the output (y), moving every point up 3.

The idea underneath. A vertical shift adds the same amount to every y-value, and adding the same amount to both points does not change their difference, so the slope (rise over run) stays the same.

The best answer is '$b$ should come from the y-axis crossing' because the line crosses the y-axis at 1, not 3.

1. 1Look at the two marked points on the line: $(0, 1)$ and $(3, 3)$.
2. 2Check the slope using rise over run.
3. 3Now find $b$. The line crosses the y-axis at 1, so $b = 1$. The y-value 3 belongs to the point $(3, 3)$, which is not the y-intercept.
4. 4Compare that to Jin's work. Step 1 is correct, but Step 2 says $b = 3$, so Jin used the y-value of $(3, 3)$ instead of the y-axis crossing.
5. 5So the correct equation is:

Common mistake. A common mistake is using the y-value of any marked point on the line as $b$. In $y = mx + b$, $b$ is only the y-value where the line crosses the y-axis.

The idea underneath. Any two points on the line can give you $m$, but $b$ is special: it comes only from where the line crosses the y-axis.

The answer is True because both triangles are right triangles built from horizontal and vertical legs, with their hypotenuses on the same line, and their rise-to-run ratios match: $\frac{2}{3} = \frac{4}{6}$.

1. 1Look at the graph. Both triangles have one horizontal leg and one vertical leg, so each is a right triangle.
2. 2Count the legs of the smaller triangle. It goes right 3 units and up 2 units.
3. 3Count the legs of the larger triangle. It goes right 6 units and up 4 units. The rise-to-run ratios match.
4. 4Both triangles are right triangles whose hypotenuses lie on the same line, so they share the same angle structure and are similar.

Common mistake. A common mistake is thinking similar means the triangles must be congruent. These triangles differ in size but have the same angle structure, so they are still similar.

The idea underneath. Right triangles are similar when their acute angles match. Horizontal and vertical legs make right angles. If the hypotenuses lie on the same line, the acute angles match too.

The answer is Mateo because the correct equation is $y = \frac{2}{3}x - 5$, and Mateo wrote exactly that.

1. 1Start with $y = mx + b$. Here, $m = \frac{2}{3}$ and $b = -5$.
2. 2Substitute the given values into $y = mx + b$ to get the correct equation.
3. 3Mateo wrote $y = \frac{2}{3}x - 5$. The slope $\frac{2}{3}$ matches $m$ and the constant $-5$ matches $b$. Mateo is correct.
4. 4Zoe wrote $y = \frac{2}{3}x + 5$. The slope matches, but the constant is $+5$ instead of $-5$. Zoe flipped the sign of $b$.
5. 5Only Mateo wrote the correct equation.

Common mistake. A common mistake is choosing Both because both equations share the slope $\frac{2}{3}$. However, $b = -5$ means the constant term must be $-5$, and Zoe changed it to $+5$.

The idea underneath. In $y = mx + b$, the coefficient of $x$ must match $m$ and the constant term must match $b$; equivalent ways of writing the coefficient are okay, but the sign of $b$ cannot change.