<p>In this lesson, students add and subtract rational numbers using a number line. Students will:&nbsp;<br>- define and discuss the concept of rational numbers.&nbsp;<br>- to add and subtract rational numbers, use a number line.&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?&nbsp;<br>- What makes a tool and/or strategy suitable for a certain task?&nbsp;<br>&nbsp;</p>

<p>- Irrational Number: A real number that cannot be expressed as a fraction, terminating decimal, or repeating decimal.<br>- 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 1 Exit Ticket (M-7-5-1_Exit Ticket and KEY)&nbsp;<br>- Lesson 1 Small-Group Practice worksheet (M-7-5-1_Small Group Practice and KEY)&nbsp;<br>- Lesson 1 Expansion Worksheet (M-7-5-1_Expansion and KEY)&nbsp;<br>- sticky notes</p>

<p>- Activity 1 can be used to examine students' understanding of rational numbers, including their relationship to irrational numbers and the real number system.<br>- Students' performance in Activity 3 can be used to assess their comprehension of how to model addition and subtraction of rational numbers.&nbsp;<br>- Use the Lesson 1 Exit Ticket (M-7-5-1_Exit Ticket and KEY) to quickly evaluate student mastery.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction&nbsp;<br>W: Students will learn to add and subtract rational numbers using a number line.&nbsp;<br>H: Begin the lesson by asking students to consider the definition, appearance, and location of rational numbers. Students will learn how rational numbers relate to other numbers before using the number line to display sums and differences between rational numbers.&nbsp;<br>E: The lesson focuses on modeling the sums and differences of rational numbers on a number line. Students will begin by calculating the sums and differences of decimal rational numbers, then move on to sums and differences of fractional rational numbers (with like denominators) and then to sums and differences of fractional rational numbers (with unlike denominators). After walking students through numerous example problems they will apply their modeling abilities in the final activity, which will involve sharing their number lines and answers with other group members.&nbsp;<br>R: Opportunities for discussion start at the beginning of the lesson, with Activity 1. Throughout the lesson, students solve problems and answer questions, prompting them to reconsider and revise their understanding of the concepts of the lesson.&nbsp;<br>E: Before the end of the class, distribute exit tickets to assess students' level of understanding. Exit tickets can be collected from students as they exit the classroom.&nbsp;<br>T: Extension lessons can be adapted to meet the needs of each class. The Routine section includes options for reviewing lesson concepts at various times throughout the year. Students who need more support can work in small groups to complete the Lesson 1 Small-Group Practice worksheet. Students who want to practice adding and subtracting rational numbers on a number line can complete the Lesson 1 Expansion Worksheet. Two challenge questions are also provided.&nbsp;<br>O: The lesson is scaffolded so that students first assign meaning to the idea of a rational number in the context of the real number system, followed by learning how to add and subtract rational numbers using the number line. This lesson serves as an introduction to addition and subtracting rational numbers. The next two lessons in the unit will expand on these topics by doing more rational number computations, including all four operations, as well as demonstrating how properties of operations can be used to solve real-world problems with rational numbers.&nbsp;</p>

<p>Start the lesson by writing the definition of a rational number on the board:&nbsp;<br><br><strong>"A rational number is any number that can be expressed as the ratio of two integers, a and b, where b is not equal to 0. In other words, a rational number is any number that can be written as \(a \over b\), b ≠ 0."</strong>&nbsp;<br><br><strong>Activity 1: Think-Pair-Share about Rational Numbers</strong>&nbsp;<br><br>Ask them the following questions: <strong>"Given the definition of a rational number, what does a rational number look like? Where do we find rational numbers? What kinds of numbers are rational vs. not rational?"</strong><br><br>Have students work in pairs. Allow students a few minutes to brainstorm their answers to the questions. Then, have each partner present his or her ideas to the other partner. After about 3 to 5 minutes, ask one member of each pair to express their thoughts on the definition of a rational number, appearance of a rational number, and locations of rational numbers. Encourage discussion and debate over each partner's presentation. If a student in the class disagrees with a particular definition (or statement), make him/her explain why. Request that the student proposing the debated definition (or statement) provide support and reasoning for his or her ideas.