<p>Students will calculate and solve problems with rational numbers. They will:&nbsp;<br>- multiply and divide rational numbers.&nbsp;<br>- multiply and divide rational numbers to solve real-world problems.&nbsp;<br>&nbsp;</p>

<p>- How can mathematics help to quantify, compare, depict, and model numbers?&nbsp;<br>- How are relationships represented mathematically?&nbsp;<br>- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?<br>- What makes a tool and/or strategy suitable for a certain task?<br>- How may detecting repetition or regularity assist in solving problems more efficiently?<br>&nbsp;</p>

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

<p>- Lesson 3 Exit Ticket (M-7-5-3_Exit Ticket and KEY)&nbsp;<br>- Lesson 3 Small-Group Practice worksheet (M-7-5-3_Small Group Practice and KEY)&nbsp;<br>- Lesson 3 Expansion Worksheet (M-7-5-3_Expansion and KEY)&nbsp;<br>- Lesson 3 Computations Worksheet (M-7-5-3_Computations and KEY)&nbsp;<br>- Lesson 3 Word-Problem Examples (M-7-5-3_Word Problem Examples and KEY)</p>

<p>- Students' conceptual comprehension of rational number multiplication and division can be assessed using the modeling activity.&nbsp;<br>- Activity 1 can be used to examine each student's ability to design a word problem involving rational number multiplication or division, as well as their understanding of the solution process.&nbsp;<br>- The Lesson 3 Exit Ticket can be used to quickly evaluate students' mastery.&nbsp;<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction&nbsp;<br>W: Students will learn how to compute rational numbers and apply these skills to real-world problems.&nbsp;<br>H: To engage students in the lesson, use the number line to demonstrate multiplication and division problems with rational numbers.&nbsp;<br>E: The lesson focuses on computing products and quotients of rational numbers. Students will then solve problems involving rational numbers. In the final class exercise, students will be given the opportunity to create an original word problem that involves the multiplication or division of rational numbers, as well as demonstrate the solution process.&nbsp;<br>R: Each computation and real-world example provides opportunities for discussion, encouraging students to reconsider and improve their understanding throughout the lesson. The PowerPoint activity allows students to review their understanding before completing the exit ticket.&nbsp;<br>E: Evaluate students' comprehension by assigning an exit ticket to the class.&nbsp;<br>T: Modify the lesson to match students' requirements by following the Extension section instructions. The Lesson 3 Small-Group Practice worksheet provides additional practice for students. The Lesson 3 Expansion Worksheet covers more complex numeric expressions as well as additional word problems.&nbsp;<br>O: The lesson is scaffolded so that students first use manipulatives to model multiplication and division problems, followed by computation of products and quotients. The second part of the class is about problem solving with rational numbers. Students will go over the steps involved in computing each product or quotient, as well as how to solve each word problem. The lesson builds on students' prior understanding of word problems involving addition and subtraction of rational numbers.&nbsp;</p>

