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

Lesson 9: Divide Whole Numbers by Unit Fractions

In this Grade 5 Illustrative Mathematics lesson from Chapter 3: Multiplying and Dividing Fractions, students learn to divide a whole number by a unit fraction using the context of cutting paper strips into fractional-length pieces. They apply standard 5.NF.B.7.b as they use tape diagrams and equations to represent quotients and explore how the size of the unit fraction affects the result. By the end of the lesson, students can solve problems such as determining how many one-fourth foot pieces can be cut from a 3-foot strip of paper.

Section 1

Modeling Quotative Division: Whole Number ÷ Unit Fraction

Property

Dividing a whole number, aa, by a unit fraction, 1b\frac{1}{b}, is a way of asking: "How many groups of size 1b\frac{1}{b} are in aa?"
This can be modeled visually to find the total number of fractional parts.

Examples

Section 2

Divide a Whole Number by a Unit Fraction

Property

To divide a whole number by a unit fraction, you can multiply the whole number by the denominator of the fraction. For a whole number ww and a unit fraction 1d\frac{1}{d}:

w÷1d=w×dw \div \frac{1}{d} = w \times d

Examples

  • 6÷14=6×4=246 \div \frac{1}{4} = 6 \times 4 = 24
  • 10÷12=10×2=2010 \div \frac{1}{2} = 10 \times 2 = 20
  • 3÷18=3×8=243 \div \frac{1}{8} = 3 \times 8 = 24

Explanation

Dividing by a unit fraction is like asking "how many of this fraction fit into the whole number?" For example, 6÷146 \div \frac{1}{4} asks how many 14\frac{1}{4}-sized pieces are in 6 wholes. Since there are 4 fourths in 1 whole, there must be 6×46 \times 4 fourths in 6 wholes. This shows that dividing by a unit fraction is the same as multiplying by its denominator.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Modeling Quotative Division: Whole Number ÷ Unit Fraction

Property

Dividing a whole number, aa, by a unit fraction, 1b\frac{1}{b}, is a way of asking: "How many groups of size 1b\frac{1}{b} are in aa?"
This can be modeled visually to find the total number of fractional parts.

Examples

Section 2

Divide a Whole Number by a Unit Fraction

Property

To divide a whole number by a unit fraction, you can multiply the whole number by the denominator of the fraction. For a whole number ww and a unit fraction 1d\frac{1}{d}:

w÷1d=w×dw \div \frac{1}{d} = w \times d

Examples

  • 6÷14=6×4=246 \div \frac{1}{4} = 6 \times 4 = 24
  • 10÷12=10×2=2010 \div \frac{1}{2} = 10 \times 2 = 20
  • 3÷18=3×8=243 \div \frac{1}{8} = 3 \times 8 = 24

Explanation

Dividing by a unit fraction is like asking "how many of this fraction fit into the whole number?" For example, 6÷146 \div \frac{1}{4} asks how many 14\frac{1}{4}-sized pieces are in 6 wholes. Since there are 4 fourths in 1 whole, there must be 6×46 \times 4 fourths in 6 wholes. This shows that dividing by a unit fraction is the same as multiplying by its denominator.