Let’s go through the worksheet step-by-step.

Part 1: Questions on the Given Graph and Slope Calculation

1. Using the values in the ordered pairs, how else could Aiden have found the rise to be 2?

   • By looking at the difference in the y-coordinates of the points $(3, 5)$ and $(9, 7)$: $7 - 5 = 2$
   • This is the rise (change in $y$) between the two points.

2. Using the values in the ordered pairs, how else could Aiden have found the run to be 6?

   • By looking at the difference in the x-coordinates of the points $(3, 5)$ and $(9, 7)$: $9 - 3 = 6$
   • This is the run (change in $x$) between the two points.

Part 2: Calculating the Slope for Each Pair of Points

For each point pair given in the table, we’ll apply the slope formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Row A: Points (2, 4) and (1, 7)

1. Calculate the difference in y-values: $7 - 4 = 3$
2. Calculate the difference in x-values: $1 - 2 = -1$
3. Slope calculation: $\text{slope} = \frac{3}{-1} = -3$

Row B: Points (-1, 3) and (5, -5)

1. Calculate the difference in y-values: $-5 - 3 = -8$
2. Calculate the difference in x-values: $5 - (-1) = 6$
3. Slope calculation: $\text{slope} = \frac{-8}{6} = -\frac{4}{3}$

Row C: Points (8, 11) and (10, 22)

1. Calculate the difference in y-values: $22 - 11 = 11$
2. Calculate the difference in x-values: $10 - 8 = 2$
3. Slope calculation: $\text{slope} = \frac{11}{2}$

Row D: Points (-1, 9) and (-1, 6)

1. Calculate the difference in y-values: $6 - 9 = -3$
2. Calculate the difference in x-values: $-1 - (-1) = 0$
3. **Since the difference in x is zero, this line has an undefined slope (vertical line).

Summary Table

Would you like more details or explanations on any of these calculations?

Here are some further questions to deepen understanding:

1. What does a positive slope indicate about the direction of the line?
2. How does a negative slope affect the line’s orientation on a graph?
3. What conditions result in a slope of zero, and what does this mean for the line's appearance?
4. How would you explain an undefined slope in simple terms?
5. Can you think of real-world scenarios where the concept of slope is useful?

Tip: Remember that the slope formula measures the "rise over run," helping us understand how steep a line is between two points.