Download 6.1 negative numbers and computing with signed

6.1 NEGATIVE NUMBERS AND COMPUTING WITH SIGNED NUMBERS
Jordan’s friend, Arvid, lives in Europe; they exchange e-mail regularly. It has been very cold lately and Jordan
wrote to Arvid, “Our high temperature was 13 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 –13 as the degrees Fahrenheit:
-13
− 32 =
-45
degrees Fahrenheit
-45
× 5 =
9
-25
Complete Jordan’s response to Arvid:
-25 degrees Celsius.
Hey, Arvid. −13 degrees Fahrenheit is __________
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
•
Multiplying and dividing signed numbers
– correct absolute value of the answer
– correct absolute value of the answer
– correct sign of the answer
– correct sign of the answer
237
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Example 1: Evaluate –19 + (–24)
Example 2: Evaluate (–26) + (–29)
Try It!
Steps in the Methodology
Step 1
Identify terms.
Example 1
Identify the terms and confirm they have
the same sign.
Special Adding signed fractions
Case: (see Model 1)
Step 2
Determine
absolute values.
Determine the absolute value of each
term.
Step 3
Add their absolute values.
Add absolute
values.
In this step, compute only with absolute
values, not signs.
Step 4
To present your answer, attach the
common sign of the terms to the sum.
Present the
answer.
Model 1
Add:
–19 and –24
Example 2
–26 and –29
both negative
both
negative
−19 = 19
−26 = 26
−24 = 24
−29 = 29
26
19
+24
+29
43
55
−55
–43
Special Case: Adding Signed Fractions
3 ⎛ 2⎞
− + ⎜⎜− ⎟⎟⎟
4 ⎜⎝ 3 ⎠
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.
3 3 ⎛⎜ 2 4 ⎞⎟
× + ⎜− × ⎟
4 3 ⎜⎝ 3 4 ⎟⎠
⎛ 8⎞
9
=−
+ ⎜⎜− ⎟⎟⎟
12 ⎜⎝ 12 ⎠
−
−9 −8
+
12
12
−9 + (−8)
=
12
=
Apply the methodology to add the terms in the numerator:
Steps 1 & 2 both are negative
−9 = 9
−8 = 8
238
absolute value of the answer
Step 3
9 + 8 = 17
Step 4
numerator sum is negative: –17
Answer :
−17
17
5
=−
= −1
12
12
12
Activity 6.1 — Negative Numbers and Computing with Signed Numbers
Model 2
Simplify: 0.17 + 2.8 + 6.42
Step 1
0.17, 2.8, and 6.42 are all positive.
Step 2
Absolute values are 0.17, 2.8, 6.42
Step 3
0.17
2.80
6.42
9.39
Step 4
Answer: +9.39 or simply 9.39
absolute value of the answer
Example 1: Evaluate 15 + (–32)
Example 2: Evaluate –28 + 31
Steps in the Methodology
Step 1
Determine the absolute value of each term.
Step 3
Subtract the smaller absolute value from
Subtract absolute the larger absolute value.
values.
In this step, compute only with absolute
values, not signs.
Present the
answer.
Example 2
–28 and +31
+15 and –32
Determine
absolute values.
Step 4
Example 1
Identify the two terms.
Identify terms.
Step 2
Try It!
opposite
signs
15 = 15
−28 = 28
−32 = 32
31 = 31
32
−15
31
−28
17
3
To present your answer, attach the sign of
the number with the larger absolute value.
+3 or 3
−32 > 15
–17
239
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Model 1
Add: –73 + 25
Step 1
The two terms are –73 and +25, with opposite signs.
Step 2
−73 = 73
25 = 25
Step 3
73
−25
absolute value of the answer
48
-73 > 25 answer will be negative
Step 4
Answer: –48
Model 2
Simplify: 42.25 + (–16.9)
Step 1
The two terms are 42.25 and –16.9, with opposite signs.
Steps 2 & 3
42.25
−16.90
25.35
Step 4
absolute value of the answer
42.25 > -16.9 answer will be positive
Answer: +25.35 or 25.35
Model 3
Evaluate: −
2 1
+
5 3
6
5
−6 + 5
+
=
15 15
15
Apply the Methodology to add the terms in the numerator:
Rewrite with a common denominator: −
Step 1
Step 2
−6 = 6,
Step 3
6 – 5 =1
Step 4
240
opposite signs
5 =5
-6 > 5 numerator sum is negative, - 1
Answer:
−1
1
=–
−
15
15
Activity 6.1 — Negative Numbers and Computing with Signed Numbers
To add more than two signed numbers, use either of the following two techniques.
