MATH 50
Lesson Plan #1
"Introduction to Real numbers and Algebraic Expressions"
By Professor Jacob Batarseh
Lesson plan #1 objectives are listed on the following pages in your textbook:
Objectives for section 1.1 "Introduction to Algebra" are listed on page 56.
Objectives for section 1.2 "The Real Numbers" are listed on page 63.
Objectives for section 1.3 "Addition of Real Numbers" are listed on page 75.
Objectives for section 1.4 "Subtraction of Real Numbers" are listed on page 83.
1.1: Introduction to Algebra:
Part a:
In the real world there are problems that can be modeled mathematically; that is, a problem can be written as a mathematical expression to be solved. In this course you will be studying the basic concepts of algebra and how to use those concepts to solve real world problems.
Example:
Find the value of the following expression:
It takes Erin 24 minutes less time to commute to work than it does George. Suppose that the variable X stands for the time it takes George to get to work. Then X-24 stands for the time it takes Erin to get to work. How long does it take Erin to get to work if it takes George:
Solution:
X is the time it takes George to get to work.
X-24 is the time it takes Erin to get to work.
Example:
Find the value of the following expression:
Find the Area of the parallelogram whose height is 15.4 cm and its base is 6.5 cm. The area of a parallelogram is the product of its base by its height.
Solution:
The area of the parallelogram (A) is the base (b) times the height (h):
A = bh or written as A=bxh or written as A=bทh
Since b=6.5cm and h is 15.4cm; therefore A= bxh = 6.5x15.4 = 100.1cm2
Example:
Find the value of the following expression:
Find the simple interest (I) incurred on a base principle (P) of $4800 at 9% interest rate (r) for a period (t) of two years. I = Prt.
Solution
I = Prt or written as I=Pืrืt
P = $4800
r = 9% = 0.09
t = 2 years.
I = Pxrxt = 4800 x 0.09 x 2 = 864.
Therefore, the simple interest on $4800 over 2 years with an interest rate of 9% is $864.
Example:
Find the value of the following expression:
Find the area of a triangle with a base of 5 cm and a height of 6 cm.
Solution:
b = 5 cm.
h = 6 cm.
The shape of the shark's tooth is like a triangle; therefore, the surface area is:
Area = (bื h)/2 = (5ื6)/2 = 15 cm2
When evaluating an expression, you simply substitute the value of every unknown variable in the expression with its corresponding value.
Example:
Evaluate the expression: 6Y when Y=7
Solution:
6y when y=7
6y = 6(7) = 42.
Example:
Evaluate the expression: P/Q when P=16 and Q=2
Solution:
p ๗ q , when p=16 and q=2
p ๗q = p/q = 16/8 = 2
Example
:Evaluate the expression: (5Y)/Z when Y=15 and Z=25
Solution:
(5y)/z, when y=15 and z=25
(5ืy)/z = (5ื15)/25 = 75/25 = 3
Example:
Evaluate the expression: (p+q)/2 when p=2 and q=16
Solution:
(p+q)/2 when p=2 and q=16
(p+q)/2 = (2 + 16)/2 = 18/2 = 9
Example:
Evaluate the expression: (m-n)/5 when m=16 and n=6
Solution:
(m-n)/5 when m=16 and n=6
(m-n)/5 = (16-6)/5 = 10/5 = 2
Part b:
Translating a phrase to an algebraic expression means that you should look for the key word or statement that will be translated into a mathematical operator
(+, -, ๗, ื).The mathematical operations and their respective operations:
|
Mathematical Operation |
Mathematical Operator |
|
Addition |
+ |
|
Subtraction |
- |
|
Division |
๗ or / |
|
Multiplication |
ื or x or * |
Example:
Translate the phrase "Nine more than t" to an algebraic expression.
Solution:
The key word is more than; more than means +; therefore the algebraic expression is: t+9
Example:
Translate the phrase "Thirteen increased by z" to an algebraic expression.
Solution:
The key word is increased by; it means + ; therefore the algebraic expression is: 13+z
Example:
Translate the phrase "c more than d" to an algebraic expression.
Solution:
The key word is more than; more than means + ; therefore the algebraic expression is: c+d
Example:
Translate the phrase "c divided by h" to an algebraic expression.
Solution:
The key word is divided by; it means / or ๗
; therefore the algebraic expression is: c/h or c๗hExample:
Translate the phrase "s added to t" to an algebraic expression.
Solution:
The key word is added to; it means +; therefore the algebraic expression is: s + t
Example:
Translate the phrase "p subtracted from q" to an algebraic expression.
