In this activity, students analyze another method of division in which one portion of the dividend is divided at a time. They see that there is more than one way to record this process and that different partial quotients can be used for any given division.

The first division problem, , involves the same whole-number dividend and divisor as seen in the Warm-up. The calculations also show the same partial quotients of 200, 10, and 9 being added to give the quotient of 219. Unlike the strategy shown in the Warm-up, however:

• All the calculations are arranged vertically in a single stack rather than written as equations.
• The process involves repeatedly subtracting some amounts from the original dividend until there are no remainders.
• Each partial quotient is recorded above the line with the dividend and divisor. Partial dividends that are being subtracted and remainders are recorded below that line.

In analyzing Andre’s calculations, students need to think about what calculations are performed, what each number means, why it is placed where it is or aligned as it is, and how the numbers are related. The reasoning here gives students opportunities to persevere in making sense of a problem (MP1) and to attend to precision (MP6).

Monitor for students who use different partial quotients to calculate .  Select them to share later.

Ask previously selected students to share their calculations for , starting from the most involved and ending with the most streamlined.

Highlight that there are no rules about how many partial quotients we could use or what partial dividends to use. That said, it is often helpful to take out the amount in each place value and distribute it into groups. Once the entirety of the dividend is accounted for, the partial quotients can be added and the division is complete.

This activity introduces the use of long division to calculate a quotient of whole numbers. Students make sense of the process of long division by studying an annotated example and relating it to the use of partial quotients and base-ten diagrams. They begin to see that long division is a variant of the method that uses partial quotients, but it is calculated and recorded differently.

When using partial quotients, the division is done in installments. The size of each installment can vary, but it is always a multiple of the divisor. Each partial quotient is written above the dividend and stacked. The sum of all partial quotients is the quotient.

In long division, the division is performed digit by digit, from the largest place to the smallest, so the resulting quotient is also recorded one digit at a time. In each step, one more digit of the quotient is calculated. Students notice that although only one digit of the quotient is written down at a time, the value that it represents is communicated through its placement.

As students examine the numbers, their placements, and the operations in the algorithm and interpret them in terms of place value and the process of dividing, they engage in abstract and quantitative reasoning (MP2).

To become proficient in long division requires time, encounters with a variety of division problems, and considerable practice. Students will have opportunities to study the algorithm more closely and to use it to divide increasingly more challenging numbers over several upcoming lessons.

Display the worked-out long divisions for all to see. Select a student to explain the steps for at least one of the division problems.

Draw students’ attention to the division , in which the first digit of the dividend is smaller than the divisor. Select 1–2 students to share how they approached this situation. If not brought up in students’ explanation, discuss how we could reason about these.

• “If we were using base-ten diagrams to represent , we would have 1 piece representing a thousand. How would we divide that piece into 4 groups?” (Decompose it into 10 hundreds, add them to the 8 pieces representing 8 hundreds, and then distribute the 18 hundreds into 4 groups.)
• “How can we apply the same idea to long division when there are not enough thousands to divide into 4 groups?” (We can think of the 1 thousand and 8 hundreds as 18 hundreds and divide that value instead.)
• “How many hundreds would go into each group if we divide 18 hundreds into 4 groups?” (4 hundreds, with a remainder of 2 hundreds.)
• “Where should we write the 4? Why?” (In the hundreds place, because it represents 4 hundreds.)
• “How do we deal with the remaining 2 hundreds?” (Decompose them into 20 tens, combine those with the 1 ten, and divide 21 tens by 4.)

In this activity, students continue to practice using long division to find quotients. Here, the presence of 0’s in the dividend and the quotient presents an added layer of complexity, prompting students to reason abstractly and quantitatively (MP2) as they make sense of the meaning of each digit in the numbers they are dealing with.

Display the solutions to the first two division problems and give students time to check their work. Then, focus the discussion on ways to tell if a long division is done correctly. Discuss questions such as:

• “How can we check if 143 is the quotient of ?” (We can multiply 143 by 7. If the division was done correctly, then the product would be 1,001, which is the case.)
• “Why does multiplying Priya’s answer, 320, by the divisor 3, give 960 instead of 906? Where did the error happen?” (Priya placed the result of dividing the 6 in 906 by 3 in the tens place, even though the 6 represents ones. It should’ve been 2 ones.)

Make sure students notice that although checking an answer can tell us that we have made a mistake, it will not necessarily show where the mistake is. This way of checking also works only if we perform the multiplication correctly.