Name For those going into LESSON 2.1 Study Guide For use with pages 64 70 Algebra 1 Honors GOAL: Graph and compare positive and negative numbers Date Natural numbers are the numbers 1,2,3, Natural numbers do not contain the number 0, fractions, decimals, or negative numbers. Another name for the set of natural numbers is counting numbers.) The symbol for the set of natural numbers is N Whole numbers are the numbers 0, 1, 2, 3, Whole numbers include the set of natural numbers plus 0. The symbol for whole numbers is W. Integers are the numbers..., 3, 2, 1, 0, 1, 2, 3,... No fractions or decimals belong to the set of integers The symbol for the set of integers is J. School years that begin with an ODD year: do the odds School years that begin with an EVEN year: do the evens DUE THE FIRST WEEK OF SCHOOL Packet counts as 2 homework grades; there will be a quiz on this material A rational number is a number that can be written as a quotient of two integers,, where b 0. When written in decimal form, a rational number either terminates, such as ⅛ = 0.125, or repeats (in a pattern), such as ⅓ = 0.333. The symbol for rational numbers is Q An irrational number is a number that cannot be written as a quotient of two integers. The decimal form of an irrational number neither terminates nor repeats. Some examples of irrational numbers include π (3.1459265358979323---) and = 30.74085 There is no symbol for the set of irrational numbers Real numbers include both sets of rational numbers and irrational numbers. The symbol for real numbers is R or R. Opposites are two numbers that are same distance from 0 on a number line but are on opposite sides of 0; for example -3 and 3. The absolute value of a number a is the distance between a and 0 on the number line. The symbol a represents the absolute value of a. The absolute value of a number is also called its magnitude. The absolute value of a number is never negative. If a is negative, then its absolute value is a. A conditional statement can be written in if-then form where the if part contains the hypothesis and the then part contains the conclusion. page 1
EXAMPLE 1 Graph and compare integers Graph 7 and 5 on a number line. Then tell which is greater. On the number line, 5 is to the right of 7. So, 5 > 7 TO DO Exercises for Example 1 Graph the numbers on a number line. Then tell which number is greater 1 4 and 2 2 0 and 3 3 1 and 1 EXAMPLE 2 Classify numbers Tell whether each of the following numbers is a whole number, an integer, or a rational number: 17, 0.3, 8, and 2 Number Whole number? Integer? Rational number? 17 No Yes Yes 0.13 No No Yes 8 Yes Yes Yes 2 No No No 2 1 EXAMPLE 3 Order rational number Order the rational numbers 2.2, 1.7,,0 and from least to greatest. 5 3 Begin by graphing the numbers on a number line. 1 2 From least to greatest, the numbers are 1.7, 3, 0, 5, and 2.2. TO DO Exercises for Examples 2 and 3 Tell whether each number in the list is a whole number, an integer, or a rational number. Then order the numbers from least to greatest 4 2, 1.5, 3, 0.5 5 0, 0.3, 0.6, 0.1 1 6 3.2, 1, 2.5, 2 1 2 7,, 2 1 1 5 3 5 2 page 2
EXAMPLE 4 Find the opposites and absolute values of numbers For the given value of a, find a and. a a. a= 3 b. a = 2 7 a. If a = 1.3, then a = ( 1.3) = 1.3 If a = 1.3, then. a = 1.3. = ( 1.3) = 1.3. 2 b. If a =, then a = 2 2 =. 7 7 7 2 2 2 If a =, then a = =. 7 7 7 TO DO Exercises for Example 4 For the given value of a, find a and a 8. a = 5 9. a = 11 10. a = 3.91 page 3
