Addition and Subtraction with Rational Numbers
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- Judith McGee
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1 Addition and Subtraction with Rational Numbers Although baseball is considered America's national pastime, football attracts more television viewers in the U.S. The Super Bowl--the championship football game held at the end of the season--is not only the most watched sporting event but also the most watched television broadcast every year. 4.1 Math Football Using Models to Understand Integers Walk the Line Adding Integers, Part I Two-Color Counters Adding Integers, Part II What s the Difference? Subtracting Integers What Do We Do Now? Adding and Subtracting Rational Numbers
2 194 Chapter 4 Addition and Subtraction with Rational Numbers
3 Math Football Using Models to Understand Integers Learning Goals In this lesson, you will: Represent numbers as positive and negative integers. Use a model to represent the sum of a positive and a negative integer. Golfers like negative numbers. This is because, in golf, the lower the score, the better the golfer is playing. Runners like negative numbers too. They often split the distances they have to run into two or more equal distances. If they are on pace to win, they will achieve what is called a negative split. What about football? What are some ways in which negative numbers can be used in that sport? 4.1 Using Models to Understand Integers 195
4 Problem 1 Hut! Hut! Hike! You and a partner are going to play Math Football. You will take turns rolling two number cubes to determine how many yards you can advance the football toward your end zone. Player 1 will be the Home Team and Player 2 will be the Visiting Team. In the first half, the Home Team will move toward the Home end zone, and the Visiting Team will move toward the Visiting end zone. Rules: Players both start at the zero yard line and take turns. On your turn, roll two number cubes, one red and one black. The number on each cube represents a number of yards. Move your football to the left the number of yards shown on the red cube. Move your football to the right the number of yards shown on the black cube. Start each of your next turns from the ending position of your previous turn. (Nets are provided at the end of the lesson so you can cut out and construct your own number cubes. Don t forget to color the number cubes black and red.) Scoring: Each player must move the football the combined value of both number cubes to complete each turn and be eligible for points. When players reach their end zone, they score 6 points. If players reach their opponent s end zone, they lose 2 points. An end zone begins on either the 110 or 210 yard line. Example: Player Starting Position Results of the Number Cubes Roll Ending Position First Turn Home Team 0 Red 3 and Black 5 12 Visiting Team 0 Red 5 and Black 6 11 Home Team 12 Red 1 and Black 6 17 Second Turn Visiting Team 11 Red 6 and Black Read through the table. After two turns, which player is closest to their end zone? 196 Chapter 4 Addition and Subtraction with Rational Numbers
5 2. Let s play Math Football. Begin by selecting the home or visiting team. Then, cut out your football. Set a time limit for playing a half. You will play two halves. Make sure to switch ends at half-time with the Home Team moving toward the Visiting end zone, and the Visiting Team moving toward the Home end zone. Home Team Black Red Player 1 Player 2 Visiting Team 4.1 Using Models to Understand Integers 197
6 198 Chapter 4 Addition and Subtraction with Rational Numbers
7 3. Answer each question based on your experiences playing Math Football. a. When you were trying to get to the Home end zone, which number cube did you want to show the greater value? Explain your reasoning. b. When you were trying to get to the Visiting end zone, which number cube did you want to show the greater value? Explain your reasoning. c. Did you ever find yourself back at the same position you ended on your previous turn? Describe the values shown on the cubes that would cause this to happen. d. Describe the roll that could cause you to move your football the greatest distance either left or right. 4.1 Using Models to Understand Integers 199
8 Problem 2 Writing Number Sentences You can write number sentences to describe the results of number cube rolls. Think of the result of rolling the red number cube as a negative number and the result of rolling the black number cube as a positive number. Consider the example from Problem 1. The number sentence for each turn has been included. Player Starting Position Results of the Number Cubes Roll Ending Position Number Sentence First Turn Home Team 0 Red 3 and Black (23) Visiting Team 0 Red 5 and Black (25) Second Turn Home Team 12 Red 1 and Black (21) Visiting Team 11 Red 6 and Black (26) Describe each part of the number sentence for the second turn of the Visiting Team player. 1 (6) 2 = 3 Starting position 200 Chapter 4 Addition and Subtraction with Rational Numbers
