Ray's New Intellectual Arithmetic · Lesson XIX · Unit 2: Multiplying and dividing

19. From one to many, and many to one

Goal: Your child solves rate and work problems by first finding the value for 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

Thirty-five mixed division problems on prices, rates and work, with short model solutions. Many are “find one, then scale up”.

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

    SayIf 9 men can finish a job in 2 days, 1 man would take 9 times 2, or 18 days. So to finish in 3 days you need as many men as 3 is contained in 18, which are 6 men.

    This “go through one” idea is the key to the rest of the book.

  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:

    Nine men can finish a job in 2 days: how many men are needed to finish it in 3 days?

    One man goes 10 miles while another goes 7: when the first has gone 90 miles, how far has the second gone?

    How many men can do in 5 days a piece of work which takes 3 men 10 days?

  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.

Tip

If your child finds this hard, act out a work problem with toy figures: each figure does one “day of work”.

Answers to the book's problems

1) 6. 2) 8. 3) 6. 4) 4 oranges. 5) 12 revolutions. 6) 12 trees. 8) 7 barrels. 9) 20 hours. 10) 4. 11) 6. 12) 36. 14) 6 men. 15) 5. 16) 34 dimes. 17) 3. 18) 3 cents. 19) 9. 20) 18. 21) 10. 22) 30 apples. 23) 63 mi. 24) 40 cents. 25) 45 lessons. 26) $56. 27) 4. 28) 6 barrels. 29) 49. 30) 9 yd. 31) $21. 32) 11 lb. 33) 10 hr. 34) 6 men. 35) 12 men.

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 ↗.

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