The goal of this Warm-up is to revisit dilations and similar triangles in preparation for understanding slope and slope triangles, which will be introduced in a following activity.
Launch
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 as the center. Monitor for students who use a variety of scale factors, such as , , 2, 2.5, and 3. Encourage students who choose a scale factor of 1 to select an additional scale factor to draw a dilation for. Pause for a partner then whole-class discussion.
Activity
None
Choose a scale factor and draw a dilation of triangle using point as the center of dilation. What scale factor did you use?
Use a piece of tracing paper to trace point and your dilated figure. Compare your dilation with your group. What do you notice?
Student Response
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Building on Student Thinking
Activity Synthesis
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 and their dilated triangle so that all of the point s and triangle s are on top of each other. Ask students what they notice. (The longest side of the triangles all line up.)
15.2
Activity
Standards Alignment
Building On
8.G.A.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Use similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation for a line through the origin and the equation for a line intercepting the vertical axis at .
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).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Arrange students in groups of 2. Assign one partner triangles and and the other partner triangles and . Give students 3–4 minutes of quiet work time to construct an argument for why their two triangles are similar. Pause for a partner discussion before having pairs complete the activity.
Engagement: Provide Access by Recruiting Interest. Provide choice. Invite students to decide which pair of triangles they will show are similar.. Supports accessibility for: Organization, Attention
Activity
None
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.
Complete the table.
triangle
length of
vertical side
length of
horizontal side
(vertical side) (horizontal side)
3
4
or 0.75
What do you notice about the last column in the table? Why do you think this is true?
Student Response
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Building on Student Thinking
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.)
Activity Synthesis
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.
15.3
Activity
Standards Alignment
Building On
Addressing
8.EE.B.6
Use similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation for a line through the origin and the equation for a line intercepting the vertical axis at .
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 .
Launch
Display this image for all to see, or draw a line with a slope of 2 and include a slope triangle.
Discuss with students:
"How can you tell that this line has a slope of 2?" (Dividing the vertical length of the slope triangle by the horizontal length gives the quotient is 2.)
"Could a different slope triangle be used? Explain your reasoning." (Yes, all slope triangles of a line are similar. So, dividing the vertical length by the horizontal length for any slope triangle of this line will give the same quotient, 2.)
"What do you know about a line that has a slope of 3?" (For any slope triangle of the line, the vertical length will be three times the horizontal length.)
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.
Representation: Access for Perception. Students may benefit from watching or hearing the demonstration from the launch of how to use a slope triangle to draw a line with a given slope more than once. Supports accessibility for: Language, Attention
Activity
None
Draw two lines with a slope of 3. What do you notice about the two lines?
Draw two lines with a slope of . What do you notice about the two lines?
Activity Synthesis
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 .)
MLR8 Discussion Supports. Display sentence frames to help students to formulate statements about what they notice about two lines with the same slope and lines with different slopes:
“Two lines with the same slope are____ because . . . .,”
“The lines with slopes equal to 3 are____than lines with slope 1/2, because . . . .”
Advances: Speaking, Conversing
Lesson Synthesis
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.
Student Lesson Summary
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 . Here is how the slope is calculated using the slope triangles:
Points and give .
Points and give .
Points and give .
Points and give .
Standards Alignment
Building On
Addressing
Building Toward
8.EE.B.6
Use similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation for a line through the origin and the equation for a line intercepting the vertical axis at .
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.