9 2. Write a number sentence for each situation. Use the game board for help. a. The Home Team player starts at the zero yard line and rolls a red 6 and a black 2. What is the ending position? Number sentence b. The Visiting Team player starts at the zero yard line and rolls a red 5 and a black 4. What is the ending position? Number sentence c. The Home Team player starts at the 5 yard line and rolls a red 2 and a black 2. What is the ending position? I calculated the result from the two cubes first and then added this to the starting number. Can I do that? Number sentence d. The Visiting Team player starts at the 25 yard line and rolls a red 4 and a black 6. What is the ending position? Number sentence e. Suppose the Home Team player is at the 18 yard line. Complete the table and write two number sentences that will put the player into the Home end zone. Starting Position Roll of the Red Number Cube Roll of the Black Number Cube Number Sentence f. Suppose the Visiting Team player is at the 28 yard line. Complete the table and write two number sentences that will put the player into the Visiting end zone. Starting Position 28 Roll of the Red Number Cube Roll of the Black Number Cube Number Sentence 28 Be prepared to share your solutions and methods. 4.1 Using Models to Understand Integers 201
10 202 Chapter 4 Addition and Subtraction with Rational Numbers
11 1 2 Remember to color one net red and the other net black before you cut them out Using Models to Understand Integers 203
12 204 Chapter 4 Addition and Subtraction with Rational Numbers
13 Walk the Line Adding Integers, Part I Learning Goals In this lesson, you will: Model the addition of integers on a number line. Develop a rule for adding integers. Corinne: I m thinking of a number between 220 and 20. What s my number? Benjamin: Is it 25? Corinne: Lower. Benjamin: 22? Corinne: That s not lower than 25. Benjamin: Oh, right. How about 211? Corinne: Higher. Benjamin: 28? Corinne: Lower. Benjamin: 29? Corinne: You got it! Try this game with a partner. See who can get the number with the fewest guesses. 4.2 Adding Integers, Part I 205
14 Problem 1 Adding on Number Lines 1. Use the number line and determine the number described by each. Explain your reasoning a. the number that is 7 more than 29 b. the number that is 2 more than 26 c. the number that is 10 more than 28 d. the number that is 10 less than 6 e. the number that is 5 less than 24 f. the number that is 2 less than Chapter 4 Addition and Subtraction with Rational Numbers
15 A number line can be used to model integer addition. When adding a positive integer, move to the right on a number line. When adding a negative integer, move to the left on a number line. Example 1: The number line shows how to determine Step 1 5 Step Example 2: The number line shows how to determine 5 1 (28). 8 5 Step 1 Step Compare the first steps in each example. a. What distance is shown by the first term in each example? b. Describe the graphical representation of the first term. Where does it start and in which direction does it move? Why? c. What is the absolute value of the first term in each example? Remember that the absolute value of a number is its distance from Adding Integers, Part I 207
16 3. Compare the second steps in each example. a. What distance is shown by the second term in each example? b. Why did the graphical representation for the second terms both start at the endpoints of the first terms but then continue in opposite directions? Explain your reasoning. c. What are the absolute values of the second terms? 4. Use the number line to determine each sum. Show your work. a b. 3 1 (27) Chapter 4 Addition and Subtraction with Rational Numbers
17 c (27) d Notice that the first term in each expression in parts (a) through (d) was either 3 or (23). a. What do you notice about the distances shown by these terms on the number lines? b. What is the absolute value of each term? 6. Notice that the second term in each expression was either 7 or (27). a. What do you notice about the distances shown by these terms on the number lines? b. What is the absolute value of each term? 4.2 Adding Integers, Part I 209
18 7. Use the number line to determine each sum. Show your work. a b. 9 1 (25) c (25) d Chapter 4 Addition and Subtraction with Rational Numbers
19 8. Notice that the first term in each expression in parts (a) through (d) was either 9 or (29). a. What do you notice about the distances shown by these terms on the number lines? b. What is the absolute value of each term? 9. Notice that the second term in each expression was either 5 or (25). a. What do you notice about the distances shown by these terms on the number lines? b. What is the absolute value of each term? How is knowing the absolute value of each term important? 4.2 Adding Integers, Part I 211
