• Activity 1 can be used as an initial assessment of students’ understanding of the concept of rational numbers and the relation of rational numbers to irrational numbers and to the real number system.
• Student performance during Activity 3 can be used to gauge students’ level of understanding of how to model addition and subtraction of rational numbers.
• Use the Lesson 1 Exit Ticket (M-7-5-1_Exit Ticket and KEY.docx) to quickly evaluate student mastery.

Begin the lesson by writing the following definition of a rational number on the board:

“A rational number is any number that can be written as the ratio of two integers, a and b, where b does not equal 0. In other words, a rational number is any number that can be written as .”

Activity 1: Think-Pair-Share about Rational Numbers

Ask students the following questions: “Given the definition of a rational number, what does a rational number look like? Where can we find rational numbers? What kinds of numbers are rational vs. not rational?”

Have each student partner with another student. Give students a couple of minutes to brainstorm their answers to the questions. Then, have each partner share his or her ideas with the other partner. After about 3 to 5 minutes, have one member from each pair share their thoughts on the definition of a rational number, appearance of a rational number, and locations of rational numbers. Encourage discussion and debate with each partner presentation. If a student in the class disagrees with a given definition (or statement), have the student give his/her reasoning. Have the student offering the debated definition (or statement) provide support and justification for his/her ideas.

Rational Numbers

“Essentially, a rational number is any number that can be written as a fraction. This does not mean, however, that rational numbers are always written as fractions—simply that they can be. Let’s talk about the kinds of numbers that can be written as fractions, even though they might not look like fractions.”

• Natural Numbers

“Who can remind us what a natural number is? Give an example of a natural number.” (The natural numbers are the set of positive counting numbers, beginning with 1; {1, 2, 3, 4, …} An example of a natural number is 4.) “Can the natural number 4 be written as a fraction?” (Yes, it can be written as .) “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.”

• Whole Numbers

“Who can remind us what a whole number is? Give an example of a whole number.” (The whole numbers are the set of positive numbers, beginning with 0; {0, 1, 2, 3, …} An example of a whole number is 12.) “Can the whole number 12 be written as a fraction?” (Yes, it can be written as .) “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.”

• Integers

“Who can remind us what an integer is? Give an example of an integer.” (Integers are the set of positive and negative “whole” numbers; {…, −3, −2, −1, 0, 1, 2, 3, …}. An example of an integer is −5.) “Can the integer −5 be written as a fraction?” (Yes, it can be written as .) “If you give integers a denominator of 1, they can all be written in fraction form, which means that integers are all rational numbers.”

• Terminating Decimals

“Who can remind us what a terminating decimal is? Give an example of a terminating decimal.” (A terminating decimal is a decimal number with a finite number of digits. An example of a terminating decimal is 0.25. ) “Can the terminating decimal 0.25 be written as a fraction?” (Yes, 0.25 is equivalent to the fraction .) “What about the terminating decimal 0.345? Can this also be written as a fraction? How?” (0.345 can be written as a fraction by remembering that 0.345 is literally read as “three hundred forty-five thousandths” or .) “By placing the digits of a terminating decimal over their place value, all terminating decimals can be written as a fraction. This means all terminating decimals are rational numbers.”

• Repeating Decimals

“Who can remind us what a repeating decimal is? Give an example of a repeating decimal.” (A repeating decimal is a decimal number whose decimal digits repeat infinitely. An example of a repeating decimal is 0.33333…. or ) “Can the repeating decimal  be written as a fraction?” (Yes,  is equivalent to the fraction ) “What about the repeating decimal 0.45454545… or ? Can this also be written as a fraction? How?” ( can be written as the fraction .)* “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.”

*The process for converting a repeating decimal to a fraction can be a bit complex. For repeating decimals whose digits repeat immediately following the decimal point, the following examples and subsequent “9 trick” can be used. You may or may not choose to discuss this with your students at this time. The important thing here is to illustrate that all repeating decimals can be written as fractions.

Irrational Numbers

“As we have discussed, the set of rational numbers includes the set of naturals, wholes, integers, terminating and repeating decimals, and, of course, fractions, because all of these types of numbers can be written in fraction form. This leads us to the question: ‘What kind of numbers are not rational numbers?’ Who can give an example?” (Allow students to try to identify an irrational number. As they do this, either show how their example is in fact rational, or applaud their ability to identify the only type of number not previously discussed—decimals that are nonterminating and nonrepeating.)

