Eighth Grade · Math · week 29

Dilations and similarity

Perform dilations from the origin and use them to understand similarity -- same shape, proportional sides, equal angles.

A dilation resizes a figure by a scale factor from a center point (the origin, for this level): multiply every coordinate by the scale factor. Unlike the last three transformations, a dilation changes SIZE, but it keeps the SHAPE -- corresponding angles stay equal and corresponding sides stay proportional.

Two figures related by a dilation are called SIMILAR. A scale factor greater than 1 enlarges, between 0 and 1 shrinks, and exactly 1 leaves the figure unchanged.

📋 For the teacher
Teach it (10–15 min):
  • The scale-factor rule at the origin is simple multiplication: multiply BOTH coordinates by the same number every time -- no addition, unlike a translation.
  • A scale factor greater than 1 enlarges, between 0 and 1 shrinks, and exactly 1 leaves the figure unchanged (still technically a dilation, just a boring one).
  • Similar figures keep every angle the same and every side length in the exact same ratio (the scale factor) -- that ratio is the whole definition of "similar," just as equal side lengths define "congruent."
  • If time allows, let them notice area scales by the SQUARE of the scale factor (double the sides, quadruple the area) -- a nice surprise connecting back to the exponents unit.
Talk about it:
  • How is a dilation different from a translation, reflection, or rotation, in terms of what stays the same?
  • If a scale factor is 1/3, does the figure get bigger or smaller? How do you know?
  • Why does area scale by the SQUARE of the scale factor instead of just the scale factor itself?

Mastery looks like: Given a point or figure and a scale factor, they correctly compute the dilated coordinates and explain what stays the same (angles, side ratios) versus what changes (size).

Common stumbles: Adding the scale factor instead of multiplying; assuming a dilated figure is congruent to the original instead of similar.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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