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Problem Solving by Using Multiplication to Solve Division Problems

Lesson Plan

Problem Solving by Using Multiplication to Solve Division Problems

Objectives

Students will investigate problem-solving situations in which problems involving division can be readily solved using multiplication. Students will:

  • practice writing expressions and equations related to solving word problems as multiplication and division problems.

Essential Questions

How are relationships represented mathematically?
How can mathematics support effective communication?
How can patterns be used to describe relationships in mathematical situations?
How is mathematics used to quantify, compare, represent, and model numbers?
What does it mean to estimate or analyze numerical quantities?
What makes a tool and/or strategy appropriate for a given task?
When is it is appropriate to estimate versus calculate?
  • How is mathematics used to quantify, compare, represent, and model numbers?
  • How can mathematics support effective communication?
  • How are relationships represented mathematically?
  • What makes a tool and/or strategy appropriate for a given task?
  • How can patterns be used to describe relationships in mathematical situations?

Vocabulary

  • Division: The operation of making equal groups (e.g., there are 3 groups of 4 in 12).
  • Factor: A whole number that divides evenly into another whole number (e.g., 1, 3, 5, and 15 are factors of 15).

Duration

45–90 minutes

Prerequisite Skills

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Materials

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Related Materials & Resources

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Formative Assessment

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    • Observe students during the partner activity to gauge their level of understanding.
    • Observation during group work will help evaluate student comprehension and clarify any misconceptions students may have.
    • The exit ticket (M-4-4-2_Lesson 2 Exit Ticket and KEY.doc) may be used to assess student mastery.

Suggested Instructional Supports

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    Scaffolding, Active Engagement, Modeling
    W: The lesson will focus on solving division problems using multiplication and on the relationship between multiplication and division. 
    H: Remind students of the prize-bag scenario from the previous lesson and solve the problem again as a group. Correlate the problem with the number sentences that represent it.  
    E: Have students use the Match Them UP! Cards in small groups to gain experience recognizing the corresponding division number sentences when given multiplication number sentences. Students should solve the problems as a group, then record the answer to each problem on the front of each card. 
    R: While still in small groups, students should select one of the division/multiplication sentence pairs and solve the problem using manipulatives. Circulate during small group work time to check answers and provide assistance as needed. 
    E: As you circulate among groups, it should become evident if any reteaching is necessary. Exit tickets may also be used to evaluate student concept mastery. 
    T: Additional lesson ideas and suggestions for modification of lesson difficulty can be found in the extension section. 
    O: The lesson was designed using simple numbers so the division process and comparison to multiplication could be explained and practiced. As concepts were mastered at the initial level, they were extended using larger numbers. 

Instructional Procedures

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    “We are going to continue our exploration of division. Who can tell me the two types of division problems?” (sharing and grouping) “During yesterday’s lesson, we worked on learning the features of each type of division problem so we could distinguish between the two types. In today’s lesson, we are going to see how we can solve these types of division problems using multiplication.”

    Review the lesson from yesterday using the think-aloud strategy. “Let’s look at the problem we used in yesterday’s lesson. Remember, I had to make four prize bags. I had 20 items to divide equally among the four bags. So, I needed to know how many items I should place in each bag. After yesterday’s activity, we know that the correct answer is five. We used the organizer to help us. Now let’s see how we could write a multiplication sentence. We already know the total, which is 20. I know that I have done problems in the past where the total would come after an equal sign. I also know that I have four bags, so I have to find out how many are in each bag. I could write an equation like this: 4 × ___ = 20. I have to figure out 4 times what will give me 20. We call this an unknown factor. The answer is 5.” Record the following equations on the overhead or board:

    • 20 ÷ 4 =___
    • 4 × ___ = 20
    • ___ × 4 = 20

    “So a division problem that is written like this, 20 ÷ 4 =___, can be written as a related multiplication sentence that looks like this: 4 × ___ = 20.

    “Do you think I could also write the equation as ___ × 4 = 20?” Allow time for students to dialogue with each other and ask students to share their thoughts.

    “Yes, we could write this equation in two ways. It reminds me of fact families and the commutative property of multiplication. The commutative property of multiplication tells us that when two factors are multiplied together, the product will remain the same regardless of their order. Why can I not write the division equation as 20 ÷ 5 =___?” Allow time for students to dialogue with each other and ask students to share their thoughts. “For those of you who explained that the problem does not include 5 as a piece of data, you are correct. The 5 in this problem represents the number we are solving for.”

    Model a few more examples on the overhead or board. Start out with problems using smaller numbers until students begin to see a pattern of how a division equation has a related multiplication sentence. Increase the value of the numbers as students become proficient.

    • 63 ÷ 9 =____; a related multiplication sentence would be 9 × ____ = 63 or
      ___× 9 = 63. This type of multiplication sentence has an unknown factor. How many groups of 9 are needed to equal 63? What multiplied by 9 equals 63?
    • 400 ÷ 20 =____; a related multiplication sentence would be 20 × ____= 400 or ____× 20 = 400. This type of multiplication sentence has an unknown factor. How many groups of 20 are needed to equal 400? What multiplied by 20 equals 400?

