Computing with Signed Numbers and Combining Like Terms
|
|
- Bonnie Campbell
- 7 years ago
- Views:
Transcription
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.
6 Chapter Evaluating Expressions Methodology Adding and Subtracting Signed Numbers Example : Subtract: 9 ( 7) Example 2: Subtract: 2 2 Try It! Steps in the Methodology Example Example 2 Step Identify the intended operation as addition or subtraction. Step 2 If the operation identified in Step is addition, skip to Step. If the operation identified in Step is subtraction, change it to addition AND change the sign of the second term (change TWO signs). Step Determine the signs of the two terms to be combined. Step 4 If the signs are not the same, skip to Step 5. If the signs are the same, add the absolute values of the two terms and attach the common sign. Then go to Step. The process hinges on whether the operation is addition or subtraction. Subtraction is the same as addition of the opposite. Once a problem has been changed to addition by also changing the sign of the second term, follow the process for addition. This step splits the process into two possibilities: either the signs are the same both positive or both negative, or the signs are different one positive and one negative. If the signs are not the same, skip to Step 5. If the signs are the same, add the absolute values of the two terms and attach the common sign, then go to Step. Subtraction is the operation. change to addition 9 ( 7) 2 change sign of second term 9 + [+7)] 9 + 7??? Why can we do this? 9 is negative and 7 is positive Signs are not the same. Go to Step 5.
7 Section. Computing with Signed Numbers and Combining Like Terms Steps in the Methodology Example Example 2 Step 5 If the signs are opposite, subtract the smaller absolute value from the larger absolute value. Attach the sign corresponding to the larger absolute value. Step Validate by performing the opposite operation. Is one positive and one negative? If so, take the absolute values of both numbers and then subtract. Reattach the sign of the number with the larger absolute value. If the original operation was addition, subtract the original second addend from the answer. If it was subtraction, add the answer to the original second addend. The result in either case should match the original first number. - 9 > Attach the sign of the 9: Answer: 2 The original was subtract, so add: 2 + ( 7) (signs are the same, so add absolute values) (attach the common sign) 9 (and compare) 9 9??? Why can we do this? Built into the reasoning of Step 2 is an application of both the Identity for Addition and the Inverse of Addition properties. The simple problem at right illustrates the use of these properties to change 7 to + ( 7); that is, add zero to the original expression in the form of 7 + ( 7) inverses sum 0 inverses sum 0 + ( 7) ( ) ( ) Models Model Add or subtract as needed: 42 + ( 9) Step Addition problem Step 2 Skip to Step Step Both numbers are negative. Step 4 Add absolute values: Attach common sign: Answer: 5 Step Validate by subtracting: 5 ( 9) Change two signs: 5 + ( + 9) Opposite signs so subtract absolute values: Attach the sign of the number with the larger absolute value: 42
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
10 0 Chapter Evaluating Expressions Steps in the Methodology Example Example 2 Step 4 Calculate the product of the absolute values of the quantitites. Attach the sign determined in Step. Step 5 Validate by following the methodology for the opposite operation. Multiply, disregarding the sign; you have already determined the sign of the answer in Step. For division, multiply the answer by the original divisor; if it was multiplication, divide the product by either factor to get the other factor. -27x : - ( 27x) : 27x Answer: +9x 9x : - Negative answer 2 Multiplication; skip to Step 4 4 Multiply absolute values and make the result negative: 9x : - 9x : 27x - 27x Models Model Multiply: 2.c 5 Step The signs are opposite, so the answer is negative. Step 2 The operation is multiplication: 2.c 5 Skip to Step 4. Step 4 Multiply absolute values: 2.c 5.5c Take a closer look 2.c 5 can change to 2. 5 c by applying the commutative property. Step 5 Validate Answer:.5c The original problem used multiplication so use the division methodology to validate: Step Step 2 Choose either factor:.5c 5 (signs are opposite; answer will be negative) Operation is division Step -. 5c : 5 Step 4 Multiply absolute values:. 5c : 2.c 5 Answer: 2.c
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
3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationAccuplacer Arithmetic Study Guide
Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationMultiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20
SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationSession 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:
Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules
More informationParamedic Program Pre-Admission Mathematics Test Study Guide
Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More information2.3 Solving Equations Containing Fractions and Decimals
2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions
More informationThe Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,
1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationMATH-0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationPolynomial Expression
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationUnit 6 Number and Operations in Base Ten: Decimals
Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationNegative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2
4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year
More informationA.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents
Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationComputation Strategies for Basic Number Facts +, -, x,
Computation Strategies for Basic Number Facts +, -, x, Addition Subtraction Multiplication Division Proficiency with basic facts aids estimation and computation of multi-digit numbers. The enclosed strategies
