Transcript Chapter 2

Algebra 1 Interactive Chalkboard
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Lesson 2-1 Rational Numbers on the Number Line
Lesson 2-2 Adding and Subtracting Rational Numbers
Lesson 2-3 Multiplying Rational Numbers
Lesson 2-4 Dividing Rational Numbers
Lesson 2-5 Statistics: Displaying and Analyzing Data
Lesson 2-6 Probability: Simple Probability and Odds
Lesson 2-7 Square Roots and Real Numbers
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
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
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:
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
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  above 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.
85 115 126 92 104 107 78
85 116 100 121 123 131 88
79 90 110 129 108 93 84
131 114 92
97 99 116
75 70 132
The greatest 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 5
13 25
Answer:
7
32
11
37
Stem
0
1
2
3
10
21
15
10
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
54
83
87
Dallas
59 68
91 95
78 66
77
95
57
50
75
81
Atlanta
55 64
85 88
72 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 stemand-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
96
81
92
76
51
98
75
71
64
Class B
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.
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 yellow gumballs, and 40 blue
gumballs. Find P(red).
Answer:
c. A gumball machine contains 40 red gumballs, 30
green gumballs, 50 yellow gumballs, and 40 blue
gumballs. Find P(green or yellow).
Answer:
d. A gumball machine contains 40 red gumballs, 30
green gumballs, 50 yellow 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
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
in order from least
to greatest.
Answer:
51
20
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
False
The inequality is true for
Answer: The correct answer is C.
Multiple-Choice Test Item
For what value of x is
A 3
B –3
Answer: A
C 0
true?
D
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