6.1 NEGATIVE NUMBERS AND COMPUTING WITH SIGNED NUMBERS

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1 6. NEGATIVE NUMBERS AND COMPUTING WITH SIGNED NUMBERS Jordan s friend, Arvid, lives in Europe; they exchange regularly. It has been very cold lately and Jordan wrote to Arvid, Our high temperature was 3 degrees below zero today! Arvid answered, Do you happen to know what temperature that is in Celsius? Jordan realized that while the U.S. uses the Fahrenheit scale for temperature, Europe uses the Celsius scale. He found a conversion formula that allowed him to convert temperature in Fahrenheit to Celsius. Here s the formula he used, entering 3 as the degrees Fahrenheit: = -45 degrees Fahrenheit = -25 Complete Jordan s response to Arvid: -25 Hey, Arvid. 3 degrees Fahrenheit is degrees Celsius. Assess your readiness to complete this activity. Rate how well you understand: Not ready Almost ready Bring it on! the terminology and notation associated with signed numbers the meaning of absolute values how to add and subtract signed numbers how to multiply and divide signed numbers how, in general, to validate signed number computations Adding and subtracting signed numbers correct absolute value of the answer correct sign of the answer Multiplying and dividing signed numbers correct absolute value of the answer correct sign of the answer 237

2 Chapter 6 Signed Numbers, Exponents, and Order of Operations Example : Evaluate 9 + ( 24) Example 2: Evaluate ( 26) + ( 29) Try It! Steps in the Methodology Example Example 2 Step Identify terms. Identify the terms and confirm they have the same sign. Special Case: Adding signed fractions (see Model ) 9 and 24 both negative 26 and 29 both negative Step 2 Determine absolute values. Determine the absolute value of each term. 9 = 9 24 = = = 29 Step 3 Add absolute values. Add their absolute values. In this step, compute only with absolute values, not signs Step 4 Present the answer. To present your answer, attach the common sign of the terms to the sum Model Special Case: Adding Signed Fractions Add: To add signed fractions, first rewrite the fractions with a common denominator. Rewrite: Attach the sign of each fraction to its numerator and use the appropriate methodology to add the numerators = = = 9 + ( 8) Apply the methodology to add the terms in the numerator: Steps & 2 both are negative 9 = 9 8 = 8 Step = 7 absolute value of the answer Step 4 numerator sum is negative: 7 Answer : 7 7 = =

3 Activity 6. Negative Numbers and Computing with Signed Numbers Model 2 Simplify: Step 0.7, 2.8, and 6.42 are all positive. Step 2 Absolute values are 0.7, 2.8, 6.42 Step Step 4 Answer: or simply absolute value of the answer Example : Evaluate 5 + ( 32) Example 2: Evaluate Try It! Steps in the Methodology Example Example 2 Step Identify terms. Identify the two terms. +5 and and +3 opposite signs Step 2 Determine absolute values. Determine the absolute value of each term. 5 = 5 32 = = 28 3 = 3 Step 3 Subtract absolute values. Subtract the smaller absolute value from the larger absolute value. In this step, compute only with absolute values, not signs Step 4 Present the answer. To present your answer, attach the sign of the number with the larger absolute value. 32 > 5 +3 or

4 Chapter 6 Signed Numbers, Exponents, and Order of Operations Model Add: Step Step 2 The two terms are 73 and +25, with opposite signs. 73 = = 25 Step absolute value of the answer Step 4-73 > 25 answer will be negative Answer: 48 Model 2 Simplify: ( 6.9) Step The two terms are and 6.9, with opposite signs. Steps 2 & absolute value of the answer Step > -6.9 answer will be positive Answer: or Model 3 2 Evaluate: Rewrite with a common denominator: = Apply the Methodology to add the terms in the numerator: Step opposite signs Step 2 6 = 6, 5 = 5 Step = Step 4-6 > 5 numerator sum is negative, - Answer: =

