Download MATH 60, MOD 3 - Linn-Benton Community College

MATH 60 Section 2.4 Order of Operations with Signed Numbers
Review factor and term:
Exponents and Signed Numbers:
Part A: Exponents: Short hand way of expressing___________________
________________
4³ = 64
(-4 )2 "means"
5 3 "means"
the _______________ tells us how many times to use the ___________
as a factor in ______________ __________________.
Example 1: Describe in words what each expression means and simplify.
a) 4³
b) (-4)³
c) 5²
Example 2: What does -3² mean?
The exponent is ______, the base is ________ and the negative sign out
front indicates ______ _________________. ("lurking one" or "ninja one")
Example 3: Describe in words what each expression means and simplify.
a) 3²
b) (-3)²
c) - 3²
M60 Sec. 2.4 p.2
Example 4: Write each of the following in exponential form:
a) 0.5)(0.5)(0.5) =
b)( -3)(-3)(-3)(-3) =
c) 6  6  6  6  w  w  w  w 
Part B: Notation and Signed Numbers
Example 5: Explain what the parentheses mean in each case and simplify
a) (-3)(4) + 2
b) -4(6) - 2
c) (-3 · 4) + 7
d) -3 + (6 - 2)
e) (-5) + 4
f) (3 - 5) - (-2)(4)
Example 6: Explain what the " - " sign means in each case and simplify
a) -4 - 3
b) - 5²
c) -(-4)
d) - (-5) + 4
e) 4 - 7
f) are these two equivalent? -4(-1) - 3 and -4 -1 -3 ?
g) are these two equivalent? -(-5) - 7 and -1(-5) - 7 ?
M60 Sec. 2.4 p.3
Part C: Order of Operations with Signed Numbers:
Four levels of Priority:
1. G
2. E
3. MD
4. AS
Ex. 7: Simplify completely:
a)
5 (-3) 2
b)
(-2 5) 3
c)
22 + 32
d)
62 -34  33
e)
(-2)(3) – 2 2
f)
-(3 2) 2 - 5
g)
-2 – 3 (-3) 3 - 2
h)
-(-5 - 4) · 3
i)
-6 ·
j)
k)
( )
+7
M60 Sec. 2.4 p.4
Example 8: Use order of operations to simplify
a)
b)
c)
d)
Example 9: Solve the following and check your solution by substituting.
a)
b)
M60 Sec. 2.5: Order of Operations and Evaluating Expressions:
Part A: Evaluating Expressions
1.
2.
3.
Example 1: Evaluate each of the following by substituting the value for the variable and
simplifying. Use parentheses to "protect" the value
a) -9 + 4y for y = 10 and y = 3
b) -5y³ - y for y = -1 and y = 2
Example 2:
Evaluate the following for
a = - 2 and
b=-3
a) - a 2
b)
-2 a 2 + a
c) -ab ( a – b ) 2
d)
a–b(a2 +b2)
M60 Sec. 2.5 p.2
Part B: Tables and Evaluating Expressions
We like to use tables when we have to evaluate an expression for many values.
It is a nice way to summarize our results.
Example 3: Complete the following table by evaluating the algebraic expression shown in the
upper right for the given values of the variable:
x
-1
0
1
2
Example 4: Evaluate the expression
t = 3. Display your results in a table:
M60 Sec. 2.5 p.3
for t = -2, t = -1, t = 0, t = 1 and
Part C: Evaluating Formulas: Most often used algebra skill. Need to be able to substitute
correctly and apply the order of operations correctly.
Example 5: The formula
can be used to convert a temperature given in degrees
Fahrenheit to degrees Celsius. What Celsius temperature corresponds to
a) 32º F
b) 212º F
c) - 15º F
M60 Sec. 2.5 p.4
Example 6: The formula
can be used to convert a temperature given in degrees
Celsius (or Centigrade) to degrees Fahrenheit. What Fahrenheit temperature corresponds to a
termperature of -15º C?
Example 7: The formula used by women to calculate their maximum heart rate is to subtract
88% of their age from 206. Written as an algebraic expression this would be: 206 - 0.88a,
where a = age of the woman. Find Suzy's maximum heart rate if she is 22 years old.
M60 Sec. 2.6: Solving Equations with Signed Numbers
Part A: Signed number equations with one operation:
Remember the Do/Undo chart to help us figure out the correct steps.
Also keep in mind "where you are going". We are ultimately trying to get to x = ______. That
is, we are trying to isolate the variable all by itself on one side of the = and find out what
number results on the other side.
