Students may need to be reminded that a product is the result of multiplication.

The purpose of this discussion is to establish that perpendicular lines have slopes that are opposite reciprocals.

Invite previously selected students to share their conjectures about the slopes. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see. Connect the different responses to the learning goals by asking questions such as:

• “Who can restate ’s reasoning in a different way?”
• “Can you say that again, using precise mathematical language?”
• “Does this conjecture work in all cases? How do you know?”
• “Is this conjecture different from the previous one? How do you know?”

Support students' use of precise language. For example, if they say the slopes are “flipped” or “upside down,” ask them if there is a mathematical term for this concept (reciprocal). Students may want to describe the slopes of perpendicular lines as negative reciprocals. Opposite reciprocals is clearer language—if the original slope is negative (such as ), we can avoid the awkward and instead jump directly to .

To emphasize that these different conjectures are really saying the same thing, invite students to consider a slope of . Ask them to provide the opposite reciprocal (). Instruct students to find the product of these fractions (, as long as neither nor is 0). In general, the value that multiplies with to create -1 is .

If it hasn’t come up, tell students that there is at least one exception that doesn’t fit the conjecture: a pair of horizontal and vertical lines. Because a vertical line has no slope, the idea of an opposite reciprocal doesn’t make sense.

Finally, challenge students to calculate the opposite reciprocal of several values, such as , -5, and .

If students struggle to find the first slope, suggest that they draw a picture of a line passing through the origin and choose a point on the line to label .

Ask students if their proof for lines through the origin can be extended to other lines that do not pass through the origin. (Yes. For a line that doesn’t pass through the origin, we could first translate it so it passes through the origin. Translation of lines results in parallel lines, so this translation would not affect the slope. The proof doesn’t apply if we start with a vertical line, though, because it has no slope.)

Then, ask students what the difference is between what they did in the previous activity and what they did in this one. (In the previous activity, we showed that the slopes of perpendicular lines were opposite reciprocals for a few specific cases. In this activity, we showed that this is true for all pairs of non-vertical and non-horizontal lines.)

Tell students that the converse is also true: If two lines have slopes that are opposite reciprocals, then they are perpendicular. Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

Lines are perpendicular if and only if their slopes are opposite reciprocals. (Theorem)

Finally, invite students to write the equation of a line that passes through the point and is perpendicular to the line given by . The result is .