<p>Students will calculate and solve problems with rational numbers. They will:&nbsp;<br>- add and subtract rational numbers.&nbsp;<br>- solve real-world situations by adding and subtracting rational numbers.&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 2 Exit Ticket (M-7-5-2_Exit Ticket and KEY)&nbsp;<br>- Lesson 2 Small-Group Practice worksheet (M-7-5-2_Small Group Practice and KEY)&nbsp;<br>- Lesson 2 Expansion Worksheet (M-7-5-2_Expansion and KEY)&nbsp;<br>- Lesson 2 Computations Worksheet (M-7-5-2_Computations and KEY)&nbsp;<br>- Lesson 2 Word-Problem Examples (M-7-5-2_Word Problem Examples and KEY)</p>

<p>- The modeling activity can be used to examine students' prior knowledge and understanding of adding rational numbers with unlike denominators.&nbsp;<br>- Activity 1 can be used to evaluate each student's ability to write a word problem involving the addition and/or subtraction of rational numbers, as well as understand the solution procedure.&nbsp;<br>- Use the exit ticket to quickly evaluate students' mastery.&nbsp;<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction&nbsp;<br>W: Students will learn to compute rational numbers and apply these skills to solve real-world problems.&nbsp;<br>H: Begin the lesson by having students model a problem involving the addition of two rational numbers using a number line.&nbsp;<br>E: The lesson focuses on computing sums and differences of rational numbers. Once students have mastered computation with rational numbers, the lesson will proceed to problem solving with rational numbers. After walking students through several example problems, they will engage in the final class activity, which culminates in a class PowerPoint file.&nbsp;<br>R: Each computation and real-world example provides opportunities for conversation, 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' level of understanding and comprehension by providing an exit ticket.&nbsp;<br>T: Modify the lesson based on student needs as suggested in the Extension section. The Small-Group Practice worksheet provides additional practice for students. The Expansion Worksheet offers more complex numeric expressions and more word problems for students who are ready for a challenge.&nbsp;<br>O: The lesson is scaffolded, with students modeling addition problems with manipulatives before attempting to compute sums and differences. The students then discuss the computing procedure for each example. The second part of the class is about problem solving with rational numbers. Students lead the solution process, with the teacher serving as a facilitator. This lesson serves as a refresher for adding and subtracting rational numbers, as well as an introduction to problem solving with rational numbers. The unit's next lesson will cover rational number multiplication and division, as well as problem solving with these operations on rational numbers.&nbsp;</p>

<p>As students enter class, have them evaluate the following expressions on a number line.<br><br>\(1 \over 6\) – \(2 \over 3\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(-<i>\(1 \over 2\)</i>)&nbsp;<br><br>0.75 + 2.95 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(<i>3.7</i>)<br><br>-\(1 {3 \over 4} \) + \(5 {1 \over 2} \) &nbsp; &nbsp; &nbsp;(<i>\(3 {1 \over 4} \)</i>)<br><br>Walk around the classroom while students work on the example problems. Before proceeding, briefly explain the answers and ensure that students are comfortable modeling addition and subtraction of rational numbers on a number line.<br><br><strong>"In Lesson 1 of this unit, we learned how to model addition and subtraction of rational numbers on a number line. Today, we will concentrate on performing these calculations without the use of a number line. We will next apply these abilities to solve some real-world problems."</strong><br><br><br><strong>Computations: Adding and Subtracting Rational Numbers</strong><br><br>Before presenting real-world problems, allow students to practice adding and subtracting rational numbers without the use of a number line. If necessary, review the following examples as a class.<br><br>Example 1: 9 + (-\(1 \over 5\)&nbsp;)<br><br><strong>9 + (-\(1 \over 5\)&nbsp;)</strong> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<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><strong>\(9 \over 1\)&nbsp;+ (-\(1 \over 5\)&nbsp;)</strong> &nbsp;&nbsp;<strong>"To add and subtract fractions, we require a common denominator. The lowest common denominator in this example would be 5."</strong><br><br><strong>\(45 \over 5\) + (-\(1 \over 5\)&nbsp;)</strong> <strong>"If the denominators are the same, simply add or subtract the numerators as indicated. The denominator will remain as is."