<p>Students are introduced to the partial-quotients algorithm for determining the solution to a division issue. The partial-quotients algorithm is based on the concept of division as repeated subtraction. The students will:&nbsp;<br>- participate in activities that demonstrate the division process using a series of estimations.&nbsp;<br>- be able to see the division process rather than just following a list of stages as in the traditional, or standard, division algorithm.&nbsp;</p>

<p>- How are relationships represented mathematically?<br>- How can mathematics help us communicate more effectively?<br>- How may patterns be used to describe mathematical relationships?<br>- How can mathematics help to quantify, compare, depict, and model numbers?<br>- What does it mean to analyze and estimate numerical quantities?<br>- What makes a tool and/or strategy suitable for a certain task?<br>- When is it appropriate to estimate versus calculate?</p>

<p>- Division: The operation of making equal groups (e.g., there are 3 groups of 4 in 12).&nbsp;<br>- Estimate: To find a number close to an exact amount.&nbsp;<br>- Factor: The number or variable multiplied in a multiplication expression.</p>

<p>- overhead projector (optional)&nbsp;<br>- base-ten blocks&nbsp;<br>- chart paper or plain newsprint&nbsp;<br>- copies of Admit Ticket (M-4-4-3_Admit Ticket and KEY)&nbsp;<br>- copies of Base-Ten Organizer (M-4-4-3_Base-Ten Organizer and KEY)&nbsp;<br>- copies of Observation Checklist (M-4-4-3_Observation Checklist-Lesson 3)&nbsp;<br>- copies of Partial Quotients Four-Square (M-4-4-3_Partial Quotients Four-Square and KEY)</p>

<p>- The entrance ticket can be used to assess students' basic knowledge of division, allowing the course to be changed as needed.&nbsp;<br>- Observation during four-square group activity will help determine the students' degree of knowledge.&nbsp;<br>- Use the Observation Checklist during chart-problem work and the four-square activity to assess, document, and track students' understanding of the partial-quotients method of division.&nbsp;</p>

<p>Scaffolding, Active Engagement, Metacognition, Modeling, and Explicit Instruction<br>W: The lesson will teach students how to solve division problems using the partial quotients method and improve their understanding of the division process.&nbsp;<br>H: Students will use the Base-Ten Organizer and base-ten blocks to solve division problems, gaining a greater understanding of the concept.&nbsp;<br>E: Students will use the partial-quotients method of division with a partner to assist each other through the process for the first few times.&nbsp;<br>R: Students will practice the partial-quotients method using the Partial Quotients Four-Square worksheet. The teacher will be available at a "help station" where students can seek assistance if they are having problems.&nbsp;<br>E: The Observation Checklist can be used to assess student preparation for the following session or to reinforce partial quotients. Students that could benefit from further practice may be divided into small groups.&nbsp;<br>T: The extension section offers suggestions for students seeking more practice or a challenging learning experience. For students who are ready, additional practice with remainders can be provided.&nbsp;<br>O: The lecture focuses on the partial-quotients approach. Students are continue learning about division, what it means, and how it works. The goal of this lesson is to provide students with a deeper conceptual understanding of the long division process rather than teaching them the usual method.</p>

