# Computing with Signed Numbers and Combining Like Terms

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1 Section. Pre-Activity Preparation Computing with Signed Numbers and Combining Like Terms One of the distinctions between arithmetic and algebra is that arithmetic problems use concrete knowledge; you know each of the components and can calculate an answer based on complete information. For example, an arithmetic problem might be stated as: What is the sum of 7 and? In algebra, however, problems often use information given in general or abstract terms. The previous problem might be changed to, What is the sum when any number is added to? Learning Objectives Begin to learn the fundamental language of algebra Learn the foundational properties of algebra Add, subtract, multiply, and divide signed numbers Apply the Distributive Property to combine like terms Terminology New Terms to Learn absolute value coefficient constant dividend divisor expression factor integer like terms negative number line opposite product quotient reciprocal signed number simplify term variable

2 Chapter Evaluating Expressions Building Mathematical Language Addition and Subtraction Expressions Addition and subtraction expressions have the following elements: addition signs, subtraction signs, constants, variables and symbols of enclosure. (Check the table on the following page for definitions of these terms.) Below are simple examples. 4 + ( 7) four plus negative seven + 7 negative eleven plus seven or negative eleven plus positive seven a ( ) a minus negative six 7 ( 2) negative seventeen minus negative twelve x + 7 x plus seven The result of adding two numbers is called a sum. Subtracting one number from another is called a difference. Multiplication and Division Expressions Multiplication and division expressions have the following elements: multiplication signs, division signs, constants, variables and symbols of enclosure. Below are simple examples. 5 2 negative fifteen times two or negative fifteen times positive two 2 ( ) 2 ( ) 0 ( 8) 0 ( 8) ( 0)( 8) 4 two times negative three or positive two times negative three 4 negative thirty times negative eight 90 ( 9) ninety divided by negative nine or the quotient of 90 and negative nine 2x twelve x or twelve times x xy x times y or xy The result of a multiplication is called the product; the numbers being multiplied are most often called factors. (factor) (factor) product The result of a division is called the quotient. The dividend is divided by the divisor. dividend divisor quotient quotient or divisor) dividend

3 Section. Computing with Signed Numbers and Combining Like Terms Common Language of Algebra Language Definition Example Key Observations Constant Variable Term Coefficient Integer Factor??? Why can we do this? A symbol (usually a number) that does not change its value A symbol (usually a letter) that represents an unspecified or unknown number (value). Each variable represents a single value, even if the variable appears more than once in the problem. Part of an expression made up of constant and variable factors. Terms are separated by + or signs; that is, terms are added or subtracted. Any constant or variable factor of another variable All whole numbers, zero, and all the opposites (negatives) of the whole numbers Constants, variables, or expressions that are multiplied 7x + 2 x, y, z, a, b, c, etc. The expression 2x + 5y 7c + 2c has four terms: 2x, 5y, 7c, and 2c. The terms 7c and 2c have the same variable components; we call them like terms. 7 in 7x, in a, π in πd For the expression 4xy 2 : 4 is the coefficient of xy 2, 4y 2 is the coefficient of x; and 4x is the coefficient of y 2, 2,, 0,, 2,, 4x ()() 7xyz (x + 2)5 z(2 + z) π is an example of a constant that is not a numeric symbol. It is a Greek letter used to represent the constant ratio of the circumference to the diameter of a circle. Two key ideas:. An unknown value that makes an equation true or 2. A placeholder in an expression that can assume any chosen value The sign preceding a term is used as the sign of the coefficient. Connect the idea of terms with addition and subtraction.??? Why can we do this? Most often the word coefficient refers to the constant factor, the numerical coefficient. Integers are zero and the signed numbers, both positive and negative. Connect the idea of factors with multiplication Terms are separated by + or signs. Since subtraction means addition of the opposite, an expression can always be assumed to be an addition of its terms. Consequently, each subtraction sign preceding a term becomes the negative sign of the coefficient of that term. That is, always that the + or - sign before a term as the sign of its numerical coefficient. For example, 4x + 7y 2c is the same as 4x + 7y + ( 2c) and has the numerical coefficients 4, 7, and 2 in its terms 4x, 7y, and 2c. Caution: The word term can also refer to either the numerator or denominator of a fraction. Reduce to lowest terms refers to dividing out common factors from the terms of the fraction.