Solution:
The key word is subtracted from; it means - ; therefore the algebraic expression is: q - p
Note that it is q-p not p-q (q-p is p subtracted from q).
Example:
Translate the phrase "The sum of a and b" to an algebraic expression.
Solution:
The key word is sum of; it means +; therefore the algebraic expression is: a + b
Example:
Translate the phrase "Thee times q" to an algebraic expression.
Solution:
The key word is times; it means x; therefore the algebraic expression is: 3
ื qExample:
Translate the phrase "The product of 8 and t" to an algebraic expression.
Solution:
The key word is product of; it means x; therefore the algebraic expression is: 8 ื t
Example:
Translate the phrase "67% of women attending" to an algebraic expression.
Solution:
Let W be the number of women attending.
The key word is % of the; it means x; therefore the algebraic expression is: 0.67 ื W
Example:
Translate the following problem to an algebraic expression: "Juan has d dollars before spending $19.95 on a DVD of the movie "Castaway". How much did Juan have after the purchase?"
Solution:
Let J be the money Juan has after the purchase.
The key word is spending or after the purchase; it means - ; therefore the algebraic expression is: J = d - 19.99
1.2: The Real Numbers:
Part a:
Natural Numbers:
The Natural numbers are the set of numbers use for counting.
Natural Numbers = {1, 2, 3, 4, 5, 6, 7, 8, 9 , 10, 11, 12, 13, }
Whole Numbers:
The whole numbers set are the set of natural numbers with the 0 included.
Whole Numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , 10, 11, 12, 13, }
They, natural numbers, can be represented on a line as shown on page 63 of your textbook.
Integers:
Integers are whole numbers including negative whole numbers.
Integers = { , -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , 10, 11, }
They, Integers, can be represented on a line as shown on page 64 of your textbook.
Example:
State the integers that correspond to the situation: "Tiger Woods score in winning the 2000 P.G.A. championship was 18 under par".
Solution:
The integers are:
Example:
State the integers that correspond to the situation: "A student deposited her tax refund of $750 in a savings account. Two weeks later, she withdrew $125 to pay sorority fees."
Solution:
The integers are:
Example:
State the integers that correspond to the situation: "Recently the world birth rate was 270 per 10000. The death rate was 97 per 10000"
Solution:
The integers are:
Example:
State the integers that correspond to the situation: "During a video game, Maggie intercepted a missile worth 20 points, lost a starship worth 150 points, and captured a landing base worth 300 points"
Solution:
The integers are:
Part b:
Rational Numbers:
A rational number is a number represented in the following format: a/b where a and b is any two integers. b can't be zero (b 0). a is referred to as the numerator and b is referred to as the denominator. A natural number is a number that exists on the number line.
The following are examples of natural numbers: 2/3, 4/5, 9/13, -5/9, -7/9, 6/11, and 3/8
Note: Any whole number can be represented as a natural number whose denominator is 1. For example: 6 = 6/1, 100 =100/1, -13 = -13/1, -783 = -783/1
Part c:
Natural Numbers:
A natural number can be converted into a decimal number by dividing the numerator by the denominator. Please look at examples 5, 6, and 7 on page 66.
Example:
Convert each of the following rational numbers to decimal notation:
Solutions:
Part d:
X is said to be greater than Y if X exists to the right of Y on the number line. That is, the magnitude of X is bigger than the magnitude Y. In other words, X is bigger/larger than Y. It is written as X > Y
If X is larger than Y (X > Y), then Y is less than X (Y < X).
If Z is less than W (Z < W), then W is larger than Z (W > Z).
Inequalities:
x is larger than y is written as x > y
x is less than y is written as x < y
x is less than or equal to y is written as x ≤
yx is larger than or equal to y is written as x ≥ y
x is not equal to y is written as x ≠ y
x is approximately equal to y is written as x ≈ y
x is indeed equal to y is written as x ≡ y
Example:
Use one of the inequalities < or > to relate two numbers:
Solutions:
| a. 3 is less than 8 | ? | 3 < 8 |
| b. -3 is larger than -8 | ? | -3 > -8 |
| c. -3 is less than 8 | ? | -3 < 8 |
| d. 3 is larger than -8 | ? | 3 > -8 |
| e. 4.55 is larger than 3.87 | ? | 4.55 > 3.87 |
| f. 9.56 is larger than 9.32 | ? | 9.56 > 9.32 |
| g. 3.87 is less than 3.91 | ? | 3.87 < 3.91 |
| h. -2.45 is larger than -5.99 | ? | -2.45 > -5.99 |
| i. -2.45 is less than -2.34 | ? | -2.45 < -2.34 |
Example:
Indicate whether the inequality that exits between the two numbers is TRUE or FALSE.
