Reteaching Subtracting Real Numbers

Name Date Class Reteaching Subtracting Real Numbers 6 You have added real numbers. Now you will subtract real numbers. Two numbers with the same absolute value but different signs are called opposites. The opposite of a number is also called the additive inverse. The sum of a number and its additive inverse is 0. To Subtract Real Numbers To subtract a number, add its inverse. Then follow the rules for adding real numbers. Find the difference (6) 12. Step 1: To subtract a number, add its inverse. (6) (12) Step 2: Add the absolute values. 6 12 18 Step 3: Use the sign of the numbers. The sign is negative. (6) 12 18 Find the difference (8) (15). Step 1: To subtract a number, add its inverse. (8) 15 Step 2: Find the difference of the absolute values. 15 8 7 Step 2: Find the sign of the number with the largest absolute value. 15 8, so the sign is positive. (8) (15) 7 Complete the steps to find each difference. 1. (6) (11) (6) (11) (6) 11 11 (6) 5 11 6, so the sign of the answer is positive. (6) (11) 5 2. (19) 3 (19) 3 (19) (3) 19 3 22 The sign of the numbers is negative, so the sign of the answer is negative. (19) 3 22 Find each difference. 3. 17 (12) 5. 25 (32) 57 5. 5 8 1 7 8 6. 5 6 1 12 7. (9.1) 2.6 11.7 8. 7 26 19 11 12 9. For safety, scuba divers usually do not dive deeper than 0 meters below sea level. A diver in a helmet suit can safely dive about 21 meters deeper than a scuba diver. What is the maximum safe depth for a helmet suit diver in relation to sea level? 61 m Saxon. All rights reserved. 11 Saxon Algebra 1

Reteaching continued 6 Closure A set is closed under a given operation if the outcome of the operation on any two members of the set is also a member of the set. Determine whether the statement is true or false. Give a counterexample if the statement is false. The set of natural numbers is closed under subtraction. Subtract two natural numbers: 3 2 1 2 2 2 3 1 The last statement is a counterexample, so the statement is false. Complete the steps to determine whether the statement is true or false. Write a counterexample if the statement is false. 10. The set A {1, 0, 1} is closed under subtraction. Subtract two numbers in the set: 1 0 1 0 1 1 1 1 2 The last statement is a counterexample so the statement is false. Determine whether each statement is true or false. Give a counterexample for false statements. 11. The set of even integers, {...,, 2, 0, 2,,...}, is closed under subtraction. true 12. The set of irrational numbers is closed under subtraction. false; sample counterexample: 2 2 0 13. The set of odd integers plus zero, {..., 5, 3, 1, 0, 1, 3, 5,...} is closed under subtraction. false; sample counterexample: 3 1 2 1. The set of integers that are a multiple of, {..., 12, 8,, 0,, 8, 12,...} is closed under subtraction. true Saxon. All rights reserved. 12 Saxon Algebra 1

Name Date Class Reteaching Simplifying and Comparing Expressions with Symbols of Inclusion 7 You have used the order of operations to simplify expressions. Now you will apply this concept to simplifying expressions within symbols of inclusion. Symbols of inclusion indicate which numbers, variables, and operations are parts of the same term. Some symbols of inclusion are fraction bars, absolute value symbols, parentheses, braces, and brackets. Simplify 3[11 (12 9) 2 ] 7. Justify each step. 3[11 (12 9) 2 ] 7 3[11 3 2 ] 7 3[11 9] 7 3 20 7 60 7 67 Simplify inside the parentheses. Evaluate the exponent. Add inside the brackets. Multiply. Add. Complete the steps to simplify the expression. 1. 6z [2z 5 7 ] 3 6z [2z 5 7 ] 3 6z [2z 2 ] 3 6z [2z 2] 3 2z 2z 2 3 Subtract inside absolute value symbols. Simplify absolute value. Simplify the numerator. 8z 2z 2 10z 2 Simplify the fraction. Add. Simplify. 2. 11 19 15 23 3. (18 3) (1 10 6 ) 6. [ (9 2) 2 2(13 ) ] 3 3 5. 5 [ 2 (11 9) 3 ] 20 6. 1 2 3(7 2) 2 5 22 7. [ (16 7) 2 3 7 6 2 ] 3 17 Saxon. All rights reserved. 13 Saxon Algebra 1

Reteaching continued 7 Compare the expressions. Use,, or. [ 21 3(9 5) 2 ] 50 (12 7) 2 [ 19 3(21 15) ] Simplify each expression. Then compare. [ 21 3 (9 5) 2 ] 50 (12 7) 2 [ 19 3 (21 15) ] [ 21 3 () 2 ] 50 (5) 2 [ 19 3 (21 15) ] [ 21 3 16 ] 50 25 [ 19 3 (21 15) ] [ 21 8 ] 50 25 [ 19 3 (6) ] [ 27 ] 50 25 [ 19 18 ] 50 25 1 2 Since 23 2, [ 21 3 (9 5) 2 ] 50 (12 7) 2 [ 19 3 (21 15) ]. Complete the steps to simplify each expression. Compare the expressions. Use,, or. 8. 2(17 8) [ 3 2 (8 6) ] [ 3 (2) 3 ] (5 2) 2 (17 8) [ 3 2 (8 6) ] [ 3 (2) 3 ] (5 2) 2 (17 8) [ 3 2 2 ] [ 3 ( 8 ) ] (5 2) 2 (17 8) [ 9 2 ] (2) (5 2) 2(17 8) 7 2 ( 7 ) 2( 9 ) 7 2 28 18 7 11 Since11, 2(17 8) [ 3 2 (8 6) ] [ 3 (2) 3 ] (5 2). Compare the expressions. Use,, or. 9. 6 2 [ (3 2) 2 3 ] 5(9 3) [ 6 (2) 3 ] 10. 3 [ 6 2 5(22 2 ) 1 ] [(1 9) 2 6 ] 2 3 11. 9 [ 3 19 3 5 ] 2 23 8 3 2 [ 2(6 3) + 3 2 ] Saxon. All rights reserved. 1 Saxon Algebra 1

Name Date Class Reteaching Using Unit Analysis to Convert Measures 8 You have used ratios to compare two quantities. Now you will use unit ratios to convert measures into different units. Unit analysis is the process of converting measures into different units. The peregrine falcon can reach speeds up to 200 miles per hour. How fast is this in yards per hour? Step 1: Identify the known and missing information. 200 mi? yd So, the conversion is 200 mi? yd. 1 hour 1 hour Step 2: Equate units. 1760 yd 1 mi 1,760 yd So, the unit ratio is, or 1mi 1 mi 1760 yd. Step 3: Write the multiplication sentence. Then multiply. 200 mi 1760 yd 1 hr 1 mi 200 mi 