4 Chapter Evaluating Expressions Signed numbers Positive numbers are greater than zero and negative numbers are less than zero. All signed numbers, positive and negative, can be represented on a number line, such as the one below on which the integers from 7 through +7 are marked: As you move from left to right on the number line, the numbers grow larger: 5 > 7 or > 2 Or you can say that as you move left, the numbers get smaller: 2 < 0 or < 4 Opposites Aside from zero, which is neither positive nor negative, every signed number has an opposite number that is equally distant from zero, but in the opposite direction. For example, 7 is the opposite of 7 and.5 is the opposite of.5, etc. Find the opposite of a number by taking the negative of the number. Operations Addition, subtraction, multiplication, and division are operations used to combine numbers, variables or expressions. Subtraction and division can be written in terms of addition and multiplication. Subtraction is addition of the negative (additive inverse) of a number Division is multiplication by the reciprocal (multiplicative inverse) of a number For example: 2a b 2a + ( b) and 5 ' x 5 \$ x Use of the Dash Mathematicians use the dash ( ) to represent both the negation of a number and the subtraction of one number from another. We know what the meaning of the dash is from the context in which it appears. For example, in 0 7 the dash indicates subtraction while in ( 7) the dash indicates negation. The two short expressions have the same value, but the dash has different meanings in the two expressions. Absolute Value Special enclosure symbols are used to indicate the distance a signed number is from zero: a a a. Since distance is a non-negative concept, the absolute value returns a non-negative value. (Nonnegative means the same as positive OR zero. ) A number and its opposite have the same absolute value: The numbers 2. and 2. are the same distance from zero but in opposite directions on the number line. Caution: The opposite of the opposite of a number n is ( n) or n, itself. Therefore, 7 and ( 7) are different representations of the same number. 7 to its opposite, 7 7 to its opposite, ( 7), or

5 Section. Computing with Signed Numbers and Combining Like Terms Properties and Principles Algebra follows the basic properties and principles of mathematics. The following are applied as needed to solve problems and develop new concepts. Property Identity for Addition Inverse of Addition Symbolic Representation Numeric Example a a a a + ( a) ( a) + a (.5) 0 Key Observations Use this fact when a particular term is missing from an expression; add a zero term as a placeholder. The additive inverse can also be called the opposite of or the negative of. Identity for Multiplication a \$ \$ a a 4 2 \$ You can multiply by in any form anytime it is needed. Inverse of Multiplication a a \$ a a 4 4 Every non-zero number has a \$ 4 4 multiplicative inverse, also called its reciprocal. Zero Factor a \$ 0 0 \$ a No exceptions Zero Divisor Associative Commutative a 0 undefined (a + b) + c a + (b + c) ( a \$ b) \$ c a \$ ( b \$ c) 2 0 undefined However, ( 7 + 2) ( 2 + ) Know the difference between division BY 0 and division INTO 0. Always validate: Is there a number multiplied by 0 that is 2? No. Is there a number multiplied by 2 that is 0? Yes (0). 2 2 Grouping does not matter for ( 2 ) 4 2 ( 4) addition or multiplication a + b b + a 7 + ( 2) ( 2) + 7 a \$ b b \$ a ( 5) 2 2 ( 5) Order does not matter for addition or multiplication. Distributive Trichotomy a(b + c) ab + ac Given any two numbers a and b, one and only one of the following statements is true: a) a is equal to b; ab b) a is less than b; a<b c) a is greater than b; a>b ( 4 + 5) Used extensively; left to right to multiply through to remove ( 9) parentheses and right to left to break out common factors. Given:.2 and 2. a).2 2. False b).2 < 2. True c).2 > 2. False If you can rule out two of the relationships, the third one is true. This property is most often used in formal proofs.

8 Chapter Evaluating Expressions Models Model 2 Add or subtract as needed: ( 8) Step Subtraction problem Step 2 Change two signs: + (+8) Step Opposite signs Step 4 Subtract absolute values: Step 5 Number with the larger absolute value is positive; Answer: 5 Step Validate by adding: 5 + ( 8) Opposite signs so subtract absolute values: Attach the sign of the number with the larger absolute value: Model Add or subtract as needed: ( 7) 5 ( ) Step Mixed addition and subtraction problem Step 2 Change two signs for all subtractions: ( 7) + ( 5) + (+ ) Steps 5 Add left to right: (- 7)+ (- 5)+ - + (- 5) Answer: 5 Step Validate: (- ) + 5 -(-7) (- ) ( + 7) + (-) (-) (-) (-) -2

