<p>Students will study problem-solving scenarios in which division difficulties can be easily handled using multiplication. Students will:&nbsp;<br>- practice writing expressions and equations used to solve word problems such as multiplication and division.&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>- Factor: A whole number that divides evenly into another whole number (e.g., 1, 3, 5, and 15 are factors of 15).</p>

<p>- overhead projector (optional)<br>- index cards (one per student)<br>- counting cubes or chips (one or two sets per group)<br>- copies of On-Level Cards—Match Them UP! Cards (M-4-4-2_Match Them UP Cards and KEY)<br>- copies of Lesson 2 Exit Ticket (M-4-4-2_Lesson 2 Exit Ticket and KEY)</p>

<p>- Observe students during the partner exercise to determine their level of understanding.&nbsp;<br>- Observation during group work will assist in assessing student comprehension and clarifying any misconceptions students may have.&nbsp;<br>- The exit ticket (M-4-4-2_Lesson 2 Exit Ticket and KEY) can be used to evaluate student mastery.&nbsp;</p>

<p>Scaffolding, Active Engagement, and Modeling&nbsp;<br>W: The lesson will cover using multiplication to solve division problems and understanding the relationship between the two operations.&nbsp;<br>H: Remind students of the prior lesson's prize-bag scenario and work together to solve the problem again. Correlate the problem to the amount of sentences that describe it.&nbsp;<br>E: Students should utilize the Match Them UP! Cards in small groups to practice recognizing the corresponding division number sentences when given multiplication number sentences. Students should work together to solve the issues, and then note their answers on the front of each card.&nbsp;<br>R: In small groups, students should solve a division/multiplication sentence pair using manipulatives. Circulate throughout small group work time to check answers and offer assistance as required.&nbsp;<br>E: As you circulate among groups, determine if reteaching is necessary. Exit tickets can also be used to assess student concept mastery.&nbsp;<br>T: The extension section contains additional lesson ideas and tips for modifying difficulty levels.&nbsp;<br>O: The lesson used simple numbers to teach and practice division and multiplication. As concepts were learned at the initial level, they were expanded to larger numbers.&nbsp;</p>

