Ray's New Elementary Algebra · Arts. 231–243 · Unit 8: Chapter VIII: Progressions and proportion

53. Ratio and proportion

Goal: Your student explains ratio as a quotient, writes and tests proportions, and names the terms of a proportion.

⏱ About 35 minutes · 6 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to the page
  • Pencil and plenty of paper (or a whiteboard)
  • A card to cover printed answers

What's in the book

Two ways of comparing quantities, ratio and its terms, how ratios are written and changed, ratios of equality and inequality, compound ratio, proportion and how to test it, extremes and means, and mean and third proportionals.

The lesson, step by step

  1. Warm-up

    Ask: How do 2 and 6 compare? Let your student say "6 is 4 more" and "6 is 3 times as much."

  2. Ratio

    Read Arts. 231 and 232.

    Ratio is the quotient which arises from dividing one quantity by another of the same kind.

    Note Ray's choice: "the ratio of 2 to 6 is 3." Many books today would say 1/3. Read Remark 1 so your student knows both.

  3. Changing the terms

    Read Art. 236 and test each statement with 4 to 12 (ratio 3): multiply the consequent by 5, divide the antecedent by 2, multiply both by 2.

  4. Proportion

    Read Art. 240.

    Proportion is an equality of ratios.

    Saya : b :: c : d is read "a is to b as c is to d." Test 3 : 8 :: 2 : 5 the book's way. (False.)

  5. Terms

    Read Arts. 241 to 243. Your student labels the extremes and means of 3 : 6 :: 5 : 10, and finds a mean proportional between 4 and 9 (6).

  6. Wrap-up

    Ask: Why can 2 yards and 6 dollars have no ratio? (Art. 232, Remark 2.)

Tip

Ray's ratio is "second divided by first." When your student meets the other convention later, the idea is the same; only the order flips.

@teddyrookbook GitHub