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# Algebra Chapter 1

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Tweet## Jenee Santos

on 17 January 2013#### Transcript of Algebra Chapter 1

Lesson 5: Adding and Subtracting

Real Numbers Words We Should All Know!

Quantity- is anything the can be measured or counted Variable- a symbol that represents a value

Algebraic Expression- a mathematical phrase that includes one or more variables

Numerical Expression- is a mathematical phrase involving numbers and operation symbols, but no variables Lets Get Started!

Lesson 1 - Variables & Expressions Words We Should All Know:

Power- the base and exponent of and expression

Exponent- tells you how many times to use the base as a factor

Base- a number that is multiplied repeatedly

Simplify- to replace an expression with its simplest name or form

Evaluate- to substitute a given number for each variable, and then simplify Lesson 2 : Order of Operations and Evaluating Expressions Lesson 4: Properties of

Real Numbers Words We Should All Know! Words We Should Know!

Absolute Value- is the numbers distance from 0

Opposites- two numbers that are the same distance away from zero but are on different sides of the number line

Additive Inverses- a number and its opposites By: Jenee' Santos Algebra Chapter 1 Algebra uses symbols to represent quantities that are unknown or that vary. You can represent mathematical phrases and real-world relationships using symbols and operations. Writing Expressions With Addition and Subtraction

Problem: 32 more than the number n

You know its addition because the phrase says "MORE"

So the expression is - n + 32

Problem: 58 less than a number n

You know its subtraction because the phrase says "LESS"

So the expression is - 58 - n Writing Expressions With Multiplication and Subtraction

Problem: 8 times a number n

You know its multiplication because the phrase says"TIMES"

So the expression is - 8n , 8 x n

Problem: the quotient of a number n and 5

You know its division because the phrase says"QUOTIENT"

So the expression is - n/5 You can use powers to shorten how you represent repeated multiplications such as 2•2•2•2•2•2•2

(2 to the 7th power) Order of Operations

P.E.MD.A.S

P-parentheses

E-exponents

M-multiplication

D-division

A-addition

S-subtraction A power has 2 parts. A Base and an exponent. The exponent tells you how many times to use the base. Lesson 3: Real Numbers

and The Number Line Words We Should Know!

Square Root- a number that produces a specified quantity when multiplied by itself

Radicand- the expression under the radical symbol

Radical- an expression made up of a radical symbol

Perfect Square- the square of an integer

Set- is a well-defined collection of objects

Element of a Set- is what each object of a set is called

Subset- consists of elements from a given set

Rational Numbers- is any number that you can write in a form a/b , where a and b are integers, and b is not equal to zero

Natural Numbers- the counting numbers

Whole Numbers- the non negative numbers

Integers- whole numbers and their opposites

Irrational Numbers- cannot be represented as the quotient of two integers

Real Numbers- a number that is either rational or irrational

Inequality- is a mathematical sentence that compares the values of two expression using and inequality symbol Here's a Video To Help! A number (a) is only a square root to a number (b) if (a^2) is equivalent to (b)

For Example: 7^2 (7 x 7) is equal to 49.Therefore 7 is a square root of 49, Also 49 is a perfect square because you can estimate the square root of the radicand. NaturalNumbers Whole Numbers& Integers Real Numbers Rational and Irrational Numbers

form the set of real numbers Lets Do Some Examples!

To Which Subsets of the real numbers does each number belong to?

1.) 15

2.) -1.4583

3.) The square root of 57 Ans.1.) 15 {natural numbers, whole numbers, integers, rational numbers}

2.) -1.4583 {rational numbers} [because -1.4583 is a terminating decimal]

3.) The square root of 57 {irrational numbers, real numbers} Equivalent Expressions- when an algebraic expression has the same value for all variables

Deductive Reasoning- the process of reasoning logically from given facts to a conclusion

Counterexample- an example showing that a statement is false Relationships that are always true for

real numbers are called properties, which

are rules used to rewrite and compare expressions.

