This activity prompts students to apply their observations about equivalent division expressions to divide a number by a decimal divisor, offering students opportunities to make use of structure (MP7).

While the goal is to see that multiplying both numbers in a division by an appropriate power of 10 produces whole numbers that can facilitate division (using long division or another method), students may choose to reason in other ways, especially when reasoning about the first expression, . Monitor for students who use different strategies to find this quotient. Here are some likely strategies, from those that are less reliant on the base-ten structure and powers of 10 to those that are more reliant on them:

• Using multiplication to find out how many groups of 0.12 in 3: , so and , which means .
• Writing an equivalent expression using a fraction, , computing , which equals , and dividing 300 by 12.
• Finding , which is 0.25, and reasoning that dividing by 0.12, which is 100 times as small as 12, will give a quotient that is 100 times 0.25, which is 25.
• Multiplying both 3 and 0.12 by 100 to get 300 and 12, and then computing .

The purpose of this discussion is to highlight that writing an equivalent expression in which the dividend and divisors are whole numbers can facilitate division. This can be done by multiplying both numbers by the same power of 10.

Ask previously selected students to share their strategies. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see. If time is limited, focus on the last two strategies listed.

Connect the different responses to the learning goals by asking questions such as:

• “What do the strategies have in common?” (Several of them involve multiplying by 100, or thinking about a number that is 100 times as large, and dividing by 12 instead of 0.12.)
• “Why might it be helpful to multiply by 100 and divide by 12 instead of 0.12?” (It gives us whole numbers, making it easier to divide.)
• “Would dividing 300 by 12 instead of 3 by 0.12 give a different quotient? Why or why not?” (No. Because both the dividend and divisor are 100 times as much, dividing them would give the same result.)

This activity deepens students’ understanding of equivalent division expressions.

First, students analyze long-division calculations of two expressions, and , and reason about how they both lead to the same quotient. Students see that all parts of the calculations below the division symbol are related by a factor of 100, but the result is unchanged. In making sense of the digits, their placements, and the values they represent, students practice reasoning abstractly and quantitatively (MP2).

Then, students are prompted to explain whether two division expressions are equivalent and to write another expression that also has the same value. The work here offers opportunities to practice communicating with precision (MP6).

The goal of this discussion is to highlight the connections between the two calculations that students analyzed and the merits of each. Ask questions such as:

• “Why is the value of the same as the value of ?” (The dividends and divisors are related by the same factor, so the quotient is the same. Both numbers in the second expression are 100 times the numbers in the first expression.)
• “What are some advantages of calculating with the decimal intact, as in Calculation A?” (It is fast, and we don’t need to deal with a bunch of zeros. If the numbers are about a situation, we could better make sense of what they mean in their original form.)
• “What are some advantages of calculating with long division? (We are familiar with how to divide whole numbers. We could express the division as a fraction and write an equivalent fraction of , which then tells us that its value is 5.43.)

End the discussion by telling students that they will next look at quotients in which 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 using any method they prefer. Then they extend their practice to calculate the division of decimals in a situation, applying the mathematics they know in order to solve a real-world problem (MP4).

Regardless of the strategy that students use to divide, they have opportunities to attend to precision as they think about the placement of the decimal point and the meaning of the digits in the numbers used in their calculations (MP6).

When only one number in a division expression is a decimal (such as in , students might multiply only that number by a power of 10, perform the division, and neglect to adjust the quotient accordingly. Consider asking students to explain their steps (for instance, “how did you arrive at ?”) and check the quotient they found (for instance, “does multiplying 355 by 3 give 106.5?” Remind students as needed that a division expression is equivalent to another only if both the dividend and the divisor are related by the same factor.

Consider displaying the solutions to the first set of problems for all to see and giving students a moment to check their answers.

Then focus the discussion on how students go about solving the word problem. Ask questions such as:

• “How did you know what division expression to write— or —to represent the situation?” (The question is asking how many pieces of string of length 24.3 are in the long string of length 170.1, which means finding out how many groups of 24.3 are in 170.1, so it is .)
• “What would the expression represent in this situation?” (What fraction the length of string for 1 bracelet is out of the length of string that Mai has.)
• “What are some ways to calculate ?” (Multiplying each number by 10 and then dividing 1,701 by 243. Multiplying 24.3 by whole numbers and seeing what factor gives a product of 170.1.)

If any student solved the last problem by reasoning about and estimating that the answer is about 6 or 7, acknowledge that it is a valid and effective way to reason. In this case, the unknown factor is a whole number, so finding it might be relatively quick. If the unknown factor is not a whole number, however, it would likely take more time to test and multiply different factors. Dividing the two numbers would be more straightforward.