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Component 2The Language of Algebra

Component 2: The Language of Algebra12Language of Algebra

Rules of AlgebraStage 1Stage 2Stage 3When 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.22Language of AlgebraPEADMSExponents

Parenthesis (and other grouping symbols)

Multiply and divide from left to right

Add and subtract from left to rightally

earuntyxcuseleaseSometimes a problem is described by a complex expression or equation. The first step that mathematicians (thats 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 this acronym by using this phrase, Please Excuse My Dear Aunt Sally.3

Before we go any further, it would be wise to use our word wall to define several terms.

Select a term to review its definition. When you are finished, click to return to the word wall and view another word.4PEADMSxcuse Exponents

lease Parenthesis (and other grouping symbols)

y

ear unt

ally Order of OperationsTool

Please Parenthesis (and other grouping symbols)

( )[ ]/{ }(123) +1

4+1

5Grouping Symbols2Language of AlgebraAdd and subtract from left to rightMultiply and divide from left to rightLets 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, braces, a fraction bar, or a square root or radical symbol. All of these are grouping symbols.

As an example, simplify within the parenthesis and know that 12 divided by 3 is 4 and then add 1 for an answer of 5.5PEADMSxcuse Exponents

lease Parenthesis (and other grouping symbols)

y

ear unt ally

Order of OperationsToolExcuse Exponents

53 means you are to use 5 as a factor 3 times

555

12542 (82)

42 16

16 16

02Language of AlgebraMultiply and divide from left to rightAdd and subtract from left to right

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 555, in this second step, you would write 125 in place of 53.

As an example, use the first two steps of PEMDAS to simplify 42 (82).

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 0.6PEADMSxcuse Exponents

lease Parenthesis (and other grouping symbols)

year

unt ally

Order of OperationsToolM

16 divided by 8 2 7

16 divided by 8 is 2

2 2 72 2 is 4

4 7 = -3

DearCaution!2Language of AlgebraMultiply and divide from left to rightyMultiply and divide from left to rightAdd and subtract from left to right

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 in one step as they occur from left to right.

As an example, if asked to simplify the expression 16 divided by 8 2 7, you should use multiplication/division as one step as you work from left to right.

First, figure that 16 divided by 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 = -3.7

16 8 2 7

16 16 7

1 7

-6

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 show division including 20 y, 20/y, or 20/y.2Language of AlgebraA common incorrect response would be to multiply 8 and 2 before dividing and getting the incorrect answer of -6.Remember, there are several ways to show multiplication and division in algebra.

8PEADMSxcuse Exponents

lease Parenthesis (and other grouping symbols)

y

ear unt

allyOrder of OperationsTool

Aunt

SAdd and subtract from left to right

12 4 2 + 2

4 2

12 8 + 2

4 + 2 = 6

2Language of AlgebraMultiply and divide from left to rightallyAdd and subtract from left to rightThe A and S 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.

As an example, if asked to simplify the expression 12 4 2 + 2, you should first realize that there are no parenthesis (grouping symbols) or 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 = 6.9

PEADMSxcuse Exponentslease Parenthesis (and other grouping symbols)yearunt ally

60 (3 + 1) 2 229+1 = 106010 = 6

2 2 = 4

6-4 = 2

2Language of AlgebraMultiply and divide from left to rightAdd and subtract from left to rightNow, lets use PEMDAS to simplify a numerical expression. Consider the expression:

60 (32 + 1) 2 2. Lets move the steps into order to show how you would use PEMDAS to simply the expression.

First, work within the parenthesis to add 9 + 1.

Now, remember that multiplication and division are done next, from left to right.

Finally, subtract 4 from 6 to get 2.

10

Use PEMDAS to find the value or evaluate an algebraic expression.You will be given an expression that contains variables and a value to use for the variable.Substitute the number value for the variable.24 (x + 2)

for x = 1

24 (1 + 2)

2Language of AlgebraPEADMSxcuse Exponentslease Parenthesis (and other grouping symbols)yearunt ally

Multiply and divide from left to rightAdd and subtract from left to rightNow that we remember how to use PEMDAS to simplify a numerical expression, lets add one more idea so that we can use PEMDAS to evaluate an algebraic expression.

When you are asked to evaluate an algebraic expression, you will be given an expression that contains variables and a value to use for the variables.

To evaluate, substitute the number value for the variable and use PEMDAS to determine the value of the expression.

An example would be to evaluate the expression 24 (x + 2) for x = 1.

Substitute the value 1 in place of the variable x and use PEMDAS to find the value of the expression. 11

24 (x + 2) for x = 1

24 (1 + 2)

24 3

8

2Language of AlgebraPEADMSxcuse Exponentslease Parenthesis (and other grouping symbols)yearunt ally

Multiply and divide from left to rightAdd and subtract from left to rightFirst, work within the parentheses and rewrite the expression as 24 (1 + 2) and then as 24 3 and finally as 8.12

x (3 + 2) 50/x

for x = 2

2 (3 + 2) 50/2

3+2 = 5

5 = 25

2222 (25) 50/2

2 25 = 50

50 2 = 25

50-25 = 25

2Language of AlgebraPEADMSxcuse Exponentslease Parenthesis (and other grouping symbols)yearunt ally

Multiply and divide from left to rightAdd and subtract from left to rightNow, move the steps into place to evaluate the expression x (3 + 2)2 50/x for x = 2.

First, replace the variable (x) with the number 2.

Next, work within the parentheses to determine that 3 + 2 = 5.

Square 5 to get 25.

Multiply and divide from left to right.

Finally, subtract to get the value 25.13Practice on your Own!

Practice on your own!

What is the value of (2a 3b)2 when a=9 and b=4?

14

Now that you remember how to simplify and evaluate numerical and algebraic expressions, we are ready to use these skills to solve equations.2Language of AlgebraNow that you remember how to simplify and evaluate numerical and algebraic expressions, we are ready to use these skills to solve equations.15

PEADMSxcuse Exponentslease Parenthesis (and other grouping symbols)y Multiply and divide from left to rightearunt