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Multiplying and Dividing Rational Numbers to Solve Problems

Lesson Plan

Multiplying and Dividing Rational Numbers to Solve Problems

Objectives

Students will compute and solve problems using rational numbers. They will:

  • multiply and divide rational numbers.
  • solve real-world problems by multiplying and dividing rational numbers.

Essential Questions

  • How is mathematics used to quantify, compare, represent, and model numbers? 
  • How are relationships represented mathematically?
  • How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
  • What makes a tool and/or strategy appropriate for a given task?
  • How can recognizing repetition or regularity assist in solving problems more efficiently?

Vocabulary

  • Rational Number: A number expressible in the form a/b, where a and b are integers, and b ≠ 0.
  • Repeating Decimal: The decimal form of a rational number in which the decimal digits repeat in an infinite pattern.

Duration

60–90 minutes

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

IXL’s Grade 7 Multiply and Divide Rational Numbers offers students additional practice with multiplication and division of rational numbers.

IXL’s Grade 8 Multiply and Divide Rational Numbers: Word Problems offers students additional practice with solving word problems involving rational numbers.

Formative Assessment

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    • The modeling activity can be used to assess students’ conceptual understanding of multiplication and division of rational numbers.
    • Activity 1 can be used to assess each student’s ability to create a word problem involving the multiplication or division of rational numbers, and to understand the solution process.
    • The Lesson 3 Exit Ticket can be used to quickly evaluate student mastery.

Suggested Instructional Supports

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    Scaffolding, Active Engagement, Modeling, Explicit Instruction
    W: Students will learn to compute with rational numbers and use these skills to solve real-world problems. 
    H: Hook students into the lesson by asking them to model problems involving the multiplication and division of rational numbers using the number line. 
    E: The focus of the lesson is on computing products and quotients of rational numbers. Students will then solve problems involving rational numbers. In the final class activity, students will be given an opportunity to write an original word problem that involves the multiplication or division of rational numbers, and also to show the solution process. 
    R: Opportunities for discussion occur with each computation and real-world example, leading students to rethink and revise their understanding throughout the lesson. The PowerPoint activity gives students an opportunity to review their understanding prior to completing the exit ticket. 
    E: Evaluate students’ level of understanding by giving the exit ticket to the class. 
    T: Using suggestions in the Extension section, the lesson can be modified to meet students’ needs. The Lesson 3 Small-Group Practice worksheet offers more practice for students. The Lesson 3 Expansion Worksheet includes more difficult numeric expressions, as well as additional word problems. 
    O: The lesson is scaffolded so that students first model a multiplication and division problem with manipulatives and then compute products and quotients. The second part of the lesson involves problem solving with rational numbers. Students will discuss the procedure for computing each product or quotient as well as the process for solving each word problem. The lesson builds on students’ prior understanding of solving word problems that involve addition or subtraction of rational numbers. 

Instructional Procedures

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    Students need to have a conceptual understanding of why the algorithms for multiplying and dividing rational numbers work. Ask students to model the multiplication and division of rational numbers using the number line.

    “In Lesson 1 of this unit, we learned to model the addition and subtraction of rational numbers using a number line. Today, we will do the same thing with multiplication and division. Let’s look at a few examples together.”

     

    Steps for Multiplying (or Dividing) Fractions on a Number Line:

    1. If the number sentence uses a division symbol, rewrite the division as multiplication by “multiplying by the reciprocal.”*
    2. Locate the first factor on the number line. Draw a line to mark the distance between the first factor and zero.
      Refer to this line as the “absolute value line.”
    1. Split the “absolute value line” equally into the number of parts indicated by the denominator of the second factor.
    2. Start at 0, and move along your “absolute value line” the number of these equal parts indicated by the numerator of the second factor.
    3. Determine where you land on the number line. This is your answer.

     

    • Example 1:          

     

     

     

    • Example 2:          

     

     

     

     

    • Example 3:          

     

                           

     

    “Now, on your paper, represent the following products and quotients using a number line.”

     

    Computations: Multiplying and Dividing Fractions

    “Now we will practice multiplying and dividing fractions without the use of a number line. Let’s look at a few examples together.”

     

    Example 1:     

    •             “Notice that one number in this problem is written as a fraction, but

    the other is not. Often, when computing with fractions, it is best to write all numbers in fraction form.”

    •               “When multiplying fractions, all we have to do is multiply the

    numerators together and the denominators together, and then reduce.”

    •         “In this case, the product   is already reduced. We can

    leave the product as an improper fraction or convert it to a mixed number.”

     

    Example 2:     

    •                  “When multiplying and dividing fractions, it is best to convert any

    mixed numbers to improper fractions first.”

    •                  “To divide two fractions, we ‘multiply by the reciprocal.’ This means

    we rewrite the division problem into a multiplication problem by changing the division symbol into a multiplication symbol, and flipping the second fraction upside-down.”

    •                   “Now we multiply the numerators and denominators together, just

    like before.”

    •        “As always, we check to see if our final quotient or product can be

    reduced. In this case, the fraction   can be reduced to 5.”

    •                   “Another option to consider when multiplying fractions is to ‘cross-

    reduce.’ This means that we look at the numbers on each diagonal. If they share a common factor, we can divide that out.”