Technique #1
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: –8 + 3 + 2 + (–2) + (–10) + 9
Using Technique #1, working left to right:
–8 + 3 = –5
–5 + 2 = –3
–3 + (–2) = –5
–5 + (–10) = –15
–15 + 9 = –6
Answer
Using Technique #2:
–8 + 3 + 2 + (–2) + (–10) + 9
Add the positive numbers:
3 + 2 + 9 = 14
Add the negative numbers:
–8 + (–2) + (–10) = –20
Add the sums
14 + (–20) = –6
Answer
241
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Example 1: Evaluate –54 – (+14)
Example 2: Evaluate 16 – (–7)
Try It!
Steps in the Methodology
Step 1
Write the problem exactly as
given.
Copy the
problem.
Step 2
Identify the second
term—the number you are
subtracting from the first
Identify the
second term.
Step 3
Change the operation sign
to addition, and change the
sign of the second term.
Change to add
the opposite.
Step 4
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.
Add
appropriately.
Example 1
Example 2
–54 – (+14)
16 – (–7)
subtracting +14
–54 + (–14)
–54 and –14 are
both negative
Add their
absolute
values.
54
+14
68
Attach a negative sign.
–68
–7
16 +(+7)
16 and 7 are both
positive
Add
absolute
values
16
+7
23
+23
Step 5
–68
Present your answer.
Present the
answer.
+23 or 23
Model 1
Subtract: 8.25 – 19.73
Steps 1, 2 & 3
8.25 – 19.73 = 8.25 + (–19.73)
Step 4
opposite signs
Step 5
242
Answer: –11.48
19.73
−8.25
11.48
Solve this addition problem.
−19.73 > 8.25 , so attach a negative sign
Activity 6.1 — Negative Numbers and Computing with Signed Numbers
Model 2
Evaluate: –20 – (–9)
subtraction sign
Step 1
–20 – (–9)
Step 2
Step 3
subtracting negative 9
–20 + (+9)
Step 4
Solve this addition problem in Step 4.
Addends are –20 and +9, opposite signs
Subtract the absolute values.
20
–9
11
Step 5
absolute value of the answer
−20 > 9 , so attach a negative sign
Answer: –11
Model 3
Simplify: –23 – 75
subtraction: “–23 minus 75”
Step 1
–23 – 75
Step 2
Step 3
subtracting +75
–23 + (–75)
Step 4
Solve this addition problem in Step 4.
same sign, both negative
Add the absolute values.
23
+75
Step 5
Answer: –98
98
Attach the common sign, negative.
Model 4
Evaluate: −
11 ⎛⎜ 1 ⎞⎟
− ⎜− ⎟
15 ⎜⎝ 3 ⎟⎠
Steps 1, 2 & 3
Step 4
11 ⎛⎜ 1 ⎞⎟
11 ⎛ 1 ⎞
− ⎜− ⎟⎟ = − +⎜⎜+ ⎟⎟⎟
15 ⎜⎝ 3 ⎠
15 ⎜⎝ 3 ⎠
First rewrite with a common denominator:
−
11 ⎛⎜ 1 ⎞⎟
11 ⎛⎜ 5 ⎞⎟ −11 + 5
+ ⎜+ ⎟ =
+ ⎜+ ⎟⎟ = −
⎜
15 ⎝ 3 ⎠
15 ⎜⎝ 15 ⎟⎠
15
−6
=
15
Step 5
−
−6 ÷ 3 −2
Reduce:
=
15 ÷ 3
5
−2
2
=−
Answer
5
5
opposite signs 11 – 5 = 6
−11 > +5
attach negative sign
numerator = –6
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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 1
Simplify: –14 – (–6) + (–2) – 1
subtraction signs
Change each subtraction:
–14 + (+6) + (–2) + (–1)
Apply addition Technique
–14 + 6 = –8
and work left to right
–8 + (–2) = –10
–10 + (–1) = –11 Answer
Model 2
Evaluate: –11 + 15 – 2 + (–21) – 7 – (–28) + 25 – 6
subtraction signs
= –11 + 15 + (–2) + (–21) + (–7) + (+28) + 25 + (–6)
244
Add the positives:
15 + 28 + 25 = +68
Add the negatives:
–11 + (–2) + (–21) + (–7) + (–6) = –47
Add the two sums:
+68 + (–47) = +21 or 21 Answer
Activity 6.1 — 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 1: –42 × 6
Example 2: –162 ÷ (–9)
Try It!