Name Date LESSON 2.2 Study Guide For use with pages 73 79 GOAL Add positive and negative numbers. Vocabulary The number 0 is the additive identity. The opposite of a is its additive inverse. Rules of Addition Same signs To add two numbers with the same sign, add their absolute values. The sum has the same sign as the numbers added. EXAMPLES: 8 + 7 = 15 (both numbers are positive) - 6 + -10 = -16 (both numbers have a negative sign) Different signs To add two numbers with different signs, subtract the lesser absolute value from the greater absolute value. The sum has the same sign as the number with the greater absolute value. EXAMPLES: a. -12 + 7 --- since -12 has the greater absolute value and it negative the sum will be negative ---subtract 7 from 12 SO -12 + 7 = - (12-7) = -5 b. 18 + (-5) --- since 18 has the greater value and is positive the sum will be positive --- subtract 5 from 18 SO 18 + (-5) = 18 5 = 13 Properties of Addition Commutative Property for addition The order in which you add two numbers does not change the sum. You can change the order but not the sum. 3 + 8 + 7 = 3 + 7 + 8 Associative Property of addition The way you group three numbers in a sum does not change the sum. (NOTE: The order of the numbers does not change; the grouping does.) ( 8 + 3) + 7 = 8 + ( 3 + 7) Additive Identity Property The sum of a number and 0 is the number. -8 + 0 = -8 0 + 6 = 6 Additive Inverses Property The sum of a number and its opposite is 0. 8 + -8 =0-2 + 2 = 0 EXAMPLE 1 Add real numbers a. 17 + ( 24) = 24 17 = 24 17 = 7 b. 1.3 + ( 5.8) = 1.3 + 5.8 = 1.3 + 5.8 = 7.1 Rule of different signs Take absolute values. Subtract. Use the sign of the number with the greater absolute value Rule of same sign Take absolute values Add and use the sign of the numbers added. page 4
TO DO Exercises for Example 1 Find the sum. 1. 3.6 + 7.1 2. 5.3 + ( 9) 3. 0.2 + ( 0.6) 1 1 4. 3 + 2 2 5 5. 11 + ( 15) + 8 6. 5 + ( 8) + 6 EXAMPLE 2 Identify the properties of addition Identify the property being illustrated. Statement Property Illustrated a. 15 + 0 = 15 Additive identity property b. 12 + ( 17) = 17 + 12 Commutative property of addition TO DO Exercises for Example 2 Identify the property being illustrated. 7. 9 + ( x) = x + 9 8. 5.1 + ( 5.1) = 0 9. [3 + ( 2)] + 1 = 3 + [( 2) + 1] EXAMPLE 3 Solve a multi-step problem Gas Prices The table shows the changes in gas prices for two companies. Which company had the greater total change in gas prices for the three weeks? Week Price change for company A Price change for company B 1 $.05 $.06 2 $.08 $.13 3 $.11 $.04 STEP 1 Calculate the total change in gas prices for each company. Company A: Total change = 0.05 + ( 0.08) + 0.11 = 0.08 + (0.05 + 0.11) = 0.08 + 0.16 = 0.08 Company B Total change = 0.06 + 0.13 + ( 0.04) = 0.13 + [ (0.06 + 0.04)] = 0.13 + ( 0.1) = 0.03 STEP 2 Compare the total change in gas prices: 0.08 > 0.03. Company A had the greater total change in gas prices. TO DO Exercise for Example 3 10. In Example 3, suppose that the changes in gas prices for week 4 are $.06 for company A and $.07 for company B and the changes in gas prices for week 5 are $.04 for company A and $.03 for company B. Which company has the greater total change in gas prices for the five weeks? page 5
Name Date LESSON 2.2 Real-Life Application: When Will I Ever Use This? For use with pages 74 79 Stockholders Stock is a right of ownership in a corporation. The stock is divided into a certain number of shares, and the corporation issues stockholders one or more stock certificates to show how many shares they hold. Stockholders may sell their stock whenever they want to, unless the corporation has some special rule to prevent it. Prices of stock change according to general business conditions and the earnings and future prospects of the corporation. If the business is doing well, stockholders may be able to sell their stock for a profit. If the business is not doing well, stockholders may have to take a loss. Stock is often traded under a contract called an option. An option allows the holder (owner) to buy or sell a certain amount of stock at a specific price within a designated time period. For example, an investor may believe that corn will increase in value. The investor can buy an option for corn at $2.22 with a call date of March 10th. Corn is currently on the market at $2.10. If the value of the corn stock rises above the price set ($2.22) by the option, the holder will profit. If the value of corn does not exceed the value of $2.22 by March 10th, the holder will lose their investment. In Exercises 1 6, use the following information. You decide to give the stock market a try. You buy one share in a company. You follow the stock market for five days, watching your specific company. 