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# Chapter 1: Foundations of Algebra

By: Bria Rouse

by

Tweet## Bria Rouse

on 11 January 2013#### Transcript of Chapter 1: Foundations of Algebra

Chapter 1: The Foundations Of Algebra * * * * * * ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ * * In this lesson, we will go over:

Variables and Expressions

Order of Operations and Evaluatiing Expressions

Real Numbers and the Number Line

Properties of Real Numbers

Adding/Subtracting Real Numbers

Multiplying/Dividing Real Numbers

The Distributive Property

An Introduction to Equations

Patterns, Equations, and Graphs Thanks for watching! 1. Variables and Expressions Vocabulary: -quantity: anything that can be measured or counted

-variable: a symbol, usually a letter, that represents the value(s) of a variable quantity

-algebraic expression: a mathematical phrase that includes one or more variables

-numerical expression- a mathematical phrase involving numbers and operation symbols, but no variables 2. Order of Operations and

Evaluating Expressions Vocabulary: -power: has a base and an exponent

(the exponent tells you how many times to use the base as a factor)

-exponent: a quantity representing the power to which a given number or expression is to be raised

-base: a number used as the basis of a numeration scale

-simplify: you do this to a numerical expression when you replace it with its single numerical value

-evaluate: replacing each variable with a given number 3. Real Numbers and

the Number Line Vocabulary: -square root: a factor of a number that when squared gives the number

-radicand: the expression under the radical

-radical: the radical symbol and the radicand together

-perfect square: the square of an integer

-set: a well-defined collection of objects

-element of a set: each object in a set

-subset: groups of elements from the same set

-rational number: any number that you can write in the form a/b

-natural numbers: all numbers from 1 and beyond

-whole numbers: all number from 0 and beyond

-integers: all numbers (both positive and negative)

-irrational number: can't be represented as the quotient of two integers

-real numbers: both rational and irrational numbers

-inequality: a mathematical sentence that compares the values of two expressions using an inequality symbol 4. Properties of

Real Numbers Vocabulary: -equivalent expressions: two expressions that have the same value for all values of the variables

-deductive reasoning: the process of reasoning logically from given facts

-counterexample: an example showing that a statement is false You're probably

thinking... When is the

presentation

going to end??? Well don't worry,

we're halfway there! 5. Adding and Subtracting

Real Numbers Vocabulary: -absolute value: a number's distance from 0 on a number line

-opposites: two numbers that are the same distance from the 0 on the number line

-additive inverse: a number and its opposites 6. Multiplying and

Dividing

Real Numbers Vocabulary: -multiplicative inverse: one of a pair of numbers whose product is 1

-reciprocal: a nonzero real number of the form a/b is b/a 7. The Distributive Property Vocabulary: -Distributive Property- a property of real numbers that helps you simplify expressions

-term: a number, a variable, or the product of a number and one or more variables

-constant: a term that has no variables

-coefficient: a numerical factor of a term

-like terms: have the same variable factors 8. An Introduction

to Equations Vocabulary: -equation: a mathematical sentence that uses an equal sign

-open sentence: an equation with one or more variables and may be true or false depending on the values of its variables

-solution of an equation: variable that makes equation true 9. Pattern, Equations,

and Graphs Vocabulary: -solution of an equation: any ordered pair (x,y) that makes the equation true

-inductive reasoning: the process of reaching a conclusion based on an observed pattern Bria Rouse

Ms. Lynch

January 11th, 2013

Algebra 1 Honors Writing Expressions With

Addition and Subtraction Writing Expressions With Multiplication and Division Writing Expressions With

Two Operations Using Words for an Expression Writing a Rule to Describe a Pattern * * * * * * * * * * * * * * * * * Writing Expressions

With Addition Writing Expressions

With Subtraction Writing Expressions

With Multiplication Writing Expressions

With Division BIBLIOGRAPHY: Simplifying Powers Simplifying a Numerical Expression Evaluating Algebraic Expressions Evaluating a Real

World Expression :) :) :) :) :) Simplifying Square

Root Expressions Estimating a

Square Root Classifying

Real Numbers Comparing

Real Numbers Graphing and Ordering

Real Numbers Identifying Properties Using Properties for

Mental Calculations Writing Equivalent

Expressions Using Deductive Reasoning

and Counterexamples Using Number Line Models Adding Real Numbers Subtracting Real Numbers Adding and Subtracting Real Numbers Multiplying Real Numbers Simplifying Square Root Expressions Dividing Real Numbers Dividing Fractions Simplifying Expressions Rewriting Fraction

