Text 2. Real Numbers

ALGEBRA PROJECT UNIT 2 REAL NUMBERS

REAL NUMBERS

Lesson 1

Rational Numbers on the Number Line

Lesson 2

Adding and Subtracting Rational Numbers

Lesson 3

Multiplying Rational Numbers

Lesson 4

Dividing Rational Numbers

Lesson 5

Statistics: Displaying and Analyzing Data

Lesson 6

Probability: Simple Probability and Odds

Lesson 7

Square Roots and Real Numbers

RATIONAL NUMBERS on the NUMBER LINE

Example 1

Identify Coordinates on a Number Line

Example 2

Graph Numbers on a Number Line

Example 3

Absolute Value of Rational Numbers

Example 4

Expressions with Absolute Value

Name the coordinates of the points graphed on the number line.

The dots indicate each point on the graph. Answer: The coordinates are {–9, –7, –6, –3}.

Name the coordinates of the points graphed on the number line.

The bold arrow on the graph indicates that the graph continues infinitely in that direction. Answer: The coordinates are {11, 12, 13, 14, …}.

Name the coordinates of the points graphed on each number line. a. Answer: {6, 9, 11, 12} b. Answer: {–0.5, 0, 0.5, 1, 1.5, …}

Graph

Answer:

.

Graph {–1.5, 0, 1.5, …}.

Answer:

Graph {integers less than –6 or greater than or equal to 1}.

Answer:

Graph each set of numbers. a.

{–5, 2, 3, 5}

Answer:

b. Answer:

c. {integers less than or equal to –2 or greater than 4} Answer:

Find

.

unit from 0 in the negative direction.

Answer:

Find

.

0.25 is 0.25 unit from 0 in the positive direction. Answer:

Find each absolute value. a.

Answer:

b.

Answer: 6.3

Replace y with 12.

Answer:

Simplify.

Answer: 10

ADDING and SUBTRACTING RATIONAL NUMBERS

Example 1

Use a Number Line to Add Rational Numbers

Example 2

Add Rational Numbers

Example 3

Subtract Rational Numbers to Solve a Problem

Use a number line to find

. –5 +8

Step 1

Draw an arrow from 0 to 8.

Step 2

Then draw a second arrow 5 units to the left to represent adding –5.

Step 3

The second arrow ends at the sum 3.

Answer:

Use a number line to find –4

. –1

Step 1

Draw an arrow from 0 to –1.

Step 2

Draw a second arrow 4 units to the left.

Step 3

The second arrow ends at the sum –5.

Answer:

Use a number line to find each sum. a.

Answer: –3

b.

Answer: –9

Find

. Subtract the lesser absolute value from the greater absolute value. Since the number with the greater absolute value is –14, the sum is negative.

Answer:

Find

.

Both numbers are negative, so the sum is negative.

Answer:

Find each sum. a.

Answer: –17

b.

Answer:

Stocks In the past year, a publishing company’s stock went from $52.08 per share to $70.87 per share. Find the change in the price of the stock. Explore

The stock price began at $52.08 per share and ended at $70.87. You need to determine the change in price for the year.

Plan

Subtract to find the change in price. ending price

70.87

minus

beginning price

52.08

Solve To subtract 52.08, add its inverse. Subtract the absolute values. The absolute value of 70.87 is greater, so the result is positive.

Answer: The price of the stock changed by $18.79. Examine The problem asks for the change in a stock’s price in the past year. Since the change in price was positive, the price increased. This makes sense since the ending price is more than the beginning price.

Stocks The stock in a company went from $46.98 to $35.09 over a one-month period. Find the change in price for the stock. Answer: –$11.89

MULTIPLY RATIONAL NUMBERS

Example 1

Multiply Integers

Example 2

Simplify Expressions

Example 3

Multiply Rational Numbers

Example 4

Multiply Rational Numbers to Solve a Problem

Example 5

Evaluate Expressions

Find (–8)(–6). same signs → positive product Answer: 48

Find (10)(–11). different signs → negative product Answer: –110

Find each product. a.

(–4)(8)

Answer: –32

b.

(–6)(–12)

Answer: 72

Simplify the expression Associative Property (×) Substitution Distributive Property Answer:

Simplify.