1. Over the five-day period your stock does the following: gains 2 cents, loses 10 cents, gains 3 cents, gains 5 cents, and loses 4 cents. Find your net profit or loss for this five-day period. 2. You paid $8.54 for your share. After the five-day period, how much is your share worth? 3. As you look back over the five-day period, when would have been the best time for you to sell? (When would you have made the greatest profit from selling your share?) 4. Suppose you do not sell your share and watch the market for another five-day period. The results are: loses 3 cents, gains 5 cents, gains 7 cents, loses 2 cents, and gains 9 cents. Find the net profit or loss for this five-day period. 5. Using your answer from Exercise 2, find the value of your share after the ten-day period. 6. After the ten-day period, did you make a profit or suffer a loss? How much? page 6
page 7 Name Date LESSON 2.3 Study Guide For use with pages 80 85 GOAL Subtract real numbers. Subtraction Rule To subtract b from a, add the opposite of b to a. a b = a + (-b) Subtracting a positive is the same as adding a negative EXAMPLE 1 Subtract real numbers Find the difference. a. 12 ( 8) b. 15 11 a. 12 ( 8) = 12 + 8 = 20 b. 15 11 = 15 + ( 11) = 26 Add the opposite of 8 Add Add opposite of 11. Add. TO DO Exercises for Example 1 Find the difference. 1. 9 ( 3) 2. 17 21 3. 23 ( 7) 4. 2.7 3.8 5. 11 ( 15) 7 6. 7 6 ( 9) EXAMPLE 2 Evaluate a variable expression Evaluate the expression x 5.1 y, when x = 3.7 and y = 2.3. x 5.1 y = 3.7 5.1 ( 2.3) Substitute 3.7 for x and 2.3 for y = 3.7 + ( 5.1) + 2.3 Add the opposites of 5.1 and 2.3. =0.9 Add. TO DO Exercises for Example 2 Evaluate the expression when x = 5 and y = 3. 7. x + y 7 8. 9 y x 9. x (5 y)
EXAMPLE 3 Evaluate change Stock A share of a stock on the New York Stock Exchange was valued at $31.26 at the opening of trading. The value of the stock at closing was $27.97. What was the change in stock value? STEP 1 Write a verbal model of the situation. STEP 2 Choose a variable to represent what you have to find. Let C = change in stock value STEP 3 Find the change in value. C = 27.97 31.26 Substitute values. = 27.97 + ( 31.26) Add the opposite of 31.26. = 3.29 Add. The change in value of the stock was $3.29. (or The change in the value of the stock was a decrease of $3.29) TO DO Exercises for Example 3 Find the change in temperature. 10. From 12 o F to 2 o F 11. From 8 o C to 21 o C 12. From 57 o F to 43 o F 13. From 11 o C to 15 o C page 8
Name Date LESSON 2.4 Study Guide For use with pages 87 93 GOAL Multiply real numbers. Vocabulary: The number 1 is called the multiplicative identity. The Sign of a Product The product of two real numbers with the same sign is positive. Examples: ( 3)( 4) = 12 4(5) = 20 The product of two real numbers with different signs is negative. Example: 6(7) = 42 NOTE: An odd number of negative factors will have a negative product [ 3( 2)( 4) = 24] while an even number of negative factors will have a positive product [ 2( 3)( 4)( 2) = 48 ] Properties of Multiplication Commutative Property The order in which two numbers are multiplied does not change the product. 2 3 4 = 6 4 = 24 while 2 4 3 = 8 3 = 24 -- the order change but the product is the same Associative Property The way you group three numbers when multiplying does not change the product. (2 3) 4 = 2 (3 4) because 6 4 = 2 12 Multiplicative identity The product of a number and 1 is that number. 6(1) = 6 1( 8) = 8 Multiplicative property of Zero The product of a number and 0 is 0. 24(0) = 0 Multiplication property of 1 The product of a number and 1 is the opposite of the number. Examples: 3( 1)=3 ; 6( 1) = 6 EXAMPLE 1 Multiply real numbers Find the product. a. ( 7)( 3) b. 4( 2)( 6) a. ( 7)( 3) = 21 Same signs; product is positive. b. 4( 2)( 6) = ( 8)( 6) Multiply 4 and 2; product is negative. = 48 Same signs; product is positive. TO DO Exercises for Example 1 Find the product. 1. 5(4) 2. 9( 8) 3. 12( 6) 4. ( 4) 1 2 5. 4( 8)(5) 6. 8( 2)( 8) page 9
