KEY TERMS integers ellipsis rational numbers Density Property
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1 What s in a Name? Rational Number System 3 WARM UP Represent each decimal or percent as a fraction in lowest terms. a. 0.3 b. 2.8 c. 3 4 % d. 212% LEARNING GOALS Classify numbers according to their number systems. Apply and extend an understanding of whole numbers and integers to the system of rational numbers. Understand ordering of rational numbers. KEY TERMS integers ellipsis rational numbers Density Property You use many different types of numbers in math class and in the world, including whole numbers, fractions, and decimals, both positive and negative. How can you organize and classify different types of numbers? LESSON 3: What s in a Name? M4-35
2 Getting Started Sort It Out! Cut out the cards found at the end of the lesson. Then, analyze and sort the numbers into different groups. You may group them in any way you feel is appropriate, but you must sort the numbers into more than one group. 1. For each of your groups, create a title that fits the numbers in that group. list the numbers included. write a rationale for why you group those particular numbers. 2. Compare your sort with your classmates sorts. Create a list of the different ways your class grouped the numbers. M4-36 TOPIC 1: Signed Numbers
3 ACTIVITY 3.1 Analyzing Number Sorts NOTES 1. Suzanne grouped these numbers together. Why do you think she put these numbers in the same group? 0, 2452, 9, 24, 23, 23, 2(29), Zane had a group similar to Suzanne s but he did not include 2452 and 23. Why do think Zane omitted these numbers from his group? 3. Amelia said that she created two groups: Group 1 contains all the numbers that can be written as fractions and Group 2 contains all the numbers that cannot be written as fractions. Analyze Amelia s sorting idea. a. Which numbers do you think Amelia placed in Group 2? b. Justine is not sure about Amelia s sort. She thinks that all of the numbers can be written as fractions. Is Justine correct? Explain why or why not. LESSON 3: What s in a Name? M4-37
4 You have used different sets of numbers, including the set of natural, or counting, numbers and the set of whole numbers. 4. Identify the numbers from the sort that are in each set. a. natural numbers b. whole numbers Notice the three periods before and after specific numbers in the set. These three periods are called an ellipsis, and they are used to represent infinity in a number set. Throughout this topic, you have been learning about the set of integers. Integers are the set of whole numbers with their opposites. The integers can be represented by the set {..., 23, 22, 21, 0, 1, 2, 3, }. 5. Identify the numbers from the sort that are included in the set of integers. You have also worked with rational numbers throughout this year. Rational numbers are the set of numbers that can be written as a b, where a and b are integers and b Þ Identify the numbers from the sort that are included in the set of rational numbers. M4-38 TOPIC 1: Signed Numbers
5 ACTIVITY 3.2 Classifying Numbers There are many ways you can classify numbers. As you saw in the previous activity, many of the classifications are subsets of other classifications. The diagram shows the different sets of numbers you have encountered in your mathematical experiences. Natural numbers are a subset of whole numbers. Whole numbers are a subset of integers. Rational Integers Whole Natural Integers are a subset of rational numbers. Pin the number on the bullseye! Your teacher will direct students to pin (or tape) a number card to its correct location in the diagram of the rational number set. 1. For each value, check all of the number sets to which it belongs. Number Natural Number Whole Number Integer Rational Number ,364,698 LESSON 3: What s in a Name? M4-39
6 2. Complete the table with the missing examples and descriptions. Natural Numbers Whole Numbers Integers Rational Numbers Examples 1, 2, 3,... Description Counting numbers Natural numbers and 0, 23, 22, 21, 0, 1, 2, 3,... ACTIVITY 3.3 Density The Density Property states that between any two rational numbers there is another rational number. The property is not true for natural numbers, whole numbers, or integers. For example, there is no integer between 25 and 26. There is no whole number or natural number between 12 and Plot the given rational numbers. Then plot and label a rational number between each pair of rational numbers. a and b. 5.5 and c and d and M4-40 TOPIC 1: Signed Numbers
7 Complete each rational number line with a partner. 2. Create a number line from 0 to 1. Your goal is to plot and label a rational number closer to 1 than your partner. Partner 1: Plot a rational number, A, between 0 and 1 that is close to 1. Partner 2: Plot a rational number, B, between A and 1. Repeat at least 2 more times. 3. Create a number line from 21 to 0. Your goal is to plot and label a rational number closer to 0 than your partner. Partner 1: Plot a rational number, A, between 21 and 0 that is close to 0. Partner 2: Plot a rational number, B, between A and 0. Repeat at least 2 more times. 4. Create a number line from 26 to 25. Your goal is to plot and label a rational number closer to 25 than your partner. Partner 1: Plot a rational number, A, between 26 and 25 that is close to 25. Partner 2: Plot a rational number, B, between A and 25. Repeat at least 2 more times. LESSON 3: What s in a Name? M4-41
8 NOTES TALK the TALK Do They Always Belong? Determine if each statement is true or false. Justify your answer using definitions and/or examples. 1. True False All whole numbers are rational numbers. 2. True False All rational numbers are whole numbers. 3. True False All rational numbers are integers. 4. True False All integers are rational numbers. 5. True False All whole numbers are integers. 6. True False All integers are whole numbers. M4-42 TOPIC 1: Signed Numbers
9 % % (29) % % 0.25% LESSON 3: What s in a Name? M4-43
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11 Assignment Write Define each term in your own words. Remember Rational numbers include all numbers that can be written in the form a, where a and b are integers and b is not zero. b 1. The set of rational numbers 2. The Density Property Practice 1. Write all the sets of numbers to which each value belongs. a. The tundra covers about 1 of Earth s surface. 5 b. The average annual temperature is 2188 Fahrenheit. c. There are 48 varieties of land mammals found in the tundra region. d. The permafrost is a layer of frozen soil that is located below Earth s surface at feet. e. During the summer months, the low temperature averages about F. 2. Nadine collects data about some animals. Determine a rational number between each pair of rational numbers. Plot all three numbers on a number line. a. A mole s runway is between 23 and 212 inches in the ground b. The musky rat kangaroo weighs between 3 4 and 3 2 pound c. The percent of change of the Alaskan polar bear population in the past year was between and LESSON 3: What s in a Name? M4-45
12 Stretch Are there more integers or more natural numbers? Even though there are infinitely many of both, it seems like there should be more integers than natural numbers. But, actually, there are just as many integers as there are natural numbers! If you can show how to assign an integer to every natural number, you will demonstrate that the two sets of numbers are equal. How do you think this can be done? Review 1. Write an absolute value expression to calculate the answer to each question. a. The temperature at 9:00 A.M. was 408. The temperature at 2:00 P.M. was What was the change in temperature? b. You began your hike at 30 feet below sea level. You are now at 200 feet. How far have you hiked? 2. Complete the table for the equation w 5 m 9.2. m w Plot each ordered pair on a coordinate plane. a. (2, 4) b. (5.5, 1.75) c. (4 2 5, ) M4-46 TOPIC 1: Signed Numbers
Fractions and decimals have opposites, just as integers do. For example, 5 8 and 2 5
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