The Shortest Distance
Overview | Activities
Introductory Activities |
Culminating Activity |
Introductory Activities - Triangles
- Ask students to IDENTIFY by color each of the shapes shown in the picture below.
- DISTRIBUTE graph paper and rulers to each student. Inform students that they will be asked to TRACE the shapes onto the graph paper. Allow students time to trace each shape from the above-referenced Web site by placing the graph paper over the screen, transparency or lesson handout.
- Ask students to MEASURE the length of the sides of each figure using the metric ruler to the nearest millimeter. RECORD the measurements in the chart below.
(Sides of the green square measure approximately 3 cm long; sides of the red square measure approximately 1.5 cm long; sides of the turquoise square measure approximately 3.3 cm long.)
- Ask students to LOG onto the Animation of the Pythagorean Theorem site. Direct students to OBSERVE how each of the shapes identified in step 1 is transformed by the animation.
- Direct students to DISCUSS with a partner what you found interesting about the animation. What does the animation show about the relationship between triangles and squares?
(Hint for teachers: EXPLAIN to students that the animation shows that the sum of the squares for each of the sides of the right triangle adds up to the sum of the length of the longest side. POINT OUT to students that the name of the longest side is the hypotenuse and each of the other sides is called a leg.)
- CHECK for UNDERSTANDING using another web-based simulation activity.
Direct students to LOG onto the NOVA Web site where they can manually repeat the animation. Ask students to IDENTIFY which sides represent the legs and hypotenuse of the right triangle in the diagram shown. Direct students to complete the activity. (a and b represent the legs; c represents the hypotenuse).
Learning Activities-Problem Solving with the Pythagorean Theorem
Culminating Activities-Putting it all together
- DISTRIBUTE graph paper (GRAPHING THE SOLUTION HANDOUT).
- Direct students to LOG onto the NOVA companion site.
- Ask students to CLICK on the "How can I use the Pythagorean Theorem to solve real problems" link. Direct students to continue to the "Breaking and Entering" link.
- Get a student volunteer to READ the scenario provided:
You're locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window?
- Ask students to DRAW a triangle consistent with the scenario provided. CHECK for UNDERSTANDING by asking students to identify which of sides the ladder represents if the ladder makes a right triangle with the side of the house. (Hypotenuse).
- SOLVE the formula for c2 given side a=25 feet and b=10 feet. (Hint for teachers: Redirect students to the earlier web page that showed the legs of a right triangle. Since the scenario describes how the window is 25 feet above the ground, it must represent the length-the long side of the right triangle-or the vertical leg. The base of the ladder is 10 feet [away] from the house so the distance between the base of the ladder and the side of the house forms the horizontal leg. When students solve the expression they will substitute 25 ft. for a in the expression and 10 ft. for b in the expression. The substitutions yield the following expression: (25 ft.)2 + (10 ft.)2 = c2. The next step in the solution shows c2 = 725 ft.2. In order to find the actual length of c, the square root of 725 ft.2 must be determined. Direct students to find the square root button on their calculator which looks like "." The final result is 26.9 ft which is approximately equal to 27 feet. Take this opportunity to also explain to students that the square root is the same thing as expressing a quantity to the ½ power.)
- On the board, WRITE down the following equivalent expressions:
Ask students to RECORD each expression in their notebook. PROVIDE additional examples for students to complete independently.
- Students can verify their measurements from Introductory Activity #3 by substituting the measurements for the legs into the Pythagorean Theorem. When students find the sum of the squares of each side (green square = side 1; red square = side 2), they will get a value equivalent to c2. When students find the square root of that sum, they should come close to the measurement they found for the turquoise side. Discuss the result with students.
Extension activities-Next steps in learning
- DISTRIBUTE graph paper to students. Ask students to find a partner with whom they can work on the culminating activities for this lesson.
- Direct students to CREATE an x-y coordinate system on the graph paper with four quadrants represented. For each quadrant, have students draw two dots that are sufficiently far apart (at least two squares in both the vertical and horizontal directions).
- Direct students to TRADE papers with their partners so that each partner has the other's coordinate system. For each of the points shown, ask students to use parenthetic notation to represent the x and y values for each point. Walk around to CHECK for UNDERSTANDING.
- Once students have accurately identified the coordinates for the points, instruct students to return the graph paper back to the originator.
- Use the straight edge of the ruler to DRAW a right triangle using the first quadrant points, labeling the legs for each point by counting the number of squares between the original point and the intersecting line segment. (Hint to teachers: to construct the right triangle, beginning students might first want to trace with their fingers where the lines should actually cross. You can check to make sure students understand exactly where the corresponding horizontal and vertical lines should originate and intersect before drawing on the graph paper.)
- Once students have drawn their right triangles, ask students to calculate the hypotenuse of the resulting right triangle using the Pythagorean Theorem. EXPLAIN to students that the hypotenuse represents the distance between those two points.
- EXPLAIN to students that for the remaining quadrants, we will not create triangles to determine the distance between points but that we will simply use the data provided by the coordinates.
- For each of the remaining points (Hint to teachers: students should have two points in each of the three remaining quadrants) ask students to IDENTIFY the x-values and the y-values for each point and record those data in the chart below.
*(Hint to teachers: The sample point shows how a point described as such would be drawn in Quadrant II and students should record -3 for the x-value and 5 for the y-value.)
- Direct students to LOG onto the Distance Formula web site. Ask students to WRITE down the actual distance formula from the Web site in the space provided below.
Distance formula (d) = ___________________________½
REMIND students that it is some times easier to express long equations in brackets expressed to the ½ power rather than write them under a square root symbol. Refer students back to Step 7 of the Learning Activities if necessary.
(Students should have recorded the expression: d = [(x2-x1)2 + (y2-y1)2]½ in the space provided.)
- Using the data generated from step 8, ask students to input the appropriate x and y values for each of their points into the radio boxes on the Web site and hit "solve." INSTRUCT students to record each distance in the same space as the point for reference.
- To CHECK for UNDERSTANDING, ask students to TRADE papers with another partner (one different from the first partner). Give students about five minutes to choose one set of points to DRAW a right triangle to determine the distance (hypotenuse).
- COLLECT student work and PROVIDE appropriate feedback.
- DIRECT students to Log onto the Distance Formula Web site to compare their results.
Cross curricular connections
Build on this lesson by exploring linear functions. Visit the NCTM Web site http://standards.nctm.org/document/eexamples/chap7/7.5/index.htm
and see the effects changes in points on the line will have on the equation of the line. Remove the checks from the "show trace" and "connect sliders" boxes. Observe details about the line (the equation for example) and a specific point on the line (the coordinates for example) as you move each of the three sliders around.
When thinking about next steps, the first thing that comes to mind is the actual daily task of walking and getting around. Chart out a destination using this "shortest distance" concept. The shortest distance between any two points is a straight line–unfortunately, it is sometimes difficult to map out linear paths when we travel in our neighborhoods and through buildings: streets and walls get in the way! However, there are a few times when we can: we call them short cuts! Think about the many places you go during the day and map out a linear path between two of your target destinations. This is an especially important exercise for emergency route planning. If there is not an emergency path posted in your classroom or work area, create a grid of the space and plot the shortest distance to the exit. Time yourself traveling using your new path plan and see just how much it makes a difference.