Ray's New Elementary Algebra · Arts. 130–132 · Unit 3: Chapter III: Algebraic fractions

19. Mixed quantities and the signs of fractions

Goal: Your student changes improper fractions to mixed quantities and back, and handles the minus sign before a fraction.

⏱ About 40 minutes · 7 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

Case II (fraction to entire or mixed quantity, exercises 1 to 7), Case III (mixed quantity to fraction), then the signs of fractions, with exercises 1 to 12 on p. 73. Answers are printed.

The lesson, step by step

  1. Warm-up

    Change 17/5 to a mixed number and 3 2/7 to an improper fraction.

  2. Case II

    Read the rule:

    Divide the numerator by the denominator for the entire part; and, if there be a remainder, place it over the denominator for the fractional part.

    Do exercise 1 together: (3ax + b²)/a = 3x + b²/a.

  3. Case III

    Read Art. 131 ("In 2 1/3, there are how many thirds?") and the rule:

    Multiply the entire part by the denominator of the fraction. Add the numerator of the fractional part to this product, or subtract it, as the sign may direct, and place the result over the denominator.

  4. Signs of fractions

    Read Art. 132.

    The signs prefixed to the terms of a fraction affect only those terms; but the sign placed before the dividing line of a fraction, affects its whole value.

    SayWhen a minus sign stands before a fraction, it changes the sign of EVERY term in the numerator when you write it out.

  5. Work an example together

    Example 2 on p. 73: 4a − (a − b)/3c. Write 4a as 12ac/3c. Then 12ac − (a − b) = 12ac − a + b, over 3c. The + b is the point of the lesson.

  6. Independent practice

    Case II exercises 2 to 7 and the exercises on p. 73. Check against the printed answers.

  7. Wrap-up

    Ask: Is −a/b the same as a/(−b)? As (−a)/(−b)? Test with a = 6, b = 2.

Tip

Put parentheses around a numerator before subtracting it. That one habit prevents most sign errors in this chapter.

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 ↗.

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