Simplify the expression Answer: 3x

Find

same signs → positive product

Answer:

Find

Answer:

Stocks The value of a company’s stock dropped by $1.25 per share. By what amount did the total value of the company’s stock change if the company has issued 500,000 shares of stock? To find the change in the total value of the company’s stock, multiply the price lost per share by the number of shares.

different signs → negative product Answer: The total value of the company’s stock changed by –$625,000.

Construction A construction project is stopped by a winter storm. For every day that they are unable to work, the company loses $35,000. If the storm keeps them from working for 4 days, how much money do they lose? Answer: –$140,000

Substitution

Answer:

same signs → positive product

Answer:

DIVIDE RATIONAL NUMBERS

Example 1

Divide Integers

Example 2

Simplify Before Dividing

Example 3

Divide Rational Numbers

Example 4

Divide Rational Numbers to Solve a Problem

Example 5

Simplify Algebraic Expressions

Example 6

Evaluate Algebraic Expressions

Find

Answer:

.

positive quotient

Find

.

Divide.

Answer:

negative quotient

Find each quotient. a.

Answer: 20

b.

Answer: –15

Simplify

Simplify the numerator first.

Multiply.

Answer:

different signs → negative quotient

Simplify Answer: 3

Find

Answer:

.

Use a calculator. different signs → negative quotient

Find

.

Multiply by

the reciprocal

of

Answer:

same signs → positive quotient

Find each quotient. a.

Answer: 15.3

b.

Answer:

Baseball The perimeter of a square baseball diamond is 360 feet. Find the length of one side of the diamond. To find the length of one side, divide the perimeter by the number of sides. same signs → positive quotient Answer: The length of one side is 90 feet.

The perimeter of a triangular building is 450 feet. Find the length of each side. Answer: 150 feet

Simplify

The fraction bar indicates division.

Multiply by the reciprocal of 13.

Distributive Property.

Answer:

Simplify.

Simplify

Answer:

Evaluate

if w = 2, x = –9.1 and y = 4. Replace w with 2, x with –9.1 and y with 4. Find the numerator and denominator separately.

Answer:

Use a calculator. different signs → negative quotient

Evaluate

if s = 2.3, t = 5 and u = –4.

Answer: –14.375

STATISTICS DISPLAYING and ANALYZING DATA

Example 1

Create a Line Plot

Example 2

Use a Line Plot to Solve a Problem

Example 3

Create a Stem-and-Leaf Plot

Example 4

Back-to-Back Stem-and-Leaf Plot

Example 5

Analyze Data

Example 6

Determine the Best Measure of Central Tendency

Draw a line plot for the data.

11 –2 10 –2 7 2 7 4 9 0 6 9 7 2 0 4 10 7 6 9

Step 1

The values of the data range from –2 to 11, so construct a number line containing these values.

Step 2

Then place an × a number for each time it occurs.

Draw a line plot for the data.

3 5 7 6 0 –4 6 4 7 0 0 –2 3 7

Answer:

Traffic The highway patrol did a radar survey of the speeds of cars along a stretch of highway for 1 minute. The speeds (in miles per hour) of the 20 cars that passed are listed below.

72 70 72 74 68 69 70 72 74 75 79 75 74 72 70 64 69 66 68 67 Make a line plot of the data. The lowest value is 64 and the highest value is 79, so use a scale that includes those values. Place an × above each value for each occurrence.

Answer:

Which speed occurs the most frequently?

Answer: Looking at the line plot, we can easily see that 72 miles per hour occurs most frequently.

Family Size Students in Mrs. Barrett’s class listed the number of family members in their households below.

6 4 8 3 3 5 4 4 3 5 5 2 5 6 3 5 6 2 4 4 4 a. Make a line plot of the data.

Answer: b. Which family size occurs the most frequently? Answer: 4

Use the data below to make a stem-and-leaf plot.

85 114 85

115 126 92 104 107 78 131 92 116 100 121 123 131 88 97 99 116 79 90 110 129 108 93 84 75 70 The greatest 132 common place value is tens, so the digits in the tens place are the stems.

Answer:

Stem 7 8 9 10 11 12 13

Leaf 0589 4558 022379 0478 04566 1369 112

Use the data below to make a stem-and-leaf plot.

