Ray's New Elementary Algebra · Arts. 128–129 · Unit 3: Chapter III: Algebraic fractions

18. Reducing fractions to lowest terms

Goal: Your student reduces algebraic fractions by canceling common factors, including polynomial factors.

⏱ About 40 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

Case I: the rule for lowest terms, with exercises where the common factor is a monomial and then where it is a polynomial, and division exercises whose quotients are fractions. Answers are printed (the fraction exercises are hard to read in the scan, so use the page image).

The lesson, step by step

  1. Warm-up

    Reduce 12/18 and 15/25. (2/3 and 3/5.)

  2. The rule

    Read Art. 128.

    Divide both terms by their greatest common divisor.

    Or, Resolve the numerator and denominator into their prime factors, and then cancel those factors common to both terms.

    The book says the second way is generally most convenient.

  3. Work the book's example

    Example 1 writes the top and bottom as products, sets the common factor apart, and cancels it. Follow it on the page step by step, saying which factor is canceled.

  4. Polynomial factors

    Read example 9 (3a³ − 3ab² over 5ab + 5b²): factor the top as 3a(a + b)(a − b) and the bottom as 5b(a + b), cancel a + b, leaving 3a(a − b)/5b.

  5. Guided practice

    Your student does exercises 2 and 3, then 11 and 13, factoring top and bottom before canceling anything. Compare with the printed answers.

  6. Division giving a fraction

    Read Art. 129. Do exercise 5 together: x² + 2x − 3 divided by x² + 5x + 6 is (x + 3)(x − 1) over (x + 3)(x + 2), which is (x − 1)/(x + 2).

  7. Independent practice

    The remaining exercises of Arts. 128 and 129. Check two by putting x = 2 (or a = 2, b = 1) into the original and the answer.

  8. Wrap-up

    Ask: Why can't you cancel the x in (x + 3)/x? (Only factors can be canceled, not terms.)

Tip

The big mistake here is canceling terms instead of factors. If you see it, go back to plain numbers: (2 + 3)/2 is not 3.

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