Solutions:
| a. 3 ≤ 8 | ? | True |
| b. -3 ≈ 7 | ? | False (-3 is not approximately equal to 7) |
| c. 7 ≠ 9 | ? | True |
| d. 7 ≠ -9 | ? | True |
| e. 7 ≠ 7 | ? | False (7 is equal to 7) |
| f. 5 ≤ 5 | ? | True |
| g. 5 ≤ -5 | ? | False (5 is larger than -5) |
| h. -5 ≤ 5 | ? | True |
| i. 4 ≥ 9 | ? | False (4 is less than 9) |
| j. 4 ≥ -9 | ? | True |
| k. -4 ≥ -9 | ? | True |
Part e:
The absolute value of a number is the positive value of that number. The absolute value of X is written mathematically as:
|X| = X
|-X| = X
Example:
Find the absolute value of the given numbers:
Solutions:
1.3 + 1.4 Addition/Subtraction of Real Numbers:
To add two or more numbers, you use the + operator. Please look at pages 75 and 76 of your textbook to view how you would add two numbers using the number line. When adding X to Y, it is written algebraically as: X + Y
Rules of adding/subtraction of Real Numbers:
Notes: Before you go over the rules, please look at the following notes:
Rule 1: When you add two positive numbers, the answer is a bigger positive number equal to the sum of the two numbers.
A + B = + (A + B)
Example:
Find the sum of the following numbers in each part:
Solutions:
Rule 2: When you add two negative numbers, the answer is a bigger negative number equal to the sum of the two numbers.
-A + (-B) = -A - B = - (A + B)
Example:
Find the result of the following mathematical operations in each part:
Solutions:
Rule 3: When you add two numbers where one number is positive and the other one is negative, the answer is negative if the larger of the two numbers is a negative number; however, the answer is positive if the larger of the two numbers is positive.
A + (-B) = A B = the answer is positive if A is larger than B and it equal to + (A-B)
A + (-B) = A B = the answer is negative if B is larger than A and its equal to (B-A)
-A + B = B - A
A A = 0
Example:
Find the result of the following mathematical operations in each part:
Solutions:
"Textbook Reading Assignments/Coverage"
Lesson plan #1 will over the following topics from the textbook:
1.1: Introduction to Algebra.
1.2: The Real Numbers.
1.3 Addition of Real Numbers.
1.4: Subtraction of Real Numbers.
Please read the above mentioned sections (1.1 through 1.4).
After you have read the assigned sections from your textbook and the "Lesson Plan Summary" above, please do as many as you can from the following assigned problems from your textbook. Don't email me the solutions. The assigned problems are to prepare you for the quizzes and exams.
From Section 1.1: Do all odd problems: #1 through #57 on Pages 60+61+62.
From Section 1.2: Do all odd problems: #1 through #79 on Pages 72+73+74.
From Section 1.3: Do all odd problems: #1 through #77 on Pages 80+81+82.
From Section 1.4: Do all odd problems: #1 through #101 on Pages 86+87+88+89.
Note:
You can check your answers at the end of the book Pages A2 and A3. I do recommend that you do a problem first and then you go ahead and check the solution to that problem.
After you have read the assigned sections from your textbook, the "Lesson Plan Summary" below, and the suggested homework assignment please answer the following 20 quiz questions by the due date by clicking on the quiz button on the bottom of this page. The quiz assignment is taken from sections 1.1, 1.2, 1.3, and 1.4.
To submit the solution to the quizzes click on the quiz button on the lesson page, fill out your name, email address and student ID number, and then submit the solutions by clicking on the letter choice for each and every problem. When you are done, click on "Grade Quiz" button. The next page gives you the option to have your quiz results sent to you via email. You must enter your correct email address and you will then receive your grade via email almost instantly. If you did not receive your grade, let me know and I will let you know your grade.
Please use this quiz submission procedure format for all the upcoming quizzes.
Please submit the solution by the due date.
Quiz #1 (20 questions):
Click the quiz button below to take your quiz. You may also click on the quiz link on the main page of the course and choose Math 50 Quiz 1.
Post a comment/answer, on the discussion board, regarding one of the following questions:
If the length of a rectangle is doubled, does the area double? Why or why not?
If the height and the base of a triangle are doubled, what happens to the area? Explain.
You must post a comment/answer to any of the two questions to receive credit for the discussion portion of this lesson plan. Please use the discussion guidelines explained in the syllabus to receive full credit.
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