# Investigation: Slope, distance, and midpoint partner activity

Lesson

### Two points, three calculations - a review

Given any two points, we can calculate many different properties of the line segment joining the two points.

We can find the slope between the points, the midpoint of the two points, and the distance between the two points. Let's quickly review each of the relevant formulas.

Slope formula

For two points, $A$A $\left(x_1,y_1\right)$(x1,y1) and $B$B $\left(x_2,y_2\right)$(x2,y2), we can find the slope of the line segment between them using the formula $m=\frac{y_2-y_1}{x_2-x_1}$m=y2y1x2x1

Midpoint formula

For two points, $A$A $\left(x_1,y_1\right)$(x1,y1) and $B$B $\left(x_2,y_2\right)$(x2,y2), we can find the midpoint between them using the formula $M_{AB}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$MAB=(x1+x22,y1+y22).

Distance formula

For two points, $A$A $\left(x_1,y_1\right)$(x1,y1) and $B$B $\left(x_2,y_2\right)$(x2,y2), we can find the distance between them using the formula $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$d=(x2x1)2+(y2y1)2.

### Partner activity

In this activity, first one partner will set up a question and the other partner will perform the calculations. Points are earned for each correct calculation. Part marks can be awarded to partially correct solutions if desired. The roles are then swapped and repeat until the first person gets to 36 points.

Calculation Number of points, if correct
Slope 1
Midpoint 2
Distance 3

1. Open the GeoGebra applet below
2. Partner 1 - Create questions for your partner by clicking and moving the two points.
3. Partner 2 - Calculate the slope, midpoint and distance between the two points chosen by your partner.
4. Partner 1 - Check your partner's answers by selecting the checkbox for each value one at a time.
5. Partner 2 - For any incorrect calculations, "Show Full Solution" and see if you can find your error. You can step forwards and backwards through the solution using the slider in the bottom left corner.
6. Partner 2 - Calculate the number of points you earned that round and record (maximum of 6 points)
7. Repeat with partner 1 and 2 swapped around.

To increase the challenge, consider using non-integer values or not showing your partner the diagram and having them just use the coordinates of the points.

#### Discussion questions

1. Which calculation overall went the best for you, the slope, midpoint, or distance? Why do you think this was?
2. Did you find yourself using the formulas or the conceptual definitions more? Did it depend on the calculation?
3. Is it easier or more challenging calculating without the diagram compared to when given the diagram?
4. If you are given one point and the slope, can you determine the other point uniquely? Explain.
5. If you are given one point and the midpoint, can you determine the other point uniquely? Explain.
6. If you are given one point and the distance, can you determine the other point uniquely? Explain.
7. If you are given one point, the slope and the distance, can you determine the other point uniquely? Explain.