5 Activity 6. Negative Numbers and Computing with Signed Numbers To add more than two signed numbers, use either of the following two techniques. Technique # Add the first two numbers, using the appropriate Methodology for Adding Signed Numbers. Then add each succeeding number as you work left to right. Technique #2 Find the sum of the positive numbers and the sum of the negative numbers. Then add the two sums. Note that the Commutative and Associative Properties of Addition make this possible. Model Simplify: ( 2) + ( 0) + 9 Using Technique #, working left to right: = = ( 2) = ( 0) = = 6 Answer Using Technique #2: ( 2) + ( 0) + 9 Add the positive numbers: = 4 Add the negative numbers: 8 + ( 2) + ( 0) = 20 Add the sums 4 + ( 20) = 6 Answer 24

6 Chapter 6 Signed Numbers, Exponents, and Order of Operations Example : Evaluate 54 (+4) Example 2: Evaluate 6 ( 7) Try It! Steps in the Methodology Example Example 2 Step Copy the problem. Write the problem exactly as given. 54 (+4) 6 ( 7) Step 2 Identify the second term. Identify the second term the number you are subtracting from the first subtracting +4 7 Step 3 Change to add the opposite. Change the operation sign to addition, and change the sign of the second term ( 4) 6 +(+7) Step 4 Add appropriately. For the expression in Step 3, determine whether you are adding two numbers with the same sign or two numbers with opposite signs and follow the appropriate Methodology for Adding Signed Numbers. 54 and 4 are both negative Add their absolute values and 7 are both positive Add absolute values Attach a negative sign Step 5 Present the answer. Present your answer or 23 Model Subtract: Steps, 2 & = ( 9.73) Solve this addition problem. Step 4 opposite signs Step 5 Answer: > 8. 25, so attach a negative sign 242

7 Activity 6. Negative Numbers and Computing with Signed Numbers Model 2 Evaluate: 20 ( 9) subtraction sign Step 20 ( 9) Step 2 subtracting negative 9 Step (+9) Solve this addition problem in Step 4. Step 4 Addends are 20 and +9, opposite signs 20 Subtract the absolute values. 9 absolute value of the answer Step 5 Answer: 20 > 9, so attach a negative sign Model 3 Simplify: subtraction: 23 minus 75 Step Step 2 subtracting +75 Step ( 75) Solve this addition problem in Step 4. Step 4 same sign, both negative Add the absolute values. Step 5 Answer: Attach the common sign, negative. Model 4 Evaluate: 5 3 Steps, 2 & 3 = Step 4 First rewrite with a common denominator: + + opposite signs 5 = 6 = = > + 5 attach negative sign = = 2 Reduce: numerator = 6 Step 5 2 = Answer 243

8 Chapter 6 Signed Numbers, Exponents, and Order of Operations Use the following technique when the expression contains both addition and subtraction signs. Technique Change each subtraction in the expression to addition of the opposite (do not change additions) and apply a Technique for adding three or more numbers. Model Simplify: 4 ( 6) + ( 2) subtraction signs Change each subtraction: 4 + (+6) + ( 2) + ( ) Apply addition Technique = 8 and work left to right 8 + ( 2) = ( ) = Answer Model 2 Evaluate: ( 2) 7 ( 28) subtraction signs = ( 2) + ( 2) + ( 7) + (+28) ( 6) Add the positives: = +68 Add the negatives: + ( 2) + ( 2) + ( 7) + ( 6) = 47 Add the two sums: ( 47) = +2 or 2 Answer 244

9 Activity 6. Negative Numbers and Computing with Signed Numbers The Methodology for Multiplying or Dividing Signed Numbers is based upon whether the signs of the numbers are the same or different. Example : 42 6 Example 2: 62 ( 9) Try It! Steps in the Methodology Example Example 2 Step Determine sign of answer. Determine the sign of the answer. If the two numbers have opposite signs, the answer will be negative. If the two numbers have the same sign, the answer will be positive opposite signs The answer will be negative. 62 ( 7) both are negative The answer will be positive. Step 2 Determine absolute value. Determine the absolute value of each term. 42 = 42 6 = 6 62 = 62 9 = 9 Step 3 Multiply or divide absolute values. Calculate the product (for multiplication) or quotient (for division) of the absolute values of the numbers. In this step, compute only with absolute values, not signs ) Step 4 Present the answer. To present your answer, attach the correct sign (as determined in Step ) to the product or quotient Model Evaluate: 8 ( 2) negative eight times negative twelve Step Step 2 The factors have the same sign. The answer will be positive. 8 = 8 2 = 2 Step = 96 absolute value of the answer Step 4 Answer: +96 or

