Dividing a Decimal by a Decimal

10 min

Teacher Prep
Setup
Introduce the activity by giving students 1–2 minutes to notice and wonder about the given calculations. Then 7-8 minutes quiet work time on problems followed by a whole-class discussion.

Narrative

In this Warm-up, students study a series of division expressions that produce the same quotient. The goal is to notice the structure in the expressions—the dividends and divisors are related by the same power of 10—and to make use of them to later reason about division of a decimal by a decimal (MP7).

Students may make connections between equivalent expressions to what they learned about equivalent fractions in grade 5: Multiplying the numerator and denominator of a fraction by the same factor creates an equivalent fraction. (Note that students are not expected to use the term “equivalent expressions” at this point.)

Launch

Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and then time to discuss their thinking and complete the activity with their partner. Follow with a whole-class discussion.

Student Task

Analyze the dividends, divisors, and quotients in the calculations, and then answer the questions.

  1. Complete each sentence. In the calculations shown:

    1. Each dividend is ‾\underline{\hspace{1in}} times the dividend to the left of it.

    2. Each divisor is ‾\underline{\hspace{1in}} times the divisor to the left of it.

    3. Each quotient is ‾\underline{\hspace{2in}} the quotient to the left of it.

  2. Select all expressions that would also have a quotient of 8. Be prepared to explain your reasoning.

    1. 80÷1080 \div 10

    2. 80÷10080 \div 100

    3. 8,000÷1,0008,000 \div 1,000

    4. 800,000÷1,000,000800,000 \div 1,000,000

    5. 0.8÷0.10.8 \div 0.1

    6. 0.08÷0.0010.08 \div 0.001

  3. Write two expressions that have the same value as 250÷10250 \div 10. One of your expressions should include decimals.

Sample Response

    1. 100 times
    2. 100 times
    3. Equal to
  2. A, C, E
  3. Sample responses:
    • 25÷125 \div 1
    • 2,500÷1002,500 \div 100
    • 2.5÷0.12.5 \div 0.1
    • 0.25÷0.010.25 \div 0.01
Activity Synthesis (Teacher Notes)

Invite students to share their responses. Highlight that the value of a quotient does not change when both the divisor and the dividend are multiplied by the same power of 10.

If no students make a connection to equivalent fractions, display the fractions 2510\frac{25}{10} and 250100\frac{250}{100}. Ask students whether these fractions are equivalent and how they know. Students may note that:

  • Both fractions are equivalent to 52\frac{5}{2}.
  • Dividing 250 by 100 and 25 by 10 both give a value of 2.5.
  • Multiplying the numerator and denominator of 2510\frac{25}{10} by 10 gives 250100\frac{250}{100}.

Emphasize the last point—that these fractions are equivalent because their numerators and denominators are related by the same factor: 2510=25⋅1010⋅10=25010\frac{25}{10} = \frac{25 \boldcdot 10}{10 \boldcdot 10} =\frac{250}{10}

Because we can interpret a fraction as division of the numerator by the denominator, we can tell that 25÷1025 \div 10 and 250÷100250 \div 100 are also equivalent or have the same value (even without calculating that value).

Tell students that their observations here will help them divide decimals in upcoming activities.

Standards
Building Toward
  • 6.NS.2·Fluently divide multi-digit numbers using the standard algorithm.
  • 6.NS.B.2·Fluently divide multi-digit numbers using the standard algorithm.

15 min

10 min