Fractions: division and multi-step stories

Practice 5 items →

Division of fractions is the last arithmetic operation to learn, and the one most often done by rote. Keep the sentence “how many of these fit in that”: 3½ ÷ ¾ asks how many three-quarters are in three and a half, which is 4⅔. The rule — multiply by the reciprocal of the divisor — follows from that picture, and inverting the wrong fraction is the classic mistake. Mixed-number arithmetic should be fluent in all four operations. Multi-step bar-model stories (a fraction of the remainder, then what is left) are where the year's fraction work pays off; draw the bar before touching the arithmetic.

Specification · for a parent or teacher

Part no.: fraction-operations-6
Course: Advanced Math 6 · Unit 2
Also in: High School Math 1
Realm: Foundations
Difficulty: 3 of 4 — Grade 6–7 core. Multi-step, first symbolic work.
Qty: 5 items

Part no. is the node's id; the realm is the kind of math; qty is how many practice items it carries.

Students can

$\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c}$ ; mixed-number arithmetic; multi-step bar-model stories.

Ready when

$3\frac{1}{2} \div \frac{3}{4}$ as “how many $\frac{3}{4}$ in $3\frac{1}{2}$ ” (which is $4\frac{2}{3}$ ) plus a story.

Formulas

Division of fractions

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Invert the divisor (the second fraction), never the first.

Fraction product

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Watch for

invert the first fraction; $\frac{1}{2}\div\frac{1}{8} = \frac{1}{16}$ .

Before · requires

Multiplying fractionsfraction-mul-div · Math 5
Adding and subtracting fractions (stories)fraction-add-sub-5 · Math 5

After · used by

Area of plane figuresarea-triangles-trapezoids · Math 6
Volume and surface areavolume-surface-nets · Math 6
Simple equations in one variablelinear-equations-7 · Math 7
Probabilityprobability-trees · Math 7

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