The following properties show expressions that are equivalent for all real numbers Commutative Property of Addition. Associative Property of Addition. Commutative Property of Multiplication Associative Property of Multiplication Identity Property Of Addition

Basically says that the sum of any real number and zero is the original number. Identity Property Of Multiplication:

Basically says that the product of any real number and one is the original number. Zero Property of Multiplication:

Simply The product of a and 0 is 0 Multiplication Property of -1:

The product of -1 and a is equal to -a What Property is illustrated by

each statement?

1.) 42 * 0 = 0

2.) (y + 2.5) + 28 = y + (2.5 + 28)

3.) 10x + 0 = 10x Ans.

1.) 42 * 0 = 0 {Zero Property of Multiplication}

2.) (y + 2.5) +28 = y (2.5 + 28) {Associative Property of Addition}

3.) 10x + 0 = 10x {Identity Property of Addition} You can add or subtract any real numbers using

a numbers line model. You can also add or subtract real numbers using rules involving absolute value Lesson 6: Multiplying

and Dividing

Real Numbers Words We Should Know!

Multiplicative Inverse- the product of a nonzero number and its reciprocal Lesson 7: The

Distributive Property Words To Know!

Distributive Property- is another property of real numbers that helps you simplify expressions

Term- is number or variable, or the product of a number and one or more variables

Constant- a term that has no variable

Coefficient- is a numerical factor of a term

Like Terms- have the same variable factors Lesson 8: An Introduction To Equations Equation- is a mathematical sentence that uses an equal sign (=)

Open Sentence- is an equation that equations one or more variables and may be true or false depending on the values of its variables

Solution of an Equation- is a variable that makes the equation true Example Example Lets Practice! What is Each Sum?

1.) -12+7= ?

2.) -18+(-2)= ?

3.) 20+12= ?

4.) 9+(-3)= ? Ans.

1.) -12+7= -5

2.) -18+(-2)= -20

3.) 20+12= 32

4.) 9+(-3)= 6 Inverse Property of Addition Subtracting Real Numbers To subtract a real number, add its opposite.

a - b = a + (-b)

Example: 3 - 5 = 3 + (-5) Both equal to ( -2 ) You can use Distributive Property to simplify the product f a number and a sum or difference. Lets a, b, and c, be real numbers. a(b + c) = ab + ac

(b + c)a = ba + bc

a(b - c) = ab - ac

(b - c)a = ba - ca Lets Try One!

3(x + 8)

3(x + 8) = 3(x) + 3(8)

= 3x + 24 You can use an equation to represent the relationship between 2 quantities that uses an equal sign (=) -An equation is true if the expressions on either side or the equal sign are equal.

-An equation is false if the expression is false if the expression on either side of the equation sign are not equal Is the equation TRUE, FALSE, or OPEN?

1.) 24 + 18 = 20 + 22

2.) 7 * 8 = 54

3.) 2x - 14 = 54 Ans.

1.) 24 + 18 = 20 + 22 {True, because both expressions equal 42}

2.) 7 * 8 = 54 {False, because the expression equals 56}

3.) 2x - 14 = 54 {Open, because there is a variable Lesson 9: Patterns, Graphs, and Equations Words To Know

Solution of an Equation- is any ordered pair (x,y) that makes the equation true

Inductive Reasoning- is the process of reaching a conclusion based on an observed pattern Sometimes the value of one quantity can be found if you know the value of another. You can represent the relationship between the quantities in different ways, including tables, equations and graphs You can represent the relationship

between two varying quantities. Identifying Solutions of a Two-Variable Equation:

Is (4,10) a solution of the equation y=4x ?

Replace (x) with the first value in the ordered pair, then add (y) with the second value in the ordered pair.

10 = 4 * 4 ? Substitute 4 for x and 10 for y

10 is not equal to 16

So ( 4, 10 ) is not a solution for

y=4x Using a Table, an Equation, and a Graph!

Problem: Both Carrie and her sister Kim were born on October 25, but Kim was born 2 years before Carrie. How can you represent the relationship between Carrie's age and Kim's age in different ways? Step 1: MAKE A TABLE Step 2: WRITE AN EQUATION

Let x = Carrie's Age. Let y = Kim's Age

From the table, you can see that (y) is always 2 greater than x.

