Download Adv Rad Unit 1 Workbook

REV: 3/06 jcl
Advanced Radiographer Course
Unit One
Technical Math
(STUDENT NAME)
Mr Lee Darby
(INSTRUCTOR)
REV: 3/06 jcl
UNIT 1 CONVENTIONS
The following conventions apply to learning objectives throughout Unit 1.
STANDARDS:
Unless otherwise stated, performance will be in accordance with the following references:
1. Ewen, D. and Topper, M. Mathematics for Technical Education.
2. Naval Education and Training Program Development Center, Mathematics, Volume 1.
3. Dolciani’s, Algebra: Structure and Method, Book 1.
CONDITIONS:
Unless otherwise stated, all calculations are to be performed without computational aids
(calculators, computers).
ABBREVIATIONS:
LCD - Least Common Denominator
LCM - Least Common Multiple
GCM - Greatest Common Multiple
2
REV: 3/06 jcl
TABLE OF CONTENTS
LESSON:
PAGE
1.1
THE REAL NUMBER SYSTEM
.
.
.
.
.
6
1.2
FRACTIONS .
.
.
.
.
.
14
1.3
ADDITION AND SUBTRACTION OF FRACTIONS
.
.
20
1.4
MULTIPLICATION AND DIVISION OF FRACTIONS
.
.
25
1.5
DECIMALS .
.
.
.
.
.
.
.
29
1.6
PERCENTS .
.
.
.
.
.
.
.
35
1.7
ADDITION AND SUBTRACTION OF SIGNED NUMBERS
.
38
1.8
MULTIPLICATION AND DIVISION OF SIGNED NUMBERS .
41
1.9
GROUPING SYMBOLS AND ORDER OF OPERATIONS
.
44
1.10
ALGEBRAIC EXPRESSIONS
.
.
46
1.11
ADDITION AND SUBTRACTION OF ALGEBRAIC
EXPRESSIONS
.
.
.
.
.
.
.
50
1.12
EXPONENTS AND RADICALS
.
.
53
1.13
MULTIPLICATION AND DIVISION OF ALGEBRAIC
EXPRESSIONS
.
.
.
.
.
.
.
57
1.14
LINEAR EQUATIONS
.
.
.
59
1.15
SCIENTIFIC NOTATION AND POWERS OF 10 .
.
.
64
1.16
FORMULAS .
1.17
RATIO AND PROPORTION
1.18
DIRECT VARIATION
1.19
INVERSE VARIATION
1.20
GRAPHING ORDERED PAIRS OF REAL NUMBERS
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
68
.
.
.
.
.
71
.
.
.
.
.
.
74
.
.
.
.
.
.
77
.
.
80
3
REV: 3/06 jcl
UNIT 1 TECHNICAL MATH
CONTACT HOURS: 41.0 DIDACTIC 0.0 LAB/PRACTICAL
TERMINAL OBJECTIVES:
1.1
Perform mathematical operations involving the set of real numbers.
1.2
Describe the different types of fractions and compute for the least common denominator
(LCD) using prime factorization.
1.3
Solve problems in addition and subtraction of fractions.
1.4
Solve problems in multiplication and division of fractions.
1.5
Perform mathematical operations involving decimals.
1.6
Convert percents to decimals, fractions, and fractions to decimals.
1.7
Perform addition and subtraction of signed numbers.
1.8
Perform multiplication and division of signed numbers.
1.9
Evaluate mathematical problems using the order of operations convention.
1.10 Determine the degrees of algebraic expressions.
1.11 Simplify indicated operations involving addition and subtraction of algebraic expressions.
1.12 Perform algebraic operations using the laws of exponents.
1.13 Simplify indicated operations involving
multiplication and division of algebraic expressions.
1.14 Evaluate equations using the “basic properties in solving equations.”
1.15 Perform mathematical operations using the laws of exponents and express each result in
scientific notation.
1.16 Solve problems using formulas where all but the unknown quantities are given.
1.17 Solve equations using the principle of proportions.
1.18 Perform mathematical operations involving variables that are directly related.
1.19 Perform mathematical operations involving variables that are inversely related.
4
REV: 3/06 jcl
UNIT 1 TECHNICAL MATH (CONT.)
1.20 Plot points in the number line corresponding to each ordered pair of numbers.
5
REV: 3/06 jcl
LESSON TOPIC 1.1: THE REAL NUMBER SYSTEM
CONTACT HOURS: 3.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform mathematical operations involving the set of real numbers.
ENABLING OBJECTIVES:
1.1.1 Define the following terms:
a. Positive integers
b. Negative integers
c. Rational numbers
d. Irrational numbers
e. Real numbers
1.1.2 List some of the common mathematical symbols.
1.1.3 State the parts of a number line.
1.1.4 Compare two points on a number line.
1.1.5 List the properties of the set of real numbers.
1.1.6 Describe the Place Value Concept.
1.1.7 Define the following:
a. The decimal system.
b. Addition, addends, and sum.
c. Subtraction, minuend, subtrahend, and difference.
d. Multiplication, multiplicand, multiplier, and product.
e. Division, dividend, divisor, and quotient.
1.1.8 Add, subtract, multiply, and divide whole numbers.
6
REV: 3/06 jcl
LESSON TOPIC 1.1: THE REAL NUMBER SYSTEM
A. Terms:
1.
Integers include positive whole numbers, negative whole numbers, and zero, no
fractions or decimals.
a. Positive integers are numbers to the right of zero on the number line.
b. Negative integers are numbers to the left of zero on the number line.
c. The integer zero is neither positive nor negative.
d. The + sign before a positive integer is generally not written. However, the negative
integers are always written with a - sign in front of them.
e. The integers are the numbers
………., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, ……….
2. Rational numbers are numbers that can be expressed as the quotient of two integers
with a denominator other than zero that is any number which could be represented by a
fraction.
a. Rational numbers can be classified as positive, negative, or zero.
1)
A rational number is positive if its numerator and denominator are both
positive or both negative.
2)
A rational number is negative if its numerator and denominator have different
signs.
3)
A rational number is zero if its numerator is zero and its denominator is
not zero.
b. Every integer can be expressed as a fraction and is a rational number.
c. Example.
1 , 2 , 5 , -3 , -6 , 2 , ……
3 5 6 +5 -7 1
3. Irrational numbers are numbers that cannot be expressed in the form of a fraction.
7
REV: 3/06 jcl
a. Numbers that possess endless non-repeating digits to the right of the decimal point.
b. Example.
 = 3.1415...., and √2 = 1.141....
4. Real numbers include whole numbers, integers, fractions, decimals, rational numbers,
and irrational numbers.
a. All real numbers except zero are either positive or negative.
b. Common Math Symbols:
1)
> - is greater than
2)
< - is less than
3)
≤ - less than or equal to
4)
≥ - greater than or equal to
5)
= - equal to
6) │ │ - absolute value
c. Parts of a Number Line.
(negative)
(positive)
D. Compare two points on a number line.
1. Given any two numbers on the number line, the one on the right is always larger,
regardless of its sign.
a. Let a and b be two integers. Then a is greater than b, denoted by a > b, if the
integer a is to the right of the integer b on the number line.
b. Let c and d be two integers. Then c is less than d, denoted by c < d, if the integer c
is to the left of the integer d on the number line.
8
REV: 3/06 jcl
2. Examples:
a. -3 < 2
b. 7 > -2
c. l > -3
d. -19 < 0
e. 0 > -5
E.
Properties of the set of Real Numbers.
1. Commutative property for addition.
a. a + b = b + a
b. Example.
1)
2+3=3+2
2. Associative property for addition.
a. (a + b) + c = a + (b + c)
b. Example.
1)
(2 + 3) + 5 = 2 + (3 + 5)
5+5=2+8
10 = 10
3. Additive identity.
a. a + 0 = a
b. Example.
1)
5+0=5
5=5
4. Commutative property for multiplication.
a. a∙b = b∙a
9
REV: 3/06 jcl
b. Example.
1)
2∙3=3∙2
6=6
5. Associative property for multiplication.
a. (a ∙ b) ∙ c = a ∙ (b ∙ c)
b. Example.
