Transcript Math 50 - University of Wisconsin–Stout

Any questions on
today’s homework?
(Section 1.3B)
Reminder: You should be doing this
homework without using a calculator,
because calculators can’t be used
for Quizzes 1-2, Test 1, or the
Gateway Quiz.
HOMEWORK REMINDERS:
• Today’s homework assignment on
Sections 1.4 and 1.5 is due at the
start of our next class session.
• There is also a second homework
assignment (Gateway HW #2) due
next class session. This homework
has a required paper worksheet
(which will be handed out in class by the
TAs now.)
NOW
CLOSE
YOUR LAPTOPS
(You may reopen them when I finish the
lecture, at which time you can start this
homework assignment. If there’s not time to
work on it during class, you’re welcome to stay
and work on it in the open lab next door.))
Section 1.4: Order of Operations
32 is a number written in exponential notation
• Base is 3
• Exponent is 2
• The notation means 3 • 3
Example
Evaluate 32.
Answer: 32 = 3 • 3 = 9
Example
Evaluate 23.
23 = 2 • 2 • 2
=2•2•2
=4•2
=8
Example from today’s homework:
Answer: 25
Example from today’s homework:
Answer: 16/49
NOTE: Many of today’s homework
problems as well as Gateway
problems 7 & 8 all require using
the order of operations.
Order of operations:
1)
First, calculate expressions within grouping symbols
(parentheses, brackets, braces).
If there are nested sets of grouping symbols, start with the
innermost ones first and work your way out.
2)
3)
4)
Exponential expressions – left to right
Multiplication and division – left to right
Addition and subtraction – left to right
8
Order of operations memory device:
“Please excuse my dear Aunt Sally”
1. Please
2. Excuse
3. My Dear
4. Aunt Sally
(Parentheses)
(Exponents)
(Multiply and Divide)
(Add and Subtract)
… or just remember
PEMDAS
9
Example
Simplify the following expressions.
6  2  2  2  6  2  2  32  6  4  32  2  32  34
5
3  6 (8  5 )
4 2
2

