Download MATH 60 Section 2.3 Multiplying and Dividing Signed Numbers

M60 Section 2.1: Adding Signed Numbers:
Part A: signed numbers:
I. Positive and Negative Numbers: Think of them as ________________
___________________________________________________________
Example 1:
a) Albany is 200 feet above sea level. It's elevation is:
b) Death Valley is 273 feet below sea level: Elevation:
c) Rhonda has $1027 in her bank account. Her balance:
d) Ryan is overdrawn by $12. His balance:
Example 2: Use thermometers to show the indicated temperatures in Celsius:
a) -5
10
b) 6
10
c) -8.5
10
5
5
5
0
0
0
-5
-5
-5
-10
-10
-10
II. Notation: Distinguish between subtraction and a negative number. What is the
meaning of the – symbol in each case?
a–b
-4
A number without a sign is understood to be:
III. Absolute value of a number:
Example 3: Which number is more extreme?
a) -13 or 18
b) -4 or -6
c) 7 or -5
d) -7.6 or 6.7
M60 Sec. 2.1 p.2
IV. ORDER AND INEQUALITY SYMBOLS:
3
<
4 means:
5
>
2 means:
Two other inequalities:
Example 4: Insert
<
a) -3
3
b) -9
-10
c) 3
-4
d) -1.5
-2
e) -2
-1
f) 4.5
5
g) -1
0
h) -7
-7.5
or
>
Example 5: True or False?
a)
-2
<
6
b)
2
≤
5
c)
-2
>
-6
d)
-34
<
-44
e)
7
≥
7
f)
3
>
-5
Part B: Adding Two Signed Numbers:
I. Think of addition as ____________________________________________
Example 6: Use thermometers to answer the questions:
a) The temperature increased by 8º yesterday and then fell 15º overnight. Use an
addition expression with signed numbers to describe this situation.
10
b) The temperature dropped 12º overnight but it rose 20º this morning.
Write an addition expression to with signed numbers to describe this.
5
0
-5
-10
M60 Sec. 2.1 p.3
II. Think of addition as movement __________________________ on the number line.
Adding a positive number would mean a movement to the__________.
Adding a negative number would mean movement to the _______________.
Case 1: Adding two positive numbers:
Example 7: Start at the first number and move in the direction of the second.
a)
(+3) + (+2)
____________________________
b)
(+2) + (+2)
____________________________
c)
(+11) + (+14)
d)
(+20) + (+15)
Rule 1: The sum of two positive numbers is a ___________ number.
Case 2: Adding One Positive Number and One Negative Number:
Example 8: Start at the first number and move in the direction of the second.
a)
(+2) + (-3)
____________________________
b)
(-4) + (+3)
____________________________
c)
(-5) + (+2)
____________________________
d)
(+5) + (-2)
____________________________
e)
(-17) + (+9)
f)
(+16) + (-18)
Rule2: To add numbers with opposite signs: subtract the two unsigned parts.
The sign of the result will be the sign of the more _________________number.
Case 3: Adding Two Negative Numbers
M60 Sec. 2.1 p.4
Example 9: Notice that adding two negative numbers gave an answer that is even more
negative!
a)
(-2) + (-3)
___________________________
b)
(-3) + (-1)
___________________________
c)
(-13) + (-24)
e)
(-72) + (-13)
Rule 3: The sum of two negative numbers is a __________ number
Part C: Adding More than Two signed Numbers:
Remember that addition is commutative so the order in which we add them doesn't matter.
So, we can put the positive numbers together and the negative numbers together before we
decide which direction is the more extreme.
Example 10: Perform the addition:
a)
16 + (-31) + 28 + (-9)
b)
(-7) + (-4) + 19
c)
(-4) + 13 + (-3)
d)
-1.2 + (-2.4) + (-1.4)
e)
33 + (-1) + (-21)
Example 11: Write an expression for the following situations using addition of signed
numbers and answer the question.
a) You deposited a $30 check. Then you wrote a check for $15. What is your net affect?
b) You gained two pounds over the holidays and then lost 5 pounds by working out. What was
the net affect?
Part D: The Opposite of a Number
Example 12: Express each of the following in words and then simplify:
a)
b)
c)
(-17)
- (-1.2)
M60 Sec. 2.2b: Subtracting Signed Numbers:
Part A: Relating Addition and Subtraction: Subtracting a number is the same as adding
the opposite of that number. They are not the same operation, but they arrive at the same
answer. Also, subtraction can be checked by writing a related addition sentence.
Part B: SUBTRACTING SIGNED NUMBERS: Use the chart to show that the following
get the same answer.
Example 1:
a) 5 - 4
is the same as
5 + (-4)
+
+
-
-
b) - 5 – 2
is the same as
-5 + (-2)
+
-
-
c) –4 – (-3)
is the same as
-4 + (+3)
+
+
-
-
d) 4 – (-2)
is the same as
4 + (+2)
+
+
-
-
Subtracting a number is the same as adding the _______________of that number.