1) (2 ∙ 3) ∙ 4 = 2 ∙ (3 ∙ 4)
6 ∙ 4 = 2 ∙ 12
24 = 24
6. Identity property for multiplication.
a. a ∙ l = a
b. Example.
1) 5 ∙ 1 = 5
5=5
7. Multiplicative inverse.
a. a ∙ 1 = 1
a
1=1
b. Example.
1) 9 ∙ 1 = 1
9
1=1
8. Additive inverse.
a. a + ( -a ) = 0
b. Example.
1) 5 + ( -5 ) = 0
5-5=0
0=0
10
REV: 3/06 jcl
9. Distributive property.
a. a (b + c) = ab + ac
b. Example.
1) 2 (3 + 4) = 2 ∙ 3 + 2 ∙ 4
2∙7=6+8
14 = 14
F.
Place Value Concept
1. Each digit has a place value. The location of a digit in a numeral indicates the value of
the digit. Starting from the right and moving to the left, the digits represent ones, tens,
thousands, and so on.
2. Example.
a. What numbers are represented by the following numeral?
1) 48,902
*4 is in the ten thousands place; it represents 4 ∙ 10,000 = 40,000.
*8 is in the thousands place; it represents 8 ∙ 1,000 = 8,000.
*9 is in the hundreds place; it represents 9 ∙ 100 = 900.
*0 is in the tens place; it represents 0 ∙ 10 = 0
*2 is in the ones place; it represents 2 ∙ 1 = 2
G. Terms:
1. Decimal system are base on tenths or the number ten. In the decimal system, each digit
position in a number has ten times the value of the position adjacent to it on the right.
Thus the number 11 is actually a coded symbol which means “one ten plus one unit.”
Since ten plus one is eleven, the symbol 11 represents the number eleven.
2. Addition is the process of finding the sum of two or more numbers.
a. Addends are the numbers to be added.
b. Sum is the answer obtained in addition.
c. The operation of addition is indicated by a plus sign (+) as in B + 4 =
12. The numbers 8 and 4 are addends and the answer (12) is their sum.
3. Subtraction is the process of finding the difference between two numbers.
11
REV: 3/06 jcl
a. Minuend is the number to be subtracted from.
b. Subtrahend is the number to be subtracted.
c. Difference is the answer obtained in subtraction.
d. The operation of subtraction is indicated by a minus sign (-) as in 9 - 3 = 6. The
number 9 is the minuend, 3 is the subtrahend, and the answer (6) is their difference.
4. Multiplication is a shortcut of addition, it is a method of adding a number to itself a
given number of times.
a. Multiplicand is one of the numbers that are to be mu1tip1ied.
b. Multiplier determines the number of times the multiplicand is to be multiplied.
c. Product is the answer obtained in multiplication.
d. Multiplication may be indicated by a multiplication sign (x) between two numbers,
a dot ( ∙ ) between two numbers, or parentheses around one or both of the numbers
to be multiplied. When a dot is used to indicate multiplication, it is distinguished
from a decimal point or a period by being placed above the line of writing, whereas
a period or decimal point appears on the line. Also when parentheses are used to
indicate multiplication, the numbers to be multiplied are closer together than they
are when the dot or is used. In the following example 6 x 8 = 48, 6 is the multiplier
and 8 is the multiplicand. Both 6 and the 8 are factors and 48 is the product.
5. Division evaluates how many times one number is present in another number.
a. Dividend is the number that gets divided.
b. Divisor is the number that does the dividing.
c. Quotient is the answer obtained after division.
d. Division is usually indicated by either by a division sign (÷) or by placing one
number over another number with a line between the numbers, as in the following
examples:
1) 8 ÷ 4 = 2
The number 8 is the dividend, 4 is the
divisor, and 2 is the quotient.
12
REV: 3/06 jcl
H. Basic math operations.
1. Addition.
a. 357 + 845 + 22 = 1224
b. 1100 + 110 + 14 = 1224
c. 30 + 13 = 43
2. Subtraction.
a. 5234 - 2345 = 2889
b. 47 - 24 = 23
c. 69 - 38 = 31
3. Multiplication.
a. 27 x 6 = 162
b. 43 x 27 = 1161
c. 27 x 40 = 1080
4. Division.
a. 56 ÷ 4 = 14
b. 252 ÷ 7 = 36
c. 1862 ÷ 38 = 49
13
REV: 3/06 jcl
LESSON TOPIC 1.2: FRACTIONS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERNINAL OBJECTIVE: Describe the different types of
fractions and compute for the least common denominator (LCD) using prime factorization.
ENABLING OBJECTIVES:
1.2.1 Define the following terms:
a. Fraction
b. Lowest term
c. Factor
d. Prime number
e. Prime factor
f. Prime factorization
g. Least Common Denominator (LCD)
h. Least Common Multiple (LCM)
i. Greatest Common Multiple (GCM)
1.2.2 List the parts of a fraction.
1.2.3 Describe the following types of fractions:
a. Proper
b. Improper
c. Mixed number
d. Simple
e. Complex
f. Equivalent
1.2.4 State the prime numbers from 1 to 100.
1.2.5 Compute LCD using prime factorization.
14
REV: 3/06 jcl
LESSON TOPIC 1.2. FRACTIONS
A. Terms:
1. A fraction is used to represent a part of a whole. The number line maybe used to show
the relationship between integers and fractions. For example, if the interval between 0
and 1 is marked off to form three equal spaces (thirds), then each space so formed is
one-third of the total interval. If we moved along the number line from 0 toward 1, we
will have covered two of the three “thirds” when we reach the second mark. Thus the
position of the second mark represents the number 2/3.
a. A fraction is made up of two parts:
1) The numerator is the top number and it signifies how many parts of the whole
number are being represented by the fraction (dividend).
2) The denominator is the bottom number and it signifies how many parts the
whole has been divided or sectioned into (divisor).
2. Lowest term
a. A fraction is in its lowest
term when its numerator and denominator
are both divisible only by the number 1.
b. A fraction is in its lowest terms when the numerator and the denominator have
no common factors except 1.
c. Examples:
1) 3/4 , 7/8 , 17/29
3. Factor
a. Is a number which will divide into another number without a remainder.
b. The numbers that are being multiplied in multiplication.
c. Examples:
1) 6 has 1, 2, 3 and 6 as its factors.
15
REV: 3/06 jcl
4. Prime numbers are natural numbers greater than 1 that have only two factors, 1 and
itself.
a. Examples:
1) 2, 3, 5, 7
5. Prime factor - a factor which is a prime
factor.
a. A positive number that has exactly two different factors.
b. Examples:
1) 11, 13, 17
6. Prime factorization is a way of writing a number as a product of prime factors.
a. Example:
Find the prime factorization of 30.
1) Is 30 divisible by 2?
30 ÷ 2 = 15
Yes (2 is a prime factor of 30)
Is 15 divisible by 2?
No (Proceed to the next prime number)
Is 15 divisible by 3?
Yes (3 is a prime factor of 30)
Note that each divisor and the final quotient make up the prime factorization.
Other examples:
12 = (2) (2) (3)
42 = (2) (3) (7)
7. Least common multiple (LCM) is the smallest multiple that is common to two or more
numbers.
16
REV: 3/06 jcl
a. Example.
1) Find the LCM of 15 and 20.
15 = (3) (5)
20 = (2) (2) (5)
LCM = (2) (2) (3) (5)
= 60
8. Least common denominator (LCD) — are also known as the lowest common multiple
of the denominator.
a. Example.
Find the LCD of 1/15 and 1/20:
LCM of the denominators 15 and 20 is 60 so the LCD is also 60.
9. Greatest common multiple (GCM) is the greatest multiple that is common to two or
more numbers. Also sometimes known as the Greatest Common Divisor (GCD).
a. Example.
Find the GCM of 650, 900, and 700.
650 = 2 • 52 • 13
900 = 22 • 32 • 52
700 = 22 •52 • 7
GCD = 2 • 52 = 50
Notice that 2 and 52 are factors of each number. The greatest common divisor is 2
x 25 = 50.
