Unit Plan 26 (Grade 8 Math): Similarity in Right Triangles (Extension/Spiral)
8th graders connect similarity and modeling by using AA similarity to solve right-triangle problems, create scale drawings, and perform indirect measurements. They apply proportions, justify scale factors, and verify results using the Pythagorean Theorem and real-world reasoning.
Focus: Connect similarity ratios to side relationships in right triangles; scale drawings and indirect measurement.
Grade Level: 8
Subject Area: Mathematics (Geometry • Modeling)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This week deepens earlier work on similarity by applying it to right triangles and to real-world measurement. Students use AA similarity to relate side ratios within and between right triangles (including the smaller triangles formed by drawing the altitude to the hypotenuse). They then create and interpret scale drawings and use indirect measurement (for example, shadows and mirror setups) to find heights or distances that are hard to measure directly. The Pythagorean Theorem is spiraled to check results.
Essential Questions
- How does AA similarity let us compare side lengths in right triangles?
- How do scale factors help us move between a real object and a scale drawing?
- How can we use similar triangles to measure something we cannot reach, and how can we check our answer?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Establish AA similarity for pairs of right triangles (including the two smaller triangles formed by the altitude to the hypotenuse) and identify corresponding sides.
- Use proportions and scale factors to determine unknown lengths in similar right triangles; explain which ratio they used and why.
- Create and read scale drawings (choose a scale, convert measurements, and label) and solve problems using the drawing.
- Model indirect measurements (shadows, mirrors, similar right triangles in the field), justify the setup, and validate with a reasonableness or Pythagorean check.
Standards Alignment — CCSS Grade 8
- 8.G.4: Understand similarity via sequences of transformations (dilations and rigid motions) and describe a sequence that exhibits similarity.
- 8.G.7 (spiral): Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems.
- MP.4 (Model with mathematics): Make assumptions, construct diagrams, and interpret results in context.
Success Criteria (student-friendly)
- I can match corresponding sides in similar right triangles and write a correct proportion.
- I can find a scale factor and use it to move between real size and scale drawing.
- I can design an indirect measurement setup, compute a missing length, and check if my answer makes sense.
- I can explain my steps in clear sentences with correct units.