• Gauge class interaction during the introduction of taxicab and “regular” distance to judge level of understanding.
• Use the Civil Engineering worksheet to determine student mastery.

On the board, show the first quadrant of a coordinate plane with points at (1, 4) and (2, 8). Label the first point “Hotel” and the second one “Arena.”

Ask students to imagine they have arrived at their hotel and are going to an event at the arena and need to get a taxi to take them there. Explain that each line on the coordinate plane represents a downtown street. “What is the least number of blocks the taxi has to travel to get from the hotel to the arena?” (5)

Have students come up and draw or explain a path the taxi could take to get from the hotel to the arena by going exactly 5 blocks. Emphasize the two routes with the least number of turns.

“Let’s call the distance between two points as long as we stay on horizontal and vertical paths the ‘taxicab distance,’ so we can say that the taxicab distance between (1, 4) and (2, 8) is 5.”

Put another point on the graph at (6, 10) and label it “Restaurant.” Ask students to find the taxicab distance between the arena and the restaurant. (6 units) Ask students to explain their methods while guiding them towards the realization that you can subtract the x-coordinates from one another to find the horizontal distance and you can subtract the y-coordinates from one another to find the vertical distance.

“Imagine you are wealthy enough that you can afford to rent a helicopter to fly you straight from the arena to the restaurant. You don’t have to stick to the streets anymore. How does the distance you travel by helicopter compare to the taxicab distance between the arena and the restaurant?” (It is less.)

“To help figure out how much less, first imagine you’re taking a very simple taxicab route.” Sketch in a vertical line from (2, 8) and (2, 10) and a horizontal line from (2, 10) to
(6, 10).

“Now, let’s sketch in the line our helicopter might take.” Sketch in a line connecting (2, 8) and (6, 10).

“By doing that, we form a right triangle. How can we determine the actual distance the helicopter has to travel?”

Guide students towards using the Pythagorean theorem and have them calculate the actual distance. (The actual distance is .)

Have students find the actual distance between the hotel and the arena. (The actual distance is .)

“Is there ever a case in which the taxicab distance between two points is the same as the actual distance between the two points?” (when the two points lie on the same horizontal or vertical line)

Activity

Have students break up into groups of 3 students. Provide each group with a coordinate plane that goes from −20 to 20 on both axes and a copy of the Civil Engineering worksheet (M-8-6-2_Civil Engineering and KEY.docx). Explain to students that they are going to design their own town on their coordinate plane based on some instructions provided to them.

Reinforce to students before beginning that “distance” means the shortest possible distance between two points—“as the crow (or helicopter) flies” and taxicab distance means traveling only along horizontal or vertical lines.

After students in each group have completed their towns, have them pass their maps to another group. The second group should verify that the plotted points match all the criteria given.

Extension:

• Once students understand that the Pythagorean theorem can be used to find the distance between two points on a coordinate plane, introduce the distance formula:

, where d is the distance between  and .

• Use The Distance Formula worksheet (M-8-6-2_Distance Formula and KEY.docx) to challenge students to derive the distance formula from the Pythagorean theorem. Then have students practice using the distance formula to calculate the distance between various pairs of points.