20 10. Use the number line to determine each sum. Show your work. a b. 8 1 (22) c (22) d Use the number line to determine each sum. Show your work. a Chapter 4 Addition and Subtraction with Rational Numbers
21 b. 4 1 (211) c (211) d In Questions 4 through 11, what patterns do you notice when: a. you are adding two positive numbers? b. you are adding two negative numbers? c. you are adding a negative and a positive number? Can you see how knowing the absolute value is important when adding and subtracting signed numbers? 4.2 Adding Integers, Part I 213
22 13. Complete each number line model and number sentence. a b c d Be prepared to share your solutions and methods. 214 Chapter 4 Addition and Subtraction with Rational Numbers
23 Two-Color Counters Adding Integers, Part II Learning Goals In this lesson, you will: Key Term additive inverses Model the addition of integers using two-color counters. Develop a rule for adding integers. Opposites are all around us. If you move forward two spaces in a board game and then move back in the opposite direction two spaces, you re back where you started. In tug-of-war, if one team pulling on the rope pulls exactly as hard as the team on the opposite side, no one moves. If an element has the same number of positively charged protons as it does of negatively charged electrons, then the element has no charge. In what ways have you worked with opposites in mathematics? 4.3 Adding Integers, Part II 215
24 Problem 1 Two-Color Counters 1. Use the number line model to determine each sum. a. 3 1 (23) b. (214) c. 8 1 (28) d. What pattern do you notice? Two numbers with the sum of zero are called additive inverses. Addition of integers can also be modeled using two-color counters that represent positive (1) charges and negative (2) charges. One color, usually red, represents the negative number, or negative charge. The other color, usually yellow, represents the positive number, or positive charge. In this book, gray shading will represent the negative number, and no shading will represent the positive number Chapter 4 Addition and Subtraction with Rational Numbers
25 You can model the expression 3 1 (23) in different ways using two-color counters: (3) 3 Three positive charges and three negative charges have no charge. 3 1 (23) 5 0 (3) 3 Each positive charge is paired with a negative charge. 3 1 (23) What is the value of each and pair shown in the second model? 3. Describe how you can change the numbers of and counters in the model, but leave the sum unchanged. 4.3 Adding Integers, Part II 217
26 Let s consider two examples where integers are added using two-color counters. Example 1: There are 13 positive charges in the model. The sum is 13. Example 2: 5 1 (28) There are five pairs. The value of those pairs is 0. There are 3, or negative charges, remaining. There are 3 negative charges remaining. The sum of 5 1 (28) is Create another model to represent a sum of 23. Write the appropriate number sentence. 218 Chapter 4 Addition and Subtraction with Rational Numbers
27 5. Share your model with your classmates. How are they the same? How are they different? 6. Write a number sentence to represent each model. a. b. c. d. e. f. 4.3 Adding Integers, Part II 219
28 7. Does the order in which you wrote the integers in your number sentence matter? How do you know? 8. Write each number sentence in Question 6 a second way. 9. Draw a model for each, and then complete the number sentence. a (24) 5 b c. 9 1 (24) 5 d Chapter 4 Addition and Subtraction with Rational Numbers
29 10. Complete the model to determine the unknown integer. a b c Describe the set of integers that makes each sentence true. a. What integer(s) when added to 27 give a sum greater than 0? Consider drawing a number line model or a two-color counter model to help you answer each question. b. What integer(s) when added to 27 give a sum of less than 0? c. What integer(s) when added to 27 give a sum of 0? 4.3 Adding Integers, Part II 221
30 12. When adding two integers, what will the sign of the sum be if: a. both integers are positive? b. both integers are negative? c. one integer is negative and one integer is positive? What happens when you add a negative and a positive integer and they both have the same absolute value? 13. Write a rule that states how to determine the sum of any two integers that have the same sign. 14. Write a rule that states how to determine the sum of any two integers that have opposite signs. 222 Chapter 4 Addition and Subtraction with Rational Numbers
