Ray's New Higher Arithmetic · Arts. 130–135 · Unit 8: Common Fractions

32. Reduction of fractions

Goal: Your student reduces fractions to lowest and to higher terms, changes mixed numbers to improper fractions and back, and reduces compound fractions to simple ones.

⏱ About 45 minutes · 8 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

Reduction defined, then Cases I to V: lowest terms, higher terms, whole or mixed number to improper fraction, improper fraction to whole or mixed number, and compound to simple fractions, each with a rule and examples (answers printed; many small fractions are hard to read in the scan).

The lesson, step by step

  1. Warm-up

    Mental: 12/16 in lowest terms; 3/4 as sixteenths.

  2. The idea

    Read:

    Reduction of Fractions consists in changing their form without altering their value.

    A fraction is in its lowest terms when the numerator and denominator are prime to each other.

  3. Lowest and higher terms

    Read:

    Reject all factors common to both terms of the fraction.

    Dividing both terms by their G. C. D. does it in one step. For higher terms:

    Divide the required denominator by the denominator of the given fraction.

    then multiply both terms by the quotient. The book's example: 5/8 = 25/40.

  4. Mixed and improper

    Read Case III's rule:

    Multiply together the whole number and the denominator of the fraction.

    then add the numerator. 3¾ = 15/4. Case IV goes back:

    Divide the numerator by the denominator; the quotient will be the whole or mixed number.

    13/5 dollars = $2⅗.

  5. Compound fractions

    Case V: multiply the numerators together and the denominators together. Read:

    Whole or mixed numbers must be reduced to improper fractions before applying the rule.

    The book suggests indicating the work and cancelling. Try 2/3 of 3/4 of 8 = 4.

  6. Independent practice

    Your student works a selection of examples from each case. Cases I and III have the most; do half today and half tomorrow.

  7. Check

    Answers are printed. Where the scan is unclear, check by changing the answer back (for example, multiply out an improper fraction).

  8. Wrap-up

    Ask: Why reduce to lowest terms? (Smaller numbers, fewer mistakes, and you can see the size at a glance.)

Afterwards

Finish Cases I to V over two days.

Tip

The scan's small printed fractions are often hard to read. Keep the paper book or a clear copy of the page open when your student works the examples.

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