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:
“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:
the other is not. Often, when computing with fractions, it is best to write all numbers in fraction form.”
numerators together and the denominators together, and then reduce.”
leave the product as an improper fraction or convert it to a mixed number.” Example 2:
mixed numbers to improper fractions first.”
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.”
like before.”
reduced. In this case, the fraction can be reduced to 5.”
reduce.’ This means that we look at the numbers on each diagonal. If they share a common factor, we can divide that out.”
together.”
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
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.”
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.”
Example 2: 9 × 0.64
Example 3: 19.44 ÷ 3.6
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
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.
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.
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