Download Unit 1 ~ Contents

Unit 1 ~ Contents
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
Algebra Beauty and Awe ~ Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Types of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Graphing Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Absolute Values and Opposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Absolute Values in Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Real Number Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Variables, Constants, and Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Symbols for Multiplication in Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Quiz 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Mersenne’s Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Adding Positive and Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Subtracting Positive and Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Translating From Word Phrases to Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Multiplying Positive and Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Quiz 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Dividing Positive and Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Division Involving Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Operations With Exponential Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Exponents Using One or Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Roots and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Negative Radicands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Review for Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Tight Around the Middle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.7
Subtracting Positive and Negative Numbers
Subtraction can seem confusing when working with negative numbers. Problems like 7 – (– 2), or
(– 3) – 5 look difficult. However, subtraction is the same as adding the opposite.
7 – 3 = 4
7 + (– 3) = 4
Changing subtraction to adding the opposite simplifies subtracting positve and negative numbers.
Example 1. 7 – (– 2) = ?
7 – (– 2) = ?
Original problem
7 + (+2) = 9
Addition completed.
7 + (+2)
Subtraction changed to adding the opposite of – 2.
Example 2. (– 3) – 5 = ?
(– 3) – 5 = ?
Original problem
–3 + (– 5) = – 8
Addition completed.
(– 3) + (– 5)
Subtraction changed to adding the opposite of 5.
Rules for Subtracting Numbers
1. Change subtraction to addition.
2. Add the opposite.
Subtraction does not have commutative or associative properties. Subtracting must be done from
left to right. However, if subtraction is changed to adding the opposite, it becomes an addition
problem and the commutative and associative can be used to rearranged the numbers.
Example 3. Combine the numbers –8 – (–5) + (+2) – (+6) = ?
– 8 – (–5) + (+ 2) – (+ 6) = ?
Original problem.
– 8 + (– 6) + (+ 5) + (+2) = ?
Negatives and positives grouped together.
– 8 + (+ 5) + (+2) + (– 6) = ?
–14 + (+7) = ?
–14 + (+7) = (–7)
20 ~ Algebra I Unit 1
Subtractions changed to “add the opposites.”
Negatives combined and positives combined.
Addition completed.
Note: On the preceding pages, the “+” sign has been used to emphasize that numbers were
positive. However, this is not necessary and will be discontinued from now on. Negative numbers,
on the other hand, must always have the “ – ” sign.
Combine the numbers using subtraction. 1.7
1. – 9 – (–2)
2. 3 – (–3)
3. 4 – 4
7. –10 – (–20)
8. 4 – (– 9)
9. – 4 – 9
4. –2 – (–1)
Review
5. 0 – (– 5)
6. 10 – 12
Combine the numbers using addition. 1.6
10. 10 + (– 5)
13. 3 + 4 + 4
11. – 9 + (–3)
14. –7 + (– 6) + (– 5)
Evaluate the expressions if x = 6, y = 12. 1.4
16. xy
3
17. 2xy
12. 7 + (– 8)
15. –12 + 17 + (–10)
18. x • 2y
Follow the directions.
19. In the old English measuring system 1 gallon = 2 pottles = 4 quarts = 8 pints = 16 cups. Draw a
number line from 0 to 20 (in increments of 2) and plot these 5 numbers: 1, 2, 4, 8, and 16. (1.2)
20. The area of a rectangle can be found by using the formula A = lw where l is the length and w is the
width. Find the area of a rectangle whose length is 6 inches and whose width is 2 inches. (1.4)
21. Find the area of a rectangular corral which is 100 meters long and 30 meters wide. (1.4)
22. A scale shows that the deli worker has cut 1.35 pounds of salami. How much more salami should
she cut to fill the customer’s order of 2.5 pounds? (1.7)
Which property is illustrated by the following statements? 1.3
23. Multiplying 6 by 8 gives the same answer as multiplying 8 by 6.
24. Adding positive 13 to negative 13 (or – 2 to +2) will always equal zero.
25. Multiplying 8 by 3 before multiplying by another 5 is the same as multiplying 3 by 5 before
multiplying by another 8.