10 Chapter 6 Signed Numbers, Exponents, and Order of Operations Model 2 Simplify: absolute value of the answer Step opposite signs; answer will be negative 24. Step = = 02. Step ) Step 4 Answer: Model 3 Evaluate: Step factors have the same sign; answer will be positive 2 2 Step 2 = = Step Step 4 Answer : + 2 or = absolute value of the answer Use the following technique when multiplying more than two signed factors. Technique Multiply the first two factors, then multiply by each succeeding number as you work left to right. Shortcut Determining the sign of the product fi rst (see Models & 2) 246

11 Activity 6. Negative Numbers and Computing with Signed Numbers Model Simplify: 4 ( 2) ( ) Work left to right, keeping track of the sign for each operation. ( ) = 4 2 opposite signs same signs = 2 =+ = same signs = opposite signs 24 Answer ( ) = Shortcut Determining the Sign of the Product First Determine the sign of the answer first by counting the negative factors. An even number of negative factors yields a positive product. An odd number of negative factors yields a negative product. 4 ( 2) ( ) three negative factors; the answer will be negative Then simply multiply the absolute values of the factors and attach the sign = 2 24 Answer: 24 Model 2 Evaluate: ( 0.5) 2 Use shortcut: two negative factors; the answer will be positive = = = 20 2 = 40 Answer: +40 or

12 Chapter 6 Signed Numbers, Exponents, and Order of Operations Validation for Adding, Subtracting, Multiplying, and Dividing Signed Numbers You can validate your answer to a signed number addition, subtraction, multiplication, or division problem as you have validated in previous Activities; that is, by using the opposite operation to work back to the first term in the original problem. When validating, it is of utmost importance to: be aware of the signs of the original terms keep in mind that the opposite operation has its own methodology Following are validation models for each of the basic operations: Addition (+) Validate by subtracting. Example 9 + ( 24) = 43 Validation: 43 ( 24) = 43 + (+24) = 9 Example ( 2) + ( 0) + 9 = 6 Validation: Example ( 32) = 7 Validation: 7 ( 32) = 7 + (+32) = 5 Work backwards and subtract all terms but the first. 6 9 ( 0) ( 2) 2 3 = 6 + ( 9) + (+0) + (+2) + ( 2) + ( 3) = [ 6 + ( 9) + ( 2) + ( 3)] + [(+0) + (+2)] = 20 + (+2) = 8 Subtraction ( ) Validate by adding. Example = 68 Validation: = negative Example =.48 Validation: = positive Addition & Subtraction (+ ) Validate by using successive opposite operations to work back to the first term. Example ( 2) 7 ( 28) = 2 Validation: ( 28) + 7 ( 2) = ( 25) + ( 28) (+2) ( 5) = [ (+7) + (+2) + 2] + [( 25) + ( 28) + ( 5)] = 57 + ( 68) = Multiplication ( ) Validate by dividing. Division ( ) Validate by multiplying. 248 Example = 252 Validation: = ) negative Example = Validation: = negative

13 Activity 6. Negative Numbers and Computing with Signed Numbers Make Your Own Model Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of. Answers will vary. Problem: 249

