Ray's New Elementary Algebra · Arts. 56–59 · Unit 1: Chapter I: Fundamental rules

5. Subtraction

Goal: Your student subtracts algebraic quantities by changing the signs of the subtrahend and adding.

⏱ About 45 minutes · 8 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to the page
  • Pencil and plenty of paper (or a whiteboard)
  • A card to cover printed answers

What's in the book

Subtraction of similar terms, of unlike terms, then the general rule (change the signs of the subtrahend and add), with forty exercises; finally how to show subtraction with a parenthesis. Answers are printed.

The lesson, step by step

  1. Warm-up

    Three quick additions: 5a + (−2a), −3x + (−4x), 6b + (−9b). (3a, −7x, −3b.)

  2. The easy cases

    Read Art. 56 (Thomas gives away 2a of his 5a cents) and Art. 57.

    Subtraction, in Algebra, is the process of finding the difference between two algebraic quantities.

    Say5a take 2a leaves 3a. But a take b can only be written a − b.

  3. Why the signs change

    Read Art. 58: subtracting 5 + 3 from 9, then 5 − 3 from 9. Work both in plain numbers.

    SayTaking away 5 − 3 takes away 5, but then we took away 3 too many, so we must add 3 back. So 9 − (5 − 3) = 9 − 5 + 3 = 7.

  4. The rule

    1. Write the quantity to be subtracted under that from which it is to be taken, placing similar terms under each other.

    2. Conceive the signs of all the terms of the subtrahend to be changed, and then reduce by the rule for Addition,

    The book suggests beginners actually rewrite the subtrahend with its signs changed. Do that for now.

  5. Guided practice

    Cover the answers. Your student does these while you watch:

    16. From x+y, take x — y. Ans. 2y.

    22. From 8a, take —3a. . Ans. 11a.

    Then exercise 34: from 6a + 5 − 3b take −2a − 9b − 8. Changed signs: +2a + 9b + 8. Sum: 8a + 6b + 13.

  6. Independent practice

    Exercises 6 to 9, 2 to 5 of Art. 57, and 12 to 40 of Art. 58, checking each against the printed answer. Then the four short examples in Art. 59.

  7. Check

    For one long exercise, add the answer to the subtrahend. You should get the minuend back. That is the arithmetic check, done in algebra.

  8. Wrap-up

    Ask: From −9 take −16. Why is the answer positive? (Taking away a debt of 16 leaves you better off.)

Tip

If this many exercises is too much for one sitting, stop at 25 and do the rest next session as a warm-up.

Answers to the book's problems

The book prints the answer after each exercise.

Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.

@teddyrookbook GitHub