26. Multiplying zero by a trillion doesn’t change the zero.
1.7 Subtracting Positive and Negative Numbers ~ 21
Write the opposite-number pairs for these absolute values. 1.2
27. 1,465
28. 11
Write the word(s) for each definition.
29. 38
30. a number factor in front of a variable (1.4)
31. numbers that cannot be expressed as a quotient of integers (1.1)
32. a group of numbers, variables, and symbols of mathematical operations (1.4)
33. the category of all whole numbers and their opposites (1.1)
34. the distance of a number from zero (1.2)
35. numbers that can be expressed as a quotient of integers (1.1)
Combine the numbers using subtraction. 1.7
36. 21 – (–10)
37. – 24 – (–11)
39. –100 – 200
40. 150 – 200
38. 38 – (– 22)
41. (– 450) – (– 50)
Extra Practice
Combine the numbers using subtraction. 1.7
42. 28 – (– 8)
43. 19 – 5
44. –10 – (– 6)
48. –16 – (–4)
49. 21 – (– 5)
50. 34 – 3
45. 42 – 2
46. –17 – 3
47. –9 – 8
Archimedes discovered how to find the volume and density of irregularly
shaped objects while bathing. Simply submerse the object being examined
and the water level will rise to a height that reflects the size of the object.
22 ~ Algebra I Unit 1
1.8
Translating From Word Phrases to Math
In order for two people who speak different languages to communicate, a translation from one
language to the other is necessary.
Mathematics, especially algebra, has a language of its own. Solving real-life problems with
algebra requires translating words and ideas into mathematical symbols.
The first step in translating problems into mathematical symbols is to recognize words and
phrases commonly used for certain mathematical operations and relationships. Notice that there are
often several different words or phrases for a mathematical idea.
add
Addition
added to
plus
sum
increased by
more than
greater than
Powers/Roots
Subtraction
product of
divide
less or less than
times
divide into
subtracted from
reduce(d) by
nth* root
nth* power of
to the nth* power of
* n = any number
divided by
quotient of
ratio of
half of, third of, etc.
decreased by
diminished by
Equality/Inequality
equals
cube root
double, triple. etc.
difference between
square root
cubed
twice
difference (of)
is
cube (of)
Division
subtract
square (of)
squared
Multiplication
is equivalent to
results in
Other Phrases
absolute value of
reciprocal of
opposite of
additive inverse of
adds up to
inverse of
the result is
is greater than
consecutive
is less than
is greater than or equal to
is less than or equal to
1.8 Translating From Word Phrases to Math ~ 23
When translating from one language to another, the order of the words does not always stay the
same. So it is with mathematics. The order of numbers in word phrases does not necessarily stay
the same when translated into mathematical symbols. When translating, the meaning of the entire
expression must be considered, especially if it includes several operations.
Example 1. Translate the following into mathematical symbols.
a. eight increased by the square of seven is fifty-seven
8
+
72
b. three less four
3
–
4
c. three less than four
4
=
57
Notice what a difference one word can make!
–
3
Variables are usually indicated with phrases such as a number, an amount, or with the actual
variable. If the variable is not specified, any letter can be chosen to represent the unknown.
Example 2. Translate the following into mathematical symbols.
a. a number squared
x
2
is less than
<
b. Six times a number less than
6x
–
sixty-five
65
four times another number
c. twice the sum of three different numbers
2
Notice how the word is in a makes a difference.
Notice how the words
another in b and different in
c indicate using different
variables.