Treat the equation like a balance. Be sure to show the operation and resulting equation at
each step. Keep the = signs lined up and work in a neat and orderly fashion. This is your
mathematical communication. As you solve your equation, you are writing a "math
paragraph". Use correct notation and punctuation!
Example 1: For each, describe the operation that has been done to the variable, what needs to
be undone and then solve.
a)
b)
c)
d)
M60 Sec. 2.6 p.2
Part B: Signed Number Equations with two operations:
Remember when there's more than one operation, to solve the equation you will need to do
the ____________________ operation in the _______________________ order.
Example 2: solve each equation and check the solution by substituting.
a)
44 = -22y + 54
b)
-7 = - 3h - 4
c)
-12 = -12 + 40p
d)
0.22 - 0.5x = 1.31
e)
27 – 7x = 14
f)
100 – x = 129
g)
h)
M60 Sec. 2.6 p. 3
M60 Sec. 2.6 p.4
Part C: Writing and solving Equations: These need to be algebraic equations. Review
Strategy 1, the Word Method and remember the 4 critical ingredients for solving an application
problem:
1. Identify the variable
2. Write an algebraic equation that represents the situation
3. Solve the equation
4. Check the answer in two ways
5. Write a sentence.
Strategy 2: Make a Guess: Make up an answer and try to work backwards to come up with the
correct operations and then the equation that follows. Play “what if” with a number.
Example 3: Bryan purchased two used textbooks. The algebra textbook cost $28 more than
the literature book. If the total cost of the two books was $132, what was the price of the
literature book?
Note: At this point we are only able to use one variable, even though there are two unknowns.
a) Make a guess: The price of the literature book = _____________ (make it a nice easy
number, we’re not trying to guess the right answer, just make a guess to help us figure out the
equation.)
b) Write out how you would calculate the total cost of both books using your guess:
c) Choose a variable to represent the cost of the literature book.
d) Substitute your guess with the variable to create an equation. Solve and check.
e) Write your sentence.
Example 4: Use the Guess Method above to solve the following:
George got a tip that a certain stock was going to increase quite a bit in a short time. A short time later
the stock price dropped $43.15. His stock was then worth $94.57. What was the original purchase price
of the stock?
M60 Sec. 2.7: Displaying Data with Bar Graphs
Part A: Bar Graphs
Bar graph or bar chart: They can be vertical bars, horizontal bars, stacked or double
Categories:
Heights or length of bars:
Four things necessary for a good graph:
1.
2.
3.
4.
Example 1: Drawing a bar graph from data in table form:
Days of the Week
Mon Tues Wed Thurs Fri
Hours Spent in Class 6
4
5
4
2
Hours Spent in Class
Bar Graph
Student Hours Spent in Class
7
6
5
4
3
2
1
0
Day of the Week
a) What is the title?
b) What are the categories?
c) What do the heights of the bars represent?
M60 Sec. 2.7 p.2
Part B: Constructing a Bar Graph:
Example 2: The daily high temperatures for the month of June in Hotz, AZ are listed below:
66, 72, 69, 84, 94, 95, 82, 80, 75, 70, 67, 90, 87, 71, 71, 75, 87, 90, 69, 73, 78
Group the temperatures by 10's. Draw a bar graph to represent the given data. Be sure to mark the
scale and label each axis on the graph clearly. Give the graph a title.
Part C: Mean, Median and Mode from a Bar Graph
M60 Sec. 2.7 p.3
Review the terms for Center of the data. These are all called averages:
Mean:
Median:
Mode:
Example 3: This shows the results of a survey asking people how many hours of television they watch
per week.
Survey Results:
Hours of TV Watched per Week
Number of People
7
6
5
4
3
2
1
0
2
4
6
8
10
Hours of Tv
a) Write out a complete list of all the data points.
b) How many people were included in the survey?
c) Find the mean two ways. You don’t need to write out the data points. There is a short cut.
Summarize using the actual data points:
Using the short cut method:
d) Find the median two ways:
Summarize using the actual data points:
By counting to the correct data point on the bar graph:
e) Find the mode two ways:
Summarize using the actual data points:
Looking at the graph to see which bar is the tallest:
f) Find the extremes of the data and calculate the range.
g) What percent of the people surveyed watch 8 hrs of TV/wk?
h) How many people watch 8 or more hours of television/wk?
i) What percent of the people surveyed watch 8 or more hours of TV/wk?
M60 Sec. 2.7 p.4