• The admit ticket may be used to evaluate students’ base-level knowledge of division so that the lesson may be adjusted, if necessary.
• Observation during four-square group work will aid in determining students’ level of understanding.
• Use the Observation Checklist during chart-problem work and the four-square activity to assess, record, and track student mastery of the partial-quotients method of division.

“Today we are going to look at different ways we can solve division problems. There are several methods, or algorithms, people can use to solve division problems with larger numbers. We are going to focus on partial quotients. The reason we are learning this method is because most of the division problems we encounter in the real world do not use simple numbers. Now that we are beginning to understand the concept of division, after doing the previous lessons in this unit, we can divide larger numbers and apply the processes we previously learned.”

Show students the Base-Ten Organizer for division of larger numbers (M-4-4-3_Base-Ten Organizer and KEY.doc). “This organizer is similar to the organizer we used in previous lessons. We still have the total in the center. The total is the dividend of a division problem. We will still use rectangles to represent groups. The divisor of a division problem tells us how many groups we have. The number that goes inside each rectangle is the answer to the division problem, or the quotient. After we do one example this way, you will see why it is important that we use a more efficient way to solve division problems.

“Using the problem 135 ÷ 5 = ___, what is the total number of pieces we have, or what is the dividend?” (135)

“How many groups do we want to make?” (5)

“That is the divisor. What are we trying to find out?” (how many will go in each group)

“The answer to our division problem is called the quotient.

“I have to put 135 pieces in the center. If I used individual pieces, it would not be practical. I could use base-ten blocks like this. I could use 1 flat, 3 skinnies, and 5 bits. A flat is 100 units, a skinny is 10 units, and a bit is 1 unit. Now I know I have to draw five rectangles because that is how many groups I need.” Use the think-aloud strategy and model the process for students to see. “I have five groups and only one flat. Each group cannot get a flat. I can break down the flat into 10 skinnies because each skinny is 10 units and 10 × 10 = 100. Now I have 13 skinnies altogether, 10 from the trade and 3 from the start. I have 5 groups and each group has to be equal, so I can place 2 skinnies in each rectangle. That means I have used up 10 skinnies. I still have 3 skinnies left, but I cannot distribute them to the rectangles because there are not enough for all of them. I need to trade again. I can trade skinnies for bits. Each skinny is 10 bits, so 3 skinnies would be 30 bits because 3 × 10 = 30. Now I have 35 bits. Thirty bits from the trade plus 5 bits from the start means I have 30 + 5 = 35. Now, if I have 35 bits and 5 groups, how many can each group get? Well, I know the basic fact of 7 × 5 = 35 so if I have 35 bits and 5 groups, each group will get 7 bits. Now I can put 7 bits in each group. I end with each group having 2 skinnies and 7 bits. That equals 27 because 2 × 10 = 20 + 7 = 27. So the answer to the problem is 135 ÷ 5 = 27.”

“Talk with the people at your table and discuss what you saw. What did you notice about the process I used? Did I get the correct answer? How was the model I used similar to one we used earlier?” Allow students time to dialogue. (Possible responses: It takes a long time. You need to have many manipulatives. Maybe you can draw the exchanges. You can easily see how the items are divided. It can get confusing if you forget where you left off. You have to make sure you do the right trades.)

“Those are all very good responses. Although this is an effective way to see the concrete process of division, it may not always be efficient.”

“The method that we are going to practice today is called the partial-quotients method. Let’s look at the following problem:”

• Five classrooms had to share 135 textbooks equally. How many textbooks should each classroom receive?

“This is a sharing type division problem because we know the total number, and we know how many groups we have to make.”

“When we use the partial-quotients method, we estimate. I know I have to figure out how many times 5 can go into 135. I see the largest place value is the hundreds place, so I will estimate that 5 goes into 135 about 20 times to start. Since 20 × 5 = 100, I have used up 100. I subtract and find out I still have 35 left.”

“I choose 7 next. I know 7 × 5 = 35, and when I subtract I see I have no more left. I add up my estimates, 20 + 7 = 27, and that is my quotient.”