4y
x+y+z
Translate into symbols. Do not simplify or solve. 1.8
1. five increased by nine
2. eight less a number
5. a number less than five
6. half of a number
3. reduce twenty-two by three
4. the product of ten and three
7. the sum of the squares of six and four
Review
Combine the numbers using subtraction. 1.7
9. 3 – 9
12. –17 – (–18)
24 ~ Algebra I Unit 1
10. – 7 – 2
13. 14 – 18
8. the sum of two different numbers
11. – 8 – (– 8)
14. 19 – (–19)
Which property is illustrated by the following statements? 1.3
15. Adding 3 to 7 gives the same answer as adding 7 to 3.
16. Multiplying 5 by 1 doesn’t change the 5.
17. Adding 2 to 4 before adding on another 9 is the same as adding 4 to 9 before adding on another 2.
18. Adding 0 to 7 doesn’t change the 7.
19. Multiplying 6 by 1 (or 1 by 9) will always be 1.
6
9
Write rational numbers, integers, whole numbers, natural numbers, or irrational numbers. 1.1
20. counting numbers plus 0
21. the category of all whole numbers and their opposites
22. numbers that can be expressed as a quotient of integers
23. counting numbers
24. numbers that cannot be expressed as a quotient of integers
Write the answer. 1.7
25. Dietrich learned from the Guinness Book of World Records that on January 23-24, 1916, the
temperature at Browning, Montana dropped 100°F from a temperature of 44°F. What was the new
temperature?
26. At Helena, Montana on January 12, 1888 E. J. Hobbs noticed that the temperature fell from 40.5°F
to – 9°F in four and a half hours. What was the change in temperature?
Evaluate. 1.2
27. 4 • | 2 – 6 | + 16
28. | 64 | ÷ | – 8 | – | – 8 | + | 16 |
Combine the numbers using addition. 1.6
29. 102 + (–22)
32. – 9 + 12 + 15
30. 200 + (–100)
33. 8 + (–7) + (–7)
31. –140 + (– 30)
34. 127 + (–103)
Translate into symbols. Do not simplify or solve. 1.8
35. the difference of eleven and nine
37. four times a number, plus three, is fifteen
36. quadruple an amount
38. twice negative twelve
1.8 Translating From Word Phrases to Math ~ 25
Extra Practice
Translate into symbols. Do not simplify or solve. 1.8
39. double a number plus five
40. the quotient of twelve and four
43. the product of two and seven plus two
44. a number less negative six
41. a number less four
45. the cube of a number less nine
26 ~ Algebra I Unit 1
42. the absolute value of negative eight
46. five plus four equals the square of a number
1.9
Multiplying Positive and Negative Numbers
Multiplying a positive number by another positive number is like adding a positive number
repeatedly. Therefore, multiplying a positive number times another positive number gives a
positive answer:
7 • 3 = 7 + 7 + 7 = 21
Multiplying a negative number by a positive number is like adding a negative number repeatedly.
Therefore, multiplying a negative number by a positive number gives a negative answer:
(–7) • 3 = (–7) + (–7) + (–7) = (–21)
Because of the commutative property of multiplication, reversing the order of the factors does
not affect the answer. Regardless of the order, whenever a positive and a negative number are
multiplied, the answer will be negative:
(–7) • 3 = (– 21)
3 • (–7) = (– 21)
or
Multiplying a negative number by a negative number gives a positive answer. The pattern
below illustrates what happens to the sign of the answer when multiplication by a negative
number crosses over 0. This pattern is created by multiplying (– 2) times a set of numbers that
decrease by one each time.
(–2) • 2 = (– 4)
(–2) • 1 = (–2)
(–2) • 0 =
0
(–2) • (–1) = (+2)
(–2) • (–2) = (+ 4)
This pattern indicates that multiplying two negative numbers together yields a positive answer.
(– 6) • (–9) = (+54)
(–2.75) • (– 0.25) = (+0.6875)
3
(– 34 ) • (– 15 ) = (+ 20
)
1.8 Multiplying Positive and Negative Numbers ~ 27
Multiplying Positive and Negative Numbers
Multiplying Same Signs = Positive Product
positive • positive = positive
negative • negative = positive
Multiplying Different Signs = Negative Product
positive • negative = negative
negative • positive = negative
Write the answers to complete the charts. 1.9
1.