9 Section. Computing with Signed Numbers and Combining Like Terms Methodology Since we cannot carry out the multiplication of two variables whose values are not known (such as x and y), we indicate their product simply as xy. Similarly, dividing x into y is shown as y/x, y x, or y x. Multiplying and Dividing Two Quantitites Containing Constants and Variables Example : Divide: ( 27x) ( ) Example 2: Divide: ( 48a) ( 7) Try It! Steps in the Methodology Example Example 2 Step Determine the sign of the answer: Positive (+) if both quantities have the same sign Negative ( ) if the two quantities have opposite signs The sign of the answer to multiplication or division is independent of what the quantities are; it hinges only on whether the signs match (a positive answer) or do not match (a negative answer). ( 27x) ( ) Both signs are negative so the answer is positive. Step 2 Identify the intended operation as multiplication or division. If multiplication, skip to Step 4. Step If the operation identified in Step 2 is division, change it to multiplication AND invert the second quantity (multiply by the reciprocal). If the process is division, there is an extra step, so make this determination from the outset. The Inverse Property of Multiplication is applied here. Two changes occur: the operation changes to multiplication AND the divisor changes to its multiplicative inverse. That is, multiplying by the multiplicative inverse is the same as dividing. ( 27x) ( ) The operation is division. ( 27x) ( ) ( 27 x) : continued on next page

11 Section. Computing with Signed Numbers and Combining Like Terms Model 2 Multiply: ( )(5x)( 2y)( 7z) Step??? The sign of the answer is negative. How do you know this? Step 2 All operations are multiplication; skip to Step 4 Step 4 Multiply absolute values from left to right: : 5x : 2y : 7z Step 5 Model Validate: Answer: 20xyz 5x : 2y : 7z 0xy : 7z 20xyz The original problem used multiplication so use the division methodology to validate: -20xyz (- 7z) 0xy 0xy (- 2y) -5x (- 5x) 5x - Take a closer look (-20xyz) ' (- 7z) 20xyz 20 xy z 0xy 7z 7 z 0xy ' (- 2y) x y -5x y ( - 5x) ' 5x 5x 5 x x 5 x Divide: Step Step 2 Step Step 4 Step The sign of the answer is negative. The operation is division. Two changes: 2 - : Multiply absolute values: 5 Answer: 4 5 Validate: : 4 5 The original problem used division so use the multiplication methodology to validate: Step Negative answer??? How do you know this? Step 2 Multiplication; skip to Step : Step 4 Multiply absolute values: : 5 7 Answer: 2 2

12 2 Chapter Evaluating Expressions??? How do you know this? Determining the sign of the answer for multiplication of signed numbers (or division, after changing to multiplication of the reciprocal) is independent of the values of the numbers. You can pair up the signs of negative factors, recalling that ( a)( b) ab. Every time an even number of negative factors is present, the answer is positive because all of the signs can pair up. If the number of negative factors is odd, the answer will always be negative, because one negative factor will not be able to pair up. Methodology Expressions containing variable factors can often be simplified by applying the Associative, Commutative and Distributive properties to combine like terms. Like terms contain the same variable factors. Below is a methodology for combining like terms. Combining Like Terms Example : Combine like terms: 2x 2y + 7x y Example 2: Combine like terms: 8x y + x 5y Try It! Steps in the Methodology Example Example 2 Step Use the Commutative Property to sort the expression so that like terms are together. Step 2 Use the Associative Property to group like terms together within parentheses. Scan through the problem to make sure like terms are identified. You can highlight each set of like terms with color or underscoring. Always convert subtraction to addition before grouping within parentheses to eliminate sign errors. 2x 2y + 7x y 2x + 7x 2y y 2x + 7x 2y y 2x + 7x + ( 2y) + ( y) (2x + 7x) + (( 2y) + ( y))

13 Section. Computing with Signed Numbers and Combining Like Terms Steps in the Methodology Example Example 2 Step For each group of like terms, use the Distributive Property to factor out the common variable(s). Step 4 Add or subtract within parentheses. Step 5 Rewrite addition of a negative as subtraction. Step Validate Factoring is un -multiplying. The Distributive Property distributes multiplication over addition, but the reverse action also applies. Follow the methodologies presented previously for combining terms within parentheses. Why do this? This is a step that technically doesn t need to happen, but it adds to the sophistication of the answer, making it simpler to read. Validation is best done after some time has elapsed. Come back to the problem and rework it from the beginning. (2 + 7)x + ( 2 + ( ))y (2 + 7)x + ( 2 + ( ))y 9x + ( )y 9x y 2x 2y + 7x y 2x + 7x + ( 2y) + ( y) (2 + 7)x + ( 2 + ( ))y 9x y

