chapter+1.ppt


Chapter 1Introduction to Algebra: Integers

1.1 Place Value

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 3

1.1 Place Value

Objectives

1. Identify whole numbers.

3. Write a whole number in words or digits.

2. Identify the place value of a digit throughhundred-trillions.


Objective 1: Identify whole numbers.

Our number system is a place value system. Each location in which a number is placed

gives it a different value.



Whole Numbers: created from the

digits 0, 1, 2, 3, …, 9


Objective 2: Identify the place value of a digit through hundred-trillions.

Example Identify the place value of

each 8 in 6,598,274,806.


Objective 3: Write a whole number in words or digits.

Example Write 6,058,120 in words.


Objective 3: Write a whole number in words or digits.

Example Write the number using digits.

Seventy-seven billion, thirty thousand, five hundred

77,000,030,500


Chapter 1Introduction to Algebra: Integers

1.2 Introduction to Signed Numbers


1.2 Introduction to Signed Numbers

Objectives

1. Write positive and negative numbers used in everyday situations.

4. Find the absolute value of integers.

3. Use the < and > symbols to compare integers.

2. Graph signed numbers on a number line.


Objective 1: Write positive and negative numbers used in everyday situations.

Numbers greater than zero are called positive numbers.

Numbers less than zero are called negative numbers.



Example Write “a loss of $500” as a

number with its appropriate sign.

– $500

Raised negative sign


Objective 2: Graph signed numbers on a number line.

A number line is like a thermometer turned sideways.


Objective 2: Graph signed numbers on a number line.

Graph each number on the

number line.

(a) –5 (b) 3 (c) 1 ½ (d) 0 (e) –1

Example

(a) (c) (b)(e)(d)


Objective 3: Use the < and > symbols to compare integers.

Integers are the numbers

…, –5, –4, –3, –2, –1, 0, 1, 2, 3 ,4, 5,… “>” means greater than “<“ means less than



Write < or > between each

pair of numbers to make a true

statement.

(a) 0 ____ 2

(b) 1 ____–4

(c) –4 ____–2

0 < 2

1 > –4

Example

–4 < –2


Objective 4: Find the absolute value of integers.


Objective 4: Find the absolute value of integers.

Find each absolute value.

(a) |4|

(b) |–4|

4 spaces, so |4| = 4

4 spaces, so |–4| = 4

Example


Chapter 1Introduction to Algebra:

Integers

1.3 Adding Integers


1.3 Adding Integers

Objectives

1. Add integers.

2. Identify properties of addition.


Objective 1: Add integers.

Numbers that you add together are called addends, and the result is called a sum.



Use a number line to find

–5 + –4 (use a football analogy).

Example



Add the integers –8 + –7.

Step 1 Add the absolute values.

|–8| = 8 and |–7| = 7

Add 8 + 7 to get 15.

Both negative Sum is negative

Step 2 Use the common sign as the sign of the sum.

–8 + –7= –15

Example



Add the integers 3 + –8.

Step 1 |3| = 3 and |–8| = 8

Subtract 8 – 3 to get 5.

Step 2 –8 has the larger absolute value and is

negative, so the sum is also

negative.

3 + –8 = –5

Example



A football team has to gain at least

10 yards during four plays in order to keep the ball.

Suppose on four plays a team lost 6 yards, gained 8

yards, lost 2 yards, and gained 7 yards. Did the

team gain enough to keep the ball?

Example



Example (continued)


Objective 2: Identify properties of addition.



Example Add 6 + 9 + –9 using the associative property.



Integers

1.4 Subtracting Integers


1.4 Subtracting Integers

Objectives

1. Find the opposite of a signed number.

2. Subtract integers.

3. Combine adding and subtracting of integers.


Objective 1: Find the opposite of a signed number.

Opposite integers are the same distance from 0 on the number line but are on opposite sides of 0.

When you add opposites, the sum is always 0. The opposites are also called additive inverses.


Objective 1: Find the opposite of a signed number.

Example Find the opposite of each number and show that the sum of the numbers is 0.

(a) 6

(b) –8

(c) 0

The opposite of 6 is –6. 6 + –6 = 0

The opposite of –8 is 8. –8 + 8 = 0

The opposite of 0 is 0. 0 + 0 = 0


Objective 2: Subtract integers.



Example Subtract the integers.

4 – 10



Example Subtract the integers.

–9 – –6


Objective 3: Combine adding and subtracting of integers.

Example Simplify by completing all the calculations.

–5 – 10 – 12 +1



Integers

1.5 Problem Solving: Rounding and Estimating


1.5 Problem Solving: Rounding and Estimating

Objectives

1. Locate the place to which a number is to be rounded.

2. Round integers.

3. Use front end rounding to estimate answers in addition and subtraction.


Objective 1: Locate the place to which a number is to be rounded.

Rounding a number means finding a number that is close to the original number, but easier to work with.



Draw a line under the place to which the number is to be rounded.Example

Round –23 to the nearest ten.



Draw a line under the place to which the number is to be rounded.

Round –54,702 to the nearest thousand.

Example


Objective 2: Round integers.



Round 349 to the nearest hundred.

Step 1

Underline the place to which the number is being rounded.

Example




Step 2

Because the next digit is 4 or less, do not change the digit in the underlined place.

Example




Step 3

Change all digits to the right of the underlined place to zeros.

Example



Round 36,833 to the nearest thousand.

Step 1

Underline the place to which the number is being rounded.

Example




Step 2

Because the next digit is 5 or more, add 1 to the underlined digit.

Example




Step 3

Change all digits to the right of the underlined place to zeros.

