Ray's New Intellectual Arithmetic · Lesson LV · Unit 7: Reasoning puzzles, ratio and proportion
55. Fractions of a price, and what is left
Goal: Your child goes from one fraction of a quantity to another, and solves “the rest is …” problems.
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
About thirty-four problems: “If ⅔ of a yard costs $2, what will ¾ cost?”, “⅔ of 1½ is ¾ of what number?”, and orchards and money divided into fractions with a remainder.
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 ⅔ of a yard costs $2, ⅓ costs $1, the whole yard $3, and ¾ of a yard three-fourths of $3, which is $2.25.
For the orchard: ¼ + ⅓ + ⅙ is ¾, so the 32 cherry trees are the other ¼, and the orchard has 128 trees.
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 ⅔ of a yard of cloth costs $2, what will ¾ of a yard cost?
A traveller went 30 miles in 3½ hours: how far at that rate in 7½ hours?
In an orchard ¼ are apple, ⅓ pear, ⅙ plum and the remaining 32 cherry: how many trees in all?
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.