Expressions Using the Multiplication

Property of -1 Using the Distributive Property for Mental Math Combining Like Terms Classifying Equations Identifying Solutions

of an Equation Writing an Equation Using Mental Math to Find Solutions Using a Table to

Find a Solution Estimating a Solution Identifying Solutions of a Two_Variable Equation Using a Table, An Equation, and a Graph Obective: To write

algebraic expressions Objectives:

- To simplify expressions involving exponents

- To use the order of operations to evaluate expressions Objectives:

-To classify, graph,

and compare real

numbers

-To find and estimate

square roots Objective:

To identify and use

properties of real numbers Objective: To find the sums and difference of real numbers Objective:

-to find products

and quotients of

real numbers Objective:

-To use the Distributive

Property to simplify

expressions Objective:

-To solve

equations

using tables

and mental math Objective:

To use tables, equations, and graphs to describe relationships What is an algebraic expression for the following word phrase? Word Phrase: 32 more than a number "n" n+32 Now you try!

Word Phrase: 20 more than a number "n" What is the algebraic expression for the following word phrase? Word Phrase: 58 less than a number "n" 58-n Now you try!

3 less than a number "n" What is the algebraic expression

for the following word phrase? Word Phrase: 8 times a number "n" Now you try!

15 times a number "n" 8n What is the algebraic expression for the following word phrase? Word Phrase: the quotient of a number "n" and 7 Now you try!

The quotient of a number "n" and 18 n/7 What is the algebraic expression

for the following word phrase? 3 more than twice a number "x" Now you try!

The product of 4 and the sum of a number "x" and 7 3+2x What word phrase can you use to represent the algebraic expression 3x? 3 times a number "x" or the product of 3 and a number "x" Now you try!

What word phrase can you use to represent the algebraic expression n/3 ? What is the simplest form of

the expression,

10 to the 7th power? 10,000,000 Now you try!

Simplify 3 to the 4th power What is the simplest form of the expression? (6-2)^3/2 Step 1: Simplify inside parenthesis

Step 2: Simplify the power

Step 3: Simplify What is the value of the

expression if x=5 and y=2 ? x^2 +x -12 /y^2 Step 1: Substitute 5 for x and 2 for y

Step 2: Simplify exponents

Step 3: Divide

Step 4: Add and subtract from left to right * * * * * * * * * * What is the square

root of 81? Answer: because 9 times 9 is 81 Now lets try!

What is the square root of 64? Now you try!

What is the value of the square root of 34 to the nearest integer? What is an inequality that compares the square root of 17 and 4 1/3? Step 1: Write the square root as a decimal

Step 2: Write the decimal as a fraction

Step 3: Compare using an inequality symbol What is the order of the square root of 4, .4, -2/3, the square root of 2, and -1.5 from least to greatest?

Step 1: Simplify all the numbers

Step 2: Order from least to greatest

Step 3: Graph on number line A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs $1.25. What is the total cost of a ticket, a drink, and popcorn? (7.75 + 2.40) + 125 = (2.40 + 7.75) + 1.25 Commutative Property of Addition

= 2.40 + (7.75 + 1.25) Association Property of Addition

= 2.40 + 9 Simplify inside parenthesis

= 11.40 Simplify Know: An expression

Need: Groups of numbers that can be simplified

Plan: Use properties to group or reorder parts of the expression Example: (4 + 7b) +8

(7b + 4) +8 Commutative Property of Addition

7b + (4+8) Association Property of Addition

7b + 12 Simplify For all real numbers j and k,

j x k = (k + 0) x j

This is true because it will be true for

every real numbers you substitute

for j and k. Now you try!

-12+7 What is the sum of 3+5. Picture using a number line. If you were at 3 on a number line, and you moved 5 spaces to the right, you would be at 8.

Therefore, 3 plus 5 is 8. Now you try!

-8 - (-13) Simplifying the square root of -25 would be -5 because the square root of 25 is 5. The absolute value of 5 is -5. Step 1: Rewrite the expression

Step 2: Substitute fractions for x and y

Step 3: Multiply by the reciprocal of the second fraction

Step 4: Simplify What sum or difference

is equivalent to 7x+2 /5 ? 1/5 (7x+2) Write division as multiplication

1/5(7x) + 1/5(2) Distribute

7/5x + 2/5 Simplify Now you try!

4x-16/3 What is the simplest form of -(2y-3x) ? -1(2y-3x) Multiplication Property of -1

-1(2y) + -1(-3x) Distribute

-2y+ 3x Simplify Now you try!

