Lesson 13

In this warm-up, students continue the decimal division work from the previous lesson and do so in the context of money. The work reinforces the idea that the value of a quotient does not change if the numerator and denominator are both multiplied by the same power of 10. 

Give students a moment to read the first question and to estimate whether the quotient will be less than 1 or more than 1. Ask them to give a signal when they have an estimate and can explain it. Ask one student from the “more than 1” group to explain their reasoning and another from the “less than 1” group to do the same. Clarify that the quotient will be less than 1 and give students a few minutes to complete the questions. If time is limited, ask students to work only on the second question. Follow with a whole-class discussion.

Focus the whole-class discussion on the second question. Select several students to explain why choices b and d are correct and why a and c are not. Students should see that \(5.04 \div 70\) and \(504,\!000 \div 700\) are not equivalent expressions to \(5.04 \div 7\) because the dividend and divisors in each pair are not results of multiplying the 5.04 and 7 by the same factor or the same power of 10. 

The goal of this task is to show that we can calculate quotients of two decimals by “moving the decimal point” (multiplying both numbers by an appropriate power of 10) and, as a result, work only with whole numbers. Students can calculate the quotient of whole numbers using long division or another method of their choice. Students also have an opportunity to evaluate and critique another’s reasoning (MP3).

Students use the structure of base-ten numbers (MP7) to move the decimal point (through multiplication by an appropriate power of 10), and they use their understanding of equivalent expressions to know that multiplying both the numbers in a division by the same factor does not change the value of the quotient. Both pieces of knowledge allow students to replace a quotient of decimal numbers with a quotient of whole numbers.

Arrange students in groups of 2. Give students 3 minutes of quiet time to consider how to find the first quotient. Encourage them to think of more than one way to do so, if possible. Then, give partners 2–3 minutes to discuss their methods and another 2–3 minutes to find the second quotient together. Follow with a brief whole-class discussion, reviewing the first two questions. If not brought up by a student, discuss the equivalent expressions \(300 \div 12\) and \(1,\!800 \div 4\). Consider bringing up the first expression and asking students to find an analogous expression for the second problem.

Ask students to finish the last problem and follow with a whole-class discussion.

The goal of this discussion is to help students recognize when division expressions are equivalent.

Ask students to write a division expression that looks like it might be equivalent to either \(3 \div 0.12\) or \(1.8 \div 0.004\) but has different decimal point locations. Select a few students to share their expression with the class.

This lesson demonstrates how the division of two equivalent expressions (e.g., \(48.78 \div 9\) and \(4878 \div 900\)) result in the same quotient. By looking at worked-out calculations, students reinforce their understanding about what each part of the calculations represent. The advantage of representing a quotient with a different equivalent expression is so students can simplify problems involving division of decimals by rewriting expressions using whole numbers.

They use the structure of base-ten numbers (MP7) to move the decimal point (through multiplication by an appropriate power of 10). They also use their understanding of equivalent expressions to know that multiplying both the numbers in a division by the same factor does not change the value of the quotient. Both pieces of knowledge allow students to replace a quotient of decimal numbers with a quotient of whole numbers.

Arrange students in groups of 2. Give partners 5 minutes to discuss the first problem and then quiet work time for the second problem. Follow with a whole-class discussion.

The goal of this discussion is for students to contrast the two division methods used in the task. Discuss:

• Why is the value of \(48.78 \div 9\) the same as the value of \(4,\!878 \div 900\)? (The numbers in the second expression are both multiplied by 100, and this does not change the value of the quotient.)
• What are some advantages of calculating \(48.78 \div 9\) with the decimal intact (the method on the left)? (It is fast, and we don’t need to deal with a bunch of 0’s. Also, if the numbers are from a contextual problem, we could better make meaning of them in their original form.)
• What are some advantages of calculating \(4,\!878 \div 900\) with long division? (These are whole numbers, and we are familiar with how to divide whole numbers. Also, we could express this as a fraction and write an equivalent fraction of \(\frac{543}{100}\), which then tells us that its value is 5.43.)

End the discussion by telling students that they will next look at quotients where both the divisor and the dividend are decimals. The method used here of multiplying both numbers by a power of 10 will apply in that situation as well.

In this activity, students practice calculating quotients of decimals by using any method they prefer. Then, they extend their practice to calculate the division of decimals in a real-world context. While students could use ratio techniques (e.g., a ratio table) to answer the last question, encourage them to use the division of decimal numbers. The application of division to solve real-world problems illustrates MP4.

As students work on the first three problems, monitor for groups in which students have different strategies used on the same question.

Arrange students in groups of 3–5. Give groups 5–7 minutes to work through and discuss the first three questions. Ask them to consult with you if there is a disagreement about a correct answer in their group. (If this happens, let them know which student’s work is correct and have that student explain their thinking so all group members are in agreement.)

After all group members have answered the first three questions and have the same answer, have them complete the last question. Follow with a whole-class discussion.

Some students might have trouble calculating because their numbers are not aligned so the place-value associations are lost. Suggest that they use graph paper for their calculations. They can place one digit in each box for proper decimal point and place-value alignment.

The purpose of this discussion is to highlight the different strategies used to answer the division questions. Select a previously identified group that used different strategies on one of the first three questions. Ask each student in the group to explain their strategy and why they chose it. For the fourth question, ask students:

• “Could the answer be found by calculating the quotient of this expression: \(24.3 \div 170.1\)?” (No, because the question is asking how many pieces of string of length 24.3 are in the long string of length 170.1. This is equivalent to asking how many groups of 24.3 are in 170.1 or \(170.1\div 24.3\).)
• “What would the quotient \(24.3 \div 170.1\) represent in the context of the problem?” (This quotient would represent what fraction the length of bracelet string is of the full length of string.)