Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
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.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
In Module 3, students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles. The module begins with the definition of dilation, properties of dilations, and compositions of dilations. One overarching goal of this module is to replace the common idea of “same shape, different sizes” with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.
Topic A: Dilation (8.G.A.3)
Lesson 1: What Lies Behind “Same Shape”?
Lesson 2: Properties of Dilations
Lesson 3: Examples of Dilations
Lesson 4: Fundamental Theorem of Similarity (FTS)
Lesson 5: First Consequences of FTS
Lesson 6: Dilations on the Coordinate Plane
Lesson 7: Informal Proofs of Properties of Dilations (optional)
Topic B: Similar Figures (8.G.A.4, 8.G.A.5)
Lesson 8: Similarity
Lesson 9: Basic Properties of Similarity
Lesson 10: Informal Proof of AA Criterion for Similarity
Lesson 11: More About Similar Triangles
Lesson 12: Modeling Using Similarity
Topic C: The Pythagorean Theorem (8.G.B.6, 8.G.B.7)
Lesson 13: Proof of the Pythagorean Theorem
Lesson 14: The Converse of the Pythagorean Theorem