Lesson 20

The purpose of this number talk is to help students use the structure of base-ten numbers and the distributive property to solve a division problem involving decimals.

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

Highlight the use of the distributive property in finding \(81.2 \div 4\). Students should recognize that since \(81.2=80+1.2\), we have \(81.2\div 4=(80\div 4)+(1.2\div 4)\). To make this clear, consider explaining that the division could be equivalently represented by \(81.2\boldcdot \frac14=(80+1.2)\boldcdot\frac14=(80\boldcdot \frac14)+(1.2\boldcdot\frac14)\).

In this activity, students study some carefully chosen quotients where the dividends are decimal numbers. The key goal here is to notice that there are other quotients of whole numbers that are equivalent to these quotients of decimals. In other words, when the dividend is a terminating decimal number, we can find an equivalent quotient whose dividend is a whole number. In combination with the previous task, this gives students the tools they need to divide a decimal number by a decimal number.

This activity strongly supports MP7. Students notice that, when working with a fraction, multiplying the numerator and denominator in a fraction by 10 does not change the value of the fraction. They use this insight to develop a way to divide decimal numbers in subsequent activities. This work develops students’ understanding of equivalent expressions by emphasizing that, for example, \(8\div 1= (8\boldcdot 10) \div (1\boldcdot 10)\). Eventually, students will recognize the equivalence of \(8\div 1\) to statements such as \((8\boldcdot y)\div (1\boldcdot y)\). However, in this activity, students only examine situations where the dividend and divisor are multiplied by powers of 10.

Display the following image of division calculations for all to see.

Ask students what quotient each calculation shows. (\(8 \div 1\), \(800 \div 100\), and \(80,\!000 \div 10,\!000\)).  Give students 1–2 minutes to notice and wonder about the dividends, divisors, and quotients in the three calculations. Ask them to give a signal when they have at least one observation and one question. If needed, remind students that the 8, 800, and 80,000 are the dividends and the 1, 100, and 10,000 are the divisors.

Invite a few students to share their observations and questions. They are likely to notice:

• Each calculation shows that the value of the corresponding quotient is 8; it is the same for all three calculations.
• All calculations have an 8 in the dividend and a 1 in the divisor.
• All calculations take one step to solve. 
• Each divisor is 100 times the one to the left of it.
• Each dividend is 100 times the one to the left of it.
• Each dividend and each divisor have 2 more zeros than in the calculation immediately to their left.

They may wonder:

• Why are the quotients equal even though the divisors and dividends are different?
• Would \(80 \div 10\) and \(8,\!000 \div 10\) also produce a quotient of 8?
• Are there other division expressions with an 8 in the dividend and a 1 in the divisor and no other digits but zeros that would also produce a quotient of 8?

Without answering their questions, tell students that they’ll analyze the sizes of dividends and divisors more closely to help them reason about quotients of numbers in base ten.

Give students 7–8 minutes of quiet work time to answer the four questions followed by a whole-class discussion.

Ask students to write a reflection using the following prompt:

What happens to the value of the quotient when both the divisor and the dividend are multiplied by the same power of 10? Use examples to show your thinking.

The goal of this discussion is to make sure students understand that the value of a quotient does not change when both the divisor and the dividend are multiplied by the same power of ten. Ask students to explain why \(\frac{25}{20} = \frac{250}{200}\). Possible responses include:

• Both the numerator and denominator of \(\frac{250}{200}\) have a factor of 10, so the fraction can be written as \(\frac{25}{20}\).
• Both fractions are equivalent to \(\frac{5}{4}\).
• Dividing 250 by 200 and 25 by 20 both give a value of 1.25.

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

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.

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.)