3 13

5 25

Answer:

7 32

11 37

10 21

15 10

Stem 0 1 2 3

Leaf 357 0011235 115 27

21 12

11

Weather Monique wants to compare the monthly average high temperatures of Dallas and Atlanta before she decides to which city she wants to move. The table shows the monthly high temperatures (°F) for both cities. Monthly Average High Temperature Dallas Atlanta 54 83 87

59 91 78

68 95 66

77 95 57

50 75 81

55 85 72

64 88 63

72 87 54

Make a stem-and-leaf plot to compare the data. To compare the data we can use a back-to-back stem-and-leaf plot. Since the data represent similar measurements, the plot will share a common stem. Answer: Dallas

9 7 8 8 7 5 5

4 6 7 3 1

Stem 5 6 7 8 9

Atlanta 0 4 5 3 4 2 2 5 1 5 7 8

What is the difference between the highest average temperatures in each city? Answer: 95 – 88 or 7°

Which city has higher average temperatures? Answer: Looking at the temperatures of 80 and above, we can see that Dallas has a higher number of average temperatures above 80°.

Ms. Smith wants to compare the final grades for two of her classes. The table shows the scores for both classes. Class A 87 81 92 76 71

Class B 96 51 98 75 64

99 62 77 72 69

76 57 83 85 91

a. Make a back-to-back stem-and-leaf plot to compare the data. Answer: Class A

1 4 6 5 1 7 1 8 6 2

Stem 5 6 7 8 9

Class B 7 2 9 2 6 7 3 5 1 9

b. What is the difference between the highest score in each class? Answer: 1 point c. Which class scored higher overall for the grading period? Answer: Class A

Which measure of central tendency best represents the data?

Stem 4 5 6 7 8

Leaf 11244458 0 257 39 1

Determine the mean, median, and mode. The mean is about 5.5.

Add the data and divide by 15.

The median is 4.8.

The middle value is 4.8.

The mode is 4.4.

The most frequent value is 4.4.

Answer: Either the median or the mode best represent the set of data since both measures are located in the center of the majority of the data. In this instance, the mean is too high.

Which measure of central tendency best represents the data?

Stem 1 2 3 4 5

Leaf 011568 378 2 6 459

Answer: The mean is about 2.9. The median is 2.5. The mode is 1.1. Either the mean or median can be used to represent the data. The mode is too low.

Politics The number of electoral college votes for the 12 most populous states in the 2000 Presidential election are listed below. Which measure of central tendency best represents the data?

21 14

22 32

18 13

23 33

15 13

25 54

The mean is about 23.6.

Add the data and divide by 12.

The median is 21.5.

The middle value is 21.5.

The mode is 13.

The most frequent value is 13.

Answer: Either the mean or median can be used to best represent the data. The mode is too low.

The number of points scored by the basketball team during each game in the season is listed below. Which measure of central tendency best represents the data?

48 51

45 81

52 62

63 73

59 68

64 82

67 73

72 70

58 65

Answer: Either the mean or the median can be used to best represent the data. The mode is too high.

SIMPLE PROBABILITY and ODDS

Example 1

Find Probabilities of Simple Events

Example 2

Odds of an Event

Example 3

Odds Against an Event

Example 4

Probability and Odds

Find the probability of rolling a number greater than 2 on a die. There are six possible outcomes. Four of the outcomes are favorable. That is, four of the six outcomes are numbers greater than two. 4 numbers greater than 2

Sample space: 1, 2, 3, 4, 5, 6 6 possible outcomes

Answer:

A class contains 6 students with black hair, 8 with brown hair, 4 with blonde hair, and 2 with red hair. Find P(black). There are 6 students with black hair and 20 total students. number of favorable outcomes number of possible outcomes Simplify. Answer: The probability of selecting a student with black hair is

A class contains 6 students with black hair, 8 with brown hair, 4 with blonde hair, and 2 with red hair. Find P(red or brown). There are 2 students with red hair and 8 students with brown hair. So there are 2 + 8 or 10 students with red or brown hair. number of favorable outcomes number of possible outcomes Simplify.

