The Art of Dilation (Cultural Connections)

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In this project, similarity and the geometric transformation dilation (also known as scaling, and colloquially referred to as “zooming in” or “zooming out”) are studied through modeling, exploration, and reasoning. Students will explore important ideas in geometry, and apply skills with functions and algebra. The primary context for this exploration is mathematical modeling and how geometry (specifically similarity and dilation) can be used to create scaled versions of pieces of art.

Topics covered include:

Similarity and Congruence

  • Further developing intuitive ideas about the meaning of “same shape” and learning the formal definition of similar (in contrast to last year’s study of congruent)
  • Discovering the special properties of triangles in connection with similarity, as well as other features of triangles as special polygons

Proportional Reasoning and the Algebra of Proportions

  • Understanding the meaning of proportionality in connection with similarity
  • Developing equations of proportionality from situations involving similar figures
  • Understanding the role of proportionality in nongeometric situations
  • Developing techniques for solving equations involving fractional expressions

Polygons and Angles

  • Developing angle sum formulas for triangles and other polygons
  • Discovering the properties of angles formed by a transversal across parallel lines
  • Discovering the triangle inequality and investigating its extension to polygons

Logical Reasoning and Proof

  • Working with the concept of counterexample in understanding the criteria 
for similarity

Experiments and Data Analysis

  • Planning and carrying out controlled experiments
  • Collecting and analyzing data
  • Identifying key features in graphs of data

Mathematical Modeling

  • Using a geometric diagram to represent a real-world situation
  • Using scale drawings to solve problems
  • Applying properties of similar triangles to real-world situations
  • Exploring how models provide insight in a variety of situations