3  6 (3)
16  2

3  18
16  2

21
18

37
36

7
6
Example from today’s homework:
Answer: 156
Sample Gateway Problem # 7:
Strategy: Calculate out the entire top expression and then the entire
bottom expression, using the order of operations on each part. Then
simplify the resulting fraction, if necessary.
TOP EXPRESSION: 24 – 4(7 + 2)
Step 1: Parentheses:
24 – 4(7 + 2) = 24 – 4(9)
Step 2: Exponents:
24 – 4(9) = 2•2•2•2 – 4(9) = 16 – 4(9)
(because 2•2•2•2 = 4•2•2 = 8•2 = 16)
Step 3: Multiply/Divide: 16 – 4(9) = 16 – 4•9 = 16 – 36
Step 4: Add/Subtract:
16 – 36 = -20
12
Now calculate the bottom expression:
2(6+2) + 4
Step 1: Parentheses: 2(6+2) + 4 = 2(8) + 4
Step 2: Exponents: There aren’t any in this part.
Step 3: Multiply/Divide: 2(8) + 4 = 2•8 + 4 = 16 + 4
Step 4: Add/Subtract: 16 + 4 = 20
Now put the top over the bottom and simplify the
resulting fraction:
TOP
= 24 – 4(7 + 2) = -20 = -1 = -1
BOTTOM
2(6+2) + 4
20
1
13
Full Solution to Sample
Problem #7:
Here is the work we expect to see on your quiz worksheet:
24 – 4(7 + 2) = 24 – 4(9) = 16 – 4(9) = 16 – 36 = -20 = -1 = -1
2(6+2) + 4
2(8) + 4
16 + 4
20
20 1
14
Sample Gateway Problem # 8:
Strategy: Deal with the expressions inside the grouping
symbols (parentheses, brackets) first, starting with the innermost
set (-3 + 6).
STEP 1: (inside the parentheses)
3[17 + 5(-3 + 6) - 10] = 3[17 + 5(3) - 10]
STEP 2: (inside the brackets; multiply first, then add and subtract)
3[17 + 5(3) -10] = 3[17 + 5•3 -10] = 3[17 + 15 - 10]
= 3[17 + 15 - 10] = 3[32 - 10] = 3[22]
STEP 3: Do the final multiplication: 3[22] = 3•22 = 66
15
Full Solution to Sample
Problem #8:
Here is the work we expect to see on your quiz worksheet:
3[17 + 5(-3 + 6) - 10] = 3[17 + 5(3) - 10] =
3[17 + 15 - 10] = 3[32 - 10] = 3[22] = 66
16
A variable is a symbol used to represent a
number.
An algebraic expression is a collection of
numbers, variables, operations, grouping
symbols, but NO equal signs (=) or
inequalities (< , > , ≤ , ≥ )
We can evaluate an algebraic expression by
assigning specific values to any variables
that might be in the expression.
Example
Evaluate 3x2 – 2y + 5 when x = 2 and y = 4.
3(2)2 – 2(4) + 5 =
3·4 – 8 + 5 =
12 – 8 + 5 =
9
Example from today’s homework:
Answer: 15
An algebraic equation is a statement that two
expressions have equal value.
Example of an equation: 2x – 4 = 5 - x
A solution to an equation is a number that you can
substitute in place of the variable that makes both
sides of the equation come out to the same answer.
Example: The number 3 is a solution of
the equation 2x – 4 = 5 – x.
We show this by replacing all x’s with 3’s, then
calculating each side:
The two sides
are equal, so 3
2∙x – 4 = 2∙3 – 4 = 6 – 4 = 2
is a solution of
5–x=5–3=2
2x – 4 = 5 – x.
Example from today’s homework:
Answer: no
Section 1.5:
Working with positive and negative signs
A number line is a line on which each point is
associated with a number.
–5 –4 –3 –2 –1
Negative
numbers
0
1
2
3
4
Positive
numbers
5
To add two numbers, especially if one (or both) of them is
negative, it often helps to picture them on a number line.
Example 1 : 3 + (-5)
•Locate the first number on the number line.
• Starting from that number, if the second number is positive,
move to the right by that many units. If it’s negative, move to
the left that many units.
(-5)
3
(Count 5 units to the left from 3)
–5
–4 –3
–2
–1
Answer is: -2
0
1
2
3
4
5
Example 2: -2 + (-3)
•Locate the first number on the number line.
• Start from that first number. The second number is negative,
so move to the left by that many units.
(-3)
2
(Count 3 units to the left from -2)
–5
–4 –3
Answer is: -5
–2
–1
0
1
2
3
4
5
Example 3: -2 + 6
•Locate the first number on the number line.
• Start from that first number. The second number is positive,
so move to the right by that many units.
+6
2
–5
–4 –3
–2
(Count 6 units to the right from -2)
–1
0
1
2
3
4
5
Answer is: 4
Note: We have handouts of number line worksheets for you to
use on your homework and during tests and quizzes.
TAs will now hand out copies for you to keep in your notebook for homework
use, and clean copies will be handed out for you to use on tests and quizzes.
Example from today’s homework:
Answer: -30
Example from today’s homework:
Answer: 1.6
Remember: In problems with parentheses (or
brackets), do the part INSIDE the
parentheses FIRST.
Examples:
(-3 + 7) – [6 + (-5)] =
(-3 + 7) – [6 + (-5) ] =
4–1
=3
Example from today’s homework:
Answer: -14
Example from today’s homework:
Answer: -10
REMEMBER: Come to the open lab if
you need help. There are more than
40 open lab hours each week.
Lab hours:
Mondays through Thursdays
8:00 a.m. to 6:30 p.m.
All of our teaching assistants have
lots of experience teaching
Gateway problems and the other
problems in your HW assignments.
You may now
OPEN
your LAPTOPS
and begin working on the
homework assignment.
This assignment is due at the
start of our next class session.