Ch 1. The Language of Algebra

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1 Ch 1 The Language of Algebra

2 1-1 Writing Expressions and Equations

3 Writing Expressions Buying CDs: 1 CD = $15 2 CD = $15 x 2 3 CD = $15 x 3 n number of CDs? $15 x n Algebraic Expression

4 Writing Expressions Variable Unknown number represented by a letter Called variable because its value can vary Algebraic Expression Contains at least one variable and at least one mathematical operation

5 Writing Expressions Numerical Expression Contains only numbers and mathematical operations Example: Multiplication Quantities being multiplied are called factors Result is the product Ways to write 3 times x:

6 Writing Expressions Division Result is the quotient Ways to write t divided by 3:

7 Verbal Expressions Words used in verbal expressions to mean: Addition Subtraction Multiplication Division

8 Examples Write an algebraic expression for each verbal expression. 1. The sum of p and 12: 2. The product of k and q: decreased by w: 4. 4 more than 8 times k:

9 Examples A python eats 4 pounds of meat each month. 1. Write a numerical expression to represent the amount it eats in 5 months. 2. Write an algebraic expression to represent the amount it eats in d months.

10 Examples Write a verbal expression for each algebraic expression s 2. 5(b 3) t 3. 15v r d 4..

11 Equation Equation A mathematical sentence that contains an equal sign Words to mean equality:

12 Examples Write an equation for each sentence. 1. The quotient of t and 8 equals Seven less than three times g is A number k divided by 4 is equal to 18.

13 Examples Write a sentence for each equation. j + 4 = 21 3z 12 = 11 4b 5 = 3

14 Assignment 1 st Assignment due today P6: nd Assignment due next time P6: 18 42

15 1-2 Order of Operations

16 Order of Operations Why must we have order of operations?

17 Examples Find the value of each expression x (6 + 7)

18 Example As a 16-year old, Trent Eisenberg ran his own consulting company called F1 Computer. Suppose he charged a flat fee of $50, plus $25 per hour. One day, Trent worked 1 hour for each of 2 new customers. Find the value of the expression 2(50) + 2(25) to find the total amount of money he earned.

19 Properties Properties Statements that are true for any number

20 Examples Name the property shown by each statement. 1. If k = 7, then k + 3 = If a + 4 = 9, then 9 = a c = 7 c 4. If 10 3 = and = 7, then 10 3 = 7

21 More Properties

22 Example Find the value of [ (12 11)] 11. Identify the property used in each step.

23 Example Find the value of (22 15) 7 9. Identify the properties used.

24 Evaluating Evaluating In an algebraic expression, replace the variables with known values and then use the order of operations to simplify

25 Example Evaluate each expression if x = 4 and y = 3. xy + 8 (2y + 10) x Evaluate each expression if m = 8 and p = 2. 6 p m p [m + 2(3 + p)] 2

26 Assignment 1st Assignment due today P11: 1, 2, nd Assignment due next time P11: 22 34, 36, 37, 40 50, 52, 53, 55 61, 63, 64

27 1-3 Commutative and Associative Properties

28 Commutative Property

29 Associative Property

30 Examples Name the property shown by each statement (3 + 4) = (3 + 4) (8 k) = (7 8) k = (n + 12) + 5 = n + (12 + 5)

31 Simplify Simplify Eliminate all parentheses and then add, subtract, multiply, or divide

32 Example Simplify the expression (4 m) 9. Identify the properties used in each step.

33 Example Simplify the expression 7 + 2a Identify the properties used in each step.

34 Example The volume of a box can be found using the expression l x w x h, where l is the length, w is the width, and h is the height. Find the volume of a box whose length is 20 inches, width is 12 inches, and height is 3 inches.

35 Closure Property Whole Numbers 0, 1, 2, 3, When added together, equal a whole number When multiplied, equal a whole number

36 Counterexample Are whole numbers closed under division? Counterexample An example what shows a statement is not true Only need 1 counterexample to show a statement is false

37 Example State whether the statement subtraction of whole numbers is commutative is true or false. If false, provide a counterexample.

38 Example State whether the statement subtraction of whole numbers is associative is true or false. If false, provide a counterexample.