So y = x + 2 Step 3: DRAW A GRAPH The rules for multiplying real numbers are related to the properties of real numbers and the definitions of operations.

Full transcriptReal Numbers Words We Should All Know!

Quantity- is anything the can be measured or counted Variable- a symbol that represents a value

Algebraic Expression- a mathematical phrase that includes one or more variables

Numerical Expression- is a mathematical phrase involving numbers and operation symbols, but no variables Lets Get Started!

Lesson 1 - Variables & Expressions Words We Should All Know:

Power- the base and exponent of and expression

Exponent- tells you how many times to use the base as a factor

Base- a number that is multiplied repeatedly

Simplify- to replace an expression with its simplest name or form

Evaluate- to substitute a given number for each variable, and then simplify Lesson 2 : Order of Operations and Evaluating Expressions Lesson 4: Properties of

Real Numbers Words We Should All Know! Words We Should Know!

Absolute Value- is the numbers distance from 0

Opposites- two numbers that are the same distance away from zero but are on different sides of the number line

Additive Inverses- a number and its opposites By: Jenee' Santos Algebra Chapter 1 Algebra uses symbols to represent quantities that are unknown or that vary. You can represent mathematical phrases and real-world relationships using symbols and operations. Writing Expressions With Addition and Subtraction

Problem: 32 more than the number n

You know its addition because the phrase says "MORE"

So the expression is - n + 32

Problem: 58 less than a number n

You know its subtraction because the phrase says "LESS"

So the expression is - 58 - n Writing Expressions With Multiplication and Subtraction

Problem: 8 times a number n

You know its multiplication because the phrase says"TIMES"

So the expression is - 8n , 8 x n

Problem: the quotient of a number n and 5

You know its division because the phrase says"QUOTIENT"

So the expression is - n/5 You can use powers to shorten how you represent repeated multiplications such as 2•2•2•2•2•2•2

(2 to the 7th power) Order of Operations

P.E.MD.A.S

P-parentheses

E-exponents

M-multiplication

D-division

A-addition

S-subtraction A power has 2 parts. A Base and an exponent. The exponent tells you how many times to use the base. Lesson 3: Real Numbers

and The Number Line Words We Should Know!

Square Root- a number that produces a specified quantity when multiplied by itself

Radicand- the expression under the radical symbol

Radical- an expression made up of a radical symbol

Perfect Square- the square of an integer

Set- is a well-defined collection of objects

Element of a Set- is what each object of a set is called

Subset- consists of elements from a given set

Rational Numbers- is any number that you can write in a form a/b , where a and b are integers, and b is not equal to zero

Natural Numbers- the counting numbers

Whole Numbers- the non negative numbers

Integers- whole numbers and their opposites

Irrational Numbers- cannot be represented as the quotient of two integers

Real Numbers- a number that is either rational or irrational

Inequality- is a mathematical sentence that compares the values of two expression using and inequality symbol Here's a Video To Help! A number (a) is only a square root to a number (b) if (a^2) is equivalent to (b)

For Example: 7^2 (7 x 7) is equal to 49.Therefore 7 is a square root of 49, Also 49 is a perfect square because you can estimate the square root of the radicand. NaturalNumbers Whole Numbers& Integers Real Numbers Rational and Irrational Numbers

form the set of real numbers Lets Do Some Examples!

To Which Subsets of the real numbers does each number belong to?

1.) 15

2.) -1.4583

3.) The square root of 57 Ans.1.) 15 {natural numbers, whole numbers, integers, rational numbers}

2.) -1.4583 {rational numbers} [because -1.4583 is a terminating decimal]

3.) The square root of 57 {irrational numbers, real numbers} Equivalent Expressions- when an algebraic expression has the same value for all variables

Deductive Reasoning- the process of reasoning logically from given facts to a conclusion

Counterexample- an example showing that a statement is false Relationships that are always true for

real numbers are called properties, which

are rules used to rewrite and compare expressions.

The following properties show expressions that are equivalent for all real numbers Commutative Property of Addition. Associative Property of Addition. Commutative Property of Multiplication Associative Property of Multiplication Identity Property Of Addition

Basically says that the sum of any real number and zero is the original number. Identity Property Of Multiplication:

Basically says that the product of any real number and one is the original number. Zero Property of Multiplication:

Simply The product of a and 0 is 0 Multiplication Property of -1:

The product of -1 and a is equal to -a What Property is illustrated by

each statement?

