Algebra I Notes Unit Two: Variables

Transcription

1 Syllabus Objectives:. The student will use order of operations to evaluate expressions.. The student will evaluate formulas and algebraic expressions using rational numbers (with and without technology). Order of Operations (PEMDAS). Parentheses perform the operations within grouping symbols first. Exponents evaluate powers. Multiply/Divide multiply or divide in order from left to right 4. Add/Subtract add or subtract in order from left to right Ex: Evaluate the expression. ( ) + 8 = = 6+ 8 Step One: Parentheses ( ) Step Two: Exponents (in brackets) [ ] Step Three: Mult/Div (in brackets) = [ 6+ 4] Step Four: Brackets (grouping symbols) = [ 40] Step Five: Add/Sub = 4 Ex: Evaluate the expression Step One: Exponents = = Step Two: Mult/Div (left to right) = = = 6 Step Three: Add/Sub (left to right) = 4 Variable: a letter that represents a number Variable (Algebraic) Expression: a collection of numbers, variables, and operations Writing an Algebraic Expression (Translating Verbal Phrases) In order to translate verbal phrases into algebra, we will use key words for each operation. Page of 9 McDougal Littell:..,.5,.,.6

2 Examples of these key words are given: Addition: sum, more than, plus, increased by Subtraction: difference, less than, minus, decreased by Multiplication: product, times, multiplied by Division: quotient, divided by Caution: Order is important for subtraction and division (these operations are NOT commutative)! Ex: Translate the verbal phrases into algebraic expressions. (Use x for the variable.). more than a number x + or + x. 5 less than a number x 5. more than 5 times a number 5x + or + 5x 4. The sum of a number and 4, multiplied by 8 ( x + 4) 8 or 8( x + 4) 5. 8 is decreased by the square of a number 6. The difference of a number and divided by 7. 4 cubed divided by the sum of a number and 8 x x 4 x + or ( x ) or 4 ( x + ) 8. 6 times the quantity of a number minus 0 6( x 0) or ( x ) 0 6 Evaluating Algebraic Expressions: to evaluate an algebraic expression, substitute the given value(s) in for the variable(s) and simplify using the order of operations. Ex: Evaluate Step One: Substitute ( ) 5x + x 8 +4when x =. ( ) x = into the expression. 5( ) + ( ) Step Two: Use the order of operations to simplify. ( ) ( ) ( ) ( ) = P = E = M/D = = 4 A/S Page of 9 McDougal Littell:..,.5,.,.6

3 Ex: Evaluate 5 4x 8 y 5 + when x = 4 and y = 9. Step One: Substitute x = 4 and 9 into the expression y = ( ) ( ) Step Two: Use the order of operations to simplify. = 44 ( ) ( 8 5) 5 P = 4( 4) [ 7 5 ] P = 4( 4) [ ] P 7 = 56 [ ] = = M/D 7 = 5 A/S Evaluating Formulas Ex: A car traveled 50 miles in hours. Find the average speed of the car using the formula Average Speed = Distance d =. Time t Solution: Average Speed = 50 = 50 mi hr You Try: Evaluate the expression ( ) + a b a ab when a = and b =. QOD: Why is it important to have a specified order to performing operations? Page of 9 McDougal Littell:..,.5,.,.6

4 Syllabus Objective:. The student will use algebraic expressions to identify and describe the nth term of a sequence. Sequence: an ordered list of numbers Finding the nth Term of a Sequence: look for a pattern within the sequence and write the pattern in terms of n, which is the term number Ex: Use the sequence, 6, 9,,. Write the next three terms and write an expression for the nth term. Next three terms: We are adding to each term to obtain the next term, so they are 5,8,. nth term: When n =, the term is ; when n =, the term is 6; when n =, the term is 9; To obtain each term, we multiply the term number by. So the expression for the nth term is n. Ex: Use the sequence,8,4,0, 4,... Write the next three terms and write an expression for the nth term. Next three terms: We are subtracting 4 to each term to obtain the next term, so they are 8,, 6. nth term: When n =, the term is ; when n =, the term is 8; when n =, the term is 4; To obtain each term, we multiply the term number by 4 and add 6. So the expression for the nth term is 4n + 6. Ex: Use the sequence 4, 8, 6,. Write the next three terms and write an expression for the nth term. Next three terms: We are multiplying each term by to obtain the next term, so they are,64,8. nth term: When n =, the term is 4; when n =, the term is 8; when n =, the term is 6; To obtain each term, we raise to the n + power. So the expression for the nth term is n+. You Try: Use the sequence 5,,,,... Write the next three terms and write an expression for the nth term. QOD: When a sequence is created by adding or subtracting a number to obtain the next term, where is this number in the expression for the nth term? Page 4 of 9 McDougal Littell:..,.5,.,.6