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationAddition and Subtraction of Integers
Addition and Subtraction of Integers Integers are the negative numbers, zero, and positive numbers Addition of integers An integer can be represented or graphed on a number line by an arrow. An arrow pointing
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More informationIntroduction to Fractions
Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying
More informationIV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationClifton High School Mathematics Summer Workbook Algebra 1
1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:
More informationAccentuate the Negative: Homework Examples from ACE
Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationfor the Bill Hanlon bill@hanlonmath.com
Strategies for Learning the Math Facts Bill Hanlon bill@hanlonmath.com The more sophisticated mental operations in mathematics of analysis, synthesis, and evaluation are impossible without rapid and accurate
More informationChapter 7 - Roots, Radicals, and Complex Numbers
Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationof surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationVerbal Phrases to Algebraic Expressions
Student Name: Date: Contact Person Name: Phone Number: Lesson 13 Verbal Phrases to s Objectives Translate verbal phrases into algebraic expressions Solve word problems by translating sentences into equations
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More informationRadicals - Multiply and Divide Radicals
8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More information7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School
7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School Page 1 of 20 Table of Contents Unit Objectives........ 3 NCTM Standards.... 3 NYS Standards....3 Resources
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationFactoring Trinomials of the Form x 2 bx c
4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently
More informationUNIT 5 VOCABULARY: POLYNOMIALS
2º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Algebraic Language Algebra is a part of mathematics in which symbols, usually letters of the alphabet, represent numbers. Letters are used to
More informationThe Crescent Primary School Calculation Policy
The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has
More informationOpposites are all around us. If you move forward two spaces in a board game
Two-Color Counters Adding Integers, Part II Learning Goals In this lesson, you will: Key Term additive inverses Model the addition of integers using two-color counters. Develop a rule for adding integers.
More informationReview of Basic Algebraic Concepts
Section. Sets of Numbers and Interval Notation Review of Basic Algebraic Concepts. Sets of Numbers and Interval Notation. Operations on Real Numbers. Simplifying Expressions. Linear Equations in One Variable.
More informationPre-Algebra - Order of Operations
0.3 Pre-Algebra - Order of Operations Objective: Evaluate expressions using the order of operations, including the use of absolute value. When simplifying expressions it is important that we simplify them
More informationRules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER
Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the
More informationCurriculum Alignment Project
Curriculum Alignment Project Math Unit Date: Unit Details Title: Solving Linear Equations Level: Developmental Algebra Team Members: Michael Guy Mathematics, Queensborough Community College, CUNY Jonathan
More informationUsing Proportions to Solve Percent Problems I
RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationTom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.
Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationUNDERSTANDING ALGEBRA JAMES BRENNAN. Copyright 2002, All Rights Reserved
UNDERSTANDING ALGEBRA JAMES BRENNAN Copyright 00, All Rights Reserved CONTENTS CHAPTER 1: THE NUMBERS OF ARITHMETIC 1 THE REAL NUMBER SYSTEM 1 ADDITION AND SUBTRACTION OF REAL NUMBERS 8 MULTIPLICATION
More informationMultiplication and Division with Rational Numbers
Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up
More informationAbsolute Value Equations and Inequalities
. Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between
More informationDirect Translation is the process of translating English words and phrases into numbers, mathematical symbols, expressions, and equations.
Section 1 Mathematics has a language all its own. In order to be able to solve many types of word problems, we need to be able to translate the English Language into Math Language. is the process of translating
More informationNegative Exponents and Scientific Notation
3.2 Negative Exponents and Scientific Notation 3.2 OBJECTIVES. Evaluate expressions involving zero or a negative exponent 2. Simplify expressions involving zero or a negative exponent 3. Write a decimal
More informationSIMPLIFYING SQUARE ROOTS
40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationYOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!
DETAILED SOLUTIONS AND CONCEPTS - DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationSample Fraction Addition and Subtraction Concepts Activities 1 3
Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationPAYCHEX, INC. BASIC BUSINESS MATH TRAINING MODULE
PAYCHEX, INC. BASIC BUSINESS MATH TRAINING MODULE 1 Property of Paychex, Inc. Basic Business Math Table of Contents Overview...3 Objectives...3 Calculator...4 Basic Calculations...6 Order of Operation...9
More informationSession 7 Bivariate Data and Analysis
Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares
More informationExponents. Exponents tell us how many times to multiply a base number by itself.
Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationObjective. Materials. TI-73 Calculator
0. Objective To explore subtraction of integers using a number line. Activity 2 To develop strategies for subtracting integers. Materials TI-73 Calculator Integer Subtraction What s the Difference? Teacher
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More information