Download Rational Numbers

Objective 10.1A
New Vocabulary
positive numbers
negative numbers
integers
negative integers
positive integers
natural numbers
Discuss the Concepts
Classify each number as a positive integer, a negative integer, or neither.
210, 17, 28, 0,
2
3
, 465
Concept Check
A is a point on the number line halfway between − 12 and 4. B is a point on the number line halfway
between A and the graph of 2 on the number line. B is the graph of what number? −1
Optional Student Activity
Graph each of the following pairs of numbers on the number line. Then place the correct symbol, < or >,
between the two numbers.
1. 24 and 3 −4 < 3
2. 5 and 22 5 > −2
3. 21 and 23 −1 > −3
4. 25 and 0 −5 < 0
5. 27 and 1 −7 < 1
Objective 10.1B
New Vocabulary
opposites
absolute value of a number
Discuss the Concepts
True or false:
1. Every negative number is to the left of every positive number on the number line. T
2. Every positive number is to the right of every negative number on the number line. T
3. 26 is to the left of 21 on the number line. T
4. 28 is to the right of 212 on the number line. T
Optional Student Activity
The following list provides temperatures either on the surface or in the atmospheres of bodies in our solar
system.
Mercury’s dark side: −346°F
Mercury’s sunlit side: 950°F
Mars, daytime: −17°F
Saturn’s moon Titan: −190°F
Neptune’s atmosphere: −360°F
Pluto, atmospheric high: −163°F
Pluto, atmospheric low: −397°F
a. List the temperatures given from coldest to warmest.
−397, −360, −346, −190, −163, −17, 950
b. How many of the temperatures listed are below −300°F?
Three of the temperatures listed are below −300°F (−397°F, −360°F, −346°F).
Answers to Writing Exercises
122a. Students might describe the opposite of a number as the number with the sign changed, or they
might rephrase the definition that states that opposite numbers are two numbers that are the same
distance from zero on the number line but on opposite sides of zero.
b.
Students might rephrase the definition that states that the absolute value of a number is its distance
from zero on the number line. You might check that students include in their description that the
absolute value of a number cannot be negative (or that the absolute value of a number is either a
positive number or zero).
Objective 10.2A
Vocabulary to Review
Integers [10.1A]
absolute value [10.1B]
addend [1.2A]
New Rules
addition of integers
Discuss the Concepts
a. Explain the rule for adding two integers with the same sign.
b. Explain the rule for adding two integers with different signs.
Concept Check
State whether or not the operation is addition. If the operation is addition, state whether the numbers being
added have the same sign or different signs.
a. 2 + (−6)
b. −4 + 8
c. 7 − 13
d. (−10) + (−9)
e. −15 + (−3)
f. −5(+13)
Yes; different signs
Yes; different signs
No
Yes; same sign
Yes; same sign
No
Optional Student Activity
On the number line, illustrate each of the following sums.
a. −3 + 5
b. −4 + 7
c. 2 + (−6)
d. 5 + (−9)
e. −1 + (−4)
f. −5 + (−2)
Objective 10.2B
Vocabulary to Review
minus [1.3B]
negative [10.1A]
opposite [10.1B]
New Rules
subtraction of integers
Discuss the Concepts
Explain why the absolute value of −16 is greater than the absolute value of 4.
Concept Check
State whether each “−” sign is a minus sign or a negative sign.
a. 7 − 6 Minus
b. 3 − (−9) Minus; negative
c. −4 − 1 Negative; minus
d. −5 − (−2) Negative; minus; negative
e. (−8) − 10 Negative; minus
Objective 10.2C
Optional Student Activity
The following list provides temperatures either on the surface or in the atmospheres of bodies in our solar
system. (This is the same list provided in Objective 10.1B.)
Mercury’s dark side: −346°F
Mercury’s sunlit side: 950°F
Mars, daytime: −17°F
Saturn’s moon Titan: −190°F
Neptune’s atmosphere: −360°F
Pluto, atmospheric high: −163°F
Pluto, atmospheric low: −397°F
a. Find the difference between the temperatures on the sunlit side and the dark side of Mercury. 1296°F