Steps in the Methodology
Example 1
Example 2
–42 × 6
–162 ÷ (–7)
both are
negative
Step 1
Determine the sign of the answer.
Determine sign
of answer.
•
If the two numbers have opposite
signs, the answer will be negative.
opposite signs
•
If the two numbers have the same
sign, the answer will be positive.
The answer
will be
negative.
The answer will
be positive.
−42 = 42
−162 = 162
Step 2
Determine
absolute value.
Step 3
Multiply or divide
absolute values.
Determine the absolute value of each
term.
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.
6 =6
42
×6
252
−9 = 9
18
9 162
)
−9
72
−72
0
Step 4
Present the
answer.
To present your answer, attach the correct
sign (as determined in Step 1) to the
product or quotient.
–252
18
Model 1
Evaluate: –8 (–12)
“negative eight times negative twelve”
Step 1
The factors have the same sign. The answer will be positive.
Step 2
−8 = 8
Step 3
8 × 12 = 96
Step 4
Answer: +96 or 96
−12 = 12
absolute value of the answer
245
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Model 2
Simplify:
−4.82
0.2
absolute value of the answer
Step 1
opposite signs; answer will be negative
Step 2
−4.82 = 4.82
Step 4
0.2 = 0.2
Step 3
2 4.1
0.2 4.8 2
−4
08
−8
)
02
−2
0
Answer: –24.1
Model 3
⎛ 2⎞ ⎛ 1⎞
Evaluate: ⎜⎜− ⎟⎟⎟ • ⎜⎜− ⎟⎟⎟
⎜⎝ 3 ⎠ ⎜⎝ 8 ⎠
Step 1
factors have the same sign; answer will be positive
Step 2
−
Step 4
Answer : +
2
2
=
3
3
−
1
1
1
=
8
8
1
or
12
Step 3
1
12
2
1
1
×4 =
3
12
8
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
246
Determining the sign of the product first (see Models 1 & 2)
Activity 6.1 — Negative Numbers and Computing with Signed Numbers
Model 1
⎛ 1⎞
4 × (−2) × ⎜⎜− ⎟⎟⎟ × 3 × (−2)
⎜⎝ 2 ⎠
Simplify:
Work left to right, keeping track of the sign for each operation.
4 × (−2)
opposite signs = −8
⎛ 1⎞
−8 × ⎜⎜− ⎟⎟⎟
⎜⎝ 2 ⎠
4
same signs
=−
+ 4×3
same signs
= +12
+12 × (−2)
Shortcut
8 ⎛⎜ 1
× ⎜−
1 ⎜⎜⎝ 1 2
⎞⎟
⎟⎟ = + 4 = +4
⎟⎟
1
⎠
opposite signs = –24 Answer
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.
⎛ 1⎞
4 × (– 2) × ⎜⎜– ⎟⎟⎟ × 3 × (– 2)
⎜⎝ 2 ⎠
three negative factors; the answer will be negative
Then simply multiply the absolute values of the factors and attach the sign.
1
4
2
1
3 2 24
Answer: –24
×
×1 × × =
1
1
1
1
1
2
Model 2
Evaluate: –5 × 4 × 2 × (–0.5) × 2
Use shortcut: two negative factors; the answer will be positive
5 × 4 × 2 × 0.5 × 2
= 20 × 2 × 0.5 × 2
= 40 × 0.5 × 2
= 20 × 2 = 40
Answer: +40 or 40
247
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 1
–19 + (–24) = –43
Example 2
15 + (–32) = –17
Validation:
–43 – (–24)
= –43 + (+24)
= –19
Validation:
–17 – (–32)
= –17 + (+32)
= 15
Example 3
–8 + 3 + 2 + (–2) + (–10) + 9 = –6
Validation:
Work backwards and subtract all terms but the first.
–6 – 9 – (–10) – (–2) – 2 – 3
= –6 + (–9) + (+10) + (+2) + (–2) + (–3)
= [–6 + (–9) + (–2) + (–3)] + [(+10) + (+2)]
=
–20
+
(+12)
= –8
Subtraction (–) Validate by adding.