<p>Students need to have a conceptual understanding of why the algorithms for multiplying and dividing rational numbers work. Use the number line to teach students how to multiply and divide rational numbers.&nbsp;<br><br><strong>"In Lesson 1 of this unit, we learned how to use a number line to model the addition and subtraction of rational numbers. Today we'll do the same thing with multiplication and division. Let's look at some examples together."</strong><br><br><br><u>Steps for Multiplying (or Dividing) Fractions on a Number Line:</u><br><br>1. If the number sentence includes a division symbol, rewrite the division as multiplication by "multiplying by the reciprocal."*&nbsp;<br><span style="color:#38761d;">*If students are unfamiliar with this strategy, go over how to rewrite division problems as multiplication problems (or "multiply by the reciprocal") before moving on.</span>&nbsp;<br>2. Locate the first factor on the number line. Draw a line to represent the distance between the first factor and zero.&nbsp;<br>Refer to this as the "absolute value line."<br>3. Split the "absolute value line" equally into the number of parts given by the denominator of the second factor.&nbsp;<br>4. Begin at 0 and move along your "absolute value line" the number of equal parts indicated by the numerator of the second factor.&nbsp;<br>5. Determine where you land on the number line. This is your answer.<br><br><strong>Example 1: \(2 \over 3\) × \(1 \over 4\)</strong></p><figure class="image"><img style="aspect-ratio:488/175;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_60.png" width="488" height="175"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>\(2 \over 3\) × \(1 \over 4\) = \(1 \over 6\)</strong></span><br><br><br><strong>Example 2: \(2 {1 \over 5} \) ÷ \(-1 \over 2\)</strong></p><figure class="image"><img style="aspect-ratio:525/175;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_61.png" width="525" height="175"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>\(2 {1 \over 5} \) ÷ \(-1 \over 2\) = \(-4 {2 \over 5} \)</strong></span><br><br><strong>Example 3: \(-5 \over 6\) × \(2 \over 5\)</strong></p><figure class="image"><img style="aspect-ratio:493/164;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_62.png" width="493" height="164"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>\(-5 \over 6\) × \(2 \over 5\) = \(-1 \over 3\)</strong></span><br><br><strong>“Now, on your paper, represent the following products and quotients using a number line.”</strong></p><ul><li><strong>\(8 \over 9\) × \(3 \over 4\)</strong></li><li><strong>\(7 {1 \over 2} \) ÷ (-4)</strong></li></ul><p>&nbsp;</p><p><strong>Computations: Multiplying and Dividing Fractions</strong><br><br><strong>"Now we'll practice multiplying and dividing fractions without using a number line. Let's look at some examples together."</strong><br><br>Example 1: 9 × \(-1 \over 5\)<br><br>9 × \(-1 \over 5\) <strong>"In this problem, one number is written as a fraction, while the other is not. When working with fractions, it is often preferable to write all numbers in fraction form."</strong><br><br>\(9\over 1\) × \(-1 \over 5\) <strong>"When multiplying fractions, all we have to do is multiply the numerators together and the denominators together, and then reduce."</strong><br><br>\(9×(-1) \over 1×5\) = -\(9 \over 5\) = \(-1 {4 \over 5} \) <strong>"In this case, the product -\(9 \over 5\) has already been reduced. We can keep the product as an improper fraction or convert it to a mixed number."</strong><br><br><br>Example 2: \(3 {1 \over 3} \) ÷ \(2 \over 3\)<br><br>\(3 {1 \over 3} \) ÷ \(2 \over 3\) <strong>"When multiplying and dividing fractions, it is best to convert any mixed numbers to improper fractions first."</strong><br><br>\(10\over 3\) ÷ \(2 \over 3\) <strong>"To divide two fractions, we 'multiply by the reciprocal.' This means, to rewrite a division problem as a multiplication problem, replace the division symbol with a multiplication symbol and turn the second fraction upside-down."</strong><br><br>\(10\over 3\) × \(3 \over 2\) <strong>"Now we multiply the numerators and denominators together, just like before."</strong><br><br>\(10×3 \over 3×2\) = \(30 \over 6\) = 5 <strong>"We always check to see if our final quotient or product can be reduced. In this case, we can reduce the fraction \(30 \over 6\) to 5."</strong><br><br>\(10\over 3\) × \(3 \over 2\) <strong>“Another option to consider when multiplying fractions is to 'cross-reduce.' This means that we'll look at the numbers on each diagonal. If they have a common factor, we can divide that out.”</strong></p><figure class="image"><img style="aspect-ratio:75/44;" src="https://storage.googleapis.com/worksheetzone/images/image.png" width="75" height="44"></figure><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>"After cross-reducing, we multiply the numerators and denominators together."</strong><br><br>\(5×1 \over 1×1\) = \(5 \over 1\) = 5 <strong>"As you can see, using cross-reduction yielded the same result as reducing at the end. It is up to you to choose which strategy you prefer."</strong><br><br><br><strong>Computations: Multiplying and Dividing Decimals</strong><br><br>Example 1: 4.56 × 1.7<br><br>4.56 × 1.7 <strong>“To multiply decimals, stack them vertically, like you would for any multidigit multiplication. However, you should not line up the decimal point because it is only used for adding and subtracting decimals. In fact, it may be best to ignore the decimal points altogether while you perform the multiplication."</strong></p><figure class="image"><img style="aspect-ratio:87/116;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_63.png" width="87" height="116"></figure><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; “After working on your multiplication, it's time to consider the decimal points. Count the number of digits that appear 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 three units to the left.”