<br><br><br><strong>Rational Numbers</strong><br><br><strong>“Essentially, a rational number is any number that can be expressed as a fraction. This does not imply that rational numbers are always expressed as fractions; rather, they can be. Let's discuss the types of numbers that can be written as fractions, even if they don't look like fractions.”</strong></p><h4><i><strong>Natural Numbers</strong></i></h4><p><strong>"Can somebody tell us what a natural number is? Provide an example of a natural number."</strong> (<i>The natural numbers are a collection of positive counting numbers that begin with 1; {1, 2, 3, 4, …} An example of a natural number is 4.</i>) <strong>"Can the natural number 4 be written as a fraction?"</strong> (<i>Yes, it can be written as \(4 \over 1\)</i>) <strong>“If you give natural numbers a denominator of 1, they can all be written in fraction form, which means that natural numbers are all rational numbers.”</strong></p><h4><i><strong>Whole Numbers</strong></i></h4><p><strong>"Can somebody remind us what a whole number is? Provide an example of a whole number."</strong> (<i>The whole numbers are the set of positive numbers that begin with 0; {0, 1, 2, 3, …}. One example of a whole number is 12.</i>) <strong>"Can the whole number 12 be written as a fraction?"</strong> (<i>It can be written as \(12 \over 1\).</i>) <strong>“If you give whole numbers a denominator of 1, they can all be written in fraction form, which means that whole numbers are all rational numbers.”</strong></p><h4><i><strong>Integers</strong></i></h4><p><strong>"Can anybody remind us what an integer is? Provide an example of an integer."</strong> (<i>Integers are positive and negative "whole" numbers: {…, −3, −2, −1, 0, 1, 2, 3, …}. An example of an integer is -5.</i>) <strong>"Can the integer −5 be written as a fraction?"</strong> (<i>Yes, it can be written as \(-5 \over 1\).</i>) <strong>“If you give integers a denominator of 1, they can all be written in fraction form, which means that integers are all rational numbers.”</strong></p><h4><i><strong>Terminating Decimals</strong></i></h4><p><strong>"Can somebody tell us what a terminating decimal is? Provide an example of a terminating decimal."</strong> (<i>A terminating decimal is a decimal number that contains a finite number of digits. An example of a terminating decimal is 0.25.</i>) <strong>"Can the terminating decimal 0.25 be written as a fraction?"</strong> (<i>Yes, 0.25 is equivalent to the fraction \(1 \over 4\).</i>) <strong>"What about the terminating decimal 0.345? Can this also be written as a fraction? How?"</strong> (<i>0.345 can be written as a fraction if you realize that it literally reads as "three hundred forty-five thousandths" or \(345 \over 1000\).</i>) <strong>“All terminating decimals can be represented as fractions by placing the digits over the place value. This means that all terminating decimals are rational numbers.”</strong></p><h4><i><strong>Repeating Decimals</strong></i></h4><p><strong>"Can anyone remind us what a repeating decimal is? Provide an example of a repeating decimal."</strong> (<i>A repeating decimal is a decimal number in which the decimal digits repeat indefinitely. An example of a repeated decimal is 0.33333…</i>). <strong>"Can the repeating decimal be written as a fraction?"</strong> (<i>Yes, 0.33333… is equivalent to the fraction \(1 \over 3\)</i>) <strong>"What about the repeating decimal 0.45454545…?" Can this also be expressed as a fraction? How?"</strong> (<i>Yes, 0.45454545… can be expressed as the fraction \(45 \over 99\).</i>)<span style="color:#38761d;">*</span> <strong>"Although the method for doing so is a bit more complex, all repeating decimals can be written as fractions, and therefore all repeating decimals are rational numbers."</strong><br><br><span style="color:#38761d;"><i>*The process of converting a repeating decimal to a fraction might be a bit complicated. The following examples and the subsequent "9 trick" can be used for repeating decimals with digits that repeat directly after the decimal point. You may or may not want to discuss this with your students right now. The main point here is to show that all repeating decimals may be expressed as fractions.