B. Types of fractions:
1. Proper fractions are fractions where the numerator is smaller than the denominator. The
value of this fraction is always less than 1.
a. Examples:
1/2, 2/3, 6/17
2. Improper fraction is fraction wherein the numerator is larger than the denominator. The
value of this fraction is always greater than or equal to 1.
17
REV: 3/06 jcl
a. Examples:
3/2, 5/3, 6/5, 4/4
3. Mixed number is a fraction that contains both a whole number and a fraction. Consist
of a whole number and a fraction written together with the understanding that they are
to be added to one another.
a. Examples:
1 2, 3 1, 10 3
3 2
2
4. Simple fraction are fractions wherein both the
numerator and the denominator are both whole numbers.
a. Examples:
5/8, 2/3
5. Complex fraction is a fraction in which the numerator or denominator is either another
fraction(s) or a mixed number(s), or both.
a. Examples:
1
1
___1___, ___2___, ___5___,
1
4
1
3
4
.5___
3
6. Equivalent fractions are fractions that name the same number. A simple method to
check if fractions are equivalent is to cross multiply and check the products. If their
products are equal then the fractions are equivalent.
Any number multiplied by 1 is equivalent to the number itself. For example 1 times 2 is
2. These facts are used in changing the form of a fraction to an equivalent form.
For example, if 1 in form 2/2 is multiplied by 3/5, the product will still have a value of
3/5 but will be in different form, as follows:
2 • 3 = 2 • 3 = _6
2 5
2•5
10
18
REV: 3/06 jcl
C. Prime numbers from 1 to 100.
A number that has no factors except itself and 1 is a
prime number. The following series shows all the prime
numbers up to 100.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 97. (Total of 24)
D. LCD computation using prime factorization.
1. Find the prime factorization of the denominators of the fractions.
2. List each factor the greatest number of times it appears in any one number.
3. The product of the factors listed in step (2) is the least common denominator (LCD).
a. Example 1.
Find the LCD of 1/6 and 1/18.
6 = (2) (3)
18 = (2) (3) (3)
LCD = (2) (3) (3) = 18
Example 2.
Find the LCD of 1/20 and 1/32.
20 = (2) (2) (5)
32 = (2) (2) (2) (2) (2)
LCD = (2) (2) (2) (2) (2) (5)
= 160
19
REV: 3/06 jcl
LESSON TOPIC 1.3: ADDITION AND SUBTRACTION OF FRACTIONS
CONTACT HOURS: 3.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Solve problems in addition and
subtraction of fractions.
ENABLING OBJECTIVES:
1.3.1 Add unlike fractions.
1.3.2 Add a fraction and a mixed number.
1.3.3 Change an improper fraction to a whole or mixed number.
1.3.4 Change a whole or mixed number t an improper fraction.
1.3.5 Add a series of fractions together.
1.3.6 Subtract unlike fractions.
1.3.7 Subtract a fraction from a whole or mixed number.
1.3.8 Subtract a whole or mixed number from a fraction.
20
REV: 3/06 jcl
LESSON TOPIC 1.3: ADDITION AND SUBTRACTION OF FRACTIONS
A. Adding like fractions.
1. Add the numerators and put the sum over the common denominator.
a. Examples:
1) 1/5 + 2/5 = 3/5
2) 1/8 + 3/8 = 4/8 or 1/2
B. Adding unlike fractions.
1. Find the Least Common Denominator (LCD).
2. Change the fractions to equivalent fractions with the same denominators.
3. Add numerators together.
4. Use the LCD as common denominator.
5. Reduce to lowest terms when possible.
a. Examples:
1) 1/8 + 1/4
LCD =8
1/8 = 1/8
+
1/4 = 2/8
3/8 ans.
2) 1/2 + 3/4
LCD = 4
1/2 = 2/4
+
3/4 = 3/4
5/4 or 1 l/4ans.
C. Changing an improper fraction to a whole or mixed number.
1. Divide the denominator into the numerator.
21
REV: 3/06 jcl
2. Take the quotient and use it as the whole number for the fraction.
3. Take the remainder and use it as the numerator.
4. Take the divisor and use it as the denominator.
a. Example:
1) Convert 19/5 to a mixed number.
a) Divide 19 by 5
b) 5 divides into 19 three times with a remainder of 4.
c) Quotient = 3 4/5 ans.
D. Changing a whole or mixed number to an improper fraction.
1. Multiply the whole number by the denominator of the fraction, add the numerator to
this product, and place the sum over the denominator.
a. Example.
1) Convert 2 7/8 to an improper fraction.
a) Multiply 2 by 8
b) Add the product (2 x 8 = 16) to 7 (numerator)
c) Place the sum (16 + 7) over the denominator (8)
d) 23/8 ans.
E.
Adding a series of fractions together.
1. Add fractions consecutively from left to right using the different rules of adding
different types of fractions.
a. Example.
1) 1/2 + 1/3 + 1/4
a) Find the LCD
b) LCD = 12
22
REV: 3/06 jcl
c) 1/2 = 6/12
1/3 = 4/12
1/4 = 3/12
13/12 or 1 1/12 ans.
F.
Subtracting like fractions.
1. Subtract the numerators and put the difference over the common denominator. Reduce
to lowest terms if possible.
a. Example.
1) 5/8 — 3/8
a) Subtract 3 from 5
b) Place the difference (2) over the denominator.
c) so 2/8 (Reduce to lowest terms)
d) Divide both terms by 2
e) 1/4 ans.
G. Subtracting unlike fractions.
1. Change all denominators to their lowest common denominators.
2. The numerators may need to be change to make sure that the fractions are still
equivalent to the originals.
3. Subtract the numerators and keep the denominator the same.
a. Example.
1) 7/8 — 3/4
2) Find the LCD
3) LCD = 8
4) 7/8 = 7/8
3/4 = 6/8
1/8 ans.
23
REV: 3/06 jcl
H. Subtract a fraction from a whole or mixed number and vice versa.
1. Convert the mixed number to an improper fraction and apply the rule in subtracting
unlike fractions.
a. Examples:
1) 2 3/4 -1/2
a) 2 3/4 = 11/4
b) Find the LCD
c) LCD = 4
d) 11/4 = 11/4
1/2 = 2/4
9/4 or 2 1/4 ans.
2) 2 3/4 – 1/2
a) 2 1/3 = 7/3
b) Find the LCD
c) LCD = 12
d) 15/14 = 45/12
7/3 = 28/12
17/12 or 1 5/12 ans.
3) 5 – 1 2/4
a) 1 2/4 = 6/4
b) Find the LCD
c) LCD = 4 (denominator of the given fraction)
d) 5/1 = 20/4
6/4 = 6/4
14/4 or 3 2/4 or 3 1/2 ans.
24
REV: 3/06 jcl
LESSON TOPIC 1.4: MULTIPLICATION AND DIVISION OF FRACTIONS
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Solve problems in multiplication and division of fractions.
ENABLING OBJECTIVES:
1.4.1 Multiply a fraction by a whole or mixed number.
1.4.2 Multiply a fraction by other fractions.
1.4.3 Multiply mixed numbers.
1.4.4 Define reciprocal.
1.4.5 Divide a fraction by a fraction.
1.4.6 Divide a fraction by a whole or mixed number.
1.4.7 Divide a whole or mixed number by a fraction.
1.4.8 Divide a mixed number by a whole number.
1.4.9 Divide a whole number by a mixed number.
25
REV: 3/06 jcl
LESSON TOPIC 1.4: MULTIPLICATION AND DIVISION OF FRACTIONS
A. Multiplication of fractions.
1. Rewrite the fractions to improper fractions if needed.
2. Multiply the numerators and the denominators together.
3. Reduce to lowest terms.
4. Examples.
a. Multiply 6 by 2/3
1) 6x2/3 = 6x2
3
= 12/3
= 4ans.
b. Find the product of 2/3, 1/5 and 3/6
1) 2/3x1/5x3/6 = 2 x 1 x 3
3x5x6
= 6/90
= 1/15 ans.
c. Find the product of 1 3/4 and 2 2/3
1) 1 3/4 = 7/4
2) 2 2/3 = 8/3
3) so 7/4 x 8/3 = 7 x 8
4x3
= 56/12 or 4 8/12 or 4 2/3
B. Reciprocal.
1. Also known as the multiplicative inverse, it is the inverse of a fraction.
a. Any number multiplied by its reciprocal equals 1.
