Ray's New Higher Arithmetic · Art. 87 · Unit 6: Division

21. General principles of multiplication and division

Goal: Your student predicts how a product or quotient changes when one of its terms is multiplied or divided.

⏱ About 30 minutes · 6 steps

Open it at the Internet Archive ↗

You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil

What's in the book

Six general principles of multiplication and division, each with a small example, then exercises in combining signs (answers not clearly printed; the scan is hard to read).

The lesson, step by step

  1. Warm-up

    Mental: 6 × 4; 12 × 4; 6 × 8. What do you notice?

  2. Products

    Read:

    Multiplying either factor of a product, multiplies the product by the same number.

    Dividing either factor of a product, divides the product by the same number.

    Multiplying one factor of a product, and dividing the other factor by the same number, does not alter the product.

    Try: 25 × 16 = 50 × 8 = 100 × 4 = 400.

  3. Quotients

    Read:

    Multiplying the dividend, or dividing the divisor, by any number, multiplies the quotient by that number.

    The book's example: 24 ÷ 6 = 4, but 48 ÷ 6 = 8 and 24 ÷ 3 = 8.

  4. The other two

    Read Principles V and VI on the next page:

    Dividing the dividend, or multiplying the divisor, by any number, divides the quotient by that number.

    Multiplying or dividing both dividend and divisor by the same number, does not change the quotient.

  5. Practice

    Your student uses the principles to do these in the head: 35 × 18 (= 70 × 9); 4500 ÷ 25 (= 18000 ÷ 100); 240 ÷ 16 (= 60 ÷ 4).

  6. Wrap-up

    Ask: Which principle lets us cancel in fractions later? (VI: dividing both terms by the same number doesn't change the quotient.)

Answers to the book's problems

Practice: 35 × 18 = 630; 4500 ÷ 25 = 180; 240 ÷ 16 = 15.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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