Example 2: Write equivalent addition problems for the following and solve them:
a) 3 – (-5)
is the same as
b) 6 – (-3)
is the same as
c) –5 – (-6)
is the same as
M60 Sec. 2.2b pg.2
d) –4 – (-5)
is the same as
e) 3 – (-3)
is the same as
f) 4 – (-4)
is the same as
Example 3: Subtract by performing an equivalent addition. Use your calculator to
check your answers. Do both the subtraction problem and the addition problem on your
calculator.
a)19 – 26
b)-2.7 – 3.5
c) -15 – (-12)
Part C: Subtracting with more than two signed numbers
Example 4: Ted decides to gable at the local casino. He begins with $95 at 5:00. By
5:15 Ted has lost $43 in a combination of craps and blackjack. Ted then wins $25 with
a hand of poker. What did he end up with? Write a subtraction problem that
represents the situation. Then write an equivalent addition problem and use it to
answer the question.
Example 5: Steve has $190 in his checking account before he purchases a $175
concert ticket with his debit card. He does not notice at the time that he is also charged
a $30 service fee. The bank sends him an alert that he is overdrawn. What is the
current balance in his account? Write a subtraction problem that represents this
situation and use it to answer the question.
Example 6: Substituting into an expression and evaluating it.
Evaluate:
m+p+q
for
m = -8 ; p = -4 and q = 5
M60 Sec. 2.2b pg.3
Back to Solving More Equations: Solve and check equations using signed numbers. It
is important to look at what is the easiest way to isolate the variable. If you have added
a negative number, instead of subtracting that negative number, add the opposite.
What will create a “zero pair” and leave the variable?
Example 7:
a)
–7 = -4 + w
b)
x – 0.4 = -0.06
c)
x – (-8) = -15 (Simplify first by rewriting as an equivalent addition sentence.)
d)
–26 + h = -26
e)
–6 + y = -3
M60 Sec. 2.2b pg.4
f)
–8.5 + k = -3.1
g)
–5.7 + n = -8.4
h)
c – (-0.8) = -0.02
MATH 60 Section 2.3 Multiplying and Dividing Signed Numbers
I. Review: Multiplication means____________ ___________
4 (3) means:
4 (-3) means:
Multiplying a positive number by a negative number gives a
________________ number.
Because of the commutative property of multiplication:
4 x (-3) = (-3) x 4 =
So, multiplying a negative number by a positive number gives a
_______________number.
Now let’s look at patterns. Mathematics is the science of patterns. It is designed and
defined so that patterns are followed (as much as possible):
(-3) x 4 =
(-3) x 3 =
(-3) x 2 =
(-3) x 1 =
(-3) x 0 =
(-3) x (
)=
(-3) x (
)=
(-3) x (
)=
(-3) x (
) =
In order for the pattern to remain consistent, we find that multiplying a
negative number times a negative number gives a ______________
number.
M60 Sec. 2.3 p.2
II. Summary of Rules for Multiplication:
1. The product of two numbers with the same signs is _______________.
Ex. 1: (5) (11) =
Ex. 2: (-4) ( -7) =
Ex. 3: 7 ( 3) =
Ex. 4: -6 ( -4) =
2. The product of two numbers with the opposite signs is ____________.
Ex. 5: (-2) (3) =
Ex. 6: (+4) (-3) =
Ex. 7: 6 (-2) =
Ex. 8: -10 (5) =
3. The product of any number and zero is ____________.
Ex. 9:
3 (0) =
Ex. 10:
0 (4) =
Part B: Relating Multiplication and Division:
Division is the ________________ _________________ of multiplication.
Division can be done by multiplying by the _____________________.
So, the same rules of signs which apply to multiplication also apply to division:
Ex. 1: 6
÷
Ex. 2: -4
÷-8=
2 =
Ex. 3:
M60 Sec. 2.3 p.3
Ex. 4:
=
Ex. 5:
=
Ex. 6: 36 ÷ - 9 =
Ex. 7: -35 ÷ 4 =
So Rules 1 and 2 can be further summarized: The product or quotient of
Any two numbers with the same sign will give a _____________ number.
And, the product or quotient of any two numbers with opposite signs will
give a _________ number.
4. The quotient of zero and any non-zero number is _______________.
That is, zero divided BY any number is ________________.
Ex. 8:
Ex. 9:
5. The quotient of any number and zero is ____________. That is,
division BY ZERO is ___________________.
Ex. 10:
Ex. 11:
6. Multiplying and dividing by (-1) changes the _______________
Of the number. It is the same as taking the ______________ of the
number.
Ex. 12:
- (-5) = -1 (-5) = (-5) ÷ (-1) =
M60 Sec. 2.3 p.4
Revisit order of operations:
Ex. 13: 12 ÷ (-6) (-3)
Ex. 14: -48 ÷ (-8) (-2) ÷ 4
Practice for the proficiency test: Compute without a calculator:
1. –5 – (-7)
6. –19 + 12
2. –11 – 23
7. –12 + 13
3. –2 – 9
8. 18 – 20
4. –15 – (-18)
9. –3 + (-8) – 14
5. 10 – (-6)
10. –20 – 5 + 7
11.
 23 1
12.
13.
27
3
14.
 4 
 8 
15.
16. 5  6  3
17.
18. 12  2  4  6  1 
19. Evaluate the expression - x + y for x = - 7 and y = - 3
20. Evaluate the expression (- 2)(x) – (5)(y) for x = 3 and y = - 4