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Chapter 1: Foundations of Algebra
Transcript of Chapter 1: Foundations of Algebra
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
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
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
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
-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
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
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
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. * * * * * * * * *