14 4 Chapter Evaluating Expressions Models Model Combine like terms: 5s + t + c 2s 7t + 2c Step Sort by like terms: 5s + t + c 2s 7t + 2c 5s 2s + t 7t + c + 2c Step 2 Group within parentheses after converting subtraction to addition: (5s + ( 2)s) + (t + ( 7)t) + (c + 2c)??? Step Factor: (5 + ( 2))s + ( + ( 7))t + ( + 2)c Why do this? Step 4 Step 5 Step Add or subtract in parentheses: s+ ( 4)t + c Rewrite as subtraction: Answer: s 4t + c Validate: (Reworked)??? Why do this? Any variable without a written numerical coefficient has an implied coefficient of. In the above model, c is understood to mean c. The is not customarily written; however, you may elect to write the implied to remember to include the term. Validate: 5 + ( 2) ( 2) 5 + ( 7) 4 4 ( 7) Model 2 Combine like terms: x + 4y 7 x + y Step Sort: x + 4y 7 x + y x x + 4y + y 7 Step 2 Group: (x + ( )x) + (4y + y) + ( 7) Step Factor: ( + ( ))x + (4 + )y + ( 7) Step 4 Combine: 2x + 7y + ( 7) Step 5 Rewrite: Answer: 2x + 7y 7 Step Validate: (Reworked) Validate: + ( ) 2 2 ( )

15 Section. Computing with Signed Numbers and Combining Like Terms 5 Addressing Common Errors Issue Incorrect Process Resolution Correct Process Validation Misinterpreting absolute value symbols 2-2 Absolute value returns a nonnegative value, not the opposite value. 2 2 and is 2 units from zero on a number line. Misunderstanding division of zero by a number 0-4 undefined 0 ' x undefined Division BY zero is undefined. Division INTO zero is ' x 0 Does 0 4 0? Yes Does x 0 0? Yes. Changing only one sign when subtracting 4 + ( 8) 0 + ( 8) 2 Always change two: apply the Inverse of Addition Property to make the changes. 4 + ( 8) (+8) ( 8) 0 4 Using the Associative Property with subtraction (2 7) (8 ) Change subtraction to addition of the opposite before using the Associative Property. Once the problem has been changed to addition, group as desired ( 7) + ( 8) + ( ) [ 2 ( 7) ] [( 8) ( ) ] Forgetting a variable factor when multiplying z 5 2y ( 2x) ( 5 2 ( 2))yx 0xy Write the variable factors in alphabetical order. Each factor must be included in the product. Mark off each factor as it is used. z 5 2y ( 2x) 5 2 ( 2)xyz 5 2 ( 2)xyz 5 2 ( 2)xyz 0( 2)xyz 0xyz Is each variable included in the product? Yes Is each numeric factor included? Check numeric factors by dividing: 0 ( 2)

16 Chapter Evaluating Expressions Issue Incorrect Process Resolution Correct Process Validation Leaving out an unmatched term when combining like terms 7x 2y + z y + x 7x + x 2y y 8x 5y Highlight or underline like terms. In order for equality to prevail, every term must be included in the simplified answer. Circle any terms that do not combine with other terms so that you don t forget to include them. 7x 2y + z y + x 7x + x 2y y + z (7x + x) + (( 2)y + ( )y) + z (7 + )x + ( 2 + ( ))y + z 8x + ( 5)y + z 8x 5y + z Come back and rework this problem after the next problem. If you rework it immediately, your eyes may be tricked into seeing combinations that are not true. Validate addition: ( ) 2 Forgetting to combine a term with no coefficient 2a + b a + 2b 2a + a + b + 2b + 5 (2a + a) + (b + 2b) + 5 (2 + )a + ( + 2)b + 5 2a + 5b + 5 Write in the implied coefficient. For a more sophisticated answer, you can rewrite it without the coefficient. 2a + b a + 2b 2a + b a + 2b 2a + a + b + 2b + 5 (2a + a) + (b + 2b) + 5 (2 + )a + ( + 2)b + 5 a + 5b + 4 Validate by reworking after working the next problem. Validate addition: Preparation Inventory Before proceeding, you should be able to do each of the following: Use and understand the basic vocabulary of algebra. Apply basic properties and principles of algebra. Perform and validate calculations with signed numbers. Combine like terms by applying the Distributive Property.

