Students should work in pairs for this activity. Each pair of students should be given a satellite map of a downtown area/city, [IS.6 - All Students] somewhere with mostly regular square blocks and preferably in the same area as the school; see sample Minneapolis map for an example (M-G-1-3_Minneapolis Map.doc). Each pair of students should also have a highlighter. Instruct each student to choose an intersection on the map and mark it with the highlighter. [IS.7 - All Students] If the map is of an area surrounding the school, recommend that students choose an intersection familiar to them (so they can better visualize the situation). Otherwise, let them choose random intersections. Then, each pair should highlight the path they would drive to get from one student’s point (intersection) to the point (intersection) their partner chooses. Have each pair calculate this distance (in blocks).
Next, have each group consider the shortest distance from one point to the other if they were not restricted to staying on the road or going around obstacles such as buildings. [IS.8 - All Students] Have them highlight this route. Then ask:
“How can we find the shortest distance between your starting and ending places?”
“What shape have you highlighted on your map?” (right triangle)
“What theorem do you know that relates the lengths of the sides of a right triangle?” (Pythagorean theorem)
Have students measure the distance between the two points using rulers, and then measure the distance it would take to get to that point by going down the streets. If students measure the sides of a triangle, have them compare and see that for their measurements, a2 + b2 = c2. Demonstrate this to the class on the board. Mark a point on the board (call it A), and then go up 3 inches and mark another point (call it B); then go right (forming a 90 degree angle in the process) 4 inches and mark that point (call it C). Measure the distance between A and C, and show the class that is it 5 inches; show them that for this example, a2 + b2 = c2, or 32 + 42 = 52.
Now challenge students to go back to their maps and see how many right angles they can find, and have them measure the straight-line distance versus the distance going down the legs of the triangle. Assist students who might need additional practice as necessary. [IS.9 - All Students]
After giving students a few minutes exploring this exercise, have students gather back together, and remind students of the Pythagorean Theorem, taking care to stress that a and b represent the lengths of the legs and c represents the length of the hypotenuse.
Have each pair of students find the shortest distance between their two points using the Pythagorean Theorem this time. [IS.10 - All Students]
Select a few groups to present their maps. [IS.11 - All Students] Have them explain:
what points they selected (and why, if applicable)
the street route they highlighted and how they found that distance
the shortest route and how they found that distance
Ensure that, by the end of the presentations, it is clear to students how to use the Pythagorean theorem to find the distance between two points. [IS.12 - All Students] Encourage the use of calculators and rounding to 1 or 2 decimal places for complicated calculations.
Distribute the Distance Worksheet (M-G-1-3_Distance Worksheet.doc and M-G-1-3_Distance Worksheet KEY.doc). Instruct students to:
1. Graph the points.
2. Draw the “street route” between the points.
3. Draw the shortest route between the points.
4. Use the Pythagorean theorem to find the distance between the points.
If students have difficulty determining the lengths of the sides of the right triangles on the Distance Worksheet, ask students to examine the two x-coordinates and determine how far it is from one x-coordinate to the other x-coordinate and illustrate that the distance between the x-coordinates is simply the length of one of the legs. Repeat this with the y-coordinates.
After students have finished the Distance Worksheet, write the distance formula on the board:
“How does the distance formula relate to the Pythagorean theorem?”
Students should note the differences between the two and discuss how the two are, algebraically, the same formula. For example, the distance formula has a square root in it, and the Pythagorean theorem does not; however, solving the Pythagorean theorem for c (rather than c2) results in a square root. Depending on the class’s algebra skills, the algebraic relationship between the two can be explored in depth.
Show this simple example: Draw a right triangle with the right angle on the origin (0, 0) and two vertices at (0, 4) and (-3, 0). Show students that the subtraction of x2 from x1 is the same as 0 – (-3), which is 3. Then the subtraction of y2 from y1 is the same as 4 – 0, which is 4. Now show that 32 + 42 is the same as a2 + b2 from the more familiar c2 = a2 + b2.
As an exit activity, have students write an explanation of how to find the shortest distance between two points (M-G-1-3_Lesson 3 Exit Ticket.doc). [IS.13 - All Students]
For each set of three ordered pairs, graph the vertices on the x/y-coordinate plane, classify each triangle according to its side lengths and angle measures, and find each perimeter.
1. A (-4, -4), B (6, 0), C (3, 4)
2. A (-4, 3), B (10, 7), C (-5, 2)
3. A (-2, 2), B (6, -2), C (2, -6)
4. A (3, 6), B (7, 2), C (3, -1)