26
REV: 3/06 jcl
C. Division of fractions.
1. Change any mixed numbers to improper fractions.
2. Take the reciprocal (invert/turn upside down) the second fraction (the one “divided
by”).
3. Proceed as in multiplication of fractions.
4. Reduce to lowest terms.
a) Examples:
1) Divide 1/2 by 2/3.
a) Multiply by 1/2 by the reciprocal of 2/3.
b) 1/2
2/3
c) l x 3
2x2
d) 3/4 ans.
2) Divide 2/3 by 5.
a) 2/3 = __2__
5
3x5
= 2/15 ans.
3) Divide 3 by 2/3.
a) _3_
2/3
b) 3 x 3/2
c) 3 x 3
2
d) 9/2 or 4 1/2 ans.
4) Divide 5 1/3 by 8.
27
REV: 3/06 jcl
a) 5 1/3 = 16/3
b) 16/3 x 1/8
c) 16 x 1
3x8
d) 16/24 or 2/3 ans.
5) Divide 15 by 1 2/3
a) 1 2/3 = 5/3
b) 15_ = 15 x 3
5/3
5
c) 45/5 or 9 ans.
28
REV: 3/06 jcl
LESSON TOPIC 1.5: DECIMALS
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform mathematical operations involving decimals.
ENABLING OBJECTIVES:
1.5.1 Define the following terms:
a. Decimal
b. Decimal point
c. Mixed decimal
d. Complex decimal
1.5.2 Describe the way to read and write decimals.
1.5.3 State the rules in converting fractions to decimals and vice versa.
1.5.4 Convert fractions to decimals.
1.5.5 Convert decimals to fractions.
1.5.6 Add and subtract decimals.
1.5.7 Multiply and divide decimals.
29
REV: 3/06 jcl
LESSON TOPIC 1.5: DECIMALS
A. DEFINITION OF TERMS.
1. Decimal and decimal point.
a. The places to the right of the ones place are decimal places.
b. The ones digit and the digit immediately to its right is separated by a dot, called a
decimal point.
c. The new numbers formed are called decimal numbers.
1) A decimal can be expressed in the form of a fraction and vice versa.
a) Example.
1) 1_ = 0.1
10
2) _2_ = 0.02
100
2. Mixed decimal - a whole number with a fraction in the form of a decimal.
a. Example.
1) 3.2
2) 160.32
3. Complex decimal — a decimal that contains a common fraction.
a. Example.
1) 0.3 1
3
2) 0.87 1
2
B. READING DECIMALS.
1. Read the digits to the left of the decimal point, if any, as you would read a whole
number.
30
REV: 3/06 jcl
2. Read the decimal point as “and.”
3. Read the digits to the right of the decimal point as though they represent a whole
number but state the place value of the digit on the extreme right.
4. Example.
a. 0.52 is read as fifty-two hundredths
b. 0.502 is read as five hundred two thousandths.
c. 327.058 is read, three hundred twenty- seven and fifty-eight thousandths.
d. 1.01 is read, on and one hundredth.
5. Another way of reading decimals is like:
a. Reading telephone numbers.
b. Starting at the left and naming the digits in order.
1) Example.
a) 3.1416 is read, three-point-one--four- one-six.
b) 204.713 is read, two-o-four-point-seven-one-three.
c. The whole number may be read as usual.
1) Example.
a) 2425.625 is read, two thousand four hundred twenty-five-point six-twofive.
C. Writing decimals.
1. Written as proper fractions with denominators beginning with 1 and ending with one or
more zeroes.
a. Example.
1) 3_
10
2) 54_
100
31
REV: 3/06 jcl
D. Converting fractions to decimals.
1. Divide the numerator by the denominator and write
the quotient in decimal form.
2. Examples:
a. 1 = 50_ = 0.50 or 0.5
2 100
b. 1 = 10_ = 0.25
4 100
E.
Converting decimals to fractions.
1. The numerator of the fraction is the decimal number without the decimal point.
2. The denominator of the fraction is a 1 followed by as many zeroes as there are decimal
places in the decimal number.
3. Simplify the fraction, if necessary.
4. Examples:
a. 0.625 = 5 (Simplified)
8
b. 23.42 = 1171 (Simplified)
50
F.
Addition of decimals.
1. Arrange the decimals numbers under each other with the decimal point lined up directly
below each other.
2. Add according to place value as you would with whole numbers.
3. Put the decimal point in the sum directly below the other decimal points.
4. Examples:
a. 2.48 + 19.2 + 21.837 = 43.517
b. 1.534 + 29.09 + 8.0287 + 0.081 = 38.7337
c. 32.1 + 4.027 + 73 + 31.2054 = 140.3324
32
REV: 3/06 jcl
G. Subtracting decimals.
1. Arrange the decimal numbers under each other with the decimal points lined up directly
below each other.
2. Subtract according to place value as you would with whole numbers.
3. Put the decimal point in the answer directly below the other decimal points.
4. Examples:
a. 29.36
-_7.342
22.018
b. 503.903
-_97.06_
406. 843
H. Multiplying decimal numbers.
1. Multiply the two decimal numbers as though they were whole numbers, disregarding
their decimal points.
2. The number of decimal places in the product is equal to the sum of the number of
decimal places in the two factors.
3. Examples:
a. 0.275 x 0.54 = 0.14580
b. 34.12 x 7.49 = 255.5588
c. 0.00321 x 16.4 = 0.052644
I.
Division of decimal numbers.
1. If the divisor is not a whole number, move the decimal point all the way to the right of
the number.
2. The divisor should now be a whole number.
3. Move the decimal point in the dividend the same number of places to the right.
4. Place the decimal point in the quotient directly above the decimal point in the dividend
(as determined in step 3 above).
33
REV: 3/06 jcl
5. Divide as you would with whole numbers.
6. Examples:
a. 336.56 ÷ 5.6 = 60.1
b. 36.8245 ÷ 2.35 = 15.67
c. 59.84 ÷ 0.044 = 1360
34
REV: 3/06 jcl
LESSON TOPIC 1.6: PERCENTS
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Convert percents to decimals, fractions, and fractions to decimals.
ENABLING OBJECTIVES:
1.6.1 Define the following terms:
a. Percent
b. Percentage
1.6.2 State the rules in converting percents to decimals, fractions and fractions to decimals.
1.6.3 Convert percents to decimals or fractions.
1.6.4 Convert decimals or fractions to percents.
35
REV: 3/06 jcl
LESSON TOPIC 1.6: PERCENTS
A. Definition of terms.
1. Percent - Is a fraction with a denominator of 100.
2. Percentage - Denotes that a whole quantity divided into a 100 equal parts is taken as the
standard of measure.
B. Converting percents to decimals.
1. Remove the % symbol.
2. Move the decimal point two places to the left.
3. Examples.
a. 37% = 0.37
b. 220% = 2.20
c. 4% = 0.04
C. Converting percents to fractions.
1. Delete the % symbol after the number.
2. Write the number over a denominator of 100.
3. Simplify the results if necessary.
4. Example.
a. 70%=7_
10
b. 275% = 11
4
c. 18.6% = 93_
500
D. Converting decimals to percents.
1. Move the decimal point two places to the right.
36
REV: 3/06 jcl
2. Place the % symbol after the resulting number.
3. Examples.
a. 0.37 = 37%
b. 0.08 = 8%
c. 6.79 = 679%
E. Converting fractions to percents.
1. If the number is a mixed number, convert it to an improper fraction.
2. Divide the numerator of the fraction by its denominator.
3. Multiply the quotient obtained by 100.
4. Add the % symbol after the result obtained in step c.
5. Examples.
a. 5 = 62.5%
8
b. 3 2 = 340%
5
c. 1 = 33 1
3
3
37
REV: 3/06 jcl
LESSON TOPIC 1.7: ADDITION AND SUBTRACTION OF SIGNED NUMBERS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform addition and subtraction of
signed numbers.
