Learn on PengiIllustrative Mathematics, Grade 5Chapter 3: Multiplying and Dividing Fractions

Lesson 3: Multiply Unit and Non-unit Fractions

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students learn to multiply a unit fraction by a non-unit fraction by writing multiplication expressions for shaded rectangular regions in area diagrams. Students explore how the product of the numerators represents the number of shaded pieces and the product of the denominators represents the total pieces in the whole, building on their prior understanding of unit fraction multiplication such as one-fifth times one-third equals one-fifteenth. The lesson uses visual models and estimation to help students make sense of expressions like six-fifths times one-third as six copies of one-fifteenth.

Section 1

Understanding Fraction Multiplication with Area Models

Property

To multiply two fractions, we can find a "fraction of a fraction." An area model visualizes this by representing the first fraction with vertical divisions and the second fraction with horizontal divisions. The product is the area where the shaded parts of both fractions overlap.

Examples

Section 2

Multiply Unit and Non-Unit Fractions

Property

To multiply a unit fraction by a non-unit fraction, multiply the numerators together and the denominators together. For fractions 1b\frac{1}{b} and cd\frac{c}{d}:

1b×cd=1×cb×d=cbd\frac{1}{b} \times \frac{c}{d} = \frac{1 \times c}{b \times d} = \frac{c}{bd}

Examples

  • 13×25=1×23×5=215\frac{1}{3} \times \frac{2}{5} = \frac{1 \times 2}{3 \times 5} = \frac{2}{15}
  • 34×12=3×14×2=38\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}

Explanation

This skill extends the area model to multiply a unit fraction by a non-unit fraction. For example, to find 13\frac{1}{3} of 25\frac{2}{5}, you first model 25\frac{2}{5} and then take 13\frac{1}{3} of that shaded area. The resulting product is found by multiplying the numerators to find the number of shaded parts and multiplying the denominators to find the total number of parts in the whole. This bridges the gap between multiplying two unit fractions and multiplying two non-unit fractions.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Understanding Fraction Multiplication with Area Models

Property

To multiply two fractions, we can find a "fraction of a fraction." An area model visualizes this by representing the first fraction with vertical divisions and the second fraction with horizontal divisions. The product is the area where the shaded parts of both fractions overlap.

Examples

Section 2

Multiply Unit and Non-Unit Fractions

Property

To multiply a unit fraction by a non-unit fraction, multiply the numerators together and the denominators together. For fractions 1b\frac{1}{b} and cd\frac{c}{d}:

1b×cd=1×cb×d=cbd\frac{1}{b} \times \frac{c}{d} = \frac{1 \times c}{b \times d} = \frac{c}{bd}

Examples

  • 13×25=1×23×5=215\frac{1}{3} \times \frac{2}{5} = \frac{1 \times 2}{3 \times 5} = \frac{2}{15}
  • 34×12=3×14×2=38\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}

Explanation

This skill extends the area model to multiply a unit fraction by a non-unit fraction. For example, to find 13\frac{1}{3} of 25\frac{2}{5}, you first model 25\frac{2}{5} and then take 13\frac{1}{3} of that shaded area. The resulting product is found by multiplying the numerators to find the number of shaded parts and multiplying the denominators to find the total number of parts in the whole. This bridges the gap between multiplying two unit fractions and multiplying two non-unit fractions.