• Use the Finding Missing Lengths in Right Triangles worksheet (M-8-6-1_Finding Missing Lengths.docx) to evaluate student understanding.
• Gauge class discussion for both activities to monitor student comprehension.

Activity 1

Give each student or small group of students a collection of sticks cut into a variety of different lengths. Each group is to construct triangles with sides of differing lengths so the sticks touch one another at the ends. The goal for each group is to construct right triangles—students should visually estimate the angle measures.

Once students create a right triangle, they should record the lengths of the legs and the hypotenuse of the triangle they created.

After the groups have had a chance to explore constructing a variety of triangles, ask for the measurements of the triangles the groups considered to be right triangles.

“How do you know the triangles with these angle measures are right triangles?” Student responses may include, “Because it looks like it,” or students may have measured the angle using a protractor or something with a known right angle like the corner of a piece of paper or note card. If students do not bring up the Pythagorean theorem, ask if a student can state it. Make sure it is stated in the right “order,” i.e., “If a triangle is a right triangle with legs with lengths a and b and a hypotenuse with length c, then a² + b² = c².”

“To use the Pythagorean theorem, what do we need to know?” Guide students toward the realization that the Pythagorean theorem “starts” with a right triangle and “ends” with knowing information about the lengths of the sides of the right triangle.

“Can we use the Pythagorean theorem to show that a triangle is a right triangle?” Make sure students recognize that the Pythagorean theorem, as it is stated, cannot be used to show that a triangle is a right triangle; however the converse of the theorem can.

Write the converse of the Pythagorean theorem on the board: “If a triangle has sides with lengths a, b, and c and a² + b² = c², then the triangle is a right triangle.”

“In order to use this theorem, what do we need to know?” Help students recognize that we need to know that the lengths of the sides of the triangle satisfy a² + b² = c².

“And if we know the lengths satisfy our equation, what do we know about our triangle based on this theorem?”

Make sure students understand that the Pythagorean theorem tells us something about the relationship among the side lengths of a triangle, while its converse tells us something about the angles of the triangle (namely, that one is a right angle).

Activity 2

Using a 3-4-5 triangle from Activity 1 as an example, ask students, “If you know the legs of a right triangle are 3 inches and 4 inches, how long is the hypotenuse?” Make sure to have students explain their reasoning using the Pythagorean theorem.

“If you know that one leg of a right triangle is 3 inches and the hypotenuse is 5 inches, how long is the other leg?” Here, make sure to emphasize the importance of letting the length of the hypotenuse be represented by c in the equation a² + b² = c². Also, point out that in both instances, you stated that the triangle is a right triangle, the prerequisite for using the Pythagorean theorem.

Have students break up into groups of 4 or 5. Give each group a length of string and a yardstick.

This activity is best done someplace where there are right angles on the floor. Possible locations include the classroom or hallway, where floor tiles are located at right angles; outdoors in a parking lot using painted parking lot lines; or indoors in a gymnasium using the lines painted to mark basketball or volleyball courts.

Have three students in each group position themselves at the vertices of a right triangle, using the right angles on the floor as a guide to make sure students form a right triangle. Each group should identify the legs of the right triangle and the hypotenuse. (For example, “The hypotenuse is the side of the triangle between Bob and Susan.”)

Students not marking the vertices of the triangle should measure the lengths of two of the three sides of the triangle. Make sure students do not always measure the same two sides in each triangle (i.e., in some triangles, students should measure both legs, while in others they should measure a leg and the hypotenuse). Students can measure with a yardstick but may find it easier to measure with a length of string and then measure the string. If they use a single piece of string, the length should be more accurate because the string can be pulled into a straight line.

Students should record the lengths of the two measured sides and then use the equation
a² + b² = c² to predict the length of the remaining side before measuring it.

After students have recorded their prediction, each group should measure the length of the remaining side and compare the actual length to their prediction.

Each group should complete at least enough triangles so each student has the opportunity to measure and make predictions.

When each group has finished, ask students:

“How accurate were your predictions?”

“What are some reasons your predictions may not have been accurate?” (Inaccuracy in measurements, the triangles were not right triangles, etc.)

“What difficulties did you encounter?”

“What were some things you had to keep in mind during the activity?” (Ideally, students will mention they had to keep in mind to let the length of the hypotenuse be represented by c in the equation a² + b² = c²; if not, remind students.)

Extension:

• It may be useful, depending on the class, to explore the relationship between true if-then statements and their converses. Students often assume that the converse of every true
  if-then statement is also true. Ask students to come up with true if-then statements with converses that are not necessarily true, i.e., “If it is raining, then there is water landing on the sidewalk.” Encourage students to come up with situations that disprove the converse, i.e., there is a sprinkler near the sidewalk. Point out to students that, while the converse of the Pythagorean theorem might seem like a somewhat trivial and obvious result, the truth of a statement does not mean its converse is necessarily true and that it is important to distinguish between a theorem and its converse.