5th Grade Dividing Unit Fractions and Whole Numbers Lesson (Free PDF)

5th Grade math lesson: Dividing Unit Fractions and Whole Numbers – Maths PDF

Dividing with unit fractions follows two patterns. A unit fraction divided by a whole number gets smaller: 1/3 ÷ 4 = 1/12, because each piece is cut again. A whole number divided by a unit fraction gets bigger: 4 ÷ 1/3 = 12, because each whole holds 3 pieces. Check every answer with multiplication.

Grade 5 · Common Core State Standards: 5.NF.B.7a, 5.NF.B.7b, 5.NF.B.7c

Key facts
  • A unit fraction has a numerator of 1, like 1/4 or 1/8.
  • A unit fraction divided by a whole number n is 1 over (denominator times n).
  • A whole number divided by a unit fraction 1/b equals the whole number times b.
  • Check any quotient by multiplying it by the divisor to get the dividend.
  • Bar models and number lines show how many pieces there are.

What does it mean to divide a unit fraction by a whole number?

A unit fraction has a 1 on top, like \(\frac{1}{2}\), \(\frac{1}{3}\) or \(\frac{1}{5}\). When you divide it by a whole number, you cut it into equal smaller pieces. Picture \(\frac{1}{3}\) of a pizza shared by 4 friends. Each friend gets \(\frac{1}{12}\) of the whole pizza.

One whole cut into 3 equal parts1/3Cut that part into 4 equal pieceseach small piece = 1/12 of the whole
Dividing \(\frac{1}{3}\) into 4 equal pieces makes twelfths.
Vocabulary

Unit fraction: a fraction with a numerator of 1. Quotient: the answer to a division problem. Dividend: the number being divided.

What does it mean to divide a whole number by a unit fraction?

Now the question is, “How many pieces fit?” The problem \(3\div\frac{1}{4}\) asks how many fourths are in 3 wholes. Each whole holds 4 fourths, so 3 wholes hold \(3\times4=12\) fourths.

1113 wholes x 4 pieces = 12 pieces of size 1/4
Three wholes contain 12 pieces that are each \(\frac{1}{4}\) of a whole.

How can a number line show the division?

Start at 0 and make jumps that are the size of the unit fraction. Count the jumps until you reach the whole number. For \(2\div\frac{1}{3}\) you make 6 jumps of \(\frac{1}{3}\) to land on 2.

0126 jumps of 1/3 make 2
Six jumps of \(\frac{1}{3}\) reach 2, so \(2 \div \frac{1}{3} = 6\).

How do you check a division answer?

Division and multiplication are partners. Multiply the quotient by the divisor and you should get the dividend back.

1/5 ÷ 21/101/10 x 21/10 x 2 = 2/10 = 1/5, so the answer checks out
Multiplying the answer by the divisor returns the starting number.
Two patterns to remember
  • \(\frac{1}{b}\div n=\frac{1}{b\times n}\): the pieces get smaller, so the quotient is less than the unit fraction.
  • \(n\div\frac{1}{b}=n\times b\): there are more pieces than wholes, so the quotient is greater than \(n\).

What are the steps to solve a problem?

  1. Read the problem. Decide: is a unit fraction being cut up, or are you counting pieces in a whole number?
  2. Draw a bar model or number line to see the pieces.
  3. Multiply the correct numbers to find the number of pieces or the new size of a piece.
  4. Check by multiplying the quotient by the divisor.

Example: Maya has \(\frac{1}{4}\) pound of raisins. She puts them equally into 2 bags. Each bag gets \(\frac{1}{4}\div2=\frac{1}{8}\) pound. Check: \(\frac{1}{8}\times2=\frac{2}{8}=\frac{1}{4}\).

Questions about dividing unit fractions and whole numbers

Why is \(\frac{1}{2} \div 3\) equal to \(\frac{1}{6}\) and not \(\frac{3}{2}\)?

Dividing \(\frac{1}{2}\) by 3 cuts one half into 3 equal pieces, so each piece is smaller than \(\frac{1}{2}\). Six of those pieces make one whole, so each piece is \(\frac{1}{6}\). The answer \(\frac{3}{2}\) comes from multiplying by 3 by mistake.

Why does dividing 5 by \(\frac{1}{4}\) give 20?

Each whole contains 4 fourths. Five wholes contain \(5 \times 4 = 20\) fourths. The question asks how many pieces of size \(\frac{1}{4}\) fit in 5, so the answer is larger than 5.

What is a unit fraction?

A unit fraction is a fraction with 1 as the numerator, such as \(\frac{1}{3}\), \(\frac{1}{5}\), or \(\frac{1}{10}\). It shows one equal part of a whole.

How can I check my division answer?

Multiply the quotient by the divisor. If you get the dividend, the answer is right. For example, \(\frac{1}{8} \times 2 = \frac{1}{4}\) confirms that \(\frac{1}{4} \div 2 = \frac{1}{8}\).

When should I use a number line or a bar model?

Use a bar model when you cut a fraction into smaller pieces. Use a number line when you count how many equal jumps reach a whole number. Both pictures lead to the same answer.

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