ENABLING OBJECTIVES:
1.7.1 Define absolute value.
1.7.2 State the rules in adding two signed numbers.
1.7.3 State the rules in adding three or more signed numbers.
1.7.4 Add signed numbers.
1.7.5 State the rules in subtracting signed numbers.
1.7.6 Subtract signed numbers.
38
REV: 3/06 jcl
LESSON TOPIC 1.7: ADDITION AND SUBTRACTION OF SIGNED NUMBERS
A. Definition of terms.
1. Absolute value - Let n be any integer. Then the absolute value of n, denoted by │n│ is
defined as follows:
a. │n│ is equal to n if n is positive or 0.
b. │n│ is equal to the opposite of n if n is negative.
B. Rules for adding signed numbers with the same sign.
1. Disregard the signs and add the numbers as you would add whole numbers.
2. For the sum, use the common sign of the numbers.
C. Rules for adding signed numbers with different signs.
1. Disregard the signs.
2. Subtract the smaller absolute value from the larger as would subtract whole numbers:
3. For the answer, take the sign of the number with the larger absolute value.
D. Example.
1. (+3) + (+9) = +12 or 12
2. (-4) + (-7) = -11
3. (+5) + (-8) = -3
4. (-17) + (+26) = +9 or 9
5. (+3) + (-4) + (-8) + (+6) + (-9) + (+4) + (-7) = -15
E. Rules for subtracting signed numbers.
1. To subtract one signed number from another signed number, replace the number to be
subtracted with its opposite and add.
F. Examples.
1. (-19) - (-7) = -12
39
REV: 3/06 jcl
2. (+13) - (+29) = -16
3. (+19) - (+7) = +12
40
REV: 3/06 jcl
LESSON TOPIC 1.8: MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform multiplication and division of signed numbers.
ENABLING OBJECTIVES
1.8.1 State the rules in multiplying and/or dividing two signed numbers.
1.8.2 State the rules in multiplying and/or dividing three or more signed numbers.
1.8.3 Multiply and divide signed numbers.
41
REV: 3/06 jcl
LESSON TOPIC 1.8: MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
A. Multiplication of two signed numbers.
1. Rule for multiplication of signed numbers involving zero (0).
a. If a is any signed number, then
1) a x 0 = 0
2) 0 x a = 0
2. Rules for multiplying two signed numbers.
a. Disregard the signs and multiply the numbers as you would multiply whole
numbers.
b. The product will be:
1) 0, if either number is 0.
2) Positive, if the two numbers have the
same sign.
3) Negative, if the two numbers have different signs.
3. Examples.
a. (+7) x 0 = 0
b. 0 x (-5) = 0
c. (+2) x (+3) = +6 or 6
d. (-13) x (+50) = -650
e. (-3) x (-6) = +18 or 18
B. Division of two signed numbers.
1. Rules for dividing signed numbers.
a. Disregard the signs and divide the numbers as you would divide whole numbers.
b. The quotient will be:
42
REV: 3/06 jcl
1) 0, if the numerator is 0 and the denominator is nonzero.
2) Positive, if the numerator and denominator have the same signs.
3) Negative, if the numerator and denominator have different signs.
2. Examples.
a. (+27) ÷ (+3) = +9 or 9
b. (-50) ÷ (-5) = +10 or 10
c. 0 ÷ (-57) = 0
d. (-54) ÷ (+3) = -18
C. Multiplying and dividing three or more signed numbers.
1. Rules for multiplying three or more signed numbers.
a. Disregard the signs and multiply as you would whole numbers.
b. The product will be:
1) 0, if at least one of the factors is zero.
2) Positive, if there is an even number of negative factors.
3) Negative, if there is an odd number of negative factors.
2. Rules for dividing three or more signed numbers:
a. Division and multiplication are inverse operations.
b. We can convert a division problem into an equivalent multiplication problem.
c. The rules for dividing signed numbers are basically the same thing as those for
multiplying signed n numbers.
3. Examples.
a. (-7) x (-8) x (+3) x (-9) x (+5) = -7560
b. 1 x (-21) = -l
(+3) X 7
43
REV: 3/06 jcl
LESSON TOPIC 1.9: GROUPING SYMBOLS ND ORDER OF OPERATIONS
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Evaluate mathematical problems using the order of operations
convention.
ENABLING OBJECTIVES:
1.9.1 List the mathematical grouping symbols.
1.9.2 State the order of operations convention.
1.9.3 Apply the order of operations convention in evaluating mathematical problems involving
grouping symbols.
44
REV: 3/06 jcl
LESSON TOPIC 1.9: GROUPING SYMBOLS AND ORDER OF OPERATIONS
A. Mathematical grouping symbols.
1. ( ) - Parentheses
2 [ ] - Brackets
3. { } - Braces
4. ──── Fraction Bar
B. Order of operations convention.
1. Rules for performing operations on whole numbers.
a. First, perform all operations within grouping symbols, if any.
b. Second, evaluate all expressions with exponents or roots.
c. Third, perform all multiplication and division in order from left to right.
d. Fourth, perform all additions and subtractions in order from left to right.
C. Application of order of operations.
1. Example.
a. (9 - 5) x (6 + 3) - 12 + 6 x 4
= (4) x (9) - 12 + 24
= 36 - 12 + 24
= 48
b. 3 x 5 - 8 ÷ 4 + 11
= 15 - 2 + 11
= 24
c. 4 x (5 + 6)
= 4 x 11
= 44
d. 8 x (5 + 3)
=8x8
= 64
45
REV: 3/06 jcl
LESSON TOPIC 1.10: ALGEBRAIC EXPRESSIONS
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Determine the degrees of algebraic expressions. –
ENABLING OBJECTIVES:
1.10.1 Define the following terms:
a. Algebraic expression
b. Variable
c. Constant
d. Term
e. Factors
f. Coefficient
g. Numerical coefficient
h. Monomial
i. Binomial
j. Trinomial
k. Multinomial (Polynomial)
1.10.2 Determine the degree of a monomial in one variable.
1.10.3 Determine the degree of a monomial in more than one variable.
1.10.4 Determine the degree of a polynomial.
1.10.5 Describe a polynomial in decreasing order.
46
REV: 3/06 jcl
LESSON TOPIC 1.10: ALGEBRAIC EXPRESSIONS
A. Definition of terms.
1. Algebraic expression
a. An expression that consists of constants and variables.
b. Operation symbols and grouping symbols may be included in the expression.
c. Variable - Letters used to represent numbers in algebra.
d. Constant - The numbers in an algebraic expression.
e. Term
1) An expression or part of an expression involving only the product of numbers
or letters.
2) It is also a part of a sum.
3) Terms are separated by + or - signs.
2. Factors
a. Numbers and/or letters that are multiplied together in a term.
b. Part of a product.
3. Coefficient - The product of the remaining factors in a term.
4. Numerical coefficient - The number factor.
5. Monomial - An algebraic expression with only one term.
6. Binomial - An algebraic expression containing exactly two terms.
7. Trinomial - An algebraic expression containing exactly three terms.
8. Multinomial (Polynomial) - An algebraic expression with two or more terms.
B. Degree of a monomial in one variable.
1. Rule.
a. The same as the exponent of the variable.
47
REV: 3/06 jcl
2. Examples:
a. 4x2 has degree of 2
b. 3y4 has degree of 4
c. -16x5 has degree of 5
d. 8 has degree of 0
C. Degree of a monomial in more than one variable.
1. Rule.
a. Equal to the sum of the exponents of its variables.
2. Examples:
a. 3x2y2 has degree of 4.
b. 12x2y3z has degree of 6
c. -2ab4 has degree of 5.
D. Degree of a polynomial.
1. Rule.
a. The same as the highest degree monomial contained in the polynomial.
2. Examples:
a. 3x2 - 4x + 7 has degree 2.
b. 2y6 - 7y4 + 4y3 - 8y + 10 has degree 6.
c. 5z8 - 9y7 - y5 + 4y3 - y has degree 8.
E. Polynomial in decreasing order.
1. Definition.
a. A polynomial is in decreasing order if each term is of some degree less than the
proceeding term.
