CH I: Connections to Algebra

CH I: Connections to Algebra 1.1) Variables in Algebra 1.2) Exponents and Powers 1.3) Order of Operations 1.4) Equations and Inequalities 1.5) Translating Words into Mathematical Symbols 1.6) A Problem Solving Plan Using Models 1.7) Tables and Graphs 1.8) An Introduction to Functions

1.1) Variables in Algebra Let’s take a look at a problem: You are driving to LA at 80 miles per hour. How many miles have you traveled after 2 hours? How about after 5 hours? The problem can be written this way: d = 80x; x = {2, 5}; d = ? Here, d and x are called variables. The answer, d, will vary depending on the number plugged in for x which in this case are 2 and 5.

1.1) Variable in Algebra (cont.) Let’s solve the problem: d = 80x; x = {2, 5}; d = ? d = 80(2) = 80 x 2 = 160 d = 80(5) = 80 x 5 = 400 The car will travel 160 miles after traveling for 2 hours at 80 miles/hr. The car will travel 400 miles after traveling for 5 hours at 80 miles/hr.

1.2) Exponents and Powers Look at this: 2³ = 2 x 2 x 2 2³ means you multiply 2 three times. x³¹=xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx x x³¹ means you multiply xthirty one times.

1.2) Exponents and Powers (cont.) BE AWARE! 2x³DOES NOT EQUAL (2x)³. 2x³ = 2 x x x x x x (2x)³ = 2x x 2x x 2x

1.3) Order of Operations 1/ First do operations that occur within grouping symbols (parentheses or brackets) 2/ Then evaluate powers 3/ Then do multiplications and divisions from left to right 4/ Finally, do additions and subtractions from left to right.

1.3) Order of Operation (cont.) Left-to-right rule- when there are same types of signs, you operate the left one first. Ex.1) in 4+2+6, you do (4+2) first. Then move on to +6. Ex.2) in 4–2+5, you do the – first. Then move on to +. Ex.3) in 4x2÷3, you do x first. Then move on to ÷.

1.4) Equations and Inequalities Equation: Any statement that has equal sign. Take a look at the following problem: 4x+1 = 9 When you see a statement, “Find the solution to the equation,” it just means to find the right values for the variable (in this case x).

1.4) equations and Inequalities (cont.) So, we got 4x +1 = 9 If you subtract 1 from both sides, 4x +1-1 = 9-1 4x = 8 Then divide both sides by 4. 4x /4 = 8/4 x = 2 Yay~!

1.4) equations and inequalities (cont.) There are 4 inequality symbols: > is greater than < is less than  is greater than or equal to • is less than or equal to Is the following statement true? 4x +1  9 when x ={2,3}

1.4) equations and inequalities (cont.) 4x +1  9 when x ={2,3} Step1) Replace x with the given value, 2 and 3. 4(2)+1  9 4(3)+1  9 Step2) Do the calculation 8+1  9 12+1  9 9 9 13 9 Step3) answer the question. Is the following statement true? 4x +1  9 when x ={2,3} {9,13} is greater than or equal to 9. TRUE.

1.5)Translating words into mathematical symbols It’s not that hard, just MEMORIZE these: Sum, more than, plus, increased => + Difference, minus, less than, decreased => - Product, times, multiplied by => x Division is worded a little differently. One fourth of 6 => 6 x ¼ The quotient of 6 and 4 => 6/4 6 divided by 4 => 6/4 Practice problems on pg.30-32. :)

1.6) A Problem Solving Plan Using Models A problem solving plan using models #1) Verbal model-Ask yourself what you need to know to solve the problem. The write a verbal model that will give you what you need to know. #2) Labels-Assign labels to each part of your verbal model. #3) Algebraic Model-Use the labels to write an algebraic model based on your verbal model. #4) Solve-Solve the algebraic model and answer the original question. #5) Check that your answer is reasonable.

1.6) A Problem Solving Plan Using Models Pg.36) You and some friends are at a Chinese restaurant. You order several $2 plates of wontons, egg rolls, and dumplings. Your bill is $25.20, which includes tax of $1.20. Use modeling to find how many plates you ordered. Step1) cost/plate x number of plates = amount of bill – tax. Step2) cost/plate = 2 Number of plates = p Amount of bill = 25.20 Tax = 1.20 Step3) 2p = 25.20 – 1.20 Step4) 2p = 24 p = 12 Step5) We ordered 12 plates

1.7) Tables and Graphs Table

Bar Graph Line Graph 1.7) Tables and Graphs (cont.)

1.8) An Introduction to Functions A function is a rule that establishes a relationship between two quantities, called the input and the output. For each input, there is exactly one output—even though two different inputs may give the same output. An input-output table (p.48) describes a function. 1 input => 1 output (Function? O) 1 input => multiple outputs (Function? X)

1.8) An Introduction to Functions (cont.) Take a look at the following equation: h = 250+20t For every value of t (input), there is exactly one value of h (output). Therefore, it is a function. Domain: a group of inputs Range: a group of outputs

Terms Variable Variable expression Value Numerical expression Evaluate Power Exponent Base Grouping symbols Order of operations Left-to-right rule Equation solution Inequality Modeling Verbal model Algebraic model Data Bar graph Line graph Function Input Output Input-output table Domain range

CH I: Connections to Algebra