Answer: The probability of selecting a student with red or brown hair is

A class contains 6 students with black hair, 8 with brown hair, 4 with blonde hair, and 2 with red hair. Find P(not blonde). There are 6 + 8 + 2 or 16 students who do not have blonde hair. number of favorable outcomes number of possible outcomes Simplify. Answer: The probability of selecting a student who does not have blonde hair is

a. Find the probability of rolling a number less than 3 on a die. Answer:

b. A gumball machine contains 40 red gumballs, 30 green gumballs, 50 yellows gumballs, and 40 blue gumballs. Find P(red). Answer:

c. A gumball machine contains 40 red gumballs, 30 green gumballs, 50 yellows gumballs, and 40 blue gumballs. Find P(green or yellow). Answer: d. A gumball machine contains 40 red gumballs, 30 green gumballs, 50 yellows gumballs, and 40 blue gumballs. Find P(not blue). Answer:

Find the odds of rolling a number greater than 2. There are six possible outcomes, 4 are successes and 2 are failures. 4 numbers greater than 2

Sample space: 1, 2, 3, 4, 5, 6 2 numbers less than or equal to 2

Answer: The odds of rolling a number greater than

Find the odds of rolling a number less than 4.

Answer:

A card is selected at random from a standard deck of 52 cards. What are the odds against selecting a 2 or 3? There are four 2s and four 3s in a deck of cards, and there are 52 – 4 – 4 or 44 cards that are not a 2 or a 3. number of ways not to pick a 2 or 3 number of ways to pick a 2 or 3

Answer: The odds against selecting a 2 or 3 are 11:2.

A card is selected at random from a standard deck of 52 cards. What are the odds against selecting a 5, 6, or 7? Answer: 10:3

Travel Melvin is waiting to board a flight to Washington, D.C. According to the airline, the flight he is waiting for is on time 80% of the times it flies. What are the odds that the plane will be on time? The probability that the plane will be on time is 80%, so the probability that it will not be on time is 20%. odds of the plane being on time Answer: The odds that the plane will be on time are 4:1.

If the probability that it will snow this weekend is 70%, what are the odds that it will snow? Answer: 7:3

SQUARE ROOTS and REAL NUMBERS

Example 1

Find Square Roots

Example 2

Classify Real Numbers

Example 3

Graph Real Numbers

Example 4

Compare Real Numbers

Example 5

Order Real Numbers

Example 6

Rational Approximation

Find

.

represents the positive and negative square roots of

Answer:

Find

.

represents the positive square root of 0.0144.

Answer:

Find each square root. a.

Answer:

b.

Answer: 0.6

Name the set or sets of numbers to which belongs.

Answer: Because , which is neither a repeating nor terminating decimal, this number is irrational.

Name the set or sets of numbers to which belongs.

Answer: Because 1 and 6 are integers and , which is a repeating decimal, the number is a rational number.

Name the set or sets of numbers to which belongs.

Answer: Because this number is a natural number, a whole number, an integer and a rational number.

Name the set or sets of numbers to which –327 belongs.

Answer: This number is an integer and a rational number.

Name the set or sets of numbers to which each real number belongs. a.

Answer: rationals

b.

Answer: naturals, whole, integers, rationals

c.

Answer: irrationals

d.

Answer: integers, rationals

Graph

.

The heavy arrow indicates that all numbers to the left of 8 are included in the graph. The dot at 8 indicates that 8 is included in the graph.

Graph

.

The heavy arrow indicates that all the points to the right of –5 are included in the graph. The circle at –5 indicates that –5 is not included in the graph.

Graph each solution set. a. Answer:

b. Answer:

Replace the • with <, >, or = to make the sentence true.

Since Answer:

the numbers are equal.

Replace the • with <, >, or = to make the sentence true.

Answer:

Replace each • with <, >, or = to make each sentence true. a.

Answer: <

b.

Answer: <

Write

in order from least

to greatest. Write each number as a decimal.

or about 2.4495

or about 2.4444

Answer: The numbers arranged in order from least to greatest are

Write to greatest. Answer:

in order from least

Multiple-Choice Test Item For what value of x is A –5

B 0

C

true? D 5

Read the Test Item The expression and the set of choices replacement set.

is an open sentence, is the

Solve the Test Item Replace x in with each given value. A

False;

and

are not real numbers.

B

False;

is not a real number.

C

Use a calculator.

0.447214 < 1 < 2.236068

True

D

Use a calculator.

2.236068 < 1 < 0.447214 The inequality is true for

Answer: The correct answer is C.

False

Multiple-Choice Test Item For what value of x is A 3

B –3

Answer: A

C 0

true? D

THIS IS THE END OF THE SESSION

BYE!