Big Idea:

Big Idea 1

A dilated figure has angles congruent to and sides proportional to the original figure.

1 week

Evidence of Understanding

  • describe the center of dilation as the vertex of an angle
    • experiment and describe relationships involving the center of dilation by constructing line segments or figures along the two rays
    • explain why dilated lines not passing through the center of dilation are parallel and dilated lines passing through the center of dilation are superimposed
  • identify and justify the center of dilation between dilated figures
    • describe angle relationships using properties of parallel lines cut by a transversal
  • analyze relationships between corresponding sides and corresponding angles of dilated figures  
    • use tools to create and describe features of figures that are enlarged or reduced
    • identify, using markings, corresponding angles that are congruent (ratio is always 1:1)
    • explain the proportional relationship between corresponding sides of a pre-image and its image
    • analyze the impacts of a dilation between dilated figures
      • determine the scale factor between a given preimage and image
      • apply the scale factor to determine measurements of a pre-image or its image
      • explain why perimeter maintains the scale factor and area squares the scale factor
  • determine the minimum criteria that proves polygons are similar (focusing on triangles)
    • transform figures to distinguish and define congruence and similarity
      • use the equality of all corresponding pairs of angles (same shape) and the proportionality of all corresponding pairs of sides (scale factor) to define similarity
      • use coordinate notation with the plane to describe transformations
    • identify the triangles, quadrilaterals, and other polygons that are always similar, sometimes similar or never similar and explain reasoning
    • use tools and counterexamples to justify Angle-Angle (AA) is valid criteria for triangle similarity
      • use the triangle angle sum theorem to describe why AA is sufficient minimum criteria
    • compare the Side-Angle-Side (SAS) and Side-Side-Side (SSS) criteria for congruence proofs versus similarity proofs
      • use tools and counterexamples to justify that two proportional sides and the congruent included angle is sufficient criteria for proving two triangles are similar
    • justify why criteria for triangle congruence are also sufficient for proving triangle similarity
      • explain congruence as a special case of similarity where the scale factor is 1

Develop conceptual understanding:

dilation, center of dilation, ratio, proportional, scale factor, perimeter, area, similarity, AA, SAS, SSS, congruence

Supporting terms to communicate:

vertex, angle, segment, ray, corresponding, enlarge, reduce, parallel, transversal, preimage, image, transformation, rotation, reflection, translation
Core Resource
A core resource supports multiple days of instruction.
  • Exploring Dilations
    This series of activities supports students in making sense of dilations and the criteria for triangle similarity.
    Exploring Dilations

    This series of activities supports students in describing the geometric properties of dilations and the criteria for triangle similarity. Students also determine if a pair of triangles or polygons is similar and describe what congruence and similarity have in common and how they are different.

    All Resources From:
Instructional Routine: Connecting Representations
These tasks are embedded within the instructional routine called Connecting Representations.