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Lesson 6 Another Way of Looking at Integer Addition Problem Solving: More Dot Graphs Another Way of Looking at Integer Addition How do we show integer addition with red and black cards? We have used the number line to show how to add positive and negative numbers. Another way to help us understand adding positive and negative numbers is to use colored cards. A couple of common phrases are in the red and in the black. The word “red” is sometimes used when we talk about losing money. It is also used with negative numbers. The word “black” is sometimes used when we talk about making money or positive numbers. We use red cards to show negative numbers and black cards to show positive numbers. Here’s a simple problem. Example 1 Show 4 + 3 using cards. 4+3 Start with four cards. Because 4 is positive, these four cards are black. Next, add three cards. Because 3 is positive, these three cards are also black. + The answer is 4 + 3 = 7. 564 Unit 8 • Lesson 6 Lesson 6 Adding positive numbers using cards is nothing new, but adding positive and negative numbers with cards is different. Let’s look at a problem that uses negative numbers. Example 2 Show 5 + −2 using cards. 5 + −2 Start with five cards. These cards are black because 5 is positive. Then add −2. Use two red cards because 2 is negative. That leaves three black cards. The answer is 3. Unit 8 • Lesson 6 565 Lesson 6 To help understand what happened, let’s look at the number line again. Example 3 Show 5 + −2 using a number line. 0 1 2 3 4 5 6 7 8 9 10 Start at 0 and move 5 in a positive direction. Then add −2. Show this by moving 2 in a negative direction. Again, as was shown using the cards, the answer is 3. In Example 2, the two red cards and the two black cards cancel each other out. What that really means is when they are added together, the sum is 0: 2 + −2 = 0. We can see this on a number line. Example 4 The sum of opposites is zero. Show this on a number line. 0 1 2 3 4 We start at 0 and move 2 places in a positive direction. Then we add −2 by shifting gears and moving 2 places in a negative direction. That takes us right back to 0. When we add opposites, we get 0. Or if we’re using cards, one red card cancels out one black card because their sum is zero. Likewise, two red cards cancel out two black cards because their sum is zero, and so on. This simple concept is one of the main properties about numbers that we use over and over again in higher-level math. Apply Skills Turn to Interactive Text, page 292. 566 Unit 8 • Lesson 6 Reinforce Understanding Use the mBook Study Guide to review lesson concepts. 5 Lesson 6 Problem Solving: More Dot Graphs How do we plot negative numbers on a dot graph? We can show negative numbers on a dot graph just like on a bar graph. We will review this concept by looking at a game in which people answer trivia questions. For every correct answer we earn 10 points and for every incorrect answer we lose 10 points. Here’s an example of the game. Tyrone and Carson are playing the trivia game in their social studies class. The table shows Tyrone’s questions, his answers, and his score. Question Answer 1. What is the longest river in the United States? 2. Where is the tallest mountain in the United States? 3. Which city is farther west—Reno, Nevada, or San Diego, California? 4. What is the capital of New York? 5. What lake is next to Chicago? Right or Wrong Points Missouri River Right 10 Colorado Wrong 0 Reno, Nevada Right 10 Albany Right 20 Lake Michigan Right 30 Here’s a dot graph of Tyrone’s points. 35 30 Points 25 20 15 10 5 0 1 2 3 4 5 Question The graph shows that Tyrone got the first answer correct. He missed only one question—Question 2—and then correctly answered the rest of the questions. Unit 8 • Lesson 6 567 Lesson 6 The table summarizes Carson’s questions, answers, and his score. Question Answer Right or Wrong Points 1. What state is east of Mississippi? Alabama Right 10 Appalachians Right 20 Miami Wrong 10 Montana Wrong 0 Oklahoma Wrong −10 2. What is the mountain range on the east coast of the United States? 3. What is the capital of Florida? 4. What state has a golden bear on its flag? 5. The Alamo is in what state? Here’s a graph of Carson’s points. 30 Points 20 10 0 10 20 1 2 3 4 5 Question We see that Carson missed enough questions to end with −10 points. The graph also shows that Carson answered the first two questions correctly and then missed all the remaining questions. As with other graphs, dot graphs help us see the change from one point to the next. They also help us see negative values easily. Problem-Solving Activity Turn to Interactive Text, page 293. 568 Unit 8 • Lesson 6 Reinforce Understanding Use the mBook Study Guide to review lesson concepts. Lesson 6 Homework Activity 1 Add the integers. 1. 2 + −2 2. 2 + −5 3. 5 + −5 4. −3 + −4 5. −10 + 10 6. −1 + 9 7. 8 + −2 8. −10 + 9 9. 3 + 2 10. −6 + 6 Activity 2 Tell what equation is represented by the cards in each of the problems, then solve the equation. Remember, red cards cancel out black cards because their sum is zero. Model Answer: 5 + −1 = 4 Notice we drew a line through one black card and one red card. They cancel each other out because their sum is zero. We are left with four black cards. That represents the number 4. The answer to the problem is +4, or just 4. 1. Write a number sentence using integers. 2. Write a number sentence using integers. 3. Write a number sentence using integers. 4. Write a number sentence using integers. Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use. Unit 8 • Lesson 6 569 Lesson 6 Homework Activity 3 Tell which number is bigger by writing > or <. 1. −10 −9 2. 0 −5 −1 3. −8 −9 4. 5 5. −9 1 6. −7 −6 Activity 4 • Distributed Practice Solve. 1. What is 10% of 100? 3 2. Convert 5 to a decimal number. 3. Convert 0.08 to a percent. 4. 2.3 · 8 5. 18.6 · 0.3 4 3 3 6. 5 · 4 5 6 4 7. 4 ÷ 3 1 8 8. 117.98 + 207.45 + 316.05 + 269.71 570 Unit 8 • Lesson 6 Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.