Math 110 online work that is due today at the start of class: Section 1.2 Homework (no worksheet to...

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 note-taking 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.

2– 2 0 1 3 4 5– 1– 3– 4– 5

Negative numbers

Positive numbers

2– 2 0 1 3 4 5– 1– 3– 4– 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.

3 (-5)

(Count 5 units to the left from 3)

Answer is: -2

2– 2 0 1 3 4 5– 1– 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.

2

(-3)

(Count 3 units to the left from -2)

Answer is: -5

Example from today’s homework:

Answer: -30

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

Subtracting 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

Multiplying or Dividing Real Numbers

•If b is a real number, 0 · b = b · 0 = 0.

•The quotient of any real number and 0 is undefined.

• The quotient of 0 and any real number = 0.

0

a

a

0 a 0

Working with zero:

Sample problem from today’s homework:

Answer: 0If this problem had been 7/0,

the answer would be “N” (undefined)

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

ExponentsWe may use exponential notation to write products in a more compact form.

Evaluate each of the following expressions.

34 = 3 · 3 · 3 · 3 = 81

(–5)2 = (– 5)(–5) = 25

–62 = – (6)(6) = –36

(2 · 4)3 = (2 · 4)(2 · 4)(2 · 4) = 8 · 8 · 8 = 512

3 · 42 = 3 · 4 · 4 = 48

Evaluating Exponential Expressions

Example:

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 (Parentheses)

2. Excuse (Exponents)

3. My Dear (Multiply and Divide)

4. Aunt Sally (Add and Subtract)

… or just remember PEMDAS18

Using the Order of Operations

Evaluate:23

396

23

396 )9(

396

9

)3(6

9

9

1

Write 32 as 9.

Divide 9 by 3.

Add 3 to 6.

Divide 9 by 9.

Example:

More examples

Simplify the following expressions.

52226

32226 3246

322 34

24

)58(632

216

)3(63

216

183

18

21

63

73

6

7

Simplify the following.

)3(29

)2(38

69

68

3

14

3

14

Example:

Sample problem from today’s homework:

324

7

Evaluating Algebraic Expressions

(a) 5x – 2 for x = 8

Evaluate each expression for the given value.

(b) 3a2 + 2a + 4 for a = – 4

5(8) – 2 = 40 – 2 = 38

3(– 4)2 + 2(– 4) + 4

= 3(16) + (– 8) + 4 = 44

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:

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 = -1BOTTOM 2(6+2) + 4 20 1

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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)

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Additional fraction problems to do if we have time:

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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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.

REMEMBER: Come to the open lab if you need help. There are more than 40 open lab hours each week.