    Be sure students are able to understand the following observations. “Remember the parts of a division problem: the dividend divided by the divisor equals the quotient. (In 35 ÷ 5 = 7, 35 is the dividend, 5 is the divisor, and 7 is the quotient.) The parts of a multiplication problem are a factor times a factor equals the product (7 × 5 = 35). When we carefully look at related division and multiplication problems, we can see 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.

    “Now that we have seen some more examples of division problems, you and a partner will create a word problem that will require the use of division to solve. Remember to include numbers that represent the dividend and the divisor in your word problem. Since the last part of your word problem will probably be a question, be sure to include a question mark.” [Note: If students need examples of word problems, review the problem-solving cards from Lesson 1 (M-4-4-1_Problem Solving Cards and KEY.doc).] While students are working, monitor the types of problems students are creating. Be sure they require the use of division. Where necessary, provide verbal prompting to redirect thinking. Once students are finished creating their word problems, have them exchange with each other and write a division equation and related multiplication equation. Students then can provide immediate feedback to each other. Have some groups share their word problems.

    Show the following problem on the overhead or chart:

    • There were 12 cookies. Six friends shared them equally. How many cookies did each friend get?

    “What would the division equation be?” (12 ÷ 6 = ___)

    “What would the related multiplication equation be?” (6 × __ = 12)

    “The number that would make both statements true is 2. We can use manipulatives to show this.” Model how to do the division problem with counting cubes or counting chips. Use the division organizer from the previous lesson to show the division process. Using the same manipulatives, show students the related multiplication equation. You can make an array with six rows of two in each row.

    With a partner, have students show the division equation and related multiplication equation that relate to the following problem. Have one student show the division equation and one student show the multiplication equation.

    • Staci was working on her scrapbook. She wanted to put three pictures on each page. She had 30 pictures. How many pages in the scrapbook would she need?

    Model the process for all students to see. Have students check their work and discuss any questions they may have and clarify any misconceptions. Write corresponding multiplication equations for each division equation. Do additional problems until students show proficiency representing the division equation and related multiplication equation with manipulatives.

    Divide the class into small groups for an activity. Give each group one set of Match Them UP! Cards (M-4-4-2_Match Them UP Cards and KEY.doc) and at least one set of counting cubes or chips. Each student will need one blank index card.

    “You will be working in small groups. You will be given a set of Match Them UP! cards. Your task as a group is to match a division sentence with its related multiplication sentence. As a group, discuss how to solve each equation and record the answer on the front of each card. Keep your cards paired together so I can come around and see your progress. Each member of the group needs to pick one division equation and its related multiplication equation and show how to solve it using manipulatives. You can use the division organizer we used in the previous lesson if it helps 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.”

    To vary the lesson for those students who are proficient, cards can be included in the set that do not have a related match. Students can create the missing cards to have a complete set and solve the problems. Use the Match Them UP! extension cards included with the on-level cards (M-4-4-2_Match Them UP Cards and KEY.doc) for this alternate activity.

    While students are working in small groups, monitor their interaction and dialogue. Ask questions similar to the ones listed below to probe student thinking. Assist those students who do not show understanding or proficiency at the task.

    “What number in the equation represents the total?”

    “How can you restate the equation in words?”

    “How are division and multiplication related?” (They are inverse operations.)

    “How does using multiplication help you solve a division problem?” (You can write an equation with an unknown factor and find the missing factor, which is the answer to the division problem. The divisor and quotient when multiplied together give you the dividend. The dividend of a division equation is the product of the related multiplication sentence.)

    “Explain how you used manipulatives to solve the division problem. Using the same manipulatives, explain how you solved the multiplication problem.”

    Have students complete an exit ticket at the end of the lesson (M-4-4-2_Lesson 2 Exit Ticket and KEY.doc). The responses on the exit tickets will help you determine who may need additional practice and who has mastered the skill.

    “We have seen how division problems can be solved using related multiplication equations. We used simpler numbers during this lesson to show the process. You can use a related multiplication sentence to solve division problems that have larger numbers as well.”

    Extension:

    • Routine: Review fact families for multiplication and division. Start out with a complete equation (5 × 4 = 20) and write the other three facts that belong to this fact family or an equation that has an unknown variable (20 ÷ ___ = 5) and write the other three facts that belong to this fact family.
    • Expansion: Vary the types of numbers used, starting with compatible numbers (like
      120 ÷ 6 = ___ or 200 ÷ 25 =___) and moving to more complex numbers (like 144 ÷ 9 =). Students can write related multiplication sentences and solve them. Students can also be given a multiplication sentence with an unknown variable and then write the related division sentence. Once that is completed, have students write a related word problem and solve it.
    • Technology Connection: There are many number tricks in which you ask a person to pick a number. Then you ask the person to perform calculations to the number according to a prescribed sequence of steps. Then “magically” you can guess the number the person started with, or the person will end up with the same number s/he started with. The truth behind the magic is that the sequence of steps used is designed in such a way that you end up at a predictable place or back where you started. Have students locate a number trick by looking through math puzzle books or online. You can locate Web sites to support this activity by doing a search for “number tricks.” Have students practice the number trick and explain the math behind the magic.

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Final 05/17/2013
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