Download 3 +

Math 110 online work that is due
today at the start of class:
• Section 1.2 Homework (no worksheet
to turn in on this one)
• The Syllabus Quiz can still be redone if you
haven’t yet gotten 100%
Any questions on the
Section 1.2 homework that
was due today?
Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Gateway Quiz Reminders:
• You will be taking an 8-question Gateway Quiz in class at the next
class session, without a calculator. The Gateway Quiz will consist
of 8 questions similar to the ones on fractions and the order of
operations that were due at the beginning of the previous class.
• We will spend some time today on how to do these kinds of
problems, and you will have another set of eight practice Gateway
problems to do at the end of the homework assignment due at the
next class session.
• You also have a Practice Gateway Quiz due at the start of next
class session, with a REQUIRED paper worksheet that must be
turned in (worksheet will be handed out now – make sure you get
one before you leave class.) You can try it up to 3 times, and only
your best score will count in the final grade. Be sure to put the
work on the worksheet for the Practice Quiz with the best score!
Section 1.3
Operations on Real Numbers
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 from today’s homework:
Answer: -30
Subtracting Real Numbers
Subtracting real numbers
• Substitute the opposite of the number being
subtracted
• Add.
• a – b = a + (– b)
Example:
Subtract the following numbers.
(– 5) – 6 – (– 3) =(– 5) + (– 6) + 3 = – 8
Multiplying or Dividing Real Numbers
Multiplying or dividing two real numbers
with same sign
• Result is a positive number
Multiplying or dividing two real numbers
with different signs
• Result is a negative number
Multiplying or Dividing Real Numbers
Examples:
Find each of the following products.
4 · (–2) · 3 = –24
(–4) · (–5) = 20
Working with zero:
•If b is a real number, 0 · b = b · 0 = 0.
•The quotient of any real number and 0 is undefined.
a
0
• The quotient of 0 and any real number = 0.
0
a
a0
Sample problem from today’s homework:
Answer: 0
If this problem had been 7/0,
the answer would be “N” (undefined)
Exponents
We may use exponential notation to write
products in a more compact form.
Exponential notation for the product of five
threes is 35
• Base is 3
• Exponent is 5
• The notation means 3 • 3 • 3 • 3 • 3, or 243
Example
Evaluate 43.
43 =4 • 4 • 4 = 64
Evaluating Exponential Expressions
Example:
Evaluate each of the following expressions.
34 = 3 · 3 · 3 · 3
(–5)2 = (– 5)(–5)
–62 = – (6)(6)
(2 · 4)3
= 81
= 25
= –36
= (2 · 4)(2 · 4)(2 · 4)
3 · 42 = 3 · 4 · 4
= 48
=8·8·8
= 512
The Order of Operations
Order of Operations
Simplify expressions using the order that follows. If
grouping symbols such as parentheses are present,
simplify expression within those first, starting with the
innermost set. If fraction bars are present, simplify the
numerator and denominator separately.
1. Evaluate exponential expressions, roots, or absolute
values in order from left to right.
2. Multiply or divide in order from left to right.
3. Add or subtract in order from left to right.
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
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Using the Order of Operations
Example:
Evaluate:
693
32
693
32




6  9 3
(9)
6  (3)
9
9
9
1
Write 32 as 9.
Divide 9 by 3.
Add 3 to 6.
Divide 9 by 9.
More
examples
Simplify the following expressions.
6  22  2 
5
6  2  2  32 
6  4  32 
2  32  34
3  6(8  5)
 3  6(3)
2
4 2
 3  18
16  2
 21
 37
16  2

18
36
7
6
Example:
Simplify the following.
8  3(2)

 9  2(3)
86

96
14

3
14

3
Sample problem from today’s homework:

324
7
Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute the
numerical value for each variable into the expression
and simplify the result using the order of operations.
Example:
Evaluate each expression for the given value.
(a) 5x – 2 for x = 8
5(8) – 2 = 40 – 2 = 38
(b) 3a2 + 2a + 4 for a = – 4
3(– 4)2 + 2(– 4) + 4
= 3(16) + (– 8) + 4 = 44
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,absolute values, fraction bars).
If there are nested sets of grouping symbols, start with the
innermost ones first and work your way out.
2) Exponential expressions – left to right
3) Multiplication and division – left to right
4) Addition and subtraction – left to right
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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
25
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
26
Full Solution to Sample Problem #7:
Here is the work we expect to see on your 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
27
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 28
Full Solution to Sample Problem #8:
Here is the work we expect to see on your worksheet:
3[17 + 5(-3 + 6) - 10] = 3[17 + 5(3) - 10] =
3[17 + 15 - 10] = 3[32 - 10] = 3[22] = 66
29
Quick review of fractions:
•
•
•
•
•
•
Terminology: numerator, denominator
Simplifying fractions
Multiplying
Dividing
Adding: needs common denominator
Subtracting: needs common denominator
ALWAYS simplify answer!!
(factor numerator and denominator into their
prime factors, then cancel common factors, if any)
If you want more tips on
Gateway Problems 1-6
(and today’s last 8 homework problems)
click on the PowerPoint Slides item in today’s
Assignment folder for Section 1.3, then click on
the link for Gateway Problems.
(click here to show link)
31
Additional fraction problems to do if we have time:
32
Reminders:
• Today’s homework assignment on section 1.3 is
due at the start of our next class session.
• The Practice Gateway Quiz is also due at the start
of the next class session, and has a required
paper worksheet that must be turned in at that
time. You can take this quiz up to three times,
and only your best score will count in your final
grade.
• Links to these online assignments are in the
Section 1.3 Assignments folder.
• At the next class session you will be taking the
in-class Gateway Quiz (no calculators allowed!)
You may now open your laptops to work
on the homework for section 1.3 and/or
the practice Gateway Quiz.
If you have finished the homework and
practice quiz already, or if you get them
both finished before the end of the class
period, show your on-screen 100% score
to the teacher or TA and you may then
leave class early.
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REMEMBER: Come to the open lab if
you need help. There are more than
40 open lab hours each week.
Math TLC Open Lab Hours:
Monday through Thursday
8:00 a.m. – 6:30 p.m.
This lab is closed on Fridays,
on weekends and during breaks.