b. Find the difference between Pluto’s atmospheric high and low temperatures. 234°F
In-Class Examples (Objective 10.2C)
1. Find the temperature after a rise of 4°C from −2°C. 2°C
2. During a card game of Hearts, you had a score of 14 points before your opponent “shot the moon,”
subtracting a score of 26 from your total. What was your score after your opponent “shot the moon”?
−12 points
Answers to Writing Exercises
101. Students should include in their description the fact that minus refers to the operation of subtraction,
whereas negative refers to the sign of a number.
Objective 10.3A
Vocabulary to Review
product [1.4A]
New Rules
multiplication of integers
Discuss the Concepts
Describe the rules for multiplying two integers.
Concept Check
True or false:
a. The product of a nonzero number and its opposite is a positive number. F
b. The product of an odd number of negative numbers is a negative number. T
c. The product of an even number of negative numbers is a negative number. F
d. To find the opposite of a number, multiply the number by 21. T
Optional Student Activity
Illustrate each of the following products on the number line.
a. 3 x (−3)
b. 2 x (−5)
c. 3 x (−2)
d. 5 x (−1)
(Hint: Remember that multiplication is repeated addition.)
Objective 10.3B
Vocabulary to Review
quotient [1.5A]
New Rules
division of integers
New Properties
zero and one in division
Discuss the Concepts
Describe the rules for dividing two integers.
Concept Check
What property of division is illustrated by each of the following?
a.
b.
c.
d.
0
0
Zero divided by any number other than zero is zero.
 18
Any number divided by 1 is the number.
is undefined.
Division by zero is undefined.
1
Any number other than zero divided by itself is 1.
12
18
1
5
0
22
22
Objective 10.3C
Optional Student Activity
A geometric sequence is a list of numbers in which each number after the first is found by multiplying the
preceding number in the list by the same number. For example, in the sequence 1, 3, 9, 27, 81, . . . , each
number after the first is found by multiplying the preceding number in the list by 3. To find the multiplier in a
geometric sequence, divide the second number in the sequence by the first number; for the example above,
3 ÷ 1 = 3.
Geometric sequences are used in the exercises below.
1.
2.
3.
4.
Find the next three numbers in the geometric sequence −5, 15, −45, . . . . 135, −405, 1215
Find the next three numbers in the geometric sequence 2, −4, 8, . . . . −16, 32, −64
Find the next three numbers in the geometric sequence −3, −12, −48, . . . . −192, −768, −3072
Find the next three numbers in the geometric sequence −1, −5, −25, . . . . −125, −625, −3125.
Answers to Writing Exercises
124. Students should rephrase the rules for multiplying two integers and for dividing two integers:
To multiply two integers with the same sign, multiply the absolute values of the factors; the product is
positive. To multiply two integers with different signs, multiply the absolute values of the factors; the
product is negative.
To divide two integers with the same sign, divide the absolute values of the numbers; the quotient is
positive. To divide two integers with different signs, divide the absolute values of the numbers; the
quotient is negative.
Be sure that students are describing both the signs of the numbers being multiplied or divided and the
sign of the resulting product or quotient.
Quick Quiz (Objective 10.3C)
1. The combined scores, in relation to par, of the top nine golfers in a golf tournament equaled -63. What
was the average score of the nine golfers? 27
2. The daily low temperatures during one week were as follows: 4°, -6°, 8°, -2°, -9°, -11°, -5°. Find the
average daily low temperature for the week. 23°
Objective 10.4A
Vocabulary to Review
integer [10.1A]
mixed number [2.2A]
New Vocabulary
rational number
terminating decimal
repeating decimal
Discuss the Concepts
Classify the numbers as natural numbers, integers, or rational numbers:
2
−3, 7, 0,
5
, and

