Lesson 11

This number talk encourages students to think about the numbers in a computation problem and rely on what they know about structure, patterns, and division to mentally solve a problem. Four expressions are given. The first three expressions are partial quotients that could help students evaluate the last expression of \(496\div8\). Be sure to allot more time to discuss the final expression and to draw out its connection to the other expressions.

Display one expression at a time, or ask students to work on one expression at a time and begin when cued. Give students 30 seconds of quiet think time per question and ask them to give a signal when they have an answer and can explain their strategy. Select 1–2 students to briefly discuss how they reasoned about the quotient. 

Some students may start from scratch when evaluating the final question. Ask them how they could use what they have already done to help them with the final question.

Ask students to share their reasoning for each quotient. Record and display their explanations for all to see; this will help students see the connections between the first three expressions and the last one. To involve more students in the conversation, consider asking:

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

If not mentioned by students, highlight how the quotients of \((400\div8) + (80\div8) + (16\div8)\) can be used to find \(496\div8\).

This activity extends division techniques used to find whole-number quotients to divide whole numbers resulting in (terminating) decimal quotients. In these problems, students see remainders in the ones place. In order to continue the division process, the ones are broken into tenths. Conceptually, this is the same unbundling idea that is used when hundreds are broken into tens or when tens are broken into ones.

If students have trouble drawing the diagrams to represent unbundling, consider providing actual base-ten blocks or paper cut-outs of base-ten units (from the blackline master used earlier in the unit) so that they can physically trade the pieces (e.g., 2 ones for 20 tenths).

Arrange students in groups of 2. Provide access to graph paper. For the first question, give students 1 minute of quiet think time to analyze Mai’s work and 2–3 minutes to discuss their observations with their partner. Pause for a whole-class discussion, making sure that all students understand how Mai dealt with the remainder.

Give students 5–7 minutes to complete the final two questions. Follow with a whole-class discussion.

The goal of this discussion is for students to relate their earlier work on division, which resulted in whole-number quotients, to division involving decimal quotients. Below is an example of how the discussion may go, along with questions to ask students and some possible responses. Begin the discussion by reminding students of the work they have previously done to evaluate \(657 \div 3\).

How is computation for \(62 \div 5\) similar to that for \(657 \div 3\)?

• The method of division is the same: we divide a given number (62 or 657) into equal groups until everything is distributed.
• We divide by using place value, unbundling one unit into ten of a smaller unit as needed. For \(62 \div 5\), the 2 ones can be broken into 20 tenths, while in \(657 \div 3\), the 2 tens were unbundled into 20 ones.

How is computation for \(62 \div 5\) different from that for \(657 \div 3\)?

• There is no remainder for \(657 \div 3\), while there is a remainder of 2 for \(62 \div 5\).
• We need to write a decimal point and work with tenths in \(62 \div 5\).

It is important to stress that the methods and steps are the same in both computations. The big new idea here is that sometimes a division problem of whole numbers does not end when we get to the ones place. In these cases, we have to add a decimal because the number being divided involves tenths, hundredths, or smaller base-ten units.

In this activity, students use long division to divide whole numbers whose quotient is not a whole number. Previously, students found the quotient of \(62 \div 5\) using base-ten diagrams and the partial quotients method. Because the long division is a particular version of the partial quotients method, and because students have been introduced to long division, they have the tools to divide whole numbers that result in a (terminating) decimal. In this activity, students evaluate and critique the reasoning of others (MP3).

Arrange students in groups of 2. Provide access to graph paper. Give students 7–8 minutes to analyze and discuss Lin’s work with a partner and then complete the second set of questions.

Pause to discuss students’ analyses and at least one of the division problems. Students should understand that, up until reaching the decimal point, long division works the same for \(62 \div 5 = 12.4\) as it does for \(657 \div 3 = 219\). In \(62 \div 5\), however, there is a remainder of 2 ones, and we need to convert to the next smaller place value (tenths), change the 2 ones into 20 tenths, and then divide these into 5 equal groups of 4 tenths.

Prepare students to do the last set of questions by setting up the long division of \(5 \div 4\) for all to see. Discuss the placement of the decimal point. Reiterate that we can write extra zeros at the end of the dividend, following the decimal point, so that remainders can be worked with. Have students work through this problem and the others. Follow with a whole-class discussion.

Students may be perplexed by the repeating decimals in the last question and think that they have made a mistake. Ask them to compare their work with a partner’s, and then clarify during discussion that some decimals do repeat. Because this work comes into focus in grade 7, the goal here is simply for students to observe that not all decimals terminate.

Focus the whole-class discussion on the third and fourth sets of questions. Ask a few students to show their long division for all to see and to explain their steps. Some ideas to bring to uncover:

• Problems like \(1 \div 25\) are challenging because the first step is 0: there are zero groups of 25 in 1. This means that we need to introduce a decimal and put a 0 to the right of the decimal. But one 0 is not enough. It is not until we add the second 0 to the right of the decimal that we can find 4 groups of 25 in 100. Because we moved two places to the right of the decimal, these 4 groups are really \(0.04\), which is the quotient of 1 by 25.
• Problems like \(1 \div 3\) are not fully treated until grade 7. At this point, we can observe that the long division process will go on and on because there is always a remainder of 1.

Some questions to ask students that highlight these points include:

• When you found \(1 \div 8\), what was your first step? (I put a 0 above the dividend 1 and added a decimal because I cannot take any 8’s from 1.)
• When you found \(1 \div 25\), what were your first few steps? (I put a 0 above the dividend 1 and added a decimal because I cannot take any 25’s from 1. I needed to add another 0 to the right of the decimal because I still cannot take any 25’s from 10.)
• Why do we not write 0’s in advance to the right of the decimal? (We do not know in advance if we will need these 0’s or how many of them we might need, so we usually write them as needed.)

Another valuable link to make is to connect the values for the decimals in Problem 4 to percentages. For example, the fact that \(\frac{4}{5} = 0.8\) means that \(\frac{4}{5}\) of a quantity is 80% of that quantity. Similarly \(\frac{1}{25}\) of a quantity is the same as 4% of that quantity.