Component Two: The Language of Algebra
1. When building a home, contractors follow a
certain sequence, such as pouring the foundation,
framing the walls, putting on the roof, and so on.
Many things in nature also follow a certain
sequence. For example, all living animals are born
(or hatched), grow to adulthood, and then die.
Because algebra is a tool we use to explain the
world around us, it would make sense that it also
follows a certain sequence. In this component,
you will review and practice the “rules of algebra”
that describe the sequence we use to
communicate our algebraic thinking and solve
problems.
2. Sometimes a problem situation is described
by a complex expression or equation. The first
step that mathematicians (that’s you!) find
helpful is to simplify the expression or equation
so it is more manageable and less likely to cause
errors in computation. One “tool” that is
commonly used for this purpose are the rules
that govern the order of operations, often
referred to as PEMDAS. Students remember,
“Please excuse my dear Aunt Sally.”
3. Before we go any farther, it would be wise to
use our word wall to define several terms. An
algebraic expression is a group of numbers,
variables, operations and grouping symbols. A
numerical expression is likewise a collection of
numbers, operations and grouping symbols.
4. An equation is a statement that two
expressions are equal. An inequality states a
relationship between two expressions such as
telling you that one is greater than or less than
the other.
5. Let’s begin by using the order of operations to
simplify a numerical expression. The “tool” that
is commonly used to assist here is called
PEMDAS. It reminds us that the first step is to
simplify any expression that is grouped by
parenthesis, brackets, a fraction bar or a square
root or radical symbol. All of these are grouping
symbols.
6. As an example, simplify within the
parenthesis and know that 12  3 is 4 and then
add 1 for an answer of 5.
7. An incorrect response
 would be to add 3 + 1
and divide 12 by 4 for an incorrect answer of 3.
The first step in using the order of operations to
simplify an expression is to simplify within the
parenthesis (or other grouping symbols). The
letter P reminds us of this first step.
8. The E in PEMDAS reminds us that the next
step is to do the multiplication necessary to
remove all exponents. Since, for example, 53
means that you are to use 5 as a factor 3 times,
or 5 * 5 * 5, in this second step, you would write
125 in place of 53.
9. As an example, use the first two steps of
PEMDAS to simplify 42 – (8 * 2). First, simplify
within the parenthesis. Rewrite the expression
as 42 – 16. Next, multiply to remove the
exponent and rewrite the expression as 16 – 16
to know that the expression simplifies to 0.
10. The M and D in PEMDAS reminds us that
the next step in simplifying an expression is to
multiply and divide from left to right. This is a
common place for student errors, as many do
not understand that they are to begin at the left
of an expression and complete multiplication
and division as one step moving from left to
right.
11. As an example, if asked to simplify the
expression 16  8 * 2 – 7, you should use
multiplication/division as one step as you work
from left to right. First, figure that 16  8 is 2,

so rewrite
the expression as 2 * 2 – 7. Next,
figure that 2 * 2 is 4, so rewrite the expression

as 4 – 7 = -3.
12. A common incorrect response would be to
multiply 8 and 2 before dividing and getting the
incorrect answer of -6.
12B. Remember that there are several ways to
show multiplication in algebra such as 3y or
(3)(y) or 3 * y. There are also several ways to
20
show division including 20  y, 20/y or .
y


13. The A and D in PEMDAS reminds
us that the
final step is to add and subtract from left to
right. As with the previous multiply/divide
step, this is a single step and must be done as
the expression is simplified from left to right.
14. As an example, if asked to simplify the
expression 12 – 4 * 2 + 2, you should first
realize that there are no parenthesis (grouping
symbols) nor exponents, but that the
multiplication of 4 * 2 must be done before
adding or subtracting. Rewrite the expression
as 12 – 8 + 2, then as 4 + 2 = 6.

15. Now, let’s use PEMDAS to simplify a
numerical expression. Consider the expression:
60  (32 + 1) – 2 * 2. Move the steps into order
to show how you would use PEMDAS to simplify
the expression.
16. Let’s check your thinking. First, work within
the parenthesis, eliminating the exponent
before adding.
17. Next, work within the parenthesis to add 9
+ 1.
18. Now, remember that multiplication and
division are done next, from left to right.
19. Finally, subtract 4 from 6 to get 2.
20. Good job! Now that we remember how to
use PEMDAS to simplify a numerical expression,
let’s add one more idea so that we can use
PEMDAS to find the value or evaluate an
algebraic expression.
21. When you are asked to evaluate an
algebraic expression, you will be given an
expression that contains variables and you will
be given a value to use for the variable or
variables. To evaluate, substitute the number
value for the variable and use PEMDAS to
determine the value of the expression.
22. An example would be to evaluate the
expression 24  (x + 2) for x = 1.
23. Substitute the value 1 in place of the
variable
 x and use PEMDAS to find the value of
the expression. First, work within the
parenthesis and rewrite the expression as
24  (1 + 2) , and then as 24  3 and finally as 8.