“The answer to my division problem is 27 textbooks for each classroom.

“When using the partial-quotients method and breaking down the dividend into a smaller number, it is easiest to estimate with multiples of 10 or 100.”

“Let’s do one more problem together.”

• A golf ball company has 694 golf balls that need to be packed in boxes. Each box will hold 5 golf balls. How many boxes will be needed?

“This is a grouping type division problem because we know the total number, and we know how many items go in each group.”

“I need to figure out how many times 5 can go into 694. I see the largest place value is the hundreds place. This time I notice that there are 6 hundreds, so my first estimate is going to be 100. Since 5 × 100 = 500, I have used up 500. I still have 194 left.”

“My next estimate will be 20. Since 5 × 20 =100, I have used up 100 more. I still have 94 left.”

“My next estimate will be 10. Since 5 × 10 = 50, I have used up 50 more. I still have 44 left.”

“My next estimate will be 8. Since 5 × 8 = 40, I have used up 40 more, so I have 4 left. These 4 are extra, or the remainder. I add up my estimates, 100 + 20 + 10 + 8 = 138, and that is my quotient.”

“So the answer to my division problem is 138 boxes of golf balls with 4 golf balls left over.” Ask if a volunteer can show the same process with different estimates.

“On the chart paper you and your partner have, please solve the following two division problems:  and  using the partial-quotients method. Be sure to show all the steps. Remember to estimate using multiples of 10 and 100. We are going to post these chart papers around the room to see the different ways we solved these problems. We should all arrive at the same answers, but using the partial-quotients method allows us to get there in different ways. I am going to keep the two examples we just saw posted, so if you hit a road block, you can look at the examples and see if they help.” While students are working, you can monitor student interaction and performance. While making observations, record what you observe on the Observation Checklist (M-4-4-3_Observation Checklist-Lesson 3.doc). Provide assistance to partners who may need some additional instruction or verbal prompting.

“This lesson taught us how to solve a division problem using the partial-quotients method. There are various methods, or algorithms, people can use to solve long division problems. The partial-quotients method allows you to keep making estimates until the dividend cannot be divided anymore. These estimates are then added up to get the quotient.”

Students who are ready for independent practice can complete the Partial Quotients Four-Square (M-4-4-3_Partial Quotients Four-Square and KEY.doc). Post answers in a separate location for students to be able to check their work. Remind students that estimates may vary among students, but the final answers should be the same. If students are unable to identify their errors, set up a location where you can provide support to those students. The other students can continue to work independently.

The Observation Checklist can help to determine who is proficient in solving division problems using the partial-quotients method. The Partial Quotients Four-Square exercise can show if the process is being used accurately to solve division problems. A quick scan of the estimates chosen by students in the partial-quotients process will aid in determining whether students are making reasonable choices for their estimates.

Extension:

• Routine: Emphasize the use of correct vocabulary during lessons and student responses. Ask students to enter these words into their notebook or vocabulary journal: dividend, divisor, factor, equation, expression, quotient, and unknown factor. Remind students to ask questions, share ideas, and explain strategies while working with partners or small groups.
• Small Group: Use partial-quotients problems as an admit ticket to class or as a spiral review (M-4-4-3_Admit Ticket and KEY.doc). Form one or more small groups for those who are still having difficulty. Walk students through two or three problems using simpler numbers to review the process.
• Tiered Problems: Based on student proficiencies, the numbers and types of problems used can vary. For students who are challenged by the process of partial quotients, smaller numbers with no remainders can be used. For students who are showing proficiency, larger numbers and remainders can be used.
• Expansion: Students who are proficient in using the partial-quotients method can create their own four-square using a topic or theme. Based on the theme or topic, students can create four real-world problems and show how to solve them using the partial-quotients method. These problems can then be used as review for the rest of the class. For example, if the theme is baseball, problems might relate to fans on a bus, buns in packages, boxes of batting helmets, and the total cost of a shipment of hot dogs.