Lesson Plan: Slope and Similar Triangles

Lesson Objectives

• Connect slope to similar triangles
• Use similar triangles to find slope
• Understand slope's geometric interpretation

TEKS Standards

• 8.4B: Graph proportional relationships, interpreting unit rate as slope.
• 8.4C: Use data to determine rate of change or slope and y-intercept.
• 8.3A: Understand ratio of corresponding sides in similar shapes.
• 8.3B: Compare attributes of a shape and its dilation on coordinate plane.

Warm-up Activity (5 min)

• Review prerequisite skills
  • Identifying right triangles. Review these definitions:
    • Right triangle: https://www.media4math.com/library/22169/asset-preview
    • Legs of a right triangle: https://www.media4math.com/library/22081/asset-preview
    • Hypotenuse: https://www.media4math.com/library/22061/asset-preview
  • Have students explore right triangles on a background grid:
    • Math Clip Art: https://www.media4math.com/library/75383/asset-preview
  • Similar triangles: https://www.media4math.com/library/42947/asset-preview
• Have students draw two right triangles on a coordinate plane with a shared hypotenuse (line segment). 

Key Vocabulary and Concepts (15 min)

Vocabulary

• Slope: https://www.media4math.com/library/22184/asset-preview
• Rise over run: https://www.media4math.com/library/42967/asset-preview
• Steepness: https://www.media4math.com/library/42980/asset-preview

Concept Development

Show the following video, which shows how similar right triangles can be used to explore slope:

https://www.media4math.com/library/75382/asset-preview 

Review (10 min)

• Use these clip art images to see if students can determine if the two right triangles are similar.

https://www.media4math.com/library/75380/asset-preview

•  For each of the lines shown, have students calculate the slope. Have them use the underlying grid to make their calculations. Then have them compare the slopes.

https://www.media4math.com/library/75381/asset-preview

Assessment (10 min)

Use this assessment to check for student understanding.

Quiz

1. What is the slope of a line?
   
    
2. If two triangles have the same shape but different sizes, what are they called?
   
    
3. How can you determine if two triangles are similar?
   
    
4. What is the relationship between the slopes of the hypotenuses of similar right triangles?
   
    
5. If one side of a triangle is doubled and the other sides are left unchanged, what happens to the slopes of the sides of the new triangle compared to the original triangle?
   
    

Answer Key

1. The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
2. If two triangles have the same shape but different sizes, they are called similar triangles.
3. To determine if two triangles are similar, you need to check if their corresponding angles are congruent (equal) and their corresponding side lengths are proportional.
4. The slopes of the hypotenuses of similar right triangles are equal.
5. If one side of a triangle is doubled and the other sides are left unchanged, the slopes of the sides of the new triangle will be half of the slopes of the corresponding sides in the original triangle.

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