</strong></p><figure class="image"><img style="aspect-ratio:87/117;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_64.png" width="87" height="117"></figure><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "Thus, the final product is 7.752."</strong><br><br><br>Example 2: 9 × 0.64</p><figure class="image"><img style="aspect-ratio:201/62;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_65.png" width="201" height="62"></figure><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 9 × 0.64 = 5.76</strong><br><br><br>Example 3: 19.44 ÷ 3.6<br><br><strong>“To divide decimals by hand, use long division as you would usually. If the divisor includes 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'll need to move the decimal point in the dividend the same number of times to the right.”</strong></p><figure class="image"><img style="aspect-ratio:103/137;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_66.png" width="103" height="137"></figure><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>"In this case, the divisor 3.6 was converted to 36 by moving the decimal point 1 unit to the right. As a result, the decimal point in dividend 19.44 also needs to move 1 unit to the right. Whenever the decimal point in the dividend ends up, copy it to the top of your division bar. Then proceed with long division as usual. When you reach a remainder of 0 or a repeating pattern, your quotient will appear on top of your division bar. Here, the quotient is 5.4."</strong><br><br><br>Example 4: 4.2 ÷ 8</p><figure class="image"><img style="aspect-ratio:133/136;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_67.png" width="133" height="136"></figure><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;4.2 ÷ 8 = 0.525</strong><br><br><br>Distribute the Lesson 3 Computations Worksheet (M-7-5-3_Computations and KEY). Instruct students to complete the worksheet independently. Walk around the room as students work to ensure they are on track and doing the computations correctly. Allow time after the worksheet for students to discuss any problems, questions, or revelations they find. First, ask students to describe the method they used to calculate each product or quotient. Then confirm their understanding by repeating the correct procedure.<br><br><br><strong>Problem Solving with Rational Numbers</strong><br><br>It is now time for students to apply their knowledge of computation to real-world problems. Discuss the examples below as a class.<br><br>Hannah cuts 5 pieces of fabric, each of which is \(2 \over 3\)&nbsp;feet long. How many feet of fabric does she cut?&nbsp;<br><strong>"The solution is the product of the number of pieces of fabric and the length of each piece of fabric. Thus, the solution can be written as 5 x \(2 \over 3\). The number of feet of fabric she cuts is equal to \(10 \over 3\) ft, or \(3 {1 \over 3} \) ft."</strong>&nbsp;<br><br>Roy eats \(1 \over 5\) of a cake. Michael eats \(1 \over 4\) of what is left. How much cake did Roy and Michael eat?<br><strong>"If Roy eats \(1 \over 5\) of the cake, then \(4 \over 5\) of the cake is left. If Michael eats \(1 \over 4\) what remains, he eats \(1 \over 4\) of \(4 \over 5\) of the cake. The amount Michael eats can be expressed as \(1 \over 4\) × \(4 \over 5\), which equals \(1 \over 5\). So, if Roy eats \(1 \over 5\) of the cake and Michael eats another \(1 \over 5\) of the cake, then they eat \(2 \over 5\) of the cake together."</strong><br><br>You may want to have students draw a diagram to represent the problem. For example, students can draw a rectangle with five equally-spaced sections. The fraction that Roy eats would be represented by one shaded section. Since Michael eats \(1 \over 4\) of what is left, one more section should be shaded (\(1 \over 4\) of 4 is 1). Because 2 out of 5 sections are shaded, Roy and Michael eat a total of \(2 \over 5\) of the cake.<br><br>Distribute the Lesson 3 Word Problem Examples (M-7-5-2_Word Problem Examples and KEY). Encourage students to explain the solution procedure for each example problem in a manner similar to the one presented above. Confirm that the students' ideas are correct. <strong>"Review the problems you just received. Consider how the example word problems can be solved. Do you need to multiply or divide rational numbers? How will you do this with fractions, decimals, or mixed numbers?"</strong><br><br><br><strong>Activity 1: Write-Pair-Share</strong><br><br>Encourage students to think about real-world scenarios involving multiplication and/or division of rational numbers. Have students create a list of five to ten scenarios. After 5 minutes, they can exchange their lists with a partner's. The partners should discuss and debate the ideas, as well as contribute new ones, to create a cumulative list. After 5 minutes, the class can resume. Ask one partner from each group to share the group's cumulative list. The lists of the groups can be consolidated into one PDF file, which can then be uploaded to the class Web site or posted as a classroom display.&nbsp;<br><br>Students should complete the Lesson 3 Exit Ticket (M-7-5-3_Exit Ticket and KEY) at the end of the lesson to assess their level of understanding.<br><br><br><strong>Extension:</strong><br><br>The lesson can be adjusted to match the needs of the students by following the guidelines below.&nbsp;<br><br><strong>Routine:</strong> Throughout the school year, have students write down real-world scenarios that need the multiplication or division of rational numbers.&nbsp;<br><br><strong>Small Groups:</strong> Students who require further practice can be divided into small groups to complete the Lesson 3 Small-Group Practice worksheet (M-7-5-3_Small Group Practice and KEY). Students can complete the matching activity together or independently, and then compare their answers when done.&nbsp;<br><br><strong>Expansion:</strong> Students who are ready for a greater challenge can be given the Lesson 3 Expansion Worksheet (M-7-5-3_Expansion and KEY).</p>