48
REV: 3/06 jcl
2. Example.
a. 5x6 - 4x4 + 5x3 - 9x2 - 6x + 1 - is in decreasing order.
49
REV: 3/06 jcl
LESSON TOPIC 1.11: ADDITION AND SUBTRACTION OF ALGEBRAIC
EXPRESSIONS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Simplify indicated operations involving addition and subtraction of
algebraic expressions.
ENABLING OBJECTIVES:
1.11.1 State the rules in adding and subtracting algebraic expressions.
1.11.2 Define “like terms.”
1.11.3 List the rules in removing grouping symbols.
1.11.4 Add and subtract algebraic expressions and simplify.
50
REV: 3/06 jcl
LESSON TOPIC 1.11: ADDITION AND SUBTRACTION OF ALGEBRAIC
EXPRESSIONS
A. Rules in adding and subtracting algebraic expressions.
1. Combine like terms.
2. Add or subtract their numerical coefficients.
3. Multiply the sum by the common variable part.
B. Terms.
1. Like terms
a. Terms that contain the same variables with the same exponents.
b. All constant terms are like terms.
c. Terms that are not like terms are called unlike terms.
C. Rules in removing grouping symbols.
1. If the grouping symbol is preceded by a + sign,
a. Drop the parentheses and the + sign.
b. Leave the sign of each term within the parentheses as is.
2. If the parentheses are preceded by a - sign,
a. Drop the parentheses and the - sign.
b. Change the sign of each term within the parentheses.
3. If two or more grouping symbols are involved in the expression,
a. Remove the innermost symbol first and work outwards.
D. Adding and subtracting algebraic expressions.
1. Examples.
a. 12a + 5a = 17a (ans.)
b. b - 5b = -4b (ans.)
51
REV: 3/06 jcl
c. 6x5 - 9x5 = -3x5 (ans.)
d. 6x2y3 - 3x3y2 + 5x2y3 = 11x2y3 - 3x3y2 (ans.)
e. -2s - {2 + 3t + [4 - 2s - (3t - 2s) + 5] - 2) + 5s = 3s - 9 (ans.)
52
REV: 3/06 jcl
LESSON TOPIC 1.12: EXPONENTS AND RADICALS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform algebraic operations using the laws of exponents.
ENABLING OBJECTIVES:
1.12.1 Define terms relating to radicals.
1.12.2 List the laws of exponents.
1.12.3 Multiply and divide expressions using the laws of exponents.
1.12.4 Solves problems using radicals.
53
REV: 3/06 jcl
LESSON TOPIC 1.12: EXPONENTS AND RADICALS
A. Terms
1. Base — The factor that is expressed as a power.
2. Exponent
a. The number that indicates the number of times the based is to be used as a factor.
b. It expresses the power to which the quantity is to be raised or lowered.
3. Radical
a. An expression of the form √a
1) √ - Radical sign
2) n - Index
3) a - Radicand
b. The index of a square root is equal to 2.
B. Laws of exponents.
1. Product rule for exponents.
a. Let x be any number and let m and n be any positive integers, then
1) (xm) (xn) = xm+n
b. Note:
1) The product rule involves multiplying powers of the same base.
2. Power rule for exponents.
a. Let x be any number and let m and n be positive integers. Then,
1) (x m)n = x mn
3. Quotient rule for exponents.
a. Let x be any nonzero number and let m and n be positive integers. Then,
54
REV: 3/06 jcl
1) xm = xm-n , if m>n
xn
2) xm = __1__ , if m<n
xn xn-m
3) xm = 1 , if m=n
xn
4. Power of a product rule for exponents.
a. Let a and b be any numbers and let m, n, and p be positive integers. Then,
1) (ab)m = ambm
2) (ambn)p = ampbnp
5. Power of a quotient rule for exponents.
a. Let a be any number, b be any nonzero number, and let n be any positive integer.
Then,
1) a n = an
b
bn
6. Zero power rule for exponents.
a. Let x be any nonzero number. Then,
1) x0 = 1
7. Negative integer power rule for exponents.
a. Let a be any nonzero number and let n be a positive integer exponent. Then,
1) a-n = _1
an
C. Multiplication and division of exponential expressions.
1. Examples.
a. y6xy8 = y14
b. (y3)4 = y12
55
REV: 3/06 jcl
c. (3x2)4 = 81x8
d. 3a2 x 4a3 = 12a5
e. 12x9 = 3x6
4x3
f. -4x 2 = 16x2
3y2
9y4
g. (2a2b3c4)3 = 8a6b9c12
D. Radicals.
1. Examples.
___
a. √ 25 = 5
__
b. √ 4 = 2
56
REV: 3/06 jcl
LESSON TOPIC 1.13: MULTIPLICATION AND DIVISION OF ALGEBRAIC
EXPRESSIONS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Simplify indicated operations involving multiplication and division
of algebraic expressions.
ENABLING OBJECTIVES:
1.13.1 State the rule in multiplying a multinomial by a monomial.
1.13.2 Multiply a multinomial by a monomial.
1.13.3 State the rule in dividing a multinomial by a monomial.
1.13.4 Divide a multinomial by a monomial.
57
REV: 3/06 jcl
LESSON TOPIC 1.13: MULTIPLICATION AND DIVISION OF ALGEBRAIC
EXPRESSIONS
A. Multiplying a multinomial by a monomial.
1. Rule.
a. Multiply each term of the multinomial by the monomial.
2. Examples:
a. 5a(6b + 3c) = 30ab + l5ac
b. 4x(3x2 - 2x + 5) = 12x3 - 8x2 + 20x
c. 6a2b(-2a3b2 + (6a2b) (3a2b) + (6a2b) (-b) + (6a2b) (1) = -12a5b3 + 18a4b2 - 6a2b2 +
6a2b
B. Dividing a multinomial by a monomial.
1. Rule.
a. Divide each term in the multinomial by the monomial.
b. Simplify if needed.
2. Examples:
a. 15x2 - 6x = 5x - 2
3x
b. 6a3 - 10a2 + 4a = 3a - 5 + 2
2a2
a
58
REV: 3/06 jcl
LESSON TOPIC 1.14: LINEAR EQUATIONS
CONTACT HOURS: 3.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Evaluate equations using the “basic properties in solving equations.”
ENABLING OBJECTIVES:
1.14.1 List the basic properties used to solve equations.
1.14.2 Transform equations using addition and subtraction.
1.14.3 Transform equations using multiplication and division.
1.14.4 Solve a first-degree equation in one unknown or variable.
59
REV: 3/06 jcl
LESSON TOPIC 1.14: LINEAR EQUATIONS
A. Basic properties for solving equations.
1. Addition rule for solving linear equations.
a. If the same quantity is added to each side of an equation the resulting equation is
equivalent to the original equation. That is, if
1) x - a = b, then
(x - a) + a = b + a, or
x=b+a
2. Subtraction rule for solving linear equations.
a. If the same quantity is subtracted from each side of an equation, the resulting
equation is equivalent to the original equation. That is, if
1) x + a = b, then
(x + a) –a = b - a, or
x=b-a
3. Multiplication rule for solving linear equations.
a. If each side of an equation is multiplied by the same (nonzero) quantity, the
resulting equation is equivalent to the original equation. That is, if
1) x = b ( a not equal to zero), then
a
2) a x = (a)(b) or
a
3) x = ab
4. Division rule for solving linear equations.
a. If each side of an equation is multiplied by the same (nonzero) quantity, the
resulting equation is equivalent to the original equation. That is, if
1) ax = b (a not equal to zero), then
60
REV: 3/06 jcl
2) ax = b or
a a
3) x = b
a
B. Solving equations using the addition rule.
1. Example.
a. y – 6 = 2;
y=2+6
y = 8 (answer)
b. t – 9 = 5;
t=5+9
t = 14 (answer)
C. Solving equations using the subtraction rule.
1. Examples.
a. x + 7 = 3 ,
x=3-7
x = -4 (answer)
b. s + 3 = 8,
s=8-3
s = 5 (answer)
D. Solving equations using the multiplication rule.
1. Examples.
a. u = 7
5
u = 35 (answer)
b. x = 1
-3 4
x = -3 (answer)
4
E. Solving equations using the division rule.
61
REV: 3/06 jcl
1. Examples.
a. 3x = 12;
x = 12/3
x = 4 (answer)
b. -2p = 30;
p = 30/-2
p = -15 (ans.)
