Learn on PengiBig Ideas Math, Course 1Chapter 2: Fractions and Decimals

Lesson 1: Multiplying Fractions

In this Grade 6 lesson from Big Ideas Math, Course 1, students learn how to multiply fractions by multiplying numerators together and denominators together, expressed algebraically as (a/b) × (c/d) = (ac)/(bd). The lesson uses visual models such as number line segments and folded paper grids to build conceptual understanding before introducing the standard procedure, including how to divide out common factors to simplify products. Students practice applying the rule to both numerical exercises and real-life problems involving fractional parts of quantities.

Section 1

Algorithm for Multiplying Unit Fractions

Property

To multiply two unit fractions, multiply their denominators. The numerator of the product is always 1.

Examples

  • 12×13=12×3=16\frac{1}{2} \times \frac{1}{3} = \frac{1}{2 \times 3} = \frac{1}{6}
  • 14×15=14×5=120\frac{1}{4} \times \frac{1}{5} = \frac{1}{4 \times 5} = \frac{1}{20}

Explanation

A unit fraction is a fraction with a numerator of 1. When you multiply two unit fractions, you are finding a part of a part. The product is found by multiplying the numerators (which is always 1×1=11 \times 1 = 1) and multiplying the denominators. This is a special case of the general rule for multiplying any two fractions.

Section 2

Multiply Fractions

Property

If aa, bb, cc, and dd are numbers where b≠0b \neq 0 and d≠0d \neq 0, then

ab⋅cd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Examples

  • To multiply 23⋅45\frac{2}{3} \cdot \frac{4}{5}, multiply the numerators (2⋅4=82 \cdot 4 = 8) and the denominators (3⋅5=153 \cdot 5 = 15). The result is 815\frac{8}{15}.
  • Multiply −47⋅1416-\frac{4}{7} \cdot \frac{14}{16}. The product will be negative. We can simplify before multiplying: −47⋅2⋅74⋅4=−11⋅24=−24-\frac{4}{7} \cdot \frac{2 \cdot 7}{4 \cdot 4} = -\frac{1}{1} \cdot \frac{2}{4} = -\frac{2}{4}, which simplifies to −12-\frac{1}{2}.
  • To multiply 8⋅348 \cdot \frac{3}{4}, first write 8 as 81\frac{8}{1}. Then multiply: 81⋅34=244\frac{8}{1} \cdot \frac{3}{4} = \frac{24}{4}. Simplifying this fraction gives 6.

Explanation

Multiplying fractions is straightforward: multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Remember to simplify the resulting fraction by canceling any common factors for the final answer.

Section 3

Multiplying a Fraction and a Mixed Number

Property

To multiply a fraction and a mixed number, first convert the mixed number into an improper fraction. Then, multiply the numerators and multiply the denominators.

abc×de=ac+bc×dea \frac{b}{c} \times \frac{d}{e} = \frac{ac+b}{c} \times \frac{d}{e}

Examples

  • 214×13=94×13=912=342 \frac{1}{4} \times \frac{1}{3} = \frac{9}{4} \times \frac{1}{3} = \frac{9}{12} = \frac{3}{4}
  • 25×312=25×72=1410=75=125\frac{2}{5} \times 3 \frac{1}{2} = \frac{2}{5} \times \frac{7}{2} = \frac{14}{10} = \frac{7}{5} = 1 \frac{2}{5}

Explanation

This process combines two key ideas: converting mixed numbers and multiplying fractions. The first step is always to change the mixed number into an improper fraction. Once you have two fractions, you can apply the standard multiplication rule: multiply the numerators together and the denominators together. Remember to simplify your final answer if possible, and convert it back to a mixed number if needed.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Algorithm for Multiplying Unit Fractions

Property

To multiply two unit fractions, multiply their denominators. The numerator of the product is always 1.

Examples

  • 12×13=12×3=16\frac{1}{2} \times \frac{1}{3} = \frac{1}{2 \times 3} = \frac{1}{6}
  • 14×15=14×5=120\frac{1}{4} \times \frac{1}{5} = \frac{1}{4 \times 5} = \frac{1}{20}

Explanation

A unit fraction is a fraction with a numerator of 1. When you multiply two unit fractions, you are finding a part of a part. The product is found by multiplying the numerators (which is always 1×1=11 \times 1 = 1) and multiplying the denominators. This is a special case of the general rule for multiplying any two fractions.

Section 2

Multiply Fractions

Property

If aa, bb, cc, and dd are numbers where b≠0b \neq 0 and d≠0d \neq 0, then

ab⋅cd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Examples

  • To multiply 23⋅45\frac{2}{3} \cdot \frac{4}{5}, multiply the numerators (2⋅4=82 \cdot 4 = 8) and the denominators (3⋅5=153 \cdot 5 = 15). The result is 815\frac{8}{15}.
  • Multiply −47⋅1416-\frac{4}{7} \cdot \frac{14}{16}. The product will be negative. We can simplify before multiplying: −47⋅2⋅74⋅4=−11⋅24=−24-\frac{4}{7} \cdot \frac{2 \cdot 7}{4 \cdot 4} = -\frac{1}{1} \cdot \frac{2}{4} = -\frac{2}{4}, which simplifies to −12-\frac{1}{2}.
  • To multiply 8⋅348 \cdot \frac{3}{4}, first write 8 as 81\frac{8}{1}. Then multiply: 81⋅34=244\frac{8}{1} \cdot \frac{3}{4} = \frac{24}{4}. Simplifying this fraction gives 6.

Explanation

Multiplying fractions is straightforward: multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Remember to simplify the resulting fraction by canceling any common factors for the final answer.

Section 3

Multiplying a Fraction and a Mixed Number

Property

To multiply a fraction and a mixed number, first convert the mixed number into an improper fraction. Then, multiply the numerators and multiply the denominators.

abc×de=ac+bc×dea \frac{b}{c} \times \frac{d}{e} = \frac{ac+b}{c} \times \frac{d}{e}

Examples

  • 214×13=94×13=912=342 \frac{1}{4} \times \frac{1}{3} = \frac{9}{4} \times \frac{1}{3} = \frac{9}{12} = \frac{3}{4}
  • 25×312=25×72=1410=75=125\frac{2}{5} \times 3 \frac{1}{2} = \frac{2}{5} \times \frac{7}{2} = \frac{14}{10} = \frac{7}{5} = 1 \frac{2}{5}

Explanation

This process combines two key ideas: converting mixed numbers and multiplying fractions. The first step is always to change the mixed number into an improper fraction. Once you have two fractions, you can apply the standard multiplication rule: multiply the numerators together and the denominators together. Remember to simplify your final answer if possible, and convert it back to a mixed number if needed.