Lesson 1

Algebra Review

Name:________________________ Pd:__________College Bound Math Mr. Guernsey

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Lesson Objectives Use the order of operations to evaluate expressions Analyze and solve verbal problems that involve the order of operations Use proper calculator techniques to compute expressions

Who's correct?????????

When you evaluate an expression, you find its numerical value. What is the value of (1 + 3 x 2 - 4)? Without a set order of how to do things, what are some other possible outcomes?

To avoid confusion, mathematicians have agreed on a set of rules called the order of operations.

ParenthesesExponents

Multiplication/DivisionAddition/Subtraction

Why are multiplication/division and addition/subtraction on the same line? _______________________________________________________

What other symbols appear in expressions and where do they fall in the order of operations?

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Examples:4[25 – (5 – 2)2]

Evaluate if m = 9, n = 2, and p = 5

Practice:

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Jenny’s Number Contest

Directions: Four dice will be rolled. Your team’s job is to use (all of) those four numbers along with the Order of Operations to get each digit in Jenny’s Number.

What will your game plan be?

Extra work space:

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Lesson Objectives Use the distributive property and combining like terms to simplify expressions Translate verbal phrases into algebraic expressions Use proper calculator techniques to compute expressions

Vocabulary ExamplesA term is a number, a variable, or a product or quotient of numbers and variables.Like-terms are terms that contain the same variables with the same exponents. An expression is a combination of terms.

An expression is in simplest form is when all like terms are combined and parentheses are removed.An equation relates two expressions with an equal sign.

The Distributive Property can be used to simplify expressions by removing parentheses.Examples:6(x2 + 3x – 5) -0.5(12 – 6x)

4(a2 + 3ab) -2(7 + x – 2x2)

14(j – 2) – 3j(4 – 7) 50(3a – b) – 20(b – 2a)

Practice:

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Writing expressions and equations is the process of translating into math symbols.

Math TranslatorAddition Subtraction

Multiplication Division

Equals

Words Variable (Unknown) Expression/Equation

Five years older than her brother

six dollars an hour times the number of hours

The difference between seven and a number

eight less than a number is 22

The product of -11 and the square of a number

The sum of x and its square is equal to y times z

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Practice:

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Lesson Objectives Solving multi-step Equations

Steps:1.____________________________________________________________________________

2.____________________________________________________________________________

3.____________________________________________________________________________

4.____________________________________________________________________________

Examples:

3x + 5 = 12 4y – 8 = 12 5m – 4 = -25

7x – 3x – 8 = 24 3x – x + 15 = 41 25x – 16x – 24 = -65

5x + 3(x + 4) = 28 4x – 3(x – 2) = 21 2x – 5(x – 9) = 27

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Practice:

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Lesson Objectives Solving multi-step Equations

Steps:1.___Perform the distributive property_______________________________________________

2.___Combine like terms (on either side of the equal sign)_______________________________

3.___Add/Subtract variables to one side, constant to the other_____________________________

4.___Multiply/divide the coefficient of the variable_____________________________________

Examples:

3(4m – 6) = 2(7m – 5) 13 + 5(2x – 7) = 6x + 13 8 – 3(4x – 11) = 15x + 1 – 7x

13(5 – x) – 4x = -5(x – 13) 12 + 6(12 + 5a) = 3(1 + 7a) - 12

x + 4 – (13x + 5) = 7(3 – 8x) + 11(x – 11) 4(3x – 5) – 2(x – 13) = 4x + 6

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Practice:

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Lesson Objectives Solving absolute value equations

The absolute value of a number is the _______________________ that number is from zero. Can distance be negative? Therefore absolute value cannot be negative. The symbol |x| is used to represent the absolute value of, x.

Evaluate: |-5| = _______ |5| = _________ |6 – 10| = __________

So if I was to ask |x| = 5, what would the value of x be? _______________

Solving Absolute Value Equations

1. Get the absolute value bars _________________________________________.2. Make sure that the absolute value is equal to a __________________ number ONLY!!! As we talked about

earlier, absolute value is NEVER negative!3. Create _________ equations.4. Solve both Equations5. Check to be sure your answers work.

Examples:1.) |t – 4| = 5 2.) |m + 3| = 12

3.) |6x| = 12 4.)

5.) 2|2x – 3| = 6 6.) -3|4x + 9| = 24

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7.) -5|3x – 7| + 8 = -102 8.) 4|3 – 5x| + 13 = 5

9.) |x – 2| = 3x + 1 10.) |4x + 3| = 3 - x

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Lesson Objectives Solving inequalities

Solve each inequality and graph its solution

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Lesson Objectives Solving compound inequalities

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Practice:

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Partner Problems:

This activity is a wonderful reinforcement for work on linear equations and inequalities.  Depending upon the sophistication of your students, this activity may take upwards of 40 minutes to complete.

Directions:

Group students in pairs.   Students are allowed to use graphing calculators.

