Ray's New Intellectual Arithmetic · Lesson XLIV · Unit 4: Working with fractions

44. Shortcuts: parts of a dozen and of a dollar

Goal: Your child uses handy fractions (12½ cents is ⅛ of a dollar, 33⅓ cents is ⅓) to work prices quickly.

⏱ 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

Two printed tables: parts of 12 (2 = ⅙, 3 = ¼, 4 = ⅓, 6 = ½, 8 = ⅔, 9 = ¾, 10 = ⅚) and parts of 100 (12½ = ⅛, 16⅔ = ⅙, 20 = ⅕, 25 = ¼, 33⅓ = ⅓, 37½ = ⅜, 50 = ½, 62½ = ⅝, 66⅔ = ⅔, 75 = ¾, 87½ = ⅞). Then thirty-one problems using them.

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

    Say33⅓ cents is one-third of a dollar. So $12 buys as many pounds of coffee as ⅓ is contained in 12, which is 36 pounds.

    And to multiply by 12½: multiply by 100 and divide by 8. 32 × 12½ = 3200 ÷ 8 = 400.

  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:

    What will 24 yards of alpaca cost, at 37½ cents a yard?

    Paid $12 for coffee at 33⅓ cents a pound: how many pounds?

    Multiply 48 by 75; 24 by 37½; 51 by 33⅓.

  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

These shortcuts are still useful at the shop: 25% off is a quarter off, 33% is about a third.

Answers to the book's problems

1) $18. 14) $2. 15) $2. 16) $9. 17) $10. 18) $5. 19) $18. 20) $25. 21) 36 lb. 24) 12 yd. 25) 400. 26) 900; 2500; 5100. 27) 3600; 900; 1700. 28) 2600; 5600; 6000. 29) 12. 30) 12; 28; 10.

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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