39 Assignment 1 st Assignment due today P17: 1, 2, nd Assignment due next time P17: 10 25, 28 33

40 1-4 Distributive Property

41 Distributive Property

42 Example Simplify each expression. 5(2 + m) 3(4x + 2) (1+3t)9

43 Simplifying Term A number, variable, or product or quotient of number and variables Examples: Coefficient The numerical part of a term with a variable Examples:

44 Simplifying Like Terms Terms that contains the same variables Examples 5b + 3b + x + 12x Number of terms: Like terms: Coefficients:

45 Simplifying Equivalent Expressions When two expressions are the same value Example: Simplest Form When there are no like terms and no parentheses

46 Example Simplify each expression. 8p 5p 10k + 6m 5k + 2m 5st + 2st 6 + y + 3z + 4y

47 Example Area of a rectangle: One wing of a building contains three rooms that are identical in size. The floor of each room is 20 meters in length and width. Find the total floor area of the wing.

48 Assignment 1 st Assignment due today P21: nd Assignment due next time P22: 15 33, 35, 38, 40, 43 48, 50

49 1-5 A Plan for Problem Solving

50 Problem-Solving Plan 1. Explore: Read the problem. Identify information. 2. Plan Select a strategy to solve i.e. Table, work backwards, formula, graph, etc 3. Solve Solve the problem. 4. Examine Check your answer.

51 Formula Formula An equation that states the rules for the relationship between quantities Simple Interest formula I = prt I = Interest p = principal, or amount deposited r = interest rate, written as a decimal t = time in years

52 Example Suppose you deposit $350 into an account that pays 2% interest. How much money would you have in the account after five years?

53 Example How many ways can you make 50 using quarters, dimes, and nickels?

54 Properties of Real Numbers

55 Assignment 1 st Assignment due today P27: nd Assignment due next time P28: 9 25

56 1-6 Collecting Data

57 Collecting Data Data Information or results Experiments A test made under controlled conditions Observational Studies Subjects studied without action by the investigator

58 Collecting Data Sampling A way to gather data and make predictions about a population Population The set of individuals, items, or data from which a statistical sample is taken Sample A small group that is used to represent a much larger population

59 Sampling Criteria Representative of larger population Selected at random Large enough to provide accurate data

60 Example One hundred cable-television subscribers are surveyed to find how much time the average American spends reading. Is this a good sample? Explain.

61 Example Two hundred students at a school basketball game are surveyed to find the students favorite sport. Is this a good sample? Explain.

62 Example Every other person leaving a supermarket is asked to name their favorite soap. Is this a good sample? Explain.

63 Frequency Tables Frequency Table A table that uses tally marks to record and display the frequency of events Data is usually displayed in intervals

64 Example Make a frequency table to organize the data in the chart.

65 Cumulative Frequency Table Cumulative Frequency Table A frequency table in which a column is added that accumulates the frequencies

66 Example The owners of a bookstore specializing in travel books are looking for a new location. They counted the number of people who passed by the proposed location during one afternoon. The frequency table shows the results of their sampling. Which group of people passed by the location most frequently? Is this a good location for the bookstore? Explain.

67 Assignment 1 st Assignment due today P34: 1, nd Assignment due next time P35: 10 12, 16 21, 24-28

68 1-7 Statistics: Displaying and Interpreting Data

69 Line Graph Line graph Show trends or changes over time Consists of Title Label on each axis Equal intervals on each axis

70 Example Construct a line graph of the data given in the table. Use the graph to predict the percent of the labor force in farming in the year 2010.

71 Histogram Histogram Uses data from a frequency table and displays it over equal intervals. Consists of Title Label on each axis Equal intervals on each axis Equal width bars No spaces between bars

72 Example The table shows the number of people in different age groups who entered a new store during the first hour of its grand opening. Construct a histogram of the data.

73 Cumulative Histograms Cumulative Histograms Uses the accumulated values of the frequency table

74 Example Construct a cumulative frequency histogram of the data.

75 Stem-and-Leaf Plot Stem-and-leaf plot Data representation that organizes data numerically by place value

76 Example The table shows the record high temperatures for several states. Make a stem-and-leaf plot of the temperatures.

77 Assignment 1st Assignment due today P41: 1, nd Assignment due next time P42: 12 28

78 Review P44: 1 47 P47: 1-20

Ch 1 The Language of Algebra. 1.1 Writing Expressions and Equations. Variable: Algebraic Expression: Numerical Expression: Multiplication: Division:

Ch 1 The Language of Algebra. 1.1 Writing Expressions and Equations. Variable: Algebraic Expression: Numerical Expression: Multiplication: Division: Ch 1 The Language of Algebra 1.1 Writing Expressions and Equations Variable: Algebraic Expression: Numerical Expression: Multiplication: Division: Words for Addition: Subtraction: Multiplication: Division: