Ray's New Intellectual Arithmetic · Lesson LVII · Unit 7: Reasoning puzzles, ratio and proportion
57. Dividing in fractional parts, and clock puzzles
Goal: Your child divides a quantity so one part is a fraction of another, and solves clock puzzles.
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-six problems such as dividing 15 so the smaller part is ⅔ of the larger, a broken tree, and “the time past noon is equal to half the time till midnight”.
The lesson, step by step
Warm up
Ask three or four problems from the last lesson again, out of order. Your child answers and says why.
Show the reasoning
SayIf the smaller part is ⅔ of the larger, the two together are 5/3 of the larger. 5/3 of the larger is 15, so ⅓ is 3, and the larger is 9 and the smaller 6.
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:
Divide 15 into two parts so that the less is ⅔ of the greater.
A tree 70 feet high broke into 3 pieces, each a fraction of the next: how long is each?
The time past noon is equal to half the time till midnight: what time is it?
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.
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) 9 and 6. 21) 4 o'clock in the afternoon.
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 ↗.