<p><strong>"Today, we'll look at alternative strategies to solve division problems. People can use a variety of approaches, or algorithms, to handle division problems with larger numbers. We'll focus on partial quotients. We're learning this strategy since most real-world division problems don't employ basic numbers. We can divide greater numbers now that we understand the concept of division and have completed the previous lessons in this course."</strong><br><br>Show students the Base-Ten Organizer for dividing larger numbers (M-4-4-3_Base-Ten Organizer and KEY). <strong>"This organizer is identical to the ones we used in earlier courses. We still have the sum in the center. The total is the dividend from a division problem. We shall continue to utilize rectangles to represent groupings. The divisor of a division problem indicates how many groups we have. The number inside each rectangle is the division problem's solution, also known as the quotient. After we complete one example in this way, you will see why it is critical that we utilize a more efficient method to solve division problems.</strong><br><br><strong>"Using the problem 135 ÷ 5 = ___, what is the total number of pieces we have, or what is the dividend?"</strong> (<i>135</i>)<br><br><strong>“How many groups do we want to make?”</strong> (<i>5</i>)<br><br><strong>“That is the divisor. What are we trying to find out?”</strong> (<i>how many will go in each group</i>)<br><br><strong>“The answer to our division problem is called the quotient.</strong><br><br><strong>"I need to put 135 pieces in the center. If I used separate components, it would be impractical. 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, and a bit is 1 unit. Now I know I need to design five rectangles because that's how many groups I require."</strong> Use the think-aloud method and demonstrate the process for students to see. <strong>"I have five groups and just one flat. Each group cannot obtain a flat. I can divide the flat into 10 skinnies since each skinny equals 10 units, and 10 × 10 = 100. I now have 13 skinnies in total, 10 from the transaction and 3 from the start. I have 5 groups, and each group must be equal, so I can fit 2 skinnies into each rectangle. That means I've gone through 10 skinnies. I still have three skinnies left, but there aren't enough to fill all of the rectangles. I need to swap again. I can exchange skinnies for parts. Each skinny is 10 bits, thus 3 skinnies would be 30 bits (3 x 10 = 30). Now I've got 35 bits. Thirty bits from the exchange plus 5 bits from the start gives me 30 + 5 = 35. So, if I have 35 bits and 5 groups, how many may each group receive? I understand that 7 × 5 = 35, thus with 35 bits and 5 groups, each will receive 7 bits. Now I can place seven bits in each group. I finish with each group having two skinnies and seven bits. This equals 27 since 2 x 10 = 20 + 7 = 27. The solution to the problem is 135 ÷ 5 = 27."</strong><br><br><strong>"Talk to the others at your table about what you saw. What did you notice about the method I used? Did I receive the proper answer? How did the model I used compare to the one we used earlier?"</strong> Allow students to engage in dialogue. (Possible answers: <i>This takes a long time. You need a lot of manipulatives. Perhaps you could sketch the exchanges. You can see how the objects are separated. If you lose track of where you left off, things might become confused. You must ensure that you make the correct trades.</i>)&nbsp;<br><br><strong>"Those are all excellent responses. Although this is a useful approach to see the concrete process of division, it may not always be efficient."</strong><br><br><strong>"Today, we will practice the partial-quotients method. Let's look at the following problem:"&nbsp;</strong><br><br>Five classrooms had to distribute 135 textbooks equally. How many textbooks should each class receive?&nbsp;<br><strong>“This is a sharing type division problem because we know the total number, and we know how many groups we have to make.”</strong><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_22.png" width="68" height="47"><br><strong>"Using the partial-quotients approach, we estimate. I know I need to figure out how many times 5 can go into 135. I observe that the largest place value is the hundreds place, therefore I'll guess that 5 goes into 135 around 20 times to begin. Since 20 × 5 = 100, I have used 100. I subtract and discover that I still have 35 remaining."</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_20.png" width="82" height="93"><br><strong>"I choose 7 next. I know 7 × 5 = 35, but when I subtract, I have no more left. I add up my estimates: 20 + 7 = 27, and that is my quotient."</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_21.png" width="102" height="121"><br><strong>"The answer to my division problem is 27 textbooks for each classroom.&nbsp;</strong><br><br><strong>"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."&nbsp;</strong><br><br><strong>"Let's do one more problem together."</strong><br><br>A golf ball company has 694 balls that need to be put into boxes. Each package will store five golf balls. How many boxes are needed?<br><strong>"This is a grouping type division problem because we know the total number, and we know how many items go in each group."</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_23.png" width="67" height="42"><br><strong>"I need to figure out how many times 5 can go into 694. I see that the hundreds place is the largest place value. This time, I see there are 6 hundreds, therefore my first guess will be 100. Since 5 × 100 = 500, I have used 500. I still have 194 left."</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_24.png" width="99" height="91"><br><strong>"My next estimate will be 20. Since 5 × 20 = 100, I have used up 100 more. I still have 94 left.”</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_25.png" width="106" height="123"><br><strong>“My next estimate will be 10. Since 5 × 10 = 50, I have used up 50 more. I still have 44 left.”</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_26.png" width="113" height="158"><br><strong>"My next estimate will be 8. Since 5 × 8 = 40, I have used up 40 more and have 4 remaining. These 4 are extra, or the remainder. I sum up my estimates: 100 + 20 + 10 + 8 = 138, which is my quotient."</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_27.png" width="128" height="199"><br><strong>"So the answer to my division problem is 138 boxes of golf balls with 4 golf balls left over." </strong>Ask whether a volunteer can demonstrate the same procedure with various estimates.<br><br><strong>"On the chart paper you and your partner have, please solve the following two division problems: 4⟌92 and 8⟌950 with the partial-quotients approach. Make sure to show every step. Remember to estimate in multiples of 100 and 100. We will display these chart papers around the room to demonstrate the various ways we solved these difficulties. We should all arrive to the same conclusions, but employing the partial-quotients method allows us to do so in different ways. I'm going to keep the two examples we just saw online so that if you run into a stumbling block, you can look at them and see if they help."</strong> You can keep track on student involvement and performance as they work. Record your observations on the Observation Checklist (M-4-4-3_Observation Checklist-Lesson 3). Offer support to partners who may require further training or verbal prompting.&nbsp;<br><br><strong>"In this lesson, we learned how to solve a division problem using the partial-quotient method. There are several ways, or algorithms, for solving long division problems. The partial-quotients method allows you to continue making estimations until the payout cannot be divided further. These estimates are then combined together to determine the quotient."</strong><br><br>Students who are prepared for individual practice can complete the Partial Quotients Four-Square (M-4-4-3_Partial Quotients Four-Square and KEY). Post the answers in a separate site so that students can check their work. Remind students that while estimations may differ, final solutions should be consistent. If students are unable to identify their errors, set up a location where they can receive support. The remaining students can continue working alone.</p><p>The Observation Checklist can help you determine who is good at solving division problems with the partial-quotients method. The Partial Quotients Four-Square exercise helps determine whether the procedure is being applied correctly to solve division problems. A quick assessment of the estimations chosen by students during the partial-quotients process will help determine whether they are making reasonable choices.<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> Emphasize the use of appropriate vocabulary in lessons and student answers. Ask students to write the following words in their notebooks or vocabulary journals: <i>dividend, divisor, factor, equation, expression, quotient,</i> and <i>unknown factor</i>. Remind students to ask questions, share ideas, and explain strategies while working in pairs or small groups.</p><p><strong>Small Group:</strong> Use partial-quotient problems as an admit ticket to class or as a spiral review (M-4-4-3_Admit Ticket and KEY). Create one or more small groups for people who are still experiencing problems. To recap the method, have students work through two or three problems with simpler numbers.&nbsp;</p><p><strong>Tiered Problems:</strong> The number and type of problems utilized can vary depending on the student's proficiency. Smaller numbers with no remainders can be used to help students who struggle with the partial quotient method. For students who demonstrate proficiency, larger numbers and remainders can be used.</p><p><strong>Expansion:</strong> Students that are proficient in the partial-quotients method can build their own four-square based on a topic or theme. Based on the theme or topic, students can develop four real-world problems and demonstrate how to solve them using the partial-quotients method. These issues can then be used to review with the remainder 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.</p>