Meet Slope
10 min
Teacher Prep
Setup
Required Preparation
Narrative
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 A as the center. Monitor for students who use a variety of scale factors, such as 31, 21, 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.
Student Task
-
Choose a scale factor and draw a dilation of triangle BCD using point A as the center of dilation. What scale factor did you use?
-
Use a piece of tracing paper to trace point A and your dilated figure. Compare your dilation with your group. What do you notice?
Sample Response
- Sample response: I used a scale factor of 2.
- Sample responses:
- The triangles are all similar.
- The triangles are all scaled copies of each other.
- Scale factors less than 1 resulted in smaller triangles closer to point A.
- Scale factors greater than 1 resulted in larger triangles farther away from point A.
- The quotient of the horizontal and vertical sides is the same for all of the triangles.
Activity Synthesis (Teacher Notes)
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 pointA and their dilated triangle so that all of the point As and triangle BCDs are on top of each other. Ask students what they notice. (The longest side of the triangles all line up.)
Standards
- 8.EE.6·Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
- 8.EE.B.6·Use similar triangles to explain why the slope <span class="math">\(m\)</span> is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation <span class="math">\(y = mx\)</span> for a line through the origin and the equation <span class="math">\(y = mx + b\)</span> for a line intercepting the vertical axis at <span class="math">\(b\)</span>.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Meet Slope
10 min
Teacher Prep
Setup
Required Preparation
Narrative
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 A as the center. Monitor for students who use a variety of scale factors, such as 31, 21, 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.
Student Task
-
Choose a scale factor and draw a dilation of triangle BCD using point A as the center of dilation. What scale factor did you use?
-
Use a piece of tracing paper to trace point A and your dilated figure. Compare your dilation with your group. What do you notice?
Sample Response
- Sample response: I used a scale factor of 2.
- Sample responses:
- The triangles are all similar.
- The triangles are all scaled copies of each other.
- Scale factors less than 1 resulted in smaller triangles closer to point A.
- Scale factors greater than 1 resulted in larger triangles farther away from point A.
- The quotient of the horizontal and vertical sides is the same for all of the triangles.
Activity Synthesis (Teacher Notes)
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 pointA and their dilated triangle so that all of the point As and triangle BCDs are on top of each other. Ask students what they notice. (The longest side of the triangles all line up.)
Standards
- 8.EE.6·Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
- 8.EE.B.6·Use similar triangles to explain why the slope <span class="math">\(m\)</span> is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation <span class="math">\(y = mx\)</span> for a line through the origin and the equation <span class="math">\(y = mx + b\)</span> for a line intercepting the vertical axis at <span class="math">\(b\)</span>.