Ray's New Elementary Algebra · Arts. 187–190 · Unit 6: Chapter VI: Powers, roots and radicals

39. Square roots of numbers

Goal: Your student extracts the square root of a whole number by the long method and can explain why it works.

⏱ 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

Evolution and the square root defined, how many figures a root has, why (t + u)² = t² + (2t + u)u explains the method, the square roots of 529 and 55225 worked, the rule, and six exercises. Answers are printed.

The lesson, step by step

  1. Warm-up

    Ask the squares of 1 to 12 quickly, then of 20, 30 and 40.

  2. Definitions

    Read Art. 187.

    The Second, or Square Root of a number, is that number which, being multiplied by itself, will produce the given number.

    Read Art. 188: one or two figures give a one-figure root; three or four give a two-figure root; and so on.

  3. Why the method works

    Read Art. 189. Show with 23: 23² = 20² + (2 × 20 + 3) × 3 = 400 + 129 = 529.

  4. Work 529 together

    Follow the book: 5 29. The greatest square in 5 is 4, root 2. Remainder 1, bring down 29: 129. Double the root: 4. 12 ÷ 4 gives 3. Write 43, times 3 = 129. Nothing left: the root is 23.

  5. The rule

    Read the five steps of the rule.

    1. Separate the given numbers into periods of two places each, beginning at the unit's place.

    Then do 55225 together (root 235).

  6. Guided practice

    1. 4225. . . . Ans. 65.

    2. 289444. . Ans. 538.

    Your student works both while you watch.

  7. Independent practice

    Exercises 3 to 6. Check each by squaring the root (the site's calculator can do the squaring, after the root is found by hand).

  8. Wrap-up

    Read Art. 190 and ask: Between 4² and 5², how many whole numbers are not perfect squares? (8.)

Tip

This method is slow at first. Two or three sessions of practice is normal before it feels easy.

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