-(4x-12) 8(4.95) = 8( 5 - 0.05) Think of 4.95 as 5 - 0.05

8(5) -8(.05) Distributive Property

40- 0.4 Multiply mentally

39.60 Subtract mentally 5x - 3 - 3x+6y + 4 = 5x + (-3) + (-3x) + 6y +4 Rewrite as a sum

5x + (-3x) + 6y + (-3) +4 Commutative Property

(5-3)x + 6y + (-3) + 4 Distribute

2x + 6y + 1 Simplify Now you try!

3y - y How to tell if an equation

is true, false, or open... -If the equation is equal on both sides, it is true

-If it isn't equal, it is false

-If a variable is present, the equation is open Now you try!

24 + 18 = 20 + 22

2x-14 = 54 Is x = 6 a solution of the equation 32 = 2x +12? You would substitute 6 for x

to find out if the equation is true. If it's not true, 6 is not a solution

for x in this equation What is the solution of the equation? Use mental math. x + 8 = 12 Think: What # plus 8 equals 12

Solution: 4

Check 4 + 8 = 12 Now you try!

a/8 = 9 If the equation is 5n + 8 = 48, you would use a table to substitute numbers for n until the equation is true. To estimate the solution, find the integer values of x between which the solution must be. Is (3,10) a solution of the equation y=4x If you substitute 3 for x and 10 for y, and the equation is true, then (3,10) is a solution. 0 - 25 + 16 -47 +29 Write an expression

0 + (-25) + 16 + (-47) + 29 Use rule for subtracting real numbers

0 + 16 + 29 + (-25) + (-47) Commutative Property of Addition

0 + (16 + 29) + [(-25) + (-47)] Group Addends with the same sign

0 + 45 + (-72) Add inside grouping symbols

45 + (-72) Identity Property of Addition

-27 Use rule for adding with different signs "LESSON: Writing Expressions and Equations." LESSON: Writing Expressions and Equations. N.p., n.d. Web. 10 Jan. 2013.

"Welcome to SciMath MN." Welcome to SciMath MN. N.p., n.d. Web. 10 Jan. 2013.

"Get Math Help, Online Math Tutoring." Free Math Tutoring. N.p., n.d. Web. 10 Jan. 2013.

YouTube. YouTube, n.d. Web. 10 Jan. 2013. * * * * * * * * *

Full transcriptVariables and Expressions

Order of Operations and Evaluatiing Expressions

Real Numbers and the Number Line

Properties of Real Numbers

Adding/Subtracting Real Numbers

Multiplying/Dividing Real Numbers

The Distributive Property

An Introduction to Equations

Patterns, Equations, and Graphs Thanks for watching! 1. Variables and Expressions Vocabulary: -quantity: anything that can be measured or counted

-variable: a symbol, usually a letter, that represents the value(s) of a variable quantity

-algebraic expression: a mathematical phrase that includes one or more variables

-numerical expression- a mathematical phrase involving numbers and operation symbols, but no variables 2. Order of Operations and

Evaluating Expressions Vocabulary: -power: has a base and an exponent

(the exponent tells you how many times to use the base as a factor)

-exponent: a quantity representing the power to which a given number or expression is to be raised

-base: a number used as the basis of a numeration scale

-simplify: you do this to a numerical expression when you replace it with its single numerical value

-evaluate: replacing each variable with a given number 3. Real Numbers and

the Number Line Vocabulary: -square root: a factor of a number that when squared gives the number

-radicand: the expression under the radical

-radical: the radical symbol and the radicand together

-perfect square: the square of an integer

-set: a well-defined collection of objects

-element of a set: each object in a set

-subset: groups of elements from the same set

-rational number: any number that you can write in the form a/b

-natural numbers: all numbers from 1 and beyond

-whole numbers: all number from 0 and beyond

-integers: all numbers (both positive and negative)

-irrational number: can't be represented as the quotient of two integers

-real numbers: both rational and irrational numbers

-inequality: a mathematical sentence that compares the values of two expressions using an inequality symbol 4. Properties of

Real Numbers Vocabulary: -equivalent expressions: two expressions that have the same value for all values of the variables

-deductive reasoning: the process of reasoning logically from given facts

-counterexample: an example showing that a statement is false You're probably

thinking... When is the

presentation

going to end??? Well don't worry,

we're halfway there! 5. Adding and Subtracting

Real Numbers Vocabulary: -absolute value: a number's distance from 0 on a number line