10
9
Concept Check
Show that the number is a rational number by writing it as the quotient of two integers. (Note: Answers may
vary. Possible answers are given.)
1.

2.

3.
3
5
5
6
7
6
7
25 25
2
29
9
9
4
4. −0.16
5. −0.75
6. 5.2
7. −2.4
25
3
4
26
5
12
8. −6.25
5
25
4
Optional Student Activity
Have students use calculators to write
5
,
17
,
42
,
63
99 99 99 99
, and
81
99
as repeating decimals. Have them
describe the pattern. Ask them to find fractions with other denominators that have interesting decimal
patterns (for example, ninths or elevenths).
Optional Student Activity
To determine whether a fraction can be written as a terminating decimal, first write the fraction in simplest
form. Then look at the denominator of the fraction. If it contains prime factors of only 2s and/or 5s, then it
can be expressed as a terminating decimal. If it contains prime factors other than 2s or 5s, it represents a
repeating decimal. For example, if the denominator of a fraction in simplest form is 20, then it can be written
as a terminating decimal because 20 = 2 *2 * 5 (prime factors of only 2 and 5). If the denominator of a
fraction in simplest form is 6, it represents a repeating decimal because it contains the prime factor 3 (a
number other than 2 or 5).
Assume that each of the following numbers is the denominator of a fraction written in simplest form. Does
the fraction represent a terminating or a repeating decimal?
a. 4 Terminating
b. 5 Terminating
c. 7 Repeating
d. 9 Repeating
e. 10 Terminating
f. 12 Repeating
g. 15 Repeating
h. 16 Terminating
i. 18 Repeating
j. 21 Repeating
k. 24 Repeating
l. 25 Terminating
m. 28 Repeating
n. 40 Terminating
Write another number that, as the denominator of a fraction in simplest form, represents a terminating
decimal.
Answers will vary. For example, 8 or 80.
Write another number that, as the denominator of a fraction in simplest form, represents a repeating
decimal.
Answers will vary. For example, 3 or 30.
Objective 10.4B
Optional Student Activity
Replace the question mark with the number that makes the equation true.
1. 4.3 x ? = −2.58 −0.6
2. 32.4 = 29 x ? −3.6
3. ? x (−0.2) = −1.6 8
?
 20 8
0.4
?
 3.1 24.8
5.
8
?
 12.5 −6
6.
0.48
4.
Optional Student Activity
Find the product of 1.0035 and 1.00079 without using a calculator. Then find the product using a calculator
and compare the two numbers.
Some calculators truncate the product, which means that the digits that cannot be displayed are discarded.
Other calculators round the answer to the rightmost place value in the calculator’s display. Determine which
method your calculator uses to handle approximate answers.
If the decimal places in a negative number are truncated, is the resulting number greater than, less than, or
equal to the original number?
1.004292765; truncated: 1.0042927; rounded: 1.0042928; greater than
Objective 10.4C
Optional Student Activity
The table below shows the total trade balance, in billions of dollars, for the United States in the 1990s.
(Source: U.S. Department of Commerce, Bureau of Economic Analysis)
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
Trade Balance
6.616
−47.724
−82.681
−118.605
−109.457
−123.318
−140.540
−217.138
−331.479
1.
2.
3.
4.
For which year was the trade balance the lowest? 1999
Calculate the difference between the trade balance in 1991 and that in 1999. $338.095 billion
Between which two consecutive years was the difference in the trade balance greatest? 1998–1999
How many times greater was the trade balance in 1999 than in 1992? Round to the nearest whole
number. 7 times greater
5. Calculate the average trade balance per quarter in 1997. −$35.135 billion
6. Examine the data. Would you expect the trade balance to have increased or decreased from 1999 to
2000? Why? Answers will vary.
Answers to Writing Exercises
95. Given any two different rational numbers, it is always possible to find a rational number between them.
One method is to add the two numbers and divide by 2.
Another method is to add the numerators and add the denominators. For example, given the fractions
2
5
and
3
,
23
4 54

5
9
and
2
5

5
9

3
4
.
Objective 10.5A
New Vocabulary
scientific notation
Discuss the Concepts
Is the number 42.5 x 107 written in scientific notation? Why or why not?
Concept Check
Which of the following numbers represent numbers greater than 10?
Which represent numbers less than 1?
a. 4.3 x 105 Greater than 10
b. 3.09 x 1026 Less than 1
c. 8 x 1017 Greater than 10
d. 6.175 x 10−22 Less than 1
Optional Student Activity
1. Our galaxy is estimated to be 6 x 1017 mi across. How many hours would it take a spaceship to cross the
galaxy traveling at 25,000 mph? Write the answer in scientific notation. 2.4 x 1013 h
2. One light year is the distance light travels in 1 year, or about 5.87 x 10 12 mi. The distance to the star
Alpha Centauri is 4.3 light years. That is, it would take 4.3 years traveling at the speed of light to reach
this star. Astronauts travel at a speed of approximately 25,000 mph. At this speed approximately how
many years would it take to reach Alpha Centauri? Round to the nearest thousand. 115,000 years
Objective 10.5B
Procedures to Review
Order of Operations Agreement [1.6B]
Concept Check
Put the following expressions in order from the smallest value to the largest value.
a.
 3  1
( 3)         −16
 2  8
b.
3 − (−4 + 2) ÷ 2 4
3
c.
 1   3 9  1 1 7
    
24
 2   4 16  6
d. 8 − (−7) x (6 − 10) + 32 −11
Optional Student Activity
1. What is the smallest prime number greater than 32 ÷ (1 − 9) + (−3)2? 7
2. What is the smallest integer greater than −22 − (−3)2 + 5(4) ÷ 10 − (−6)? −4
3. What is the smallest prime number that divides evenly into the sum of 3 9 and 511? 2
Answers to Writing Exercises
97. Because the first statement is false (it was either Becky or Diana), neither Becky nor Diana could have
done it. Therefore, it was either Abdul or Carl. Because the second statement is false (it was neither
Becky nor Carl), either Becky or Carl did it. Because the first statement ensured that it was Abdul or
Carl, it must have been Carl.
98. Check that students state that, for a number to be written in scientific notation, it must be the product of
a number between 1 and 10 and a power of 10, where the exponent on 10 is an integer.
Answers to Projects and Group Activities—Deductive Reasoning
4 Ós
2 Λs
6 Ós
24 ∫s
15 Ôs
Chris: compact car
Dana: sports car
Leslie: station wagon
Pat: sedan
7. Anna: cucumbers
Kay: beans
Michelle: tomatoes
Nicole: squash
8. Ontkeans: French
1.
2.
3.
4.
5.
6.
Kedrovas: Mexican
McIvers: Chinese
Levinsons: Italian