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Arrange students in groups of 3–4. Provide access to geometry toolkits, making sure tracing paper is available for each student. Display the image from the task for all to see.
Give students 2–3 minutes to choose a scale factor and draw the dilation using that scale factor and point
Choose a scale factor and draw a dilation of triangle
Use a piece of tracing paper to trace point
The goal of this discussion is to show how dilations of a triangle with the same center but different scale factors will result in a series of similar triangles, all having their longest side along the same line.
Display 3–4 dilated triangles from previously selected students who used different scale factors. Ask students to share what they noticed in their groups and record the observations for all to see.
If not mentioned by students that the triangles are similar, suggest it now. Ask students how they would be able to tell that the triangles are similar. (Since the triangles are all dilations of triangle BCD, they are all similar to each other.)
Have students stack their tracing papers containing point
The purpose of this activity is to show that right triangles with their longest side along the same line are similar. This fact about the triangles is used to define slope.
To show that two triangles are similar, students need to use the structure of the grid. They can use it to describe a sequence of transformations or to show that the triangles share two pairs of congruent angles because the vertical or horizontal grid lines are parallel (MP7).
Arrange students in groups of 2. Assign one partner triangles
The grid shows three right triangles, each with its longest side on the same line. Your teacher will assign you two of the triangles. Explain why the two triangles are similar.
| triangle | length of vertical side |
length of horizontal side |
(vertical side) |
|---|---|---|---|
| 3 | 4 |
|
|
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
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
In this activity, students use slope triangles to draw a line with a given slope and observe two important properties (MP8):
Lines with the same slope are parallel.
As the slope of a line increases from 0, so does its steepness (from left to right).
Monitor for students who draw slope triangles of different sizes when drawing lines with slope
Discuss with students:
Now display this image for all to see, or draw and label two legs of a triangle as shown.
Discuss with students how this information could be used to draw a line with a slope of 3. (Use a ruler or straightedge to draw a line through what would be the third side of the triangle.) Remind students that any slope triangle whose vertical length divided by its horizontal length equals 3 could also be used, such as a triangle with a vertical side 6 units long and a horizontal side 2 units long.
Provide access to straightedges. Give students 3–4 minutes of quiet work time to complete the task followed by a whole-class discussion.
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
“What did you notice about the two lines you drew with a slope of 3? With a slope of
“What did you notice about the lines with a slope of 3 compared to the lines with a slope of
The goal of this discussion is to review the definition of slope. Discuss:
“What is a slope triangle for a line?” (A triangle whose longest side is on the line and whose other sides are horizontal and vertical.)
“How can you use a slope triangle to find the slope of a line?” (Divide the length of the vertical side by the length of the horizontal side.)
“Does it matter which two points on a line you use to create a slope triangle? Why?” (No. Any two slope triangles for a line are similar. So the quotient of the two corresponding sides will always give the same value, regardless of which two points you use to draw the slope triangle.)
“Why are any two slope triangles for the same line similar?” (They can be created by performing a dilation, so they will always be similar.)
Refer to the classroom display created in an earlier activity as needed.
Here is a line drawn on a grid. There are also four right triangles drawn.
<p>Four right triangles each with hypotenuse on the same line. First horizontal side 6, vertical side 4. Second horizontal side 3, vertical side 2. Third horizontal side 1, vertical side fraction 2 over 3. Fourth horizontal side 6, vertical side 4.</p>
These four triangles are all examples of slope triangles. The longest side of a slope triangle is on the line, one side is vertical, and another side is horizontal. The slope of the line is the quotient of the vertical length and the horizontal length of the slope triangle. This number is the same for all slope triangles for the same line because all slope triangles for the same line are similar.
In this example, the slope of the line is