EXAMPLE 2 Identify the properties of multiplication Identify the property being illustrated. Statement Property Illustrated a. x 5 = 5 x Commutative property of multiplication b. 6 ( 1) = 6 Multiplicative property of 1 c. ( 3 x) 2 = 3 (x 2) Associative property of multiplication TO DO Exercises for Example 2 Identify the property being illustrated. 7. 5 1 = 5 8. [5 ( 2)] ( 3) = 5 [( 2) ( 3)] 9. 0 11 = 0 10. 3 ( 12) = 12 ( 3) EXAMPLE 3 Use properties of multiplication Find the product ( 3x) ( 2). Justify your steps. ( 3x) ( 2) = ( 2) ( 3x) Commutative property of multiplication =[2 ( 3)]x Associative Property of multiplication = 6x Product of 2 and 3 is 6. TO DO Exercises for Example 3 Find the product. Justify your steps. 11. 9( x) 12. w( 3)(12) 13. ( 8)(5)( z) page 10
Name LESSON 2.5 Study Guide For use with pages 96 101 Date GOAL Apply the distributive property. Vocabulary Equivalent expressions: Two expressions that have the same value for all values of the variable. The distributive property can be used find the product of a number and a sum or the product of a number and a difference: Words The product of a number a and the sum of (b + c): The product of a number a and the difference (b c): NOTE: Each part of the expression in the parenthesis is multiplied by the number in front of the parenthesis. The parts of an expression that are added together are called terms - x 2 ; 2x ; 8 Algebra a(b + c) = ab + ac (b + c)a = ba + ca a(b c) = ab ac (b c)a = ba ca Terms The number part of a term with a variable part is called the coefficient of the term -1 ; 2 - x 2 + 2x + 8 A constant term has a number part but no variable part. 8 - x 2 + 2x + 8 Like terms are terms that have the same variable parts raised to the same power. (All constant terms are also like terms.) There are no like terms in the given expression Coefficients Constant EXAMPLE 1 Apply the distributive property Use the distributive property to write an equivalent expression. a. 7(x + 3) b. 2y(3y + 8) a. 7(x + 3) = 7(x) + 7(3) =7x + 21 b. 2y(3y + 8) = 2y(3y) + ( 2y)(8) = 6y 2 16y. Distribute 7. Simplify. Distribute 2y Simplify TO DO Exercises for Example 1 Use the distributive property to write an equivalent expression. 1. 8(x 3) 2. 3(4z 5) 3. (7 3m) 1 4. (9n + 12) 3 5. ( 2p + 1)( 3p) page 11
EXAMPLE 2 Identify parts of an expression Identify the terms, like terms, coefficients, and constant terms of the expression 11 8y + 6 + 3y. Suggestion: Write the expression as a sum: 11 + ( 8y) + 6 + 3y first, then identify the parts Terms: 11, 8y, 6, 3y Like terms: 8y and 3y, 11 and 6 Coefficients: 8, 3 Constant terms: 11, 6 TO DO Exercises for Example 2 Identify the terms, like terms, coefficients, and constant terms of the expression. 6. 7p 12 3p 8 7. 5t 2 + 7q 11q + 9t 2 EXAMPLE 3 Solve a multi-step problem Beverages Every weekday after tennis practice, you either buy a bottle of milk for $.75 or a bottle of juice for $.85. You practiced 20 days in the past month. Find the cost of beverages if you buy 12 bottles of milk. Let b be the number of beverages bought. STEP 1 Write a verbal model. Then write an equation. C = 0.75 b + 0.85 (20 b) C = 0.75b + 0.85(20 b) = 0.75b + 17 0.85b = 17 0.1b STEP 2 Find the value of C when b = 12. C = 17 0.1b Write equation. = 17 0.1(12) Substitute 12 for b. = 15.80 Simplify. Since you practiced for 20 days (and purchased beverages each day) and you bought b bottle of milk, then the number of bottles of juice would equal 20 minus the amount of bottles of milk purchased OR 20 - b You spend $15.80 on beverages after tennis practice. Exercise for Example 3 8. Using the same scenario in Example 3, suppose you practice 25 days the next month and buy 18 bottles of milk. How much do you spend on drinks? page 12