17 Section. Activity Computing with Signed Numbers and Combining Like Terms Performance Criteria Demonstrating the use of appropriate algebraic language correct use of terms in oral and written communication correct spelling of terms in written communication Computing with signed numbers accuracy correct sign for the result Simplifying expressions by combining like terms correct use of Distributive Property accuracy in signed number operations Activity A Model for Multiplication Multiplication on a Grid Supplies: square tiles, grid paper, problem set, instructions If you have tiles provided by your instructor, use them to make models of the following integer multiplication problems. If you do not have tiles provided, you can make some by using squares of paper or other material. (Alternatively, you can draw more grids and color in the tiles for each problem.) negative times a positive - + II I positive times a positive III IV negative times a negative positive times a negative Locate the two crossed number lines. They intersect at zero, dividing the grid into four parts, called quadrants. The white portion indicates positive regions: quadrants I and III; light gray indicates negative regions: quadrants II and IV. To model a problem, the first number indicates how many tiles in a row, either in the positive (right) direction or the negative (left) direction. The second number indicates the number of rows, in either the positive (up) or negative (down) direction. In the example, five tiles in the positive direction for each of three rows in the negative direction. The tiles are in the light gray area and indicate a negative product. Example: Show the product of +5 First number: +5 five tiles in a row towards the right Second number: three rows of 5 tiles each, downward +5 5 as shown by 5 tiles in the negative region. 7

18 8 Chapter Evaluating Expressions Use tiles to show the following products: Note: Additional blank grids may be found at the end of this activity Critical Thinking Questions. Can 0 be the coefficient of a term? Illustrate your answer with an example. 2. What is a variable factor?. What information does the coefficient of a variable impart?

19 Section. Computing with Signed Numbers and Combining Like Terms 9 4. How do you find the absolute value for a variable? 5. What is the difference between the opposite of a number and the negative of a number?. Is there an associative property for subtraction? (Illustrate your answer with an example.) 7. When would you insert a + (positive sign) and when would you insert ( ) (parentheses) without changing the value of an expression? 8. Why is reworking the problem the preferred method to validate combining like terms? 9. How do you validate that every term is accounted for in combining like terms? Tips for Success Mathematicians agree that two signs together, such as 4 + 7, may be ambiguous. Therefore, unless it is the first term in the expression or the denominator of a division problem, a negative number is written within parentheses: 4 + ( 7). When combining like terms, count the number of terms to be sure no term is overlooked. Underline or highlight every term in the original problem and make sure each term is accounted for as either combining with another term or left unchanged in the final answer.

20 20 Chapter Evaluating Expressions Demonstrate Your Understanding. Perform the indicated computation and identify the property used or illustrated. Problem Validation Property a) 0 + ( ) b) 9 + ( 9) c) 0 d) ( 9) 0 e) ( 209) f) 2 : 2 g) 0

21 Section. Computing with Signed Numbers and Combining Like Terms 2 2. Add or Subtract the following: Problem Validation Problem Validation a) 2 5 g) b) 8 ( ) h) 2 2 c) 0 ( 7) i) 9 4 d) 8 + ( 7) j) + ( 2) e) + ( 0) k) 2 ( 5) f) 2 4 l) 8 ( 2). Perform the indicated multiplication or division: Problem Validation Problem Validation a) 25 ( 5) b) ( 2)( ) c) 2 7 d) 7 4 e) ( ) f) 54 ( 9) g) 42 ( 7) h) 2 4 ( 5)

22 22 Chapter Evaluating Expressions 4. Apply the distributive property. Example: 5(a + b) 5a + 5b a) (m + n) b) 2(x ) c) 7(x + 2) answer: answer: answer: e) (2x y) f) ( 2x + ) g) ( 2x 2) answer: answer: answer: 5. Use the distributive property to combine like terms: a) 7x + + 4x 9 Problem Like Terms Worked Solution Validation b) ab + c + 2ab c c) x y x y d) x + 5x 2x x 4x + x e) 7x + x x + 4x x + 9

23 Section. Computing with Signed Numbers and Combining Like Terms 2 7. Multiplication and division are opposite operations. That means that every multiplication problem can be written as a division problem and every division problem can be written as a multiplication problem. In fact, we validate multiplication with division and validate division with multiplication. So 4 2 and 2 4 and 2 4 are all related. a. Use this idea to explain why a positive number, 00, divided by a negative number, 25, yields a negative number, 4. b. Show why a negative number ( 5) divided by a negative number ( ) is a positive number (5). c. Show why a negative number ( 22) divided by zero (0) is undefined and not equal to zero.

24 24 Chapter Evaluating Expressions Identify and Correct the Errors Identify and correct the errors in the following problems. Incorrect Translation List the Errors Correct Process Validation ) If , then ) 0 0 undefined, then ) ( 5) + ( 5) Validate: ( 5) 4) (+5) (2+7) 8 9 Validate: ) x2yz( ) 2( )xy 8 xy Validate: 8 ( ) and / ) x + 5y + x + 2y x + x + 5y + 2y ()x + (5+2)y x + 7y

25 Section. Computing with Signed Numbers and Combining Like Terms

26 2 Chapter Evaluating Expressions

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