The Converse

5 min

Teacher Prep
Setup
Groups of 2. 1 minute of quiet think time followed by partner and then whole-class discussion. 

Narrative

This Warm-up prepares students for the argument of the converse of the Pythagorean Theorem that will be constructed in the next activity. The Warm-up relies on the Pythagorean Theorem and geometrically intuitive facts about how close or far apart the two hands of a clock can get from one another.

Launch

Arrange students in groups of 2. Give students 1 minute of quiet think time, and follow with a whole-class discussion. 

Student Task

Consider the tips of the hands of an analog clock that has an hour hand that is 3 centimeters long and a minute hand that is 4 centimeters long.

The image of a circle that represent an analog clock. 
The image of a circle that represent an analog clock. On the circle are 12 evenly spaced tick marks. There are two hands on the clock. One hand is labeled 3, begins in the center of the circle and extends upward and to the right, and points to the third tick mark from the top. The other hand is labeled 4, begins in the center of the circle and extends upward and to the left. It points to the eleventh tick mark from the top.

Over the course of a day:

  1. What is the farthest distance apart the two tips get?

  2. What is the closest distance the two tips get?

  3. Are the two tips ever exactly five centimeters apart? Explain your reasoning.

Sample Response

  1. If the two hands are pointing in opposite directions, the tips will be 7 centimeters apart.
  2. If the two hands are pointing in the same direction (for example, at noon), the tips will be 1 centimeter apart.

  3. Yes. Sample reasoning: Whenever the two hands make a right angle (for example, at 3:00), then by the Pythagorean Theorem, the two tips will be 5 centimeters apart, since 32+42=523^2+4^2=5^2.

    <p>Right triangle inside of circle </p>

Activity Synthesis (Teacher Notes)

The focus of this discussion is to prepare students to follow a specific line of reasoning that is needed to make sense of the converse of the Pythagorean Theorem. First, invite 1–2 students to briefly share their answers to the three questions. Then ask students to consider the following line of reasoning:

Imagine two hands starting together, where one hand stays put and the other hand rotates around the face of the clock. As one hand rotates, the distance between its tip and the tip of the stationary hand continually increases until they are pointing in opposite directions. So from one moment to the next, the tips get farther apart.

(Proving this requires mathematics beyond grade 8, so for now we will just accept the results of the thought experiment as true.)

Standards
Building Toward
  • 8.G.6·Explain a proof of the Pythagorean Theorem and its converse.
  • 8.G.B.6·Explain a proof of the Pythagorean Theorem and its converse.

15 min

15 min