3
(–7) •
c __ = (–7)
(–7) •
(–7) •
(–7) •
(–7) •
(–7) •
=
2
=
0
=
(–1) =
(–2) =
2.
a __
(–7) •
b __
(–9) •
(–3) = 27
(–9) •
(–1) = 9
a __ = 18
(–9) •
(–9) •
0
d __
e __ = 21
= 0
2
=
(–9) •
b __= (–9)
(–9) •
3
(–9) •
14
0
=
c __
d __
Since multiplication is repeated addition, it is also commutative and associative. A series of
numbers can be multiplied in any order. However, each time the multiplication is by a negative
number it reverses the sign of the product. The reversing caused by negative numbers must be
observed carefully to obtain the correct sign in the answer.
Example 1. Multiply: (– 4) • 3 • (–2) • (–1) • (– 5)
(– 4)
•
negative • positive = negative
3
•
(– 2)
•
(–1)
•
(– 5)
(–12)
negative • negative = positive
24
positive • negative = negative
(– 24)
negative • negative = positive
28 ~ Algebra I Unit 1
120
Multiply. 1.9
3. –3 • (– 2) • 7 • (–1)
4. 8 • (–1) • (–3) • 5
7. –1 • (–1) • (–1) • (–1) • (–1)
8. – 2 • 2 • (– 2) • 2 (– 2)
5. –10 • 4 • (– 5) • (–1)
6. – 2 • (– 5) • (–3) • (–1)
Review
Translate into symbols. Do not simplify or solve. 1.8
9. a number divided by three
10. a number subtract four
11. the opposite of two
Combine the numbers using subtraction. 1.7
13. 3 – 10
16. 98 – 10
Write the answer. 1.7
12. the product of twelve and the square of a number
14. –7 – (–13)
15. 2 – 12
17. –20 – (– 16)
18. – 89 – (– 89)
19. The Sea of Galilee has an altitude of around – 685 ft. The Dead Sea has an altitude of around
–1,300 ft. As the water flows down the Jordan River from the Sea of Galilee to the Dead Sea, how
many feet does the water drop?
20. In the Parable of the Good Samaritan, the traveling man was robbed on his way from Jerusalem to
Jericho. The elevation of Jerusalem is around 2,500 ft. The elevation of Jericho is around – 800 ft. If
a man travels down from Jerusalem to Jericho, how many feet lower is Jericho than Jerusalem?
Write the word(s) for each definition.
21. a group of numbers, variables, and symbols of mathematical operations (1.4)
22. the category of all whole numbers and their opposites (1.1)
23. a numeral or symbol whose value never changes (1.4)
24. letters representing number values (1.4)
25. a number factor in front of a variable (1.4)
26. To subtract a number, add its
. (1.7)
Write an equation to illustrate the properties. 1.3
27. commutative property of addition
29. associative property of addition
28. commutative property of multiplication
30. associative property of multiplication
1.9 Multiplying Positive and Negative Numbers ~ 29
Which property is illustrated by the statement? 1.3
31. Adding 3 to 7 gives the same answer as adding 7 to 3.
32. Multiplying 6 by 8 gives the same answer as multiplying 8 by 6.
Evaluate the expressions if x = 2, y = 3 and z = 5. 1.4
33. x y • z
34. 2x • 3z
35. xy + 2z – y
Answer the questions.
36. A cake recipe calls for an 8 x 10 pan. Can you substitute a 9 x 9 pan of the same depth without
having batter flow over the sides of the pan? (1.4)
37. The temperature was 28°F at 10:30 Friday evening. At 7:35 the next morning it was –7°F. What was
the temperature drop? (1.7)
Write the opposite-number pairs for these absolute values. 1.2
38. 55
Multiply. 1.9
41. – 3 • 7
39. 128
40. 7
42. 9 • (– 8)
43. 2 • (– 6) • 2 • (– 2)
The Greeks used complex geometrical drawings to study
and prove some of the mathematical concepts that today
we can do much more easily by using algebra.
30 ~ Algebra I Unit 1