</i></span></p><figure class="image"><img style="aspect-ratio:652/357;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_49.png" width="652" height="357"></figure><p>&nbsp;</p><p><strong>Irrational Numbers</strong><br><br><strong>"As previously explained, the set of rational numbers contains naturals, wholes, integers, terminating and repeating decimals, and, of course, fractions, because all of these types of numbers can be expressed in fraction form. This leads us to the question: 'What kinds of numbers are not rational numbers?' Who can give an example?"</strong> (Let students try to identify an irrational number. As they do so, either demonstrate how their example is in fact rational, or applaud them on identifying the only type of number not previously discussed—nonterminating, nonrepeating decimals.)<br><br><strong>"Let's consider the number π. In decimal form, π looks like this: 3.141592654… Can we refer to π as a terminating decimal?"</strong> (<i>No, the digits do not end.</i>) <strong>"Can we call π a repeating decimal?"</strong> (<i>No, the digits do not follow a repeating pattern.</i>) <strong>"Can we write π as a fraction?"</strong> (<i>No, there is no finite place value to make the denominator, and it is not a repeating decimal.</i>) <strong>"As a nonterminating, nonrepeating decimal, π cannot be expressed as a fraction. This is precisely why we gave the number pi a symbol so that we can talk about it without having to write or say all of those decimal values. More importantly, this shows that π is an example of a number that is not rational. Numbers that are not rational are called irrational numbers. "Who can give me another example of an irrational number?"</strong> (<i>Lead students to offer any decimal values whose digits continue infinitely in a random fashion, such as 0.35767823345… or 5.6121970283…</i>)&nbsp;<br><br><br><strong>Activity 2: The Real Number System&nbsp;</strong><br><br>Show the following diagram to students. If the image cannot be projected directly onto the whiteboard or onto an interactive whiteboard, immediately draw a larger version of the diagram on the board.&nbsp;<br><br><strong>“This diagram can be used to visually represent the real number system. Real numbers are made up of both rational and irrational numbers. As we have discussed, the set of rational numbers includes the subsets of naturals, wholes, and integers.”</strong></p><figure class="image"><img style="aspect-ratio:504/288;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_50.png" width="504" height="288"></figure><p>Give all students a sticky note and invite them to write down any number they choose. (Encourage a variety of number forms!) Then have students exchange sticky notes with a partner. After students have received their sticky note numbers, have them take turns coming to the board and sticking their numbers in the correct location on the Real Number System diagram. (Promote class discussion and "help" with numbers that do not end up in the correct location.)&nbsp;<br><br><strong>"A number line is another common way for demonstrating the real number system. A number line, like the diagram, is made up of both rational and irrational numbers, representing the real number system."</strong>&nbsp;<br><br>Display a number line.</p><figure class="image"><img style="aspect-ratio:387/74;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="387" height="74"></figure><p>Ask students to find two rational numbers on the number line. Then ask them to discover at least one irrational number in between the two rational numbers. Ask students to explain how many irrational numbers may be found between these rational numbers. Have them explain their thinking.&nbsp;<br><br><strong>"Believe it or not, a simple number line can help us add and subtract all types of rational numbers. This part of the lesson teaches you how to use a number line to represent the addition and subtraction of rational numbers. Let's quickly go over the procedure of adding on a number line. Then we'll look at some examples."</strong><br><br><br><br><u>Steps to Add (or Subtract) on a Number Line:</u>&nbsp;<br><br>1. If the number sentence contains a subtraction symbol, rewrite the subtraction as addition by "adding the opposite."<span style="color:#38761d;">*</span><br>2. Locate the first number on the number line.&nbsp;<br>This is where you start.&nbsp;<br>3. Add the second number.&nbsp;<br>4. If the second number is positive, move that many spaces to the right.&nbsp;<br>5. If the second number is negative, move that many spaces to the left.&nbsp;<br>6. Determine the number you land on. This is your answer.<br><br><span style="color:#38761d;"><i>* If students are unfamiliar with this strategy, go over how to rewrite subtraction problems as addition problems (or "add the opposite") before moving on.