-opposites: two numbers that are the same distance from the 0 on the number line

-additive inverse: a number and its opposites 6. Multiplying and

Dividing

Real Numbers Vocabulary: -multiplicative inverse: one of a pair of numbers whose product is 1

-reciprocal: a nonzero real number of the form a/b is b/a 7. The Distributive Property Vocabulary: -Distributive Property- a property of real numbers that helps you simplify expressions

-term: a number, a variable, or the product of a number and one or more variables

-constant: a term that has no variables

-coefficient: a numerical factor of a term

-like terms: have the same variable factors 8. An Introduction

to Equations Vocabulary: -equation: a mathematical sentence that uses an equal sign

-open sentence: an equation with one or more variables and may be true or false depending on the values of its variables

-solution of an equation: variable that makes equation true 9. Pattern, Equations,

and Graphs Vocabulary: -solution of an equation: any ordered pair (x,y) that makes the equation true

-inductive reasoning: the process of reaching a conclusion based on an observed pattern Bria Rouse

Ms. Lynch

January 11th, 2013

Algebra 1 Honors Writing Expressions With

Addition and Subtraction Writing Expressions With Multiplication and Division Writing Expressions With

Two Operations Using Words for an Expression Writing a Rule to Describe a Pattern * * * * * * * * * * * * * * * * * Writing Expressions

With Addition Writing Expressions

With Subtraction Writing Expressions

With Multiplication Writing Expressions

With Division BIBLIOGRAPHY: Simplifying Powers Simplifying a Numerical Expression Evaluating Algebraic Expressions Evaluating a Real

World Expression :) :) :) :) :) Simplifying Square

Root Expressions Estimating a

Square Root Classifying

Real Numbers Comparing

Real Numbers Graphing and Ordering

Real Numbers Identifying Properties Using Properties for

Mental Calculations Writing Equivalent

Expressions Using Deductive Reasoning

and Counterexamples Using Number Line Models Adding Real Numbers Subtracting Real Numbers Adding and Subtracting Real Numbers Multiplying Real Numbers Simplifying Square Root Expressions Dividing Real Numbers Dividing Fractions Simplifying Expressions Rewriting Fraction

Expressions Using the Multiplication

Property of -1 Using the Distributive Property for Mental Math Combining Like Terms Classifying Equations Identifying Solutions

of an Equation Writing an Equation Using Mental Math to Find Solutions Using a Table to

Find a Solution Estimating a Solution Identifying Solutions of a Two_Variable Equation Using a Table, An Equation, and a Graph Obective: To write

algebraic expressions Objectives:

- To simplify expressions involving exponents

- To use the order of operations to evaluate expressions Objectives:

-To classify, graph,

and compare real

numbers

-To find and estimate

square roots Objective:

To identify and use

properties of real numbers Objective: To find the sums and difference of real numbers Objective:

-to find products

and quotients of

real numbers Objective:

-To use the Distributive

Property to simplify

expressions Objective:

-To solve

equations

using tables

and mental math Objective:

To use tables, equations, and graphs to describe relationships What is an algebraic expression for the following word phrase? Word Phrase: 32 more than a number "n" n+32 Now you try!

Word Phrase: 20 more than a number "n" What is the algebraic expression for the following word phrase? Word Phrase: 58 less than a number "n" 58-n Now you try!

3 less than a number "n" What is the algebraic expression

for the following word phrase? Word Phrase: 8 times a number "n" Now you try!

15 times a number "n" 8n What is the algebraic expression for the following word phrase? Word Phrase: the quotient of a number "n" and 7 Now you try!

The quotient of a number "n" and 18 n/7 What is the algebraic expression

for the following word phrase? 3 more than twice a number "x" Now you try!

The product of 4 and the sum of a number "x" and 7 3+2x What word phrase can you use to represent the algebraic expression 3x? 3 times a number "x" or the product of 3 and a number "x" Now you try!

What word phrase can you use to represent the algebraic expression n/3 ? What is the simplest form of

the expression,

10 to the 7th power? 10,000,000 Now you try!

Simplify 3 to the 4th power What is the simplest form of the expression? (6-2)^3/2 Step 1: Simplify inside parenthesis

Step 2: Simplify the power

Step 3: Simplify What is the value of the

expression if x=5 and y=2 ? x^2 +x -12 /y^2 Step 1: Substitute 5 for x and 2 for y

Step 2: Simplify exponents

Step 3: Divide

Step 4: Add and subtract from left to right * * * * * * * * * * What is the square

root of 81? Answer: because 9 times 9 is 81 Now lets try!

