: Using the Number Line to Model the Addition of Integers Classwork Exercise 1: Real-World Introduction to Integer Addition Answer the questions below. a. Suppose you received $10 from your grandmother for your birthday. You spent $4 on snacks. Using addition, how would you write a number sentence to represents this situation? b. How would you model your equation on a number line to show your answer? Example 1: Modeling Addition on the Number Line Complete the steps to finding the sum of 2 + 3 by filling in the blanks. Model the number sentence using straight arrows called vectors on the number line below. a. Place the tail of the arrow on. b. Draw the arrow 2 units to the left of 0, and stop at. The direction of the arrow is to the since you are counting down from 0. c. Start the next arrow at the end of the first arrow, or at. d. Draw the second arrow units to the right since you are counting up from -2. e. Stop at. Date: 9/20/13 S.7
f. Repeat the process from part (a) for the expression 3 + ( 2). g. What can you say about the sum of 2 + 3 and 3 + ( 2)? Does order matter when adding numbers? Why or why not? Example 2: Expressing Absolute Value as the Length of an Arrow on the Number Line a. How does absolute value determine the arrow length for 2? b. How does the absolute value determine the arrow length for 3? c. How does absolute value help you to represent 10 on a number line? Date: 9/20/13 S.8
Exercise 2 Create a number line model to represent each of the expressions below. a. 6 + 4 b. 3 + ( 8) Example 3: Finding Sums on a Real Number Line Model Find the sum of the integers represented in the diagram below. Write an equation to express the sum. 5 3 2 2 a. What three cards are represented in this model? How did you know? b. In what ways does this model differ from the ones we used in Lesson 1? c. Can you make a connection between the sum of 6 and where the third arrow ends on the number line? Date: 9/20/13 S.9
Lesson Summary On a number line, arrows are used to represent integers; they show length and direction. The length of an arrow on the number line is the absolute value of the integer. Adding several arrows is the same as combing integers in the Integer Game. The sum of several arrows is the final position of the last arrow. Problem Set For Questions 1 4, represent each of the following problems using both a number line diagram and an equation. 1. David and Victoria are playing the Integer Card Game. David drew three cards, 6, 12, and 4. What is the sum of the cards in his hand? Model your answer on the number line below. 2. In the Integer Card Game, you drew the cards, 2, 8, and 11. Your partner gave you a 7 from his hand. What is your new total? Model your answer on the number line below. 3. What cards would you need to get your score back to zero? Explain. Use and explain the term "additive inverse" in your answer. 4. If a football player gains 40 yards on a play, but on the next play, he loses 10 yards, what would his total yards be for the game if he ran for another 60 yards? What did you count by to label the units on your number line? 5. Find the sums. a. 2 + 9 b. 8 + 8 c. 4 + ( 6) + 10 d. 5 + 7 + ( 11) Date: 9/20/13 S.11
6. Mark an integer between 1 and 5 on a number line, and label it point. Then, locate and label each of the following points by finding the sums: a. Point : + 5 b. Point B: + ( 3) c. Point : ( 4) + ( 2) + d. Point : 3 + + 1 7. Write a story problem that would model the sum of the arrows in the number diagram below. 8. Do the arrows correctly represent the equation 4 + ( 7) + 5 = 2? If not, draw a correct model below. Date: 9/20/13 S.12
Lesson 3 Lesson 3: Understanding Addition of Integers Classwork Exercise 1: Addition Using the Integer Game Play the Integer Game with your group without using a number line. Example 1: Counting On to Express the Sum as Absolute Value on a Number Line Model of Counting Up Model of Counting Down 2 + 4 = 6 2 + ( 4) = 2 Remember that counting up 4 is the same as the opposite of counting up 4, and also means counting down 4. a. For each example above, what is the distance between 2 and the sum? b. Does the sum lie to the right or left of 2 on a horizontal number line? Vertical number line? c. Given the expression 54 + 81, can you determine, without finding the sum, the distance between 54 and the sum? Why? d. Is the sum to the right or left of 54 on the horizontal number line? On a vertical number line? Lesson 3: Understanding Addition of Integers Date: 9/20/13 S.13
Lesson 3 e. Given the expression 14 + ( 3), can you determine, without finding the sum, the distance between 14 and the sum? Why? f. Is the sum to the right or left of 14 on the number line? On a vertical number line? Work with a partner to create a horizontal number line model to represent each of the following expressions. Describe the sum using distance from the -value along the number line. 1. 5 + 3 2. 6 + ( 2) 3. 7 + ( 8) Lesson 3: Understanding Addition of Integers Date: 9/20/13 S.14
Lesson 3 Lesson Summary Addition of integers is represented on a number line as counting up, where counting up a negative number of times is the same as counting down. Arrows show the sum of two integers on a number line. The sum is the distance from the -value (the first addend) to the right if is positive and to the left if is negative. Problem Set 1. Below is a table showing the change in temperature from morning to afternoon for one week. a. Use the vertical number line to help you complete the table. As an example, the first row is completed for you. Morning Temperature Change in Temperatures from Morning to Afternoon Change Afternoon Temperature Number Sentence 1 rise of 3 4 1 + 3 = 4 10 5 2 rise of 8 2 4 fall of 6 rise of 7 0 6 fall of 9 5 fall of 5 7 fall of 7-5 -10 b. Do you agree or disagree with the statement: A rise of 7 C means a fall of 7 C? Explain. (Note: No one would ever say, "A rise of 7 degrees"; however, mathematically speaking, it is an equivalent phrase.) Lesson 3: Understanding Addition of Integers Date: 9/20/13 S.16