“Let’s consider the number π. In decimal form, π looks like this: 3.141592654… Can we call π a terminating decimal?” (No, the digits do not end.) “Can we call π a repeating decimal?” (No, the digits do not form a repeating pattern.) “Can we write π as a fraction?” (No, there is no finite place value to make the denominator and it is not a repeating decimal.) “Because π is a nonterminating, nonrepeating decimal, it cannot be written as a fraction. This is precisely why we have assigned the number pi a symbol so we can talk about it without writing/saying all those decimal values. More importantly, this means 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 an example of another irrational number?” (Lead students to offer any decimal values whose digits continue infinitely in a random fashion, such as 0.35767823345… or 5.6121970283…)

Activity 2: The Real Number System

Display the following diagram for students. If the image cannot be projected directly onto the whiteboard or onto an interactive whiteboard, quickly draw a large version of the diagram on the board.

“This diagram can be used as a visual representation of the real number system. The real numbers are made up of the set of rational numbers and irrational numbers. As we have discussed, the set of rational numbers includes the subsets of naturals, wholes, and integers.”

Give all students a sticky note and ask them to write down any number they wish. (Encourage a variety of number forms!) Then instruct students to exchange sticky notes with a partner. Once students receive their sticky note numbers, have them take turns coming to the board and sticking their numbers in the appropriate location on the Real Number System diagram. (Encourage class discussion and “help” for numbers that do not end up in the correct location.)

“Another common way to illustrate the real number system is by using a number line. Like the diagram, a number line is also made up of the set of rational numbers and irrational numbers, thus representing the real number system.”

Display a number line:

Ask students to locate two rational numbers on the number line. Then have them find at least one irrational number in between the two rational numbers. Ask students to explain the total number of irrational numbers that can be found between these rational numbers. Have them explain their thinking.

“Believe it or not, a simple number line can be used to help us add or subtract with all types of rational numbers. In this part of the lesson, you will learn how to represent the addition and subtraction of rational numbers using a number line. Let’s quickly review the process of adding on a number line. Then we will look at some examples.”

Steps for Adding (or Subtracting) on a Number Line:

1. If the number sentence uses a subtraction symbol, rewrite the subtraction as addition by “adding the opposite.”*
2. Locate the first number you see on the number line.

This is where you start.

1. Add the second number.
   1. If the second number is positive, move that many spaces to the right.
   2. If the second number is negative, move that many spaces to the left.
   3. Identify the number you land on. This is your answer.

Examples with Decimals

• Example 1:           8 – 3.5

8 − 3.5 = 4.5

• Example 2:           −2.6 + −3.1

−2.6 + −3.1 = −5.7

• Example 3:           −7.25 + 2.75

−7.25 + 2.75 = −4.5

“Now, on your paper, represent the following sums and differences using number lines:”

• 8.25 + 1.75
• −9.5 + (−3.5)
• 6.5 + (−4.5)
• 5.4 − 2.1

Examples with Fractions

• Example 4:          

• Example 5:          

• Example 6:          

• Example 7:          

“Now, on your paper, represent the following sums and differences using number lines:”

Activity 3: Adding and Subtracting on a Number Line

Write one expression involving addition or subtraction of rational numbers on each of 30 slips of paper, and place the slips into a bag. Have each student draw out one slip of paper and model the problem using a number line. Then arrange students in groups of three and ask them to share their modeling process and answer with the other students in their group. The students’ number lines can be scanned into a PDF document and uploaded to the class Web site or posted as a classroom display.

Distribute the Lesson 1 Exit Ticket (M-7-5-1_Exit Ticket and KEY.docx) at the close of the lesson to evaluate students’ level of understanding.

Extension:

Use the suggestions in the Routine section to review lesson concepts throughout the school year. Use the Small Group suggestions for any students who might benefit from additional instruction. Use the Expansion section to challenge students who are ready to move beyond the requirements of the standard.

• Routine: During the school year, have students identify rational numbers in the real world, including circumstances where rational numbers need to be added or subtracted. Students will also get additional practice with this skill in Lessons 2 and 3 of this unit, as they find sums and differences of rational numbers, given real-world contexts.
• Small Groups: Students who need additional practice can be pulled into small groups to work through the Lesson 1 Small-Group Practice worksheet (M-7-5-1_Small Group Practice and KEY.docx). Students can work on the worksheet together or work individually and compare answers when done.
• Expansion: Students who are prepared for a greater challenge can be given the Expansion Worksheet (M-7-5-1_Expansion and KEY.docx). The worksheet includes more practice with addition and subtraction of rational numbers and includes challenge questions.