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.
    Resource:
    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.