<p><strong>"We will continue our research of division. Who can explain the two types of division problems?"</strong> (<i>sharing and grouping</i>) <strong>"In yesterday's class, we focused on studying the features of each type of division problem so that we could differentiate between the two. In today's lesson, we'll look at how to solve these types of division problems with multiplication."</strong><br><br>Use the think-aloud approach to review yesterday's lesson. <strong>"Let's look at the problem from yesterday's lesson. Remember, I needed to make four prize bags. I had 20 items to split equally among four bags. So I needed to know how many items I should put in each bag. Following yesterday's activities, we know that the correct answer is five. We used the organizer to help us. Let's examine how we can write a multiplication sentence. We already know the total, which is 20. I know I've solved problems in the past where the total came after an equal sign. I also know I have four bags, so I need to figure out how many are in each one. I could write an equation like this. 4 × ___ = 20. I need to figure out four times what will give me 20. We refer to this as an unknown factor. The answer is 5."</strong> Note the following equations on the overhead or board:</p><p>20 ÷ 4 = _____<br>4 × _____ = 20<br>_____ × 4 = 20</p><p><strong>"So, a division problem written as 20 ÷ 4 =___ can be rewritten as a related multiplication sentence that looks like this: 4 × ___ = 20."</strong><br><br><strong>"Do you think I could also write the equation as ___ × 4 = 20?"</strong> Allow time for students to interact with one another and encourage them to share their opinions.<br><br><strong>"Yes, we might express this equation in two ways. It makes me think of fact families and the commutative property of multiplication. The commutative property of multiplication states that when two elements are multiplied together, the product remains the same regardless of the order. "Why can't I write the division equation as 20 ÷ 5 =___?"</strong> Allow time for students to interact with one another and encourage them to share their opinions. <strong>"Those of you who stated that the problem does not include 5 as a piece of data are correct. The 5 in this problem symbolizes the number we're looking for."</strong><br><br>Create a few more examples for the overhead or board. Begin with problems using smaller numbers until students notice a pattern in how a division equation is related to a multiplication statement. As students become proficient, the value of the numbers increases.<br><br>63 ÷ 9 = ____; an equivalent multiplication sentence would be 9 × ____ = 63. This type of multiplication statement includes an unknown factor. How many groups of 9 are required to equal 63?<br>400 ÷ 20 =____; a related multiplication statement would be 20 × ____= 400 or ____× 20 = 400. This type of multiplication statement includes an unknown component. How many groups of 20 are required to equal 400? What number multiplied by 20 equals 400?&nbsp;<br>Make sure students understand the following observations. <strong>"Remember the pieces of a division problem: divide the dividend by the divisor to get the quotient. (In the equation 35 ÷ 5 = 7, the dividend is 35, the divisor is 5, and the quotient is 7.) A multiplication problem consists of a factor multiplied by another factor, resulting in the product (7 × 5 = 35). When we closely examine similar division and multiplication problems, we can see that the dividend of the division problem is the product of the related multiplication problem. The divisor and quotient of the division problem are the factors of the related multiplication problem.</strong><br><br><strong>"Now that we've seen some additional division problems you and a partner will build a word problem that requires division to solve. Remember to include numbers representing the dividend and divisor in your word problem. Because the last part of your word problem will most likely be a question, insert a question mark."</strong> [Note: If students require examples of word problems, go over the problem-solving cards from Lesson 1 (M-4-4-1_Problem Solving Cards and KEY).] While students are working, keep track of the types of difficulties they are producing. Make sure they require division. When required, use verbal cues to refocus thoughts. Once students have completed their word problems, have them share ideas and produce a division equation and a related multiplication equation. Students can then offer rapid feedback to one another. Let some groups exchange their word problems.&nbsp;<br><br>Show the following issue on the overhead or chart:&nbsp;<br><br>There were 12 cookies. Six friends distributed them equally. How many cookies did each friend receive?&nbsp;<br><strong>"What would the division equation be?"</strong> (<i>12 ÷ 6 = ___</i>)&nbsp;<br><br><strong>"What would the related multiplication equation be?"</strong> (<i>6 × ___ = 12</i>)<br><br><strong>"The number that makes both propositions true is 2. We can use manipulatives to demonstrate this."</strong> Model how to solve the division problem by counting cubes or chips. Use the division organizer from the previous lesson to demonstrate the division process. Using the same manipulatives, demonstrate to students the corresponding multiplication equation. You can create an array with six rows, two in each row.&nbsp;<br><br>With a partner, students should demonstrate the division equation and corresponding multiplication equation that relate to the following problem. Have one student demonstrate the division equation and another illustrate the multiplication equation.<br><br>Staci was working on her scrapbook. She wanted three photos on each page. She had 30 photographs. How many scrapbook pages would she need?&nbsp;<br>Demonstrate the method for all students to see. Check students work, discuss any questions they have, and clear up any misconceptions. Create appropriate multiplication equations for each division equation. Do additional problems until students demonstrate skill in representing the division equation and related multiplication equation with manipulatives.<br><br>Divide the students into small groups for an exercise. Provide each group with one set of Match Them UP! Cards (M-4-4-2_Match Them UP Cards and KEY) and at least one set of counting cubes or chips. Each student will require a blank index card.<br><br><strong>"You will work in small groups. You will be given a set of Match Them Up! cards. Your group's task is to match a division sentence with the corresponding multiplication sentence. Discuss how to solve each equation with your group and write the answer on the face of each card. Keep your cards paired together so I can follow your progress. Each member of the group needs to pick one division equation and its related multiplication equation and demonstrate how to solve them using manipulatives. You can utilize the division organizer from the previous lesson if it is useful to you. Your last step will be to write a word problem on your blank index card that could be solved using the division equation from each pair.”</strong><br><br>To alter the lesson for proficient students, include cards in the set that do not have a connected match. Students can add the missing cards to complete the collection and solve the issues. For this alternate activity, use the Match Them UP! extension cards (M-4-4-2_Match Them UP Cards and KEY), which come with the on-level cards.&nbsp;<br><br>Keep an eye on the interaction and discourse amongst students while they work in small groups. To investigate student thinking, ask questions like those listed below. Assist students who do not demonstrate understanding or proficiency with the activity.<br><br><strong>"What number in the equation represents the total?"&nbsp;</strong><br><br><strong>"How can you restate the equation in words?"</strong><br><br><strong>"How are division and multiplication related?"</strong> (<i>These are inverse operations.</i>)&nbsp;<br><br><strong>"How does using multiplication help you solve a division problem?"</strong> (<i>You can use an equation with an unknown factor to determine the missing factor, which is the solution to the division problem. The dividend is calculated by multiplying the divisor and quotient. A division equation's dividend is equal to the product of the accompanying multiplication statement.</i>)<br><br><strong>"Please describe how you used manipulatives to solve the division problem. Use the same manipulatives to demonstrate how you solved the multiplication problem."</strong><br><br>Have students fill out an exit ticket at the end of the lecture (M-4-4-2_Lesson 2 Exit Ticket and KEY). The responses on the exit tickets will help you identify who needs more practice and who has mastered the skill.&nbsp;<br><br><strong>"We've seen how to solve division problems using related multiplication equations. Throughout this session, we used simpler numbers to demonstrate the method. You can also use a related multiplication sentence to solve division problems with higher numbers."</strong><br><br><strong>Extension:</strong><br><strong>Routine:</strong> Review the fact families for multiplication and division. Begin with a full equation (5 × 4 = 20) and write the other three facts in this fact family. Alternatively, use an equation with an unknown variable (20 ÷ ___ = 5) to write the other three facts in this fact family.</p><p><strong>Expansion:</strong> Begin with suitable numbers (e.g., 120 ÷ 6 = ___ or 200 ÷ 25 =___) then go to more complex numbers (e.g., 144 ÷ 9). Students can write and answer related multiplication sentences. Students can also be given a multiplication statement with an unknown variable and asked to compose the corresponding division sentence. Once that completed, have students write and solve a related word problem.</p><p><strong>Technology Connection:</strong> Many number tricks include asking someone to pick a number. Then you ask the user to make calculations on the number using a predetermined sequence of steps. Then "magically" you can guess the person's starting number, or the person will end up with the same number. The truth behind the magic is that the steps are planned such that you wind yourself in a predictable spot or back where you started. Ask students to hunt for a numerical trick in math puzzle books or online. You might find Web sites that encourage this pastime by searching for "number tricks." Encourage students to practice the number trick and explain the mathematical concept underlying the magic.</p>