31 15. Use your rule to determine each sum. a (24) 5 b (215) 5 c (212) 5 d e (213) 5 f g (225) 5 h (237) Determine each unknown addend. a. 1 (225) 5 34 b c d e f Talk the Talk Represent the sum of additive inverses in the graphic organizer provided. First, write a number sentence. Then, represent your number sentence in words, using a number line model, and using a two-color counter model. Be prepared to share your solutions and methods. 4.3 Adding Integers, Part II 223
32 Number Sentence In Words Additive Inverses and Zero, 0 Number Line Model Two-Color Counter Model 224 Chapter 4 Addition and Subtraction with Rational Numbers
33 What s the Difference? Subtracting Integers Learning Goals In this lesson, you will: Model subtraction of integers using two-color counters. Model subtraction of integers on a number line. Develop a rule for subtracting integers. Key Term zero pair I don t want nothing! We don t need no education. I can t get no satisfaction. You may have heard or even said these phrases before. In proper English writing, however, these kinds of phrases should be avoided because they contain double negatives, which can make your writing confusing. For example, the phrase I don t need none contains two negatives : the word don t and the word none. The sentence should be rewritten as I don t need any. In mathematics, double negatives can be confusing as well, but it s perfectly okay to use them! In this lesson, you will learn about subtracting integers, which sometimes involves double negatives. 4.4 Subtracting Integers 225
34 Problem 1 Temperatures 1. Complete the table to determine the difference between the maximum and minimum temperatures in each row. Subtract the minimum temperature from the maximum temperature, not the other way around. United States Extreme Record Temperatures and Differences State Maximum Temp. ( F) Minimum Temp. ( F) Georgia Hawaii Florida Alaska California North Carolina Arizona Difference ( F) Texas ºF a. Which state shows the least difference between the maximum and minimum temperature? b. Which state shows the greatest difference between the maximum and minimum temperature? 226 Chapter 4 Addition and Subtraction with Rational Numbers
35 2. You overheard a radio announcer report that from 12:00 pm to 3:00 pm the temperature went from 25 F to 210 F. He said, It is getting warmer. Was he correct? Explain your reasoning. Problem 2 Models for Subtracting Integers Subtraction can mean to take away objects from a set. Subtraction can also mean a comparison of two numbers, or the difference between them. The number line model and the two-color counter model used in the addition of integers can also be used to investigate the subtraction of integers. Using just positive or just negative counters, you can show subtraction using the take away model. Example 1: First, start with seven positive counters. Then, take away five positive counters Two positive counters remain. Example 2: 27 2 (25) First, start with seven negative counters. Then, take away five negative counters (25) 5 22 Two negative counters remain. 4.4 Subtracting Integers 227
36 1. How are Examples 1 and 2 similar? How are these examples different? To subtract integers using both positive and negative counters, you will need to use zero pairs Recall that the value of a and pair is zero. So, together they form a zero pair. You can add as many pairs as you need and not change the value. Example 3: (25) Start with seven positive counters. The expression says to subtract five negative counters, but there are no negative counters in the first model. Insert five negative counters into the model. So that you don t change the value, you must also insert five positive counters. This value is 0. Now, you can subtract, or take away, the five negative counters. Take away five negative counters, and 12 positive counters remain (25) Chapter 4 Addition and Subtraction with Rational Numbers
37 Example 4: Start with seven negative counters. 2. The expression says to subtract five positive counters, but there are no positive counters in the first model. a. How can you insert positive counters into the model and not change the value? b. Complete the model. c. Now, subtract, or take away, the five positive counters. Sketch the model to show that This is a little bit like regrouping in subtraction. 4.4 Subtracting Integers 229
38 3. Draw a representation for each subtraction problem. Then, calculate the difference. a. 4 2 (25) b (25) c Chapter 4 Addition and Subtraction with Rational Numbers
39 d How could you model 0 2 (27)? a. Draw a sketch of your model. Finally, determine the difference. b. In part (a), would it matter how many zero pairs you add? Explain your reasoning. 4.4 Subtracting Integers 231