Name Date LESSON 2.6 Study Guide For use with pages 103 108 GOAL Divide real numbers. 1 Vocabulary The reciprocal of a nonzero number a, written a, is called the multiplicative inverse of a. The mean of the average of a set of data; add the data and divide by the total number of terms. Multiplicative inverses Property The product of a nonzero number and its multiplicative inverse is 1. 1 Algebra: a = 1, a 0 a EXAMPLES: Division Rule To divide a number a by a nonzero number b, multiply a by the multiplicative inverse of b. Example: Keep the first number the same, Flip the number after the sign, and Change the operation sign The Sign of a Quotient The quotient of two real numbers with the same sign is positive. The quotient of two real numbers with different signs is negative. The quotient of 0 and any nonzero real number is 0. 0 is in the numerator of a fraction; when it is in the denominator you cannot divide by it! EXAMPLE 1 Find multiplicative inverses of numbers Find the multiplicative inverse of the number. 7 a. 8 b. 2 7 8 7 8 a. The multiplicative inverse of is because = 1. 8 7 8 7 1 1 b. The multiplicative inverse of 2 is because 2 = 1. 2 2 page 13
TO DO Exercises for Example 1 Find the multiplicative inverse of the number. 5 11 1. 7 2. 8 3. 14 4. 1 EXAMPLE 2 Divide real numbers Find the quotient. 2 a. 24 3 3 b. 7 1 2 7 2 3 a. 24 = 24 = = 36 3 2 3 1 6 b. = 3 ( 2) = 7 2 7 7 TO DO Exercises for Example 2 Find the quotient. 5. 27 ( 3). 3 6. 8 7 4 2 7. 7 5 ( 3) EXAMPLE 3 Simplify an expression Simplify an expression 48x 12 4 48x +12 4 = ( 48x +12) 4 Rewrite fraction as division expression. = ( 48x +12) 1 4 Division rule 1 1 = 48x + 12 Distributive property 4 4 = 12x + 3 Simplify. TO DO Exercises for Example 3 Find the quotient. 8. 9. 10. 7x 28 14 20x +15 5 2x 21 page 14
Chapter 2 Practice page 15
Chapter 2 Practice Identify what set of numbers the following belong to. Give all possible correct answer. 39. 16 40. 8 41. 10 42. page 16
The following are properties of multiplication and addition. State the property that each example best illustrates. 43. 5(x) = x(5) 44. (3 + 21) + 9 = 3 + (21 + 9) 45. 46. 8 + 15 + 12 = 8 + 12 + 15 47. ( ) (6x + 21) = 4x + 14 48. -18 + 18 = 0 49. 25(1) = 25 50. -1(16) = -16 51. 3(4 5) = (3 4)5 52. 25(0) = 0 The following terms have descriptions provided in random order. Match each term with its best fit answer. 53. absolute value A. Any number that can be written as a/b (the quotient of two intergers),where b 0 54. Additive inverses B. Two expressions that have the same value for all values of the variable 55. Irrational numbers C. The number part of a term with a variable part 56. Integers D. The parts of an expression that are added together 57. Real numbers E. Both the set of irrational and rational numbers 58. Natural numbers F. Whole numbers and their opposites 59. Opposites G. Two numbers whose sum if zero 60. Rational numbers H. Two numbers that are the same distance from zero on a number line, but on different sides of zero 61. Equivalent expressions I. Not negative, not zero, no fractional parts, no decimal parts 62. Coefficient J. A number that cannot be written as the quotient of two integers 63. Whole number K. The distance from zero to the number on a number line 64. Constant term L. Terms with the same variables to the same powers 65. Terms M. Numerical term with no variable 66. Like terms O. Contains no decimals, no fractions, no negatives, but does contain 0 page 17