Some students may struggle to get started. Prompt them by asking how to show that two triangles are similar (There is a sequence of translations, rotations, reflections, and dilations taking one to the other, or the two figures share 2 pairs of congruent angles.)

The goal of this discussion is for students to see that slope is a natural consequence of triangle similarity.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to why dividing the vertical side by the horizontal side results in quotients that are all equivalent. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

Display these prompts for feedback: 

• “_____ makes sense, but what do you mean when you say . . . ?”

• “Can you describe that another way?”

• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, explain that slope triangles like these can be constructed for every  non-vertical, non-horizontal line. Choose any two points on the line. Then draw horizontal and vertical lines from the points to form a right triangle.. The quotient of the lengths of each slope triangle’s vertical side and horizontal side will always be the same. This ratio is called the slope of the line, and it tells how steep the line is. In this activity, the slope can be written as 0.75 or , or any equivalent value.

Make clear to students that the mathematical convention is to define slope using the vertical change divided by the horizontal change and not the other way around. Display the following diagram, or create a similar display. Post this diagram for reference along with accompanying text:

Slope is a number that tells how steep a line is. To find the slope, divide the vertical change by the horizontal change for any two points on the line. The slope of line can be written as , , 0.75, or any equivalent value. 

Some students may draw lines with slope and . Display these alongside lines of slope 3 or slope , and ask students to describe how they are alike and different.  For now, leave the question open about whether such lines have the same or different slopes. Future lessons will focus on distinguishing positive from negative slopes.

The goals of this discussion are for students to see that since all slope triangles for a line are similar, any slope triangle can be used to draw the line. Students should also notice that lines with the same slope are parallel, and that as the slope increases from 0, the line appears steeper from left to right. 

Ask previously selected students, as described in the Activity Narrative, to share how they drew their lines with a slope of . Demonstrate how it does not matter if you draw a slope triangle with a vertical length of 1 and a horizontal length of 2, or a triangle with vertical and horizontal lengths of 3 and 6, or 5 and 10. Explain that the quotient of side lengths is the important feature, since any triangle drawn to match a given slope will be similar to any other triangle drawn to match the same slope. Here are some questions for discussion:

• “What did you notice about the two lines you drew with a slope of 3? With a slope of ?” (Lines with the same slope are parallel. Slope triangles for the lines with the same slope are similar.)

• “What did you notice about the lines with a slope of 3 compared to the lines with a slope of ? (The lines with a slope of 3 look “steeper” than the lines with a slope of .)