40 5. Does the order in which you subtract two numbers matter? Does have the same answer as 3 2 5? Draw models to explain your reasoning. 6. Write a rule for subtracting positive and negative integers. Problem 3 Subtracting on a Number Line Cara thought of subtraction of integers another way. She said, Subtraction means to back up, or move in the opposite direction. Like in football when a team is penalized or loses yardage, they have to move back. Analyze Cara s examples. Example 1: _ 6 _ (2) _ 6 opposite of First, I moved from zero to _ 6, and then I went in the opposite direction of the 2 because I am subtracting. So, I went two units to the left and ended up at _ 8. _ 6 _ (2) = _ Chapter 4 Addition and Subtraction with Rational Numbers
41 Example 2: _ 6 _ ( _ 2) _ 6 opposite of _ In this problem, I went from zero to _ 6. Because I am subtracting ( _ 2), I went in the opposite direction of the _ 2, or right two units, and ended up at _ 4. _ 6 _ ( _ 2) = _ 4 Example 3: 6 _ ( _ 2) 6 opposite of _ Explain the model Cara created in Example 3. Example 4: 6 _ (2) 6 opposite of Explain the model Cara created in Example Subtracting Integers 233
42 3. Use the number line to complete each number sentence. a (23) 5 Use Cara's examples for help b (24) c d e (23) f g h (24) Chapter 4 Addition and Subtraction with Rational Numbers
43 4. What patterns did you notice when subtracting the integers in Question 3? a. Subtracting two negative integers is similar to b. Subtracting two positive integers is similar to c. Subtracting a positive integer from a negative integer is similar to d. Subtracting a negative integer from a positive integer is similar to 5. Analyze the number sentences shown a. What patterns do you see? What happens as the integer subtracted from 28 decreases? b. From your pattern, predict the answer to 28 2 (21). For a subtraction expression, such as 28 2 (22), Cara s method is to start at zero and go to 28, and then go two spaces in the opposite direction of 22 to get 26. Dava says, I see another pattern. Since subtraction is the inverse of addition, you can think of subtraction as adding the opposite number. That matches with Cara s method of going in the opposite direction (-2) is the same as -8 -(2) = -6 opposite of - 2 = - (- 2) Subtracting Integers 235
44 An example of Dava s method is shown (-4) = 10 -(-4) 10 4 = Apply Dava s method to determine each difference. a (22) 5 b (23) 5 c d e f g (230) 5 h So, I can change any subtraction problem to show addition if I take the opposite of the number that follows the subtraction sign. 7. Determine the unknown integer in each number sentence. a b c d e. 2 (25) f g h i Chapter 4 Addition and Subtraction with Rational Numbers
45 8. Determine each absolute value. a (23) b c d. 7 2 (23) 9. How does the absolute value relate to the distance between the two numbers in Question 8, parts (a) through (d)? 10. Is equal to 6 2 8? Is equal to 6 2 4? Explain your thinking. Talk the Talk 1. Tell whether these subtraction sentences are always true, sometimes true, or never true. Give examples to explain your thinking. a. positive 2 positive 5 positive b. negative 2 positive 5 negative c. positive 2 negative 5 negative d. negative 2 negative 5 negative 4.4 Subtracting Integers 237
46 2. If you subtract two negative integers, will the answer be greater than or less than the number you started with? Explain your thinking. 3. What happens when a positive number is subtracted from zero? 4. What happens when a negative number is subtracted from zero? 5. Just by looking at the problem, how do you know if the sum of two integers is positive, negative, or zero? 6. How are addition and subtraction of integers related? Be prepared to share your solutions and methods. 238 Chapter 4 Addition and Subtraction with Rational Numbers
47 What Do We Do Now? Adding and Subtracting Rational Numbers Learning Goal In this lesson, you will: Add and subtract rational numbers. You might think that as you go deeper below the Earth s surface, it would get colder. But this is not the case. Drill down past the Earth s crust, and you reach a layer called the mantle, which extends to a depth of about miles. The temperature in this region is approximately F. Next stop is the outer core, which extends to a depth of about miles and has a temperature of approximately F. The last stop is the very center, the inner core. At approximately miles, the inner core may have a temperature as high as 12,000 F as high as the temperature on the surface of the Sun! What do you think makes the temperature increase as elevation decreases? 4.5 Adding and Subtracting Rational Numbers 239
48 Problem 1 Adding Rational Numbers Previously, you learned how to add and subtract with positive and negative integers. In this lesson, you will apply what you know about your work with integers to the set of rational numbers. Consider this problem and the two methods shown ? Kaitlin s Method Omar s Method 1. Describe each method and the correct answer. 240 Chapter 4 Addition and Subtraction with Rational Numbers