24. Now, move the steps into place to evaluate
50
the expression x (3 +2)2 –
for x = 2.
x
25. Let’s check your thinking. First, replace the
variables (x) with the
number 2.

26. Next, work within the parenthesis to
determine that 3 + 2 is 5.
27. Square 5 to get 25.
28. Multiply and divide from left to right.
29. Finally subtract to get the value 25.
30. Now, try one on your own. What is the
value of ½ (2a – 3b)2 when a = 9 and b = 4?
31. Did you find the value to be 18? How does
your thinking compare to PEMDAS?
32. Now that you remember how to simplify
and evaluate numerical and algebraic
expressions, we are ready to use these skills to
solve equations.
33. If the order of operations (remember that
we called this PEMDAS) is the first tool we have
use to simplify expressions, then the
distributive property is the second.
34. Before using this property, study this
example to learn why the distributive property
is true. Imagine that you want to find the area
of a rectangle with length (x + 2) and width 5.
There are two ways to solve this problem.
35. The first is to find the area of one rectangle.
The second is to find the sum of the areas of the
two smaller rectangles.
36. Since the area is obviously the same
regardless of which method you use, you realize
that 5(x + 2) = 5(x) + 5(2) or 5(x + 2)= 5x + 10.
37. Let’s use our word wall to study what is
meant by the “distributive property”.
38. A third “tool” you will use to solve
equations is combining like terms. The word
wall will help you remember the meaning of like
terms.
39. Now you are ready to use these “tools” to
solve an algebraic equation. Copy the equation
onto your paper and solve it for x. x=2(y+2)+2y.
40. Did you figure that x = 4y + 4? Compare
your work to the following. Use the distributive
property to write that x = 2y + 4 + 2y. Then,
combine like terms to simplify the answer.
41. In the example above, how would you solve
for x if y = 3?
42. Did you figure that x = 16? To check your
work, substitute the value 3 for the variable y
and 16 for the variable x. Use your algebra
skills to decide if your answer is correct.
43. Sometimes, the variable you are solving for
is isolated on one or the other side of the = sign.
44. More often, it is not. Two guidelines will
help you when you meet an equation such as:
5x + 3(x + 4) = 28. First, simplify both sides of
the equation (if needed) then, use inverse
operations to isolate the variable.
45. The word wall will remind us about inverse
operations. Subtraction and addition are
inverse operations. Multiplication and division
are also inverse operations.
46. Use what you have learned so far to solve
the equation for x. Copy the equation onto your
paper and solve. Check your thinking with the
following.
47. First, use the distributive property and
combine like terms to simplify the left side of
the equation.
48. Subtract 12 from both sides to isolate the
variable term. Finally, divide both sides by the
coefficient of the variable to isolate the variable
and solve the equation.
49. Check the word wall to remember the
meaning of “coefficient”.
50. Use what you have learned to solve for x.
51. Did you see that you were asked to multiply
(x-1) by -5? Does your thinking agree with the
following?
52. Check your work by substituting your
answer (your value for x) into the original
equation.
53. Use what you have learned to solve for m. .
This time, you will have to collect variable terms
on one side and constant terms (numbers with
no variables) on the other.
54. Check your answer by substituting into the
original equation. Does your thinking agree
with the following?
55. When you can write problems in your real
life as algebraic equations, you now remember
the skills needed to solve them. First, write the
following problem as an equation. It may help to
draw a picture or a graphic with words as you
think.
56. Cinderella has $50 in her savings account
and works to add an additional $5 each week.
Her ugly step-sister inherited $170 but spends
all she earns plus an additional 10$ every week.
How long will it be before Cinderella will have
more money than her ugly step-sister?
57. Now, write an equation to represent the
problem.
58. Did you write 50 + 5w= 170 – 10w? We
omitted the $ to keep our work simple. Now,
use your skills to solve the equation.
59. Did you learn that at the end of week 8, both
girls would have the same amount of money?
What does the question ask?
60. Good reading! At the end of week 9,
Cinderella will be the richer of the two!
61. Congratulations! You now have reviewed
the basics of solving equations to find the value
of variables. Although paper and pencil
methods will always serve you well, it would be
wise for us to spend some time using the
multiple representations available through
graphing calculators to solve problems we face
in our algebraic world. This will be our focus in
Component Three.