One student will solve the 10 questions on the left side of the worksheet, while his/her partner solves the 10 questions on the right side of the worksheet.

Students will check answers with their partners as problems that are numbered the same have the same answers.

Students should assist one another in finding the correct solutions.

All work must be shown.

It was intended that this assignment be graded.

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Card #1 Number Correct:1. Sketch the graph of:

2. Sketch the graph of:

3. Sketch the graph of: or

4. Solve and sketch the graph of:

5. Solve and sketch the graph of:

Card #1 Number Correct:1. Sketch the graph of:

2. Sketch the graph of:

3. Sketch the graph of: or

4. Solve and sketch the graph of:

5. Solve and sketch the graph of:

Card #2 Number Correct:

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1. Solve: 2. Solve:

3. Solve:

4. Solve: or

5. Solve:

Card #2 Number Correct:1. Solve: 2. Solve:

3. Solve:

4. Solve: or

5. Solve:

Card #3 Number Correct:

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1. Solve: 2. Solve:

3. Solve:

4. Solve:

5. Solve:

Card #3 Number Correct:1. Solve: 2. Solve:

3. Solve:

4. Solve:

5. Solve:

Card #4 Number Correct:

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1. Solve:

2. Solve:

3. Solve:

4. Solve:

5. Solve:

Card #4 Number Correct:1. Solve:

2. Solve:

3. Solve:

4. Solve:

5. Solve:

Lesson Objectives:Writing algebraic expressions from word problemsSolving algebraic expressions from word problems

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The sum of two numbers is 90.The larger number is three more than twice the other.

Find the larger number.

Sometimes we will have MORE than one variable to define. We may have to define a variable and then USE that variable to define our other unknown(s).

1.) The sum of two numbers is 90. The larger number is three more than twice the other. Find the larger number.

Define your variable(s) Write your equation and solve:Let x =Let ________________=

2) A ribbon 56 centimeters long is cut into two pieces. One of the pieces is three times longer than the other. Find the lengths, in centimeters, of both pieces of ribbon.

Define your variable(s) Write your equation and solve:Let x =Let ________________=

3) In triangle ABC, the measure of angle B is 21 less than four times the measure of angle A, and the measure of angle C is 1 more than five times the measure of angle A. The measures of the angles of a triangle add up to 180 degrees. Find the measure, in degrees, of each angle.

Define your variable(s) Write your equation and solve:Let x =Let ________________=Let ________________=

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4) Jackie has $200 to spend. She buys a clock for $66, and wants to spend the rest on picture frames marked at $15 each. How many picture frames can she afford?

Define your variable(s) Write your equation and solve:Let x =Let ________________=Let ________________=

5) The sum of 3 numbers is 207. The second number is 8 times the first, while the third is 3 less than the first. Find the number.

Define your variable(s) Write your equation and solve:Let x =Let ________________=Let ________________=

6) The length of a rectangle is 2 less than twice the width. If the perimeter is 11m, find the length and width.

Define your variable(s) Write your equation and solve:Let x =Let ________________=

Practice:

1) The length of the base of an isosceles triangle is 10 less than twice the length of one of its legs. If the perimeter of the triangle is 50, find the length of the base.

2)

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Lesson Objectives:Writing algebraic expressions from word problemsSolving algebraic expressions from word problems

1) The larger of two numbers exceeds six times the smaller by 10. If the larger number is 76 find the smaller.

Define your variable(s) Write your equation and solve:Let x =Let ________________=

What does “consecutive integers” mean?___________________________________________________________________________________________________________________________________________________

Give an example of 3 consecutive integers:_______________________________________________________

How would you write these as algebraic expressions?______________________________________________2) The sum of the ages of the three Romano brothers is 63. If their can be represented as consecutive

integers, what is the age of the middle brother?

3) Find three consecutive integers such that the sum of the first two integers is 24 more than the third.

What does “consecutive EVEN integers” mean?___________________________ _________________________________________________________________________________________________________________

Give an example of 3 consecutive even integers:__________________________________________________

How would you write these as algebraic expressions?______________________________________________

4) The sum of three consecutive even integers is 126. Find the integers.

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What does “consecutive ODD integers” mean?___________________________ _________________________________________________________________________________________________________________

Give an example of 3 consecutive odd integers:__________________________________________________

How would you write these as algebraic expressions?______________________________________________

5) The sum of four consecutive odd integers is -48. Find the integers.

6) Find three consecutive even integers such that the sum of the smallest and twice the second is 20 more than the third.

7) The perimeter of a rectangle is 40 feet. The length is 2 more than 5 times the width. Find the dimensions of the rectangle.

PRACTICE:1) The sum of three consecutive integers is 71 less than the smallest of the integers. Find the integers.

2) The sum of four consecutive odd integers is three more than five times the smallest. Find the integers.

3) Find four consecutive odd integers such that the sum of the first three exceeds the fourth by 18.

4) Is it possible to find three consecutive odd integers whose sum is 59? Why?

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