What is the square root of 64? Now you try!

What is the value of the square root of 34 to the nearest integer? What is an inequality that compares the square root of 17 and 4 1/3? Step 1: Write the square root as a decimal

Step 2: Write the decimal as a fraction

Step 3: Compare using an inequality symbol What is the order of the square root of 4, .4, -2/3, the square root of 2, and -1.5 from least to greatest?

Step 1: Simplify all the numbers

Step 2: Order from least to greatest

Step 3: Graph on number line A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs $1.25. What is the total cost of a ticket, a drink, and popcorn? (7.75 + 2.40) + 125 = (2.40 + 7.75) + 1.25 Commutative Property of Addition

= 2.40 + (7.75 + 1.25) Association Property of Addition

= 2.40 + 9 Simplify inside parenthesis

= 11.40 Simplify Know: An expression

Need: Groups of numbers that can be simplified

Plan: Use properties to group or reorder parts of the expression Example: (4 + 7b) +8

(7b + 4) +8 Commutative Property of Addition

7b + (4+8) Association Property of Addition

7b + 12 Simplify For all real numbers j and k,

j x k = (k + 0) x j

This is true because it will be true for

every real numbers you substitute

for j and k. Now you try!

-12+7 What is the sum of 3+5. Picture using a number line. If you were at 3 on a number line, and you moved 5 spaces to the right, you would be at 8.

Therefore, 3 plus 5 is 8. Now you try!

-8 - (-13) Simplifying the square root of -25 would be -5 because the square root of 25 is 5. The absolute value of 5 is -5. Step 1: Rewrite the expression

Step 2: Substitute fractions for x and y

Step 3: Multiply by the reciprocal of the second fraction

Step 4: Simplify What sum or difference

is equivalent to 7x+2 /5 ? 1/5 (7x+2) Write division as multiplication

1/5(7x) + 1/5(2) Distribute

7/5x + 2/5 Simplify Now you try!

4x-16/3 What is the simplest form of -(2y-3x) ? -1(2y-3x) Multiplication Property of -1

-1(2y) + -1(-3x) Distribute

-2y+ 3x Simplify Now you try!

-(4x-12) 8(4.95) = 8( 5 - 0.05) Think of 4.95 as 5 - 0.05

8(5) -8(.05) Distributive Property

40- 0.4 Multiply mentally

39.60 Subtract mentally 5x - 3 - 3x+6y + 4 = 5x + (-3) + (-3x) + 6y +4 Rewrite as a sum

5x + (-3x) + 6y + (-3) +4 Commutative Property

(5-3)x + 6y + (-3) + 4 Distribute

2x + 6y + 1 Simplify Now you try!

3y - y How to tell if an equation

is true, false, or open... -If the equation is equal on both sides, it is true

-If it isn't equal, it is false

-If a variable is present, the equation is open Now you try!

24 + 18 = 20 + 22

2x-14 = 54 Is x = 6 a solution of the equation 32 = 2x +12? You would substitute 6 for x

to find out if the equation is true. If it's not true, 6 is not a solution

for x in this equation What is the solution of the equation? Use mental math. x + 8 = 12 Think: What # plus 8 equals 12

Solution: 4

Check 4 + 8 = 12 Now you try!

a/8 = 9 If the equation is 5n + 8 = 48, you would use a table to substitute numbers for n until the equation is true. To estimate the solution, find the integer values of x between which the solution must be. Is (3,10) a solution of the equation y=4x If you substitute 3 for x and 10 for y, and the equation is true, then (3,10) is a solution. 0 - 25 + 16 -47 +29 Write an expression

0 + (-25) + 16 + (-47) + 29 Use rule for subtracting real numbers

0 + 16 + 29 + (-25) + (-47) Commutative Property of Addition

0 + (16 + 29) + [(-25) + (-47)] Group Addends with the same sign

0 + 45 + (-72) Add inside grouping symbols

45 + (-72) Identity Property of Addition

-27 Use rule for adding with different signs "LESSON: Writing Expressions and Equations." LESSON: Writing Expressions and Equations. N.p., n.d. Web. 10 Jan. 2013.

"Welcome to SciMath MN." Welcome to SciMath MN. N.p., n.d. Web. 10 Jan. 2013.

"Get Math Help, Online Math Tutoring." Free Math Tutoring. N.p., n.d. Web. 10 Jan. 2013.

YouTube. YouTube, n.d. Web. 10 Jan. 2013. * * * * * * * * *