49 2. Now, consider this problem: ( ) 5? a. Why might it be difficult to use either a number line or counters to solve this problem? b. What is the rule for adding signed numbers with different signs? c. What will be the sign of the sum for this problem? Explain your reasoning. d. Calculate the sum ( ) 5 Now that I am working with fractions, I need to remember to find a common denominator first. 3. What is the rule for adding signed numbers with the same sign? 4.5 Adding and Subtracting Rational Numbers 241
50 4. Determine each sum. Show your work. a b ( ) 5 c d Remember that when you add or subtract with decimals, you should first align the decimal points. e f Chapter 4 Addition and Subtraction with Rational Numbers
51 Problem 2 Subtracting Rational Numbers 1. What is the rule for subtracting signed numbers? 2. Determine each difference. Show your work. a b ( ) 5 c ( ) 5 d e (13.7) 5 f (21.7) = 4.5 Adding and Subtracting Rational Numbers 243
52 Problem 3 Adding and Subtracting with an Algorithm Add and subtract using your algorithms An algorithm is a procedure you can use to solve lots of similar problems ( ) (26.2) (213.2) Chapter 4 Addition and Subtraction with Rational Numbers
53 (20.6) (20.33) (216.3) ( ) (23.1) (2775) Be prepared to share your solutions and methods. 4.5 Adding and Subtracting Rational Numbers 245
54 246 Chapter 4 Addition and Subtraction with Rational Numbers
55 Chapter 4 Summary Key Terms additive inverses (4.3) zero pair (4.4) Writing Number Sentences to Represent the Sum of Positive and Negative Integers Integers are useful for representing some sort of progress from a starting quantity or position. Sequential events can often be modeled by a number sentence involving both positive and negative integers. Example During a model boat race, a boat is in the lead by two boat lengths at the halfway point of the race. However, during the second half of the race, the boat loses five boat lengths to the eventual winner. The boat s progress in relation to the boat race winner is shown through the additional sentence. (12) 1 (25) 5 23 Modeling Integer Addition on a Number Line A number line can be used to model integer addition. When adding a positive integer, move to the right on the number line. When adding a negative integer, move to the left on the number line. Example Any time you learn something new, whether a new math skill, or juggling, or a new song, your brain grows and changes within a few days! Chapter 4 Summary 247
56 Modeling Integer Addition Using Two-Color Counters Let a red counter represent 21 and a yellow counter represent 11. Each pair of positive and negative counters has a value of zero. Example A model representing 7 1 (24) using two-color counters is shown. The zero pairs are circled showing the sum. 7 1 (24) 5 3 Adding Integers When adding two integers with the same sign, add the integers and keep the sign. When adding integers with opposite signs, subtract the integers and keep the sign of the integer with the greater absolute value. Example 29 1 (212) 7 1 (215) 5 2(9 1 12) Chapter 4 Addition and Subtraction with Rational Numbers
57 Modeling Integer Subtraction Using Two-Color Counters Subtraction can be modeled by taking away objects of a set. Positive and negative counters can be used to represent this take away model. Because a pair of positive and negative counters has a value of zero, as many zero pairs as are needed can be added without changing the value. Example Two-color counters can be used to model subtraction. Begin by adding the number of counters to represent the first term, and then add enough zero pairs to be able to subtract the second term (25) (25) 5 3 Modeling Integer Subtraction on a Number Line A number line can be used to model integer subtraction. Subtraction means to move in the opposite direction on the number line. When subtracting a positive integer, move to the left on the number line. When subtracting a negative integer, move to the right on the number line. Example (26) (6) (10) (26) 5 24 Chapter 4 Summary 249
58 Subtracting Integers Because subtraction is the inverse of addition, it is the same as adding the opposite number. Examples (219) (221) Adding Rational Numbers When adding positive and negative rational numbers, follow the same rules as when adding integers. When adding rational numbers with the same sign, add the numbers and keep the sign. When the rational numbers have different signs, subtract the numbers and keep the sign of the number with the greater absolute value. Examples (23.4) ( ) 5 2( ) Subtracting Rational Numbers When subtracting positive and negative rational numbers, follow the same rules as when subtracting integers. Because subtraction is the inverse of addition, it is the same as adding the opposite number. Examples (210 5 ) (110 5 ) (23.4) Chapter 4 Addition and Subtraction with Rational Numbers
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