Ray's New Intellectual Arithmetic · Lesson LXII · Unit 7: Reasoning puzzles, ratio and proportion

62. Proportion by reasoning

Goal: Your child solves proportion problems (more men, fewer days; higher price, smaller loaf) by reasoning through one.

⏱ About 20 minutes · 5 steps

Open it at the Internet Archive ↗

You'll need

  • The book page (press “📖 Book page”, or use a printed copy)
  • Paper for jotting, though the work is meant to be done in the head

What's in the book

Twenty-three problems on men and days, a bankrupt paying cents on the dollar, loaf sizes and wages. Models: 5 men for 18 days is 90 days of work for 1 man, so 9 days needs 10 men.

The lesson, step by step

  1. Warm up

    Ask three or four problems from the last lesson again, out of order. Your child answers and says why.

  2. Show the reasoning

    Say5 men take 18 days, so 1 man would take 5 times 18, or 90 days. To do it in 9 days needs as many men as 9 is contained in 90, which is 10 men.

    Ask each time: does the answer get bigger or smaller? More men means fewer days.

  3. Work the problems aloud

    Read each problem on the page aloud. Your child answers, then explains in a full sentence, the way the book's model solutions do. The problems go like this:

    If 5 men can do a piece of work in 18 days, how many men can do it in 9 days?

    If a loaf weighs 8 oz. when flour is $3 a barrel, what should it weigh when flour is $4?

    If 6 men can do a piece of work in 5 days, how long will it take if 3 more men help when it is half done?

  4. Ask “Why?”

    After each answer ask “Why?” or “How do you know?” Ray's wants the reasoning said aloud, not just the number. If your child stalls, go back to the model and say the first half of the explanation together.

  5. Wrap up

    Pick the two problems that were hardest today. Tomorrow, start with those as the warm-up.

Afterwards

Do the rest of the lesson's problems over the next day or two, a few at a time. Move on when your child can explain them without help.

Answers to the book's problems

1) 10 men. 2) 10 men. 3) 8 days. 5) $48. 6) $12. 7) $48. 8) 4 loaves. 9) 10 loaves. 10) 15; 12 loaves. 11) 6 oz; 4⅘ oz. 12) 12 oz. 15) 4⅙ days. 16) 5½ days. 17) $56. 18) $45. 19) 7½ rd. 20) 27 bu. 22) $880. 23) 126 sheep.

Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in Ray's own Key ↗.

@teddyrookbook GitHub