Example 4
–54 – 14 = –68
Validation:
–68 + 14
= –54
68
−14
54 negative
Example 5
8.25 – 19.73 = –11.48
Validation:
–11.48 + 19.73
= +8.25
19.73
−11.48
8.25 positive
Addition & Subtraction (+ –) Validate by using successive opposite operations to work back to the first term.
Example 6
Validation:
–11 + 15 – 2 + (–21) – 7 – (–28) + 25 – 6 = 21
21 + 6 – 25 + (–28) + 7 – (–21) + 2 – 15
= 21 + 6 + (–25) + (–28) + 7 + (+21) + 2 + (–15)
= [21 + 6 + (+7) + (+21) + 2] + [(–25) + (–28) + (–15)]
=
57 + (–68) = –11
Multiplication (×) Validate by dividing.
Example 7
–42 × 6 = –252
Validation:
–252 ÷ 6
= –42
42 negative
6 252
−24
)
12
−12
0
248
Division (÷) Validate by multiplying.
Example 8
–4.82 = –24.1
0.2
Validation:
–24.1 × 0.2
= –4.82
24.1
×0.2
4.82 negative
Activity 6.1 — 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 difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
249
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
1. 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 − 7.
2. 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 first 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
Activity 6.1 — 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 find 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
10. 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.
11. What aspect of the model you created is the most difficult 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
Answer
a) –9 + 12
3 or +3
____________
f) 80 + (–90)
–10
____________
b) (–12) + (–3)
–15
____________
g) 25 ÷ (–5)
–5
____________
c) –16 + 8
–8
____________
h) 7 (–8)
–56
____________
d) 36 +10
46 or +46
____________
i) (–2) (–9)
18 or +18
____________
e) 15 + (–10)
5 or +5
____________
j) –12 ÷ 3
–4
____________
251
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
1. Evaluate each of the following:
Worked Solution
252
a)
–49 + (–18)
b)
–22.4 + 48.7
c)
20 – 32
d)
–37 – (–14)
e)
–26 – 14 + (–13) + 12
Validation (optional)
Activity 6.1 — Negative Numbers and Computing with Signed Numbers
Worked Solution
f)
–24 + 5 – 8
g)
33 + (–23) – 17 – (–2)
h)
2 ⎛⎜ 7 ⎞⎟
+ ⎜− ⎟
5 ⎜⎝ 8 ⎟⎠
Validation (optional)
2 1
−
5 3
i)
−
j)
–3 (–8)
253
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Worked Solution
k)
l)
254
Validation (optional)
–55 ÷ 2.5
12 × (–1) × (–3)
m)
–162 ÷ 9
n)
12 ⎛⎜ 3 ⎞⎟
÷ ⎜− ⎟
35 ⎜⎝ 7 ⎟⎠
o)
–5 (–4) (2) (0) (–10)
You can’t divide by 0 so the last
value needs to be zero.
Activity 6.1 — Negative Numbers and Computing with Signed Numbers
Worked Solution
p)
Validation (optional)
2
× (−2) × 6 × (−4)
3
Evaluate each of the following expressions.
1. 18 + (–95)
2. –29 + 18
96
5. –4.82 ÷ 0.2
–24.1
28
7. –6 • 2 • (–3) • (–10)
–11
3. 2 + (–4) – 6 – 1 + 8
4. –8 (–12)
6. –84 ÷ (–3)
–113
–1
–360
8. 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?
1) 36 – 48
2)
–14 (–6)
Identify the Errors
Did not rewrite as an addition
problem: 36 + (– 48)
Answer should be negative.
Correct Process
36 - 48
= 36 + (-48)
= -12
48
- 36
12
Answer: –12
- 4 8 > 36
This is a multiplication
problem, not subtraction.
255
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Worked Solution
What is Wrong Here?
3)
Identify the Errors
2 ⎛⎜ 1 ⎞⎟
+ ⎜− ⎟
3 ⎜⎝ 2 ⎟⎠
CORRECT
4)
–47.9 + (–1.1)
−5.4
5)
−0.2
6)
256
–1 (–2) (3) (–5)
This is an addition of two
negatives. Add the numbers
and the result is negative.
In division, a negative
number divided by a negative
number is a positive number.
In multiplication, an odd
number of negative numbers
multiplied together results in
a negative number.
Correct Process