    •             “After cross-reducing, we multiply the numerators and denominators

    together.”

    •           “As you can see, we got the same answer by cross-reducing as we did

    when we reduced at the end. It will be up to you to decide which strategy you prefer.”

     

    Computations: Multiplying and Dividing Decimals

    Example 1:      4.56 × 1.7

    • 4.56 × 1.7              “To multiply decimals, simply stack the numbers vertically as you

    would for any multidigit multiplication. You should not, however, line up the decimal point, as this is only for adding and subtracting decimals. In fact, it may be best to ignore the decimal points altogether as you work the multiplication.”

    •         “After working your multiplication, it is now time to consider the

    decimal points. Count the number of digits that come after a decimal point in your original factors. In this case, there are three: 5, 6, and 7. This means you move the decimal point of your final product over three units to the left.”

    •        “Thus, the final product is 7.752.”

     

    Example 2:      9 × 0.64

    •             9 × 0.64 = 5.76

     

    Example 3:      19.44 ÷ 3.6

    • “To divide decimals by hand, set up long division as you normally would. If the divisor has a decimal point, move the decimal over as far to the right as possible. Keep track of how many times you need to move a decimal point in the divisor, because you will then need to move the decimal point in the dividend the same number of times to the right.”
    •      “In this case, the divisor 3.6 was converted to 36 by moving the

    decimal point 1 unit to the right. Therefore, the decimal point in the dividend 19.44 also needs to move 1 unit to the right. Wherever the decimal point in the dividend ends up, copy it to the top of your division bar. Then proceed to do long division as normal. When you reach a remainder of 0 or a repeating pattern, your quotient will be sitting on top of your division bar. Here, the quotient is 5.4”

     

    Example 4:      4.2 ÷ 8

    •   4.2 ÷ 8 = 0.525

     

    Distribute the Lesson 3 Computations Worksheet (M-7-5-3_Computations and KEY.docx). Instruct students to complete the worksheet individually. Walk around the room as students work to be sure they are on task and performing the computations accurately. Following the worksheet, provide time for students to discuss any problems they encountered, questions they have, or revelations they discovered. First, ask students to describe the computation process they used to find each product or quotient. Then confirm their understanding by restating the correct process.

     

    Problem Solving with Rational Numbers

    Now it is time for students to apply their understanding of computation to solving real-world problems. Discuss the following examples together as a class.

    • Hannah cuts 5 pieces of fabric, each of which is  feet long. How many feet of fabric does she cut?
      • “The solution is equal to the product of the number of pieces of fabric and the length of each piece of fabric. Thus, the solution can be written as . The number of feet of fabric she cuts is equal to   ft, or  ft.”
    • Roy eats  of a cake. Michael eats  of what is left. How much of the cake did Roy and Michael eat?
      • “If Roy eats  of the cake, then  of the cake is left. If Michael eats   of what is left, then he eats   of   of the cake. The amount Michael eats can be represented as , which equals . So, if Roy eats  of the cake and Michael eats another  of the cake, then, together, they eat  of the cake.”
      • You may wish to have students draw a diagram that represents the problem. For example, they can draw a rectangle, with five equally-spaced sections. The fraction that Roy eats would be represented by one shaded section. Since Michael eats  of what is left, one more section should be shaded ( of 4 is 1). Since 2 out of 5 sections are shaded, Roy and Michael eat a total of  of the cake.

     

    Distribute the Lesson 3 Word-Problem Examples (M-7-5-2_Word Problem Examples and KEY.docx). Have students discuss the solution process for each example problem in a manner similar to the process demonstrated above. Confirm student ideas that are correct. “Look through the problems you just received. Think of how the example word problems can be solved. Do you need to multiply or divide the rational numbers? How will you go about doing this for fractions, decimals, or mixed numbers?”

     

    Activity 1: Write-Pair-Share

    Have students brainstorm some real-world scenarios that involve multiplication and/or division of rational numbers. Have students make a list of five to ten scenarios. After 5 minutes, they can swap their list with a partner’s list. The partners should discuss and debate the ideas and offer new ideas, forming a cumulative list. After 5 more minutes, the class can reconvene. Ask one partner from each group to share the group’s cumulative list. The groups’ lists can be combined into one PDF file that is uploaded as a reference file to the class Web site or posted as a classroom display.

    Have students complete the Lesson 3 Exit Ticket (M-7-5-3_Exit Ticket and KEY.docx) at the close of the lesson to evaluate students’ level of understanding.

     

    Extension:

    The lesson can be tailored to meet the needs of students using the following suggestions.

    • Routine: Throughout the school year, have students write down real-world situations they encounter that involve the multiplication or division of rational numbers.
    • Small Groups: Students who need additional practice can be pulled into small groups to work on the Lesson 3 Small-Group Practice worksheet (M-7-5-3_Small Group Practice and KEY.docx). Students can work on the matching activity together or work individually and compare answers when done.
    • Expansion: Students who are prepared for a greater challenge can be given the Lesson 3 Expansion Worksheet (M-7-5-3_Expansion and KEY.docx).

Related Instructional Videos

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Final 04/12/13
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