F. Solving first degree equations in one unknown or variable.
1. Procedure.
a. Eliminate any fractions.
1) Multiply each side of the equation by the lowest common denominator of all
fractions in the equation.
b. Remove any grouping symbols accordingly.
c. Combine like terms on each side of the equation.
d. Isolate all the unknown terms on one side of the equation and all other terms on the
other side of the equation.
e. Combine like terms where possible.
f. Divide each side by the coefficient of the unknown.
g. Check your solution by substituting it in the original equation.
2. Examples.
a. Solve for the unknown quantity.
1) x - 6 = 11;
x = 11 + 6
x = 17 (answer)
2) 3x = 15;
x = 5 (answer)
x = 15/3
3) 3x + 4 = 28;
3x = 28 – 4;
x = 8 (answer)
62
3x = 24;
x = 24/3
REV: 3/06 jcl
4) 1 x = 1 x - 2
3
4
x = -24 (answer)
5) 4(x + 1) = 6 - (3x - 12);
7x = 18 – 4;
4x + 4 = 6 – 3x + 12;
7x = 14;
x = 14/7;
63
4x + 3x = 6 + 12 – 4
x = 2 (answer)
REV: 3/06 jcl
LESSON TOPIC 1.15: SCIENTIFIC NOTATION AND POWERS OF 10
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform mathematical operations using the laws of exponents and
express each result in scientific notation.
ENABLING OBJECTIVES:
1.15.1 Define scientific notation.
1.15.2 List the laws of exponents involving Powers of 10.
1.15.3 Change a number from decimal form to scientific notation.
1.15.4 Change a number from scientific flotation to decimal form.
1.15.5 Multiply and divide numbers and write the results in scientific notation
64
REV: 3/06 jcl
LESSON TOPIC 1.15: SCIENTIPIC NOTATION ND POWERS OP 10
A. TERMS:
1. Scientific notation
a. A positive number N is said to be in scientific notation when it is written in the
form:
1) N = p x 10k
k
p
is an integer
is a positive number such that p is greater or equal to 1 and less than 10.
B. Laws of exponents involving powers of 10.
1. 10m x 10n = 10m+n
2. 10m = 10m-n
10n
3 (10m)n = 10mn
4. 10-n = _1_ and _1_ = 10n
10n
10-n
5. 100 = 1
C. Changing decimal numbers to scientific notation.
1. Procedure.
a. Move the decimal point to a position immediately after the first nonzero digit
reading from left to right.
b. If the decimal point is moved to the left,
1) The exponent of 10 is the number of places that the decimal point has been
moved.
c. If the decimal point is moved to the right,
1) The exponent of 10 is the negative of the number of places that the decimal
point has been moved.
d. If the original position of the decimal point is after the first nonzero digit,
65
REV: 3/06 jcl
1) The exponent of 10 is zero.
2. Examples.
a. Change to scientific notation.
1) 2,380 = 2.38 x 103
2) 52,600 = 5.26 x 104
3) 1.04 = 1.04 x 100
4) 681.4 = 6.814 x 102
5) 0.63 = 6.3 x 10-1
D. Changing scientific notations to decimal form.
1. Procedure.
a. Multiply the decimal part by the power of 10.
1) If the power is positive.
a) Move the decimal point to the right the same number of decimal places
as indicated by the power of 10.
2) If the power is negative.
a) Move the decimal point to the left the same number of decimal places
as indicated by the power of 10.
2. Examples.
a. Change to decimal form.
1) 3.45 x 102 = 345
2) 1.06 x 105 = 106,000
3) 2.77 x 10-2 = 0.0277
4) 8.15 x 10-5 = 0.0000815
5) 4.92 x 100 = 4.92
66
REV: 3/06 jcl
E. Multiplying and dividing numbers in scientific notation.
1. Examples.
a. (6.43 x 108) (5.16 x 1010) = 3.32 x 1019
b. (1.456 x 1012) (-4.69 x 10-18) = -6.83 x 10-6
c. (7.46 x 108) ÷ (8.92 x 1018) = 8.36 x 10-11
d. (-6.19 x 1012) ÷ (7.755 x 10-8) = -7.98 x 1019
67
REV: 3/06 jcl
LESSON TOPIC 1.16: FORMULAS
CONTACT HOURS: 2.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Solve problems using formulas where all but the unknown quantities
are given.
ENABLING OBJECTIVES:
1.16.1 Define the following terms:
a. Formula
b. “Solving a Formula”
1.16.2 List the steps in using a formula to solve problems.
1.16.3 Solve a formula for the given letter and substitute data to find the value of the given
letter.
68
REV: 3/06 jcl
LESSON TOPIC 1.16: FORMULAS
A. Terms:
1. Formula
a. Is a relationship between two or more quantities or amounts
b. States a rule or method for doing something.
2. “Solving a Formula”
a. Also known as “evaluating a formula.”
b. To isolate a given letter or variable on one side of the equation.
c. Expressing that letter or variable in terms of all the remaining letters by having all
the other letters appear on the opposite side of the equation.
B. Solving problems using a formula.
1. Steps:
a. Solve the formula for the unknown quantity.
b. Substitute each known quantity with its units.
c. Use the order of operations procedures to find the numerical quantity and to simplify
the units.
C. Solving formulas.
1. Examples:
a. Solve P = VI for V =?
V = P (answer)
I
b. Solve P = F for F =?
A
F = PA = AP (answer)
c. Solve P = F for A =?
A
A = F (answer)
P
69
REV: 3/06 jcl
d. Given the formula, A = bh, in which A = 150m2, and b = 25m. Find h =?
h = A/b
= 150/25
h = 6m (answer)
e. Given the formula, A = (l/2)bh, in which A = 144m2, and b = 8m. Find h =?
h = 2A/b
= 2(144)/8
h = 36m (answer)
70
REV: 3/06 jcl
LESSON TOPIC 1.17: RATIO AND PROPORTION
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Solve equations using the principle of proportions.
ENABLING OBJECTIVES:
1.17.1 Define the following terms:
a.
b.
c.
d.
Ratio
Proportion
Extremes
Means
1.17.2 State how to write or represent ratios and proportions.
1.17.3 State the principle of proportions.
1.17.4 Solve proportions using the formula involving means and extreme
71
REV: 3/06 jcl
LESSON TOPIC 1.17: RATIO AND PROPORTION
A. Terms:
1. Ratio
a. A comparison of numbers of the same kind (units), using the operation of division.
b. Given two numbers a and b of the same kind (units), the ratio a to b is the quotient
of the two numbers.
2. Proportion
a. Statement that two ratios are equal.
b. If a:b and c:d are two equal ratios, then the equation a = c is called a proportion.
b d
3. Extremes
a. In the proportion a = c, the quantities a and d are called the extremes.
b d
4. Means
a. In the proportion a = c, the quantities b and c are called the means.
b d
B. Writing ratios.
1. Ratios can be represented by:
a. Division sign – (/, ÷)
b. Colon - ( : )
c. The word “to”
2. Ratios should always be expressed in simplest form.
3. The ratio a to b is written as a or a:b or a to b.
C. Writing proportions.
1. The proportion a:b and c:d is read as “a is to b as c is to d” and can also be written as
a:b::c:d.
72
REV: 3/06 jcl
D. Principle of proportions.
1. Given the proportion a = c, the product of the extremes (a) (d), is equal to the
b d
product of the means, (b) (c).
2. That is , (a) (d) = (b) (c).
E. Problem solving involving means and extremes.
1. Examples:
a. Solve for the missing variable using the principle of proportions?
1) 3:5::y:25
y = 15 (answer)
2) x = _8
5 20
x = 2 (answer)
3) 9:u::3:21
u = 63 (answer)
b. Determine if the following ratios form a proportion:
1) 5 ? 35
8 56
280 = 280 (This ratio forms a proportion)
2) 5 ? _9
8 12
60 ≠ 72 (This ratio does not form a proportion)
3) 22 ? 16.5
8
6
132 = 132 (This ratio forms a proportion)
73
REV: 3/06 jcl
LESSON TOPIC 1.18: DIRECT VARIATION
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform mathematical operations involving variables that are
directly related.