</strong><br><br><strong>\(44 \over 5\)&nbsp;</strong> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>"Now we must ensure that our fraction is in the lowest terms. In this case it is, but we may want to rewrite the fraction as a mixed number may provide a more accurate answer."</strong><br><br><strong>\(44 \over 5\) = \(8 {4 \over 5} \)&nbsp;</strong><br><br><br>Example 2: -4.64 + 9.85<br><br><strong>-4.64 + 9.85 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;"Think about the number line. Based on the signs of each addend, do you think the final answer will be positive or negative?"</strong> (<i>Positive, the absolute value of 9.85 is larger than the absolute value of −4.64.</i>)<br><br><strong>Think: 9.85 - 4.64</strong></p><figure class="image"><img style="aspect-ratio:58/57;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_59.png" width="58" height="57"></figure><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "Set up the problem vertically, make sure to line up the decimal point. Subtract by hand, as normal."</strong><br><br>Distribute the Lesson 2 Computation Worksheet (M-7-5-2_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 process they used to calculate each sum or difference. Then confirm their understanding by repeating the correct process.<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.&nbsp;<br><br>A log measures \(8 {1 \over 2} \)&nbsp; feet long. Kevin cuts \(4 {1 \over 4} \)&nbsp; feet from the log. How long is the log now?&nbsp;<br><br><strong>"Because Kevin cuts a particular length from the original length, the rational number 4 1/4 should be subtracted from the rational number \(8 {1 \over 2} \). To solve the problem, write \(8 {1 \over 2} \) - \(4 {1 \over 4} \). In order to solve the problem, write the mixed numbers can be written with common denominators. Using the least common denominator, the number statement can be expressed as \(8 {1 \over 2} \) - \(4 {1 \over 4} \). Thus, the log's current length is \(4 {1 \over 4} \) feet."</strong><br><br>Last year, Steven saved a total of $1,018.20. This year, he has $920.45 in his savings account. How much has Steven's savings decreased?&nbsp;<br><strong>"The amount by which his savings have decreased is equal to the difference between 1018.20 and 920.45, expressed as 1018.20 – 920.45 or 97.75. Thus, Steven's savings decreased by $97.75."</strong>&nbsp;<br><br><br>Distribute Lesson 2 Word Problem Examples (M-7-5-2_Word Problem Examples and KEY). Encourage students to explain the solution process for each example problem in a manner similar to the one presented above. Confirm the accurate concepts that students convey. Then say: <strong>"Review the problems you just received. Consider how the example word problems can be solved. Do you need to add or subtract rational numbers? How will you handle this for fractions with unlike denominators or mixed numbers?"</strong><br><br><strong>Activity 1: Write-Pair-Share</strong><br><br>Ask the whole class to come up with some real-world examples of rational number addition and subtraction. Students should make a list of at least five real-world scenarios and provide one word problem. Ask students to discuss their ideas with a partner. Allow students approximately 5 minutes to exchange contexts and word problems. During this time, each partner may ask questions to the other partner. Then the whole class can rejoin. One member from each partner group will share the list of real-world contexts and word problems to the class. The teacher may want to post real-world scenarios and word problems in a file on the class Web page or use them as a classroom display. The student examples would then serve as a reference tool.&nbsp;<br><br>Students should complete the Lesson 2 Exit Ticket (M-7-5-2_Exit Ticket and KEY) at the end of the lesson to assess their level of understanding.&nbsp;<br><br><strong>Extension:&nbsp;</strong><br><br>Use the suggestions in the Routine section to go over lesson concepts throughout the school year. Use the small-group suggestions to any students who could benefit from additional teaching. Use the Expansion section to challenge students who are ready to go above and beyond the requirements of the standard.<br><br><strong>Routine:</strong> Throughout the school year, encourage students to look for real-world scenarios in which rational numbers are added or subtracted. Students can present their problems to the teacher, who will help the class participate in solving the rational number problem.<br><br><strong>Small Groups:</strong> Students who require further practice can be divided into small groups to complete the Lesson 2 Small-Group Practice worksheet (M-7-5-3_Small Group Practice and KEY). Students can work on matching together or individually, 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 2 Expansion Worksheet (M-7-5-2_Expansion and KEY). The worksheet offers increasingly challenging numeric expressions that use rational numbers.</p>