Ray's New Intellectual Arithmetic · Lesson LVIII · Unit 7: Reasoning puzzles, ratio and proportion
58. A number and a part of itself
Goal: Your child finds a number when it is increased or decreased by a fraction of itself.
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
Fifteen problems. The model: a number increased by its half is 3/2 of the number; 3/2 of the number is 15, so ½ is 5 and the number is 10.
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
SayThe number and its half make three-halves of the number. Three-halves are 15, so one-half is 5, and the number is 10.
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 number, increased by its half, will equal 15?
A father is 40 years older than his son, and the son's age is a fraction of the father's: how old is each?
A piece of flannel shrank by a fraction of its length and then measured 28 yards: how long was 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) 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 ↗.