</i></span><br><br><br><u>Examples with Decimals</u><br><br><strong>Example 1: 8 – 3.5</strong></p><figure class="image"><img style="aspect-ratio:487/125;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_52.png" width="487" height="125"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>8 – 3.5 = 4.5</strong></span><br><br><br><strong>Example 2: -2.6 + - 3.1</strong></p><figure class="image"><img style="aspect-ratio:482/136;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_53.png" width="482" height="136"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>-2.6 + - 3.1 = -5.7</strong></span><br><br><br><strong>Example 3: -7.25 + 2.75</strong></p><figure class="image"><img style="aspect-ratio:484/121;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_54.png" width="484" height="121"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>-7.25 + 2.75 = - 4.5</strong></span><br><br><br><strong>“Now, on your paper, represent the following sums and differences using number lines:”</strong></p><ul><li>8.25 + 1.75</li><li>−9.5 + (−3.5)</li><li>6.5 + (−4.5)</li><li>5.4 − 2.1</li></ul><p><br><br><u>Examples with Fractions</u><br><br><strong>Example 4: \(1 \over 6\) + \(4 \over 6\)</strong></p><figure class="image"><img style="aspect-ratio:485/137;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_55.png" width="485" height="137"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>\(1 \over 6\) + \(4 \over 6\) = \(5 \over 6\)</strong></span><br><br><br><strong>Example 5: \(1 \over 5\) – \(3 \over 5\)</strong></p><figure class="image"><img style="aspect-ratio:467/138;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_56.png" width="467" height="138"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>\(1 \over 5\) – \(3 \over 5\)</strong> = <strong>\(-2 \over 5\)</strong></span><br><br><br><strong>Example 6: -\(2 \over 3\) + \(11 \over 12\)</strong></p><figure class="image"><img style="aspect-ratio:483/193;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_57.png" width="483" height="193"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>-\(2 \over 3\) + \(11 \over 12\)</strong> = <strong>\(1 \over 4\)</strong></span><br><br><br><strong>Example 7: 9 + (-\(1 {5 \over 8} \))</strong></p><figure class="image"><img style="aspect-ratio:468/136;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_58.png" width="468" height="136"></figure><p><span style="color:hsl( 0, 100%, 50% );"><strong>9 + (-\(1 {5 \over 8} \))</strong> = <strong>\(7 {3 \over 8} \)</strong></span><br><br><br><strong>“Now, on your paper, represent the following sums and differences using number lines:”</strong></p><ul><li>\(2 \over 3\) – \(2 {5 \over 6} \)</li><li>4 + \(1 {3 \over 8} \)</li><li>-\(3 {1 \over 5} \) – \(7 \over 15\)</li></ul><p>&nbsp;</p><p><strong>Activity 3: Adding and Subtracting on a Number Line</strong><br><br>On each of 30 slips of paper, write one expression involving addition or subtraction of rational numbers and place the slips into a bag. Have each student create one slip of paper and model the problem with a number line. Then, divide students into groups of three and ask them to discuss their modeling process and answers with the other students in their group. The students' number lines can be scanned and saved as a PDF document, which can then be uploaded to the class website or posted as a classroom display.<br><br>Distribute the Lesson 1 Exit Ticket (M-7-5-1_Exit Ticket and KEY) at the end of the lesson to assess students' understanding.<br><br><br><strong>Extension:</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 identify students who could benefit from further teaching. Use the Expansion section to challenge students who are ready to move beyond the requirements of the standard.&nbsp;<br><br><strong>Routine:</strong> Throughout the school year, ask students to recognize rational numbers in the real world, especially situations where rational numbers need to be added or subtracted. Students will also gain more practice with this concept in Lessons 2 and 3 of this unit, when they will find sums and differences of rational numbers in real-world scenarios.<br><br><strong>Small Groups:</strong> Students who require further practice can be divided into small groups to work on the Lesson 1 Small-Group Practice worksheet (M-7-5-1_Small Group Practice and KEY). Students can work on the worksheet 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 Expansion Worksheet (M-7-5-1_Expansion and KEY). The worksheet provides more practice with addition and subtraction of rational numbers, as well as challenging problems.</p>