5 Syllabus Objective:.4 The student will identify and apply real number properties using variables, including distributive, commutative, associative, identity, inverse, and absolute value to expressions or equations. Real Number Properties Commutative Property of Addition: a+ b = b+ a Real Number Example: + 5= 5+ = Commutative Property of Multiplication: a b= b a Real Number Example: = = Associative Property of Addition: ( a+ b) + c = a+ ( b+ c) Real Number Example: ( ) + = 4. + ( 5 + ) =.8 Associative Property of Multiplication: ( a b) c = a ( b c) Real Number Example: 6= 6 = 9 Identity Property of Addition: a+ 0 = a Note: The additive identity is 0. Real Number Example: + 0= Identity Property of Multiplication: a = a Note: The multiplicative identity is. Real Number Example: 6. = 6. Inverse Property of Addition: ( a) 0 a+ = Note: The opposite of a number is the additive inverse. Real Number Example: 4+ 4= 0 Inverse Property of Multiplication: a = Note: The reciprocal of a number is the multiplicative a inverse. Real Number Example: = Page 5 of 9 McDougal Littell:..,.5,.,.6

6 Distributive Property: a ( b+ c) = a b+ b c Real Number Example: Use the distributive property to multiply. We can rewrite as 0 + in order to use the distributive property so that we can perform the multiplication in our heads. ( 0 + ) = 0 + = = 84 Absolute Value: the distance a number is away from the origin on the number line Ex: What two numbers have an absolute value of 4? Solution: There are two numbers that are 4 units away from the origin, as shown in the number line. They are 4 and 4. Notation: x = x if x 0 and x = xif x < 0 Ex: Evaluate the expression. 4+ Note: Absolute value bars are like grouping symbols in the order of operations. Substitute Evaluate the expression inside the absolute value bars. = Evaluate the absolute value and simplify. = ( ) = 6 Ex: Evaluate the variable expression x = and Use the order of operations to simplify. x x 4 x + when x = and y y = into the expression. ( ) 4 ( ) ( ) + = ( ) ( ) 5 + = 5 + = 5 = 5 = 6 = 9 y =. Page 6 of 9 McDougal Littell:..,.5,.,.6

7 You Try:. Identify the property shown. ( c+ d) + e= ( d + c) + e. Show that the distributive property does not work over absolute value bars by showing that QOD: Determine if the statement is true or false. Explain your answer. The reciprocal of any number is greater than zero and less than. Page 7 of 9 McDougal Littell:..,.5,.,.6

8 Syllabus Objective:.5 The student will simplify algebraic expressions by adding and subtracting like terms. Terms: algebraic expressions separated by a + or a sign Constant Term: a term that is a real number (no variable part) Coefficient: the numerical factor of a term Like Terms: terms that have the exact same variable part (must be the same letter raised to the same exponent) Note: Only like terms can be added or subtracted (combined). To add or subtract like terms, add or subtract the coefficients and keep the variable part. Ex: Simplify + 4. This means we are adding smiley faces plus 4 smiley faces. Therefore, we have a total of 7 smiley faces, which we can write as 7. Ex: Use the expression x 4xy x 5y. How many terms are in the expression? Solution: There are 5 terms.. Name the constant term(s). Solution: There is one constant term,.. How many like terms are in the expression? to answer the following questions. Solution: There are like terms. x and x 4. What is the coefficient of the xy term? What is the coefficient of the fourth term? Solution: The xy term has a coefficient of 4. The third term has a coefficient of. 5. Simplify the algebraic expression. Solution: Combine like terms. x x xy y x xy y = How many terms are in the simplified form of the algebraic expression? Solution: There are 4 terms. Page 8 of 9 McDougal Littell:..,.5,.,.6

9 Simplifying an Algebraic Expression Ex: Simplify the expression ( ) 4 x + 8 5x +. Step One: Eliminate the parentheses using the distributive property. 4x + 5x + Step Two: Combine like terms. = 4x 5x+ + = x + = x + 5 n n+ 8 4n Ex: Simplify the expression ( ) Step One: Eliminate the parentheses using the distributive property. 5 6n 6n 4n Note: The expression in parentheses is being multiplied by distributed. n, which is what was Step Two: Combine like terms. Note: The answer may be written as = 5 6n 4n 6 = 5 0n 6 0n 6n+ 5 by the commutative property. n n Ex: Write a simplified expression for the perimeter of a rectangle with length ( x + 7 and width ( x. Note: The formula for the perimeter of a rectangle is P = l+ w. ) x+ 7 + x ) Substitute the length and width into the formula. ( ) ( ) Simplify using the distributive property and combining like terms. = x + 4+ x 4 = 4x + 0 You Try: Simplify the expression x ( x). Write a simplified expression for the area of a triangle with a base of 4x and a height of ( x + ). QOD: Explain in your own words why only like terms can be added or subtracted. Page 9 of 9 McDougal Littell:..,.5,.,.6