Example



Round 13,961 to the nearest hundred.

Step 1

Example




Step 2

Example




Step 3

Example


Objective 3: Use front end rounding to estimate answers in addition and subtraction.

If you purchase a sofa for $988 and a chair for $209, you may estimate the total cost.



In front end rounding, each number is rounded to the highest possible place, so all the digits become 0 except the first digit.


Meisha’s paycheck showed gross pay of $823. It also listed deductions of $291. Estimate her net pay.

Example


Front end rounding:

Net pay estimate: $800 – $300 = $500.


Meisha’s paycheck showed gross pay of $823. It also listed deductions of $291. Find her exact net pay.

Example


Exact pay:

$823 – $291 = $532



Integers

1.6 Multiplying Integers


1.6 Multiplying Integers

Objectives

1. Use a raised dot or parentheses to express multiplication.

2. Multiply integers.

3. Identify properties of multiplication.

4. Estimate answers to application problems involving multiplication.


Objective 1: Use a raised dot or parentheses to express multiplication.

Numbers being multiplied are called factors and the answer is called the product.



Rewrite the multiplication in three different ways.

10 × 7

Example

The factors are 10 and 7 and the product is 70.


Objective 2: Multiply integers.



Multiply the integers.

–2 • 8

Example




–10 (–6 )

Example




–3 • (4 • –5)

Example

Multiply numbers in parentheses first.

Then multiply the resulting pair of numbers.




–2 • –2 • –2

Example

Multiply the first pair of numbers.Then multiply the resulting pair of numbers.


Objective 3: Identify properties of multiplication.



Illustrate the commutative property for the product below.

–7 • –4

Example



Illustrate the associative property for the product below.

5 • 10 • 2

Example



Illustrate the distributive property for the product below.

–2(–5 + 1)

Example


Objective 4: Estimate answers to application problems involving multiplication.

Last year a video store had to replace 392 defective videos at a cost of $19 each. Estimate the amount of money the store lost on the videos.

Example

Front end rounding: 392 rounds to 400 and –$19 rounds to –$20.



Integers

1.7 Dividing Integers


1.7 Dividing Integers

Objectives

1. Divide integers.

2. Identify properties of division.

3. Combine multiplying and dividing of integers.

4. Estimate answers to application problems involving division.

5. Interpret remainders in division application problems.


Objective 1: Divide integers.

The answer to a division problem is called the quotient.



Divide the integers.

(a)

(b)

Example

Different signs,quotient is negative. 4

5

20

5

20

4

24

Same sign,quotient is positive. 6

4

24


Objective 2: Identify properties of division.


Objective 3: Combine multiplying and dividing of integers.

Simplify.

6(–10) ÷ (–3 • 2)

Example

Do operations inside parentheses first.Start at the left and perform operations from left to right.


Objective 3: Combine multiplying and dividing of integers.

Simplify.

–24 ÷ –2(4) ÷ –6

Example

No operations inside parentheses. Start at the left and perform operations from left to right.


Objective 4: Estimate answers to application problems involving division.

During a 24-hour lab experiment, the temperature of a solution dropped 96 degrees. Estimate the average drop in temperature each hour.

Example

Front end rounding: –96 degrees rounds to –100

and 24 hours rounds to 20.


Objective 5: Interpret remainders in division application problems.

The math department a a college has $360 in its budget to buy calculators. If the calculators cost $25 each, how many can be purchased? How much money will be left over?

Example


Objective 5: Interpret remainders in division application problems.

Luke needs to rent tents for 135 Scouts going on a camping trip. Each tent sleeps 6 people. How many tents should he rent?

Example

Luke will need to rent 23 tents.



Integers

1.8 Exponents and Order of Operations


1.8 Exponents and Order of Operations

Objectives

1. Use exponents to write repeated factors.

2. Simplify expressions containing exponents.

3. Use the order of operations.

4. Simplify expressions with fraction bars.


Objective 1: Use exponents to write repeated factors.

An exponent can be used to represent repeated multiplication.

The base is the number being repeatedly multiplied.


Objective 2: Simplifying expressions containing exponents.

Simplify each expression.

(a) (–5)2

(b) (–5)3

Example


Objective 2: Simplifying expressions containing exponents.

Simplify the expression.

23(–3)2

Example


Objective 3: Use the order of operations.



Simplify.

9 + 3(20 – 4) ÷ 8

Example

9 + 3(20 – 4) ÷ 8

9 + 3(16) ÷ 8

9 + 48 ÷ 8

9 + 6

15

Work inside parentheses first.

Work left to right performing multiplication and division.

Add last.



Simplify.

3 + 2(6 – 8) • (15 ÷ 3)

Example

3 + 2(6 – 8) • (15 ÷ 3)

3 + 2(–2) • (5)

3 + –4 • 5

3 + –20 –17


Work left to right performing multiplications.

Add last.



Simplify.

(–4)3 – (4 – 6)2(–3)

Example

(–4)3 – (4 – 6)2(–3)

(–4)3 – (–2)2(–3)

–64 – 4(–3)

–52


Simplify exponents.

Subtract last.–64 – –12

Multiply.


Objective 4: Simplify expressions with fraction bars.

Simplify.Example

–8 + 5(4 – 6)–8 + 5(–2)–8 + (–10)

–18

Simplify the numerator.844

)64(582

Simplify the denominator.

4 – 42 ÷ 84 – 16 ÷ 8

4 – 2

2

Simplify the fraction.

chapter+1.ppt

Documents

pearson education

numbers copyright

rights reservedsec

negative numbers

positive numbers

numbers greater

everyday situations

number system