Algebra I. Lesson 1
Definitions and Properties (Grades 9-12)

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Algebra

Instruction 1-1

Variables and Expressions | Open Sentences | Distributive and Additive Identity, Multiplicative Identity Property and Properties of Equality | Commutative and Associative Property | Summary
VARIABLES AND EXPRESSIONS
http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut4_vari.htm 
CA GR7 AF 1.3 CA HS Algebra 1.1,  25.2
 
There are many terms, definitions, and other topics in algebra. We will begin with the two most basic terms. They are “variable” and “algebraic expressions.” These terms are the building blocks upon which all other vocabulary and concepts rest. There are very few areas or statements in algebra in which these terms are not used.
Everything that can be measured has a quantity. Sometimes the quantity is unknown. For example, we know that the number of hours in a day is 24. Such a number is called a constant. There are other quantities which do not have known measures. Not only are their measures unknown, but they are also changing depending on the situation.

For example, the profit of a store on a daily basis not only is unknown, but can also change day by day. Such quantities are called variables. Therefore, a variable is used to denote a quantity whose measure is changing depending on the situation. To find the value of a variable in a certain situation, we must have more data. Let’s look at some examples.
 

 

How we denote variables?
 

We usually use letters to denote the variables. The end letters such as x, y, and z are the most common letters used to denote the variables. However, this is not a rule. We can use any letter including the beginning letters such as a, b, c, and so on for variables.
 
 
When and how to use variables?
 

As previously stated, we use variables to denote anything that is unknown or changing. To do this, we choose a letter, and represent the unknown or changing value by that letter. Below are some examples.


 
In all these examples, the values of the variables are not known. By using algebraic techniques coupled with necessary information ( such as the total number of college students and the percentage of female college students), we can determine these values.
 
Any combination of variables and numbers using different operations is called an algebraic expression or simply an expression. Let's look at some examples.

 


All the expressions above are formed by some variables, numbers, and algebraic operations.
 
Expressions are the basic tools in translating the wording of any problem into algebraic language. In the table below, some examples of algebraic expressions are given. Each expression is a translation of a verbal expression.
 
Definitions:
  1. sum – the result of an addition problem
  2. difference – the result of a subtraction problem
  3. product - the result of a multiplication problem
  4. quotient - the result of a division problem.
  5. ways to indicate addition: and. sum, more than, added to, greater than, increased by
  6. ways to indicate subtraction: less than, minus, taken from, taken away from, decreased by, difference
  7. ways to indicate multiplication: times, multiplied by, product
  8. ways to indicate division: divided by, into (as 5 goes into 15 three times), splits into y equal parts, quotient

VERBAL EXPRESSION
ALGEBRAIC EXPRESSION
Sum of a number and 12
               x + 12
Difference between x and 24
               x – 24
Twice a number added to three times another number.
               2m + 3n
26 subtracted from three times a number.
                3p – 26
Square of a number added to the cube of another number.
                a2 + b3
Add 24 to one-half a number.
               algl1_1_files/i0020001.jpga + 24
Multiply twice of a number by its square.
                (2x)(x2)

Exercise 1. Write each verbal expression in an algebraic form.
                    a. Three times x more than three times 33
 
                    b. 45 less than y
 
                    c. The ratio of m to twice n
 
                    d. The quotient of S and 33
 
                    e. M splits into 12 equal parts
 
                    f. The product of x and twice m
 
                   g. The sum of m and 3n
 
                   h. M decreased by 12
 
                   i. N increased by 34
 
Answer.
a. 3x + 3(33)
b. y – 45
c. algl1_1_files/i0020002.jpg
d. algl1_1_files/i0020003.jpg
e. algl1_1_files/i0020004.jpg
f. x(2m)
g. m + 3n
h. M – 12
i. N + 34

Exercise 2. Write the following expressions in verbal forms.
                   a. 3x – 45
 
                   b. 4y + 23
 
                   c. 5x(m + n)
 
                   d. algl1_1_files/i0020005.jpg
 
                   e. algl1_1_files/i0020006.jpg

                   f. algl1_1_files/i0020007.jpg

 
Answer.
a. Three times x decreased by 45
b. Four times y increased by 23
c. Five times x multiplied by the sum of m and n
d. 11 divided by 2m
e. Sum of m and n divided by 2
f. One half the difference of x and y
 
Exercise 3. In the figure below, measures of some of the parts of each shape are given.

algl1_1_files/i0020008.jpg

What formulas representing the shapes are defined by the following expressions?
a.      (m – n)(p – m)
b.      2m + 2p + n
c.      algl1_1_files/i0020009.jpg(m + p)(m – n)

Answer.
a. Area of the rectangle
b. Perimeter of the triangle
c. Area of the rhombus


Practice 1. In the table below, each row should contain verbal and algebraic forms of the same expression. Fill all the blank cells.

VERBAL FORM
ALGEBRAIC FORM
Three times x divided by h
 
 
3(m + n + p)
Four times m less than three times x
 
49 more than one half 3x
 
 
algl1_1_files/i0020010.jpg
 
algl1_1_files/i0020011.jpg + 28
 

Answer
   

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