Domain 1 Lesson 3 Integers Common Core Standards: 6.NS.5, 6.NS.6.a, 6.NS.6.c Getting the Idea Integers include the counting numbers (1, 2, 3, ), their opposites (21, 22, 23, ), and zero. The number line below shows the integers from 25 to 5. Negative integers have values less than zero, so they are to the left of zero on the number line. Positive integers have values greater than zero, so they are to the right of zero on the number line. Zero is neither negative nor positive. Negative Positive 5 4 3 2 1 0 1 2 3 4 5 You can use integers to describe opposite situations. Here are some uses for integers: Positive integers A bank deposit (adding money to an account) An elevation above sea level A rise in temperature Negative integers A bank withdrawal (taking money out of an account) An elevation below sea level A drop in temperature Example 1 A bird is flying 25 feet above sea level and a fish is swimming 10 feet below sea level. Use integers to represent the elevation of the fish and the bird. Strategy Use an integer to describe each situation. Step 1 Step 2 What elevation would the number 0 represent? Zero represents sea level, or the surface of the water. Find a signed number for the elevation of the bird. The bird is above sea level, so use a positive number (a number greater than 0). 125 or simply 25 30
Step 3 Solution Find a signed number for the elevation of the fish. The fish is below sea level, so use a negative number (a number less than 0). 210 The elevation of the bird is 25 feet. The elevation of the fish is 210 feet. You can show negative integers by extending to the left a number line that shows the numbers (0, 1, 2, 3,...). Number lines showing positive and negative integers can be either horizontal or vertical, such as a thermometer. 5 4 3 2 1 0 1 2 3 4 5 100 F 90 80 70 60 50 40 30 20 10 0 10 20 31
Example 2 What temperatures are indicated on the Fahrenheit thermometers below? 100 F 90 80 70 60 50 40 30 20 10 0 10 20 100 F 90 80 70 60 50 40 30 20 10 0 10 20 Thermometer A Thermometer B Strategy Use integers to describe the temperature. Step 1 Find the temperature on thermometer A. The temperature is above 0, so use a positive number. Thermometer A shows a temperature of 408 F. Step 2 Find the temperature on thermometer B. The temperature is below 0, so use a negative number. Thermometer B shows a temperature of 2108F. Solution The temperature on thermometer A is 408F. The temperature on thermometer B is 2108F. Integers that are the same distance from 0 on a number line are called opposites. For example, 5 and 25 are opposites of each other. They are each the same distance from 0 on a number line, as shown below. opposites So, the opposite of 5 is 25, and the opposite of 25, written as 2(25), is 5. The opposite of 0 is 0. 32 Domain 1: The Number System
Lesson 3: Integers Example 3 Find the opposites of 6 and of 22. Strategy Step 1 Use a number line. Plot a point for 6 on a number line. Step 2 Find the integer that is the same distance from 0 in the opposite direction. The integer 6 is 6 units to the right of 0. Count 6 units to the left of 0. Plot a point. The opposite of 6 is 26. Step 3 Plot a point for 22 on a number line. Step 4 Find the integer that is the same distance from 0 in the opposite direction. The integer 22 is 2 units to the left of 0. Count 2 units to the right of 0. Plot a point. The opposite of 22 is 2. Solution The opposite of 6 is 26. The opposite of 22 is 2. 33
Coached Example On its first play of the game, a football team gained 6 yards. On its next two plays, the team lost 2 yards and then gained 7 yards. Use integers to describe these three plays. What integer represents a play in which the team neither gains yards nor loses yards? A play that gains yards would be represented by a integer. A play that loses yards would be represented by a integer. On the first play, the team gained yards. A gain of 6 yards is represented by the integer. On the second play, the team lost yards. A loss of 2 yards is represented by the integer. On the third play, the team gained yards. A gain of 7 yards is represented by the integer. The three plays can be described by the integers,, and. 34 Domain 1: The Number System
Lesson 3: Integers Lesson Practice Choose the correct answer. Use the number line for questions 1 and 2. A B C D E 10 5 0 5 1. Which point on the number line is represented by 27? A. point A B. point B C. point C D. point D 2. Which integer is represented by point E? A. 23 B. 22 C. 2 D. 3 4. Which situation would you represent with a negative integer? A. a mountain climber descending a mountain B. a price increase C. a person winning a sum of money D. an elevator going from the 2nd floor to the 5th floor 5. New Orleans has an elevation of 7 feet below sea level. How is that elevation, in feet, represented as an integer? A. 217 B. 27 C. 7 D. 177 6. What is the opposite of 213? 3. Which situation can be represented by the integer 10? A. a temperature drop of 108F B. 10 seconds before takeoff C. a growth of 10 centimeters D. a fall of 10 feet A. 31 B. 13 C. 23 D. 231 35
7. What is the opposite of 40? A. 240 B. 24 C. 4 D. 40 8. What is another way to write 2(29)? A. 299 B. 29 C. 0 D. 9 9. Use the number line below. 10 0 10 A. Plot and label point J at 28, point K at 6, point L at 21, and point M at 4. B. Point N is the opposite of point J. Plot and label point N on the number line above. Explain how you found the opposite of point J. 10. Look at the number line. A B C D E 10 5 0 Draw a line from each location to its point. A. 26 point A B. 1 point B C. 28 point C D. 4 point D E. 22 point E 5 36 Domain 1: The Number System
Lesson 3: Integers 11. Select True or False for each statement. A. A temperature drop of 8 F is represented by the integer 28. True False B. Eight miles per hour is represented by the integer 28. True False C. A growth of 8 inches is represented by the integer 28. True False D. Descending 8 feet is represented by the integer 28. True False E. Eight feet below sea level is represented by the integer 28. True False 12. Select every true statement. A B C D E 10 A. Point A is located at 29. B. Point B is located at 27. C. Point C is located at 25. D. Point D is located at 23. E. Point E is located at 1. 5 0 5 13. Minnesota has a temperature of 18 F below zero. Circle the integer that represents the temperature. 18 8 8 18 F represents the temperature. 37
14. Use a number from the box to complete each sentence. The opposite of 5 is. The opposite of 225 is. The opposite of 2(25) is. The opposite of 25 is. The opposite of 2(225) is. The opposite of 25 is. 25 5 5 25 15. Look at each statement. Is the statement true? Select Yes or No. A. The speed of a biker is represented by a positive integer. Yes No B. A price decrease is represented by a positive integer. C. The amount of money that a person earns is represented by a negative integer. D. The altitude of a scuba diver under sea level is represented by a negative integer. Yes Yes Yes No No No 16. Use a number from the box to complete each sentence. 32 The opposite of 32 is. The opposite of 223 is. 23 23 32 38 Domain 1: The Number System