ENABLING OBJECTIVES:
1.18.1 Define direct variation.
1.18.2 State the formula for direct variation.
1.18.3 List quantitative expressions in which two variables are directly related.
1.18.4 Solve equations and word problems using the direct variation formula.
74
REV: 3/06 jcl
LESSON TOPIC 1.18: DIRECT VARIATION
A. Definition.
1. Direct variation
a. Is indicated when two quantities are related, such that an increase in one causes an
increase in the other and vice versa.
B. Direct variation formula.
1. In general this is written in the form y = kx, where k is the proportionality constant or
constant of proportionality.
C. Direct variation expressions:
1. The area of a circle varies directly with the square of its radius.
2. The faster the speed, the greater the distance covered.
3. Radiographic density is directly proportional to mAs.
4. The amount of milliamperage and time needed to produce a given radiographic density
varies directly with the square of the distance.
D. Word problems using direct variation formula.
1. Examples:
a. y varies directly with x; y = 8 when x = 24. Find y when x = 36.
8/24 = y/36
y = 36(8)
24
y = 12 (answer)
b. y varies directly with the square root of x; y = 24 when x = 16. Find y when x = 36.
24/ √16 = y/ √36
y =6(24)
4
y = 36 (answer)
75
REV: 3/06 jcl
c. A distance of 300 meters is represented on a map by 2 cm. How many centimeters
on the map would represent a distance of 1800 meters?
2/300 = cm/1800
cm = 1800 (2)
300
12 centimeters (answer)
d. Radiographic density is directly proportional to mAs. This relationship is linear.
Since mAs is the product of mA and time (mA • T = mAs) an increase or decrease in
either factor results in a corresponding density change on the radiograph.
1) If 100 mA is used for 1/10 (0.1) second to produce an acceptable radiographic
density, How much time is required to produce the same density at 50 mA (all
other parameters being constant?
mAs1 = mAs1
100(1/10) = 50(time)
time = 100 (1/10)
50
time = 1/5 second (0.2) (answer)
e. The amount of milliamperage and time needed to produce a given radiographic
density varies directly with the square of the distance.
If 100 mA were used at 40-in. FFD (film focal distance) and it were necessary to
decrease the FFD to 36 in., what mA would be needed to maintain the same density?
Ma1/ (distance1)2 = mA2/ (distance2)2
mA2 = mA1 (distance2)2
(distance1)2
mA2 = 100(36)2
(40)2
mA2 = 81 mA (answer)
76
REV: 3/06 jcl
LESSON TOPIC 1.19: INVERSE VARIATION
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVE: Perform mathematical operations involving variables that are
inversely related.
ENABLING OBJECTIVES:
1.19.1 Define inverse variation.
1.19.2 State the formula for inverse variation.
1.19.3 List quantitative expressions in which two variables are inversely related.
1.19.4 Solve equations and word problems using the inverse variation formula.
77
REV: 3/06 jcl
LESSON TOPIC 1.19: INVERSE VARIATION
A. Definition.
1. Inverse variation
a. Is indicated when two quantities are related such that an increase in one causes a
decrease in the other and vice versa.
B. Inverse variation formula.
1. If two quantities, y and x, change and their product remains constant (yx = k), the
quantities vary inversely or y is inversely proportional to x.
2. In general this is written in the form yx = k, where k is the proportionality constant or
constant of proportionality.
C. Inverse variation expressions.
1. The greater the speed, the less time to travel.
2. Boyle’s law states that if the temperature of a gas is constant, its volume V, varies
inversely with its pressure p.
3. The area, A, of a circle varies directly with the square of its radius, r.
4. The change in beam intensity and/ or radiographic density varies inversely with the
square of the distance, and is expressed as: I1 = (D2)2
I2 = (D1)2
5. In a wire of given length, the electrical resistance, R, varies inversely with the square
of its diameter, D. this is expressed as : R = _k , where k is the proportionality
constant.
D2
D. Inverse variation word problems.
1. If a 12 gram mass is 60 cm from the fulcrum of a lever, how far from the fulcrum
should a 45 gram mass be to keep it at an equilibrium?
mass1/mass2 = dist.2/dist.1
dist.2 = mass1 (dist. 1)
mass2
dist.2 = 12 (60)
45
dist.2 = 16 cm (answer)
78
REV: 3/06 jcl
2. A 20 in. pulley running at 180 rounds per minute (rpm) drives an 8 in. pulley. Find
how fast the 8 in. pulley is turning?
pulley1:pulley2 :: rpm2:rpm1
rpm2 = rpm1(pulley1)
pulley2
rpm2 = 180(20)
8
rpm2 = 450 rpm (answer)
3. If the amount of radiation reaching a patient at 40 in. is 3 R, if the distance were
increased to 60 in., what would be the new patient dosage?
Exposure1/Exposure2 = (distance2)2/ (distance1) 2
Exposure2 = Exposure1 (distance1) 2
(distance2) 2
Exposure2 = 3 (40) 2
(60) 2
Exposure2 = 1.33 R
79
REV: 3/06 jcl
LESSON TOPIC 1.20: GRAPHING ORDERED PAIRS OF REAL NUMBERS
CONTACT
CONTACT HOURS: 1.0 Didactic 0.0 Lab/practical
TERMINAL OBJECTIVES: Plot points in the number line corresponding to each ordered pair of
numbers.
ENABLING OBJECTIVES:
1.20.1 Define the following terms:
a.
b.
c.
d.
e.
f.
g.
h.
i.
Graph
X-axis
Y-axis
Origin
Quadrants
Coordinates
Abscissa
Ordinate
Ordered pairs of numbers
1.20.2 Describe the Cartesian Co-ordinate System.
1.20.3 State the other name for the Cartesian Co-ordinate System.
1.20.4 List the steps in plotting a point corresponding to an ordered pair of numbers.
1.20.5 Plot ordered pair of numbers.
80
REV: 3/06 jcl
LESSON TOPIC 1.20: GRAPHING ORDERED PAIRS OF REAL NUMBERS
A. Terms:
1. Graph - A diagram that represents the variation of a variable in comparison with that of
one or more other variables.
2. X-axis - The horizontal number line of a Cartesian coordinate system.
3. Y-axis - The vertical number line of a Cartesian coordinate system.
4. Origin - The point of intersection or the zero point of X and Y axis.
5. Quadrants - The four planes or regions crested by the intersection of the X and Y axis.
6. Coordinates - The ordered pair of real numbers that corresponds to a point in the
graph.
7. Abscissa - The first number in the ordered pair, also known as the x-coordinate.
8. Ordinate - The second number in the ordered pair, also known as the y-coordinate.
9. Ordered pairs of numbers - The coordinate of a point in a graph.
B. The Cartesian Coordinate System.
1. Also known as the Rectangular Coordinate System.
2. A plane in which two number lines intersect at right angles.
3. The horizontal line is called the X-axis and the vertical line is called the Y-axis.
4. The point of intersection (0,0) is called the origin.
5. The plane is divided by the two axes into four planes or regions called quadrants.
6. In the plane there is a point which corresponds to each ordered pair of real numbers
(a, b) which corresponds to each point in the plane.
7. The points (a, b) are called coordinates of the point.
a. a is called the abscissa.
b. b is called the ordinate.
8. The graph of a number is the point on a number line paired with a number.
81
REV: 3/06 jcl
C. Steps in plotting ordered pair of numbers.
1. Procedure:
a. Count right or left from 0 (origin) the number of units along the x-axis indicated by
the first number of the ordered pair.
1) Right if positive.
2) Left if negative.
b. Count up or down the number of units indicated by the second number of the
ordered pair.
1) Up if positive.
2) Down if negative.
D. Plotting ordered pair of numbers.
1. Examples:
a. Plot the point corresponding to each pair in the number line.
1. A (3, 1)
2. B (2, -3)
3. C (-4, -2)
82
4. D (-3, 0)
5. E (-4, 2)