14 Chapter 6 Signed Numbers, Exponents, and Order of Operations. What do we mean when we say that a number is negative? A number is negative, if it is less than zero. A negative sign ( ) must be placed in front of the number to indicate that it is negative. ie. Negative 7 is written What is the absolute value of a number? The absolute value of a number is the distance that the number is from zero on a number line. The absolute value of any number is positive. 3. What is the result of adding any number to its opposite? The result is zero. 4. How do you determine the sign of the answer to an addition problem? Determine the sign of the answer to an addition problem by identifying the sign of the addends. If the signs are the same, simply add the numbers and attach their sign. If the addends have opposite signs, subtract the absolute values and attach the sign of the number with the larger absolute value. 5. What does it mean to convert a subtraction problem into an addition problem? The process of subtraction of signed numbers cannot be completed without an understanding of the addition of signed numbers. When subtracting signed numbers, it is necessary to change to an addition problem, by adding the opposite of the second number to the fi rst number. (Using the addition rules for signed numbers). 6. How do you determine the sign of the answer to a multiplication or division problem? If the problem is strictly multiplication and division, an odd number of negative numbers produces a negative result. An even number of negative numbers produces a positive result. 7. In an addition problem with more than two numbers, why can you add all the positive numbers and all the negative numbers first and then find the sum of those two numbers? Use the Commutative Property to add and rearrange terms, then the Associative Property to group addends to simplify the computation. 250

15 Activity 6. Negative Numbers and Computing with Signed Numbers 8. In a multiplication problem with more than two factors, why does an even number of negative factors produce a positive answer and an odd number produce a negative answer? If you repeatedly apply the rules for multiplication you will fi nd that the results will be as stated in the question. For example: What is the answer to the following problem? ( )( )( )( ) =? We should think: ( )( )= + (+)( )= ( )( ) = + and so on. 9. When adding signed fractions, where should you attach their signs for ease of computation? Attach the sign of each fraction to its numerator and apply the methodology (for adding numbers with opposite 0. What will be your strategy to ensure that your answer to a signed number problem is correct? Validate each part of the problem as I work it through. Be careful to use the correct order of operations.. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why. Answers will vary. Evaluate each of the following (a) through (j) by doing the calculation in your head. Answer 3 or +3 Answer a) f) 80 + ( 90) 5 b) ( 2) + ( 3) g) 25 ( 5) 8 c) h) 7 ( 8) 46 or +46 d) i) ( 2) ( 9) 5 or +5 0 e) 5 + ( 0) j) or

16 Chapter 6 Signed Numbers, Exponents, and Order of Operations. Evaluate each of the following: Worked Solution Validation (optional) a) 49 + ( 8) b) c) d) 37 ( 4) e) ( 3)

17 Activity 6. Negative Numbers and Computing with Signed Numbers Worked Solution Validation (optional) f) g) 33 + ( 23) 7 ( 2) h) i) 5 3 j) 3 ( 8) 253

18 Chapter 6 Signed Numbers, Exponents, and Order of Operations Worked Solution Validation (optional) k) l) 2 ( ) ( 3) m) 62 9 n) o) 5 ( 4) (2) (0) ( 0) You can t divide by 0 so the last value needs to be zero. 254

19 Activity 6. Negative Numbers and Computing with Signed Numbers Worked Solution Validation (optional) p) ( ) 6 ( 4) Evaluate each of the following expressions ( 95) ( 3) ( 4) ( 2) ( 3) ( 0) The current temperature is 2 F. Ten hours ago, it was 6 degrees below zero ( 6 F). What is the change from the previous temperature until now? 8º (Hint: change = current previous). Identify the error(s) in the following worked solutions. If the worked solution is correct, write Correct in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. Worked Solution What is Wrong Here? Identify the Errors ) Did not rewrite as an addition problem: 36 + ( 48) Answer should be negative. Correct Process =36 + (-48) =-2 Answer: > 36 2) 4 ( 6) This is a multiplication problem, not subtraction. 255

20 Chapter 6 Signed Numbers, Exponents, and Order of Operations Worked Solution What is Wrong Here? Identify the Errors Correct Process 3) CORRECT 4) (.) This is an addition of two negatives. Add the numbers and the result is negative. 5) In division, a negative number divided by a negative number is a positive number. 6) ( 2) (3) ( 5) In multiplication, an odd number of negative numbers multiplied together results in a negative number. 256