Ray's New Elementary Algebra · Arts. 244–255 · Unit 8: Chapter VIII: Progressions and proportion
54. The propositions of proportion; final review
Goal: Your student uses "product of means equals product of extremes" and can show the other propositions with numbers.
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
Propositions I to XII of proportion (means and extremes, its converse, continued proportion, alternation, inversion, composition, division, powers, products and sums), each proved and illustrated with numbers, then the book's General Review.
The lesson, step by step
Warm-up
Ask: Is 3 : 6 :: 5 : 10 true? Multiply 6 × 5 and 3 × 10.
Proposition I
Read Art. 244.
In every proportion, the product of the means is equal to the product of the extremes.
Do the book's exercise: the first three terms are ac, bd and acxy; the fourth is bd × acxy ÷ ac = bdxy.
Proposition II
Read Art. 245 and the table of eight proportions from 5 × 8 = 4 × 10. Your student writes three of them.
Propositions III to VI
Read Arts. 246 to 249, checking each with the book's number illustration: 4 : 6 :: 6 : 9; 2 : 7 :: 6 : 21 becomes 2 : 6 :: 7 : 21; and so on.
Propositions VII to XII
Read Arts. 250 to 255. For each, your student makes up a fresh number example and checks it. For composition: 3 : 4 :: 6 : 8 gives 7 : 4 :: 14 : 8.
Final review
Use the General Review on pp. 239 and 240 as a last oral quiz over the whole book. Go through it over two or three days.
Wrap-up
Ask your student which chapter of the book they found hardest, and plan a short review of it.
Afterwards
Spread the closing General Review over several short sessions, revisiting any chapter that comes up weak.
Tip
Every proposition here can be checked with numbers in a few seconds. Make that the habit, as with every rule in the book.