Tenth Grade · Math · week 19

Ratios, proportions, and dilations

Solve proportions using the Cross Products Property, and apply the dilation coordinate rule.

A ratio compares two quantities by division, written a:b or a/b, and a proportion states that two ratios are equal: a/b = c/d. Proportions solve real problems (scale drawings, map distances, recipe scaling) through the Cross Products Property: if a/b = c/d, then a*d = b*c -- multiply across the equals sign and an equation with no fractions falls out. This one algebra move underlies almost every similarity computation for the rest of the year.

A dilation is a transformation that scales a figure larger or smaller from a fixed center point by a scale factor k: every distance from the center multiplies by k, so on the coordinate plane with center at the origin, (x, y) maps to (kx, ky). A dilation is the first transformation we have met that is NOT a rigid motion -- it changes size unless k = 1. What a dilation DOES preserve is shape: angle measures stay exactly the same, and every length scales by the same factor k, so the image is a perfect scaled copy of the preimage. A scale factor greater than 1 enlarges the figure; a scale factor between 0 and 1 reduces it.

📋 For the teacher
Teach it (10–15 min):
  • Connect proportions to something concrete first, like a recipe (if 2 cups of flour need 3 eggs, how many eggs for 5 cups?), before jumping to cross products.
  • Have them say the Cross Products Property out loud as "multiply the outside pair, multiply the inside pair, set them equal" while drawing the two crossing diagonal arrows on the proportion.
  • Use a photocopier enlarge/reduce setting or a phone camera zoom as a real-world dilation: same shape, bigger or smaller, same angles throughout.
  • Stress the one-word distinction for the rest of the unit: rigid motions preserve SIZE; dilations preserve only SHAPE.
Talk about it:
  • Why does the Cross Products Property work -- what are you actually doing to both sides of a/b = c/d?
  • Why is a dilation not considered a rigid motion?
  • What happens to a figure when the scale factor is exactly 1?

Mastery looks like: They solve proportions using the Cross Products Property and apply the dilation coordinate rule, correctly identifying that dilations preserve shape but not size (unless k = 1).

Common stumbles: Cross-multiplying incorrectly (multiplying same-side terms instead of diagonal pairs); assuming a dilation is a rigid motion because it looks like a simple scaling rule.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

Coordinate PlanePlot points and explore the four quadrants.try →MultiplicationEqual groups and arrays make products finally click.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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