The Language of Algebra

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The Language of Algebra
Component 2 The Language of Algebra Component 2: The Language of Algebra

“Rules of Algebra” Stage 2 Stage 1 Stage 3 2 Language of Algebra
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.

P E M D A S 2 Parenthesis (and other grouping symbols) lease xcuse
Language of Algebra P Parenthesis (and other grouping symbols) lease E Exponents xcuse M y Multiply and divide from left to right D ear A Sometimes a problem 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 this acronym by using this phrase, “Please Excuse My Dear Aunt Sally.” unt S Add and subtract from left to right ally

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.

( ) [ ] { } / √ P E M P D A S (12÷3) +1 4+1 5 Order of Operations Tool
Language of Algebra Order of Operations Tool P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally Grouping Symbols (12÷3) +1 4+1 5 ( ) [ ] P lease Parenthesis (and other grouping symbols) { } / Multiply and divide from left to right 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, 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. Add and subtract from left to right

P E M E D A S Order of Operations Tool 53 means you are to use
2 Language of Algebra Order of Operations Tool P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally 53 means you are to use 5 as a factor 3 times 5•5•5 125 42 – (8•2) 42 – 16 16 – 16 E Multiply and divide from left to right xcuse Exponents 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. 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. Add and subtract from left to right

P E M M D D A S ! Order of Operations Tool 2 • 2 is 4 4 – 7 = -3
Language of Algebra Order of Operations Tool P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally Caution ! 16 divided by 8 • 2 – 7 16 divided by 8 is 2 2 • 2 – 7 2 • 2 is 4 4 – 7 = -3 M y Multiply and divide from left to right D Multiply and divide from left to right ear 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. Add and subtract from left to right

2 Language of Algebra 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. 16 ÷ 8 • 2 – 7 16 ÷ 16 – 7 1 – 7 -6 A 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.

P E A M D S A S Order of Operations Tool 12 – 4 • 2 + 2 4 • 2
Language of Algebra Order of Operations Tool P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally 12 – 4 • 2 + 2 4 • 2 12 – 8 + 2 4 + 2 = 6 A unt Add and subtract from left to right S Multiply and divide from left to right ally The 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 = 6. Add and subtract from left to right

P E M D A S 60 ÷(3 + 1) – 2 • 2 9+1 = 10 60÷10 = 6 2 • 2 = 4 6-4 = 2 2
Language of Algebra 60 ÷(3 + 1) – 2 • 2 2 P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally 9+1 = 10 60÷10 = 6 2 • 2 = 4 6-4 = 2 Multiply and divide from left to right Now, let’s use PEMDAS to simplify a numerical expression. Consider the expression: 60 ÷(32 + 1) – 2 • 2. Let’s move the steps into order to show how you would use PEMDAS to simply the expression. First, work within the parenthesis to add Now, remember that multiplication and division are done next, from left to right. Finally, subtract 4 from 6 to get 2. Add and subtract from left to right

P E M D A S 24 ÷ (x + 2) for x = 1 24 ÷ (1 + 2)
Language of Algebra P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally 24 ÷ (x + 2) for x = 1 24 ÷ (1 + 2) You will be given an expression that contains variables and a value to use for the variable. Use PEMDAS to find the value or evaluate an algebraic expression. Substitute the number value for the variable. Multiply and divide from left to right 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 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. Add and subtract from left to right

P E M D A S 24 ÷ (x + 2) for x = 1 24 ÷ (1 + 2) 24 ÷ 3 8 2
Language of Algebra 24 ÷ (x + 2) for x = 1 24 ÷ (1 + 2) 24 ÷ 3 8 P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally Multiply and divide from left to right First, work within the parentheses and rewrite the expression as 24 ÷ (1 + 2) and then as 24 ÷ 3 and finally as 8. Add and subtract from left to right

P E M D A S x (3 + 2) – 50/x 2 (25) – 50/2 for x = 2 2• 25 = 50
Language of Algebra 2 x (3 + 2) – 50/x for x = 2 2 (3 + 2) – 50/2 3+2 = 5 5 = 25 P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y ear unt ally 2 (25) – 50/2 2• 25 = 50 50 ÷ 2 = 25 50-25 = 25 2 Multiply and divide from left to right Now, 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 = 5. Square 5 to get 25. Multiply and divide from left to right. Finally, subtract to get the value 25. 2 Add and subtract from left to right

What is the value of ½ (2a – 3b)2 when a=9 and b=4?

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

Distributive Property
2 Language of Algebra P E A D M S xcuse Exponents lease Parenthesis (and other grouping symbols) y Multiply and divide from left to right ear unt Add and subtract from left to right ally Distributive Property If the order of operations (PEMDAS) is the first tool we use to simplify expressions, then the distributive property is the second. 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. 5 (x + 2)

A = 5 (x+2) A = 5(x) + 5(2) 5(x + 2) = 5(x) + 5(2) or
Language of Algebra A = 5 (x+2) A = 5(x) + 5(2) 5(x + 2) = 5(x) + 5(2) or 5(x + 2) = 5x +10 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. 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. 5 5 (x + 2) x

Let’s use our word wall to study what is meant by the term “distributive property” and “like terms”.
A third “tool” you will use to solve equations is combining “like terms” so look at the definitions carefully.

Copy this equation on your paper and solve for x.
2 Language of Algebra Solve for x. x = 2(y+2)+2y Copy this equation on your paper and solve for x. 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 Check your Work

In this example, how would you solve for x if y = 3?
2 Language of Algebra Distributive Property- a property indicating a way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac Did you find that x = 4y + 4? x = 2y y In this example, how would you solve for x if y = 3? Did you find that x = 4y + 4? Compare your work to the following. Use the distributive property to write that x = 2y y. Then, combine like terms to simplify the answer. In this example, how would you solve for x if y = 3? Continue

Did you find that x = 16? Substitute 3 for y. 16 for x. 2
Language of Algebra Distributive Property- a property indicating a way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac Did you find that x = 16? Substitute 3 for y. 16 for x. 16 = 2(3) (3) 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. Check your Work

First, simplify both sides of the equation
2 Language of Algebra First, simplify both sides of the equation (if needed) then, use inverse operations to isolate the variable. More often, it is not. Two guidelines will help you when you meet an equation such as 5x + 3(x + 4) = 28. Sometimes, the variable you are solving for is isolated on one or the other side of the = sign. Sometimes, the variable you are solving for is isolated on one or the other side of the = sign. 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.

The word wall will remind us about inverse operations.
Subtraction and addition are inverse operations. Multiplication and division are also inverse operations.

5x + 3(x+4) = 28 2 Language of Algebra
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. 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. Check your Work

2 Language of Algebra 5x + 3(x+4) = 28 8x + 12 = 28 8x = 16 x = ? 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. First, use the distributive property and combine like terms to simplify the left side of the equation. First, use the distributive property and combine like terms to simplify the left side of the equation. 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. Coefficient- a number or quantity placed (generally) before and multiplying another quantity, as 3 in the expression 3x. Distributive Property- a property indicating a way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac Check your Work

Solving for m 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.

2 Language of Algebra ¼ ( ) = 10-3( ) 2 ¼ ( ) = 10-3( ) m Check your answer by substituting into the original equation. Does your thinking agree with our answer?

Component 2 Quiz 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. 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? Work through the questions in the quiz and then click below to continue.

2 Language of Algebra 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.

This concludes Component 2.
Linear Algebra Rules of Algebra Using Technology Patterns of Change 1 2 3 4 This concludes Component 2. Close this window and locate Component 3: Using Technology to Picture Change on the main course page. Click on the course name link to begin the component.