1.) 42 * 0 = 0

2.) (y + 2.5) + 28 = y + (2.5 + 28)

3.) 10x + 0 = 10x Ans.

1.) 42 * 0 = 0 {Zero Property of Multiplication}

2.) (y + 2.5) +28 = y (2.5 + 28) {Associative Property of Addition}

3.) 10x + 0 = 10x {Identity Property of Addition} You can add or subtract any real numbers using

a numbers line model. You can also add or subtract real numbers using rules involving absolute value Lesson 6: Multiplying

and Dividing

Real Numbers Words We Should Know!

Multiplicative Inverse- the product of a nonzero number and its reciprocal Lesson 7: The

Distributive Property Words To Know!

Distributive Property- is another property of real numbers that helps you simplify expressions

Term- is number or variable, or the product of a number and one or more variables

Constant- a term that has no variable

Coefficient- is a numerical factor of a term

Like Terms- have the same variable factors Lesson 8: An Introduction To Equations Equation- is a mathematical sentence that uses an equal sign (=)

Open Sentence- is an equation that equations one or more variables and may be true or false depending on the values of its variables

Solution of an Equation- is a variable that makes the equation true Example Example Lets Practice! What is Each Sum?

1.) -12+7= ?

2.) -18+(-2)= ?

3.) 20+12= ?

4.) 9+(-3)= ? Ans.

1.) -12+7= -5

2.) -18+(-2)= -20

3.) 20+12= 32

4.) 9+(-3)= 6 Inverse Property of Addition Subtracting Real Numbers To subtract a real number, add its opposite.

a - b = a + (-b)

Example: 3 - 5 = 3 + (-5) Both equal to ( -2 ) You can use Distributive Property to simplify the product f a number and a sum or difference. Lets a, b, and c, be real numbers. a(b + c) = ab + ac

(b + c)a = ba + bc

a(b - c) = ab - ac

(b - c)a = ba - ca Lets Try One!

3(x + 8)

3(x + 8) = 3(x) + 3(8)

= 3x + 24 You can use an equation to represent the relationship between 2 quantities that uses an equal sign (=) -An equation is true if the expressions on either side or the equal sign are equal.

-An equation is false if the expression is false if the expression on either side of the equation sign are not equal Is the equation TRUE, FALSE, or OPEN?

1.) 24 + 18 = 20 + 22

2.) 7 * 8 = 54

3.) 2x - 14 = 54 Ans.

1.) 24 + 18 = 20 + 22 {True, because both expressions equal 42}

2.) 7 * 8 = 54 {False, because the expression equals 56}

3.) 2x - 14 = 54 {Open, because there is a variable Lesson 9: Patterns, Graphs, and Equations Words To Know

Solution of an Equation- is any ordered pair (x,y) that makes the equation true

Inductive Reasoning- is the process of reaching a conclusion based on an observed pattern Sometimes the value of one quantity can be found if you know the value of another. You can represent the relationship between the quantities in different ways, including tables, equations and graphs You can represent the relationship

between two varying quantities. Identifying Solutions of a Two-Variable Equation:

Is (4,10) a solution of the equation y=4x ?

Replace (x) with the first value in the ordered pair, then add (y) with the second value in the ordered pair.

10 = 4 * 4 ? Substitute 4 for x and 10 for y

10 is not equal to 16

So ( 4, 10 ) is not a solution for

y=4x Using a Table, an Equation, and a Graph!

Problem: Both Carrie and her sister Kim were born on October 25, but Kim was born 2 years before Carrie. How can you represent the relationship between Carrie's age and Kim's age in different ways? Step 1: MAKE A TABLE Step 2: WRITE AN EQUATION

Let x = Carrie's Age. Let y = Kim's Age

From the table, you can see that (y) is always 2 greater than x.

So y = x + 2 Step 3: DRAW A GRAPH The rules for multiplying real numbers are related to the properties of real numbers and the definitions of operations.