Fractions and decimals have opposites, just as integers do. For example, 5 8 and 2 5
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1 Domain 1 Lesson Rational Numbers Common Core Standards: 6.NS., 6.NS.6.c, 6.NS.7.c Getting the Idea A rational number is a number that can be expressed as the ratio of two integers in the form a, where b is not equal to 0. A rational number can be positive or negative. The set b of rational numbers includes integers, fractions, mixed numbers, percents, terminating decimals, and repeating decimals. Some examples of rational numbers are shown below. 8% Fractions and decimals have opposites, just as integers do. For example, 8 and 8 are opposites, and so are. and.. You can see rational numbers in many real-world situations, such as a sheet of paper that is 8 1 inches wide or a dog that weighs 9.1 kilograms. Example 1 Explain why 4,, and 0.9 are rational numbers. Strategy Step 1 Step Step Solution Express the numbers in the form a b. Show that 4 is a rational number , which is in the form a b. Show that is a rational number. is in the form a b. Show that 0.9 is a rational number , which is in the form a b. The numbers 4,, and 0.9 are rational numbers because each can be written in the form a b. 47
2 Remember that the absolute value of a number is the distance of that number from 0 on a number line. You can use absolute value to help you locate a rational number on a number line. Example Plot the rational numbers and on the number line shown below Strategy Step 1 Step Step Identify what fractional units the number line is divided into. Then use absolute value to plot each point. Determine what each mark on the number line stands for. There are spaces between 0 and 1. The number line is divided into fifths. Each mark stands for 1. Find the absolute value of. or fifths Plot on the number line. is three units away from 0 on this number line divided into fifths. Since is positive, count units to the right of 0. Plot the point. Step 4 Step Solution Find the absolute value of. or fifths Plot on the number line. is two units away from 0 on this number line divided into fifths. Since is negative, count units to the left of 0. Plot the point The rational numbers and are shown on the number line in Step above. 48 Domain 1: The Number System
3 Example Lesson : Rational Numbers The floor of the valley in which Griffin lives is 7 1 feet below sea level. Write that elevation as a rational number. Then plot a point for it on the number line below Strategy Step 1 Step Step Step 4 Write a rational number representing 7 1 feet below sea level. Then identify the value of each mark on the number line. Determine whether the elevation is a positive or negative number. In this case, the number 0 represents sea level. The valley floor is below sea level, so the number will be less than 0, or negative. The rational number 7 1 represents the elevation. Determine what each mark on the number line stands for. There are 0 spaces between 0 and 10 and 0 spaces between 0 and 10. Each mark between integers stands for 1. Find 7 on the number line. Since 7 is a negative number, it will be to the left of 0 on the number line. Plot 7 1 on the number line , so 7 1 will be farther from 0 than 7. Plot a point for 7 1 at the mark to the left of 7. Solution The rational number 7 1 represents the elevation. The location of 7 1 on a number line is shown in Step 4 above. 49
4 Example 4 What decimals do points P and Q represent on the number line shown? P Q Strategy Step 1 Step Identify what units the number line is divided into. Determine what each mark on the number line stands for. There are 10 spaces between 0 and 1. The number line is divided into tenths. Each mark stands for 0.1. Find the number of tenths that points P and Q represent. Count from 0. It may help to label the marks as shown below. P Q Solution Point P represents 0.8 and point Q represents 0.. Coached Example The number line below shows points for several rational numbers. Which points represent 1, 4,, and 1? A B C D There are spaces between 0 and 1. The number line is divided into. Each mark stands for. 1 is between and. It is located at point. 4 is between and. It is located at point. is between and. It is located at point. 1 is between and. It is located at point. Point represents 1, point represents 4, point represents, and point represents 1. 0 Domain 1: The Number System
5 Lesson : Rational Numbers Lesson Practice Choose the correct answer. Use the number line for questions 1 and. E F G H Which point on the number line represents 0.? A. point E B. point F C. point G D. point H. Which rational number is represented by point E? A B. 0. C D. 0.8 Use the number line for questions and 4. J K L M Which point on the number line represents _ 1 4? A. point J B. point K C. point L D. point M 4. Which rational number is represented by point J? A. 1 _ 4 B. 1 _ 1 _ C. 4 D. 1 _ 1
6 Use the number line for questions and 6. P Q R S Which point on the number line represents an integer? A. point P B. point Q C. point R D. point S 6. Which rational number is represented by point R? A. 1.6 B. 0.6 C. 0.6 D What is the opposite of _ 8? A. 8 B. _ 8 _ C. 8 D. 8 _ 8. Use the sets counting numbers, integers, and rational numbers to answer Part A. A. Name the set or sets that each of the numbers below belongs to. 4, _,, and.9 8 B. Explain why a mixed number is a rational number. Domain 1: The Number System
7 Lesson : Rational Numbers 9. Draw a line from each point to its location. A B C D A. point A 0.7 B. point B 0.1 C. point C 0.7 D. point D Circle every rational number that is represented by a letter on the number line below. W X Y Z A. 0.7 B. 0.6 C. 0. D. 0.4 E. 0. F. 0.1 G. 0. H Select True or False for each statement. F G H A. Point F is located at 1 1 _. True False B. Point G is located at _. 4 True False C. Point H is located at _ 1. 4 True False D. Point J is located at _ 1. True False J
8 1. Write the letter for each point in the correct box. M N P Q R S M N P Q R S Integer Not an Integer 1. Look at each statement. Is the statement true? Select Yes or No. A. The set of rational numbers includes the set of counting numbers. Yes No B. A decimal is a rational number. Yes No C. A rational number cannot be a fraction. Yes No D. All rational numbers are integers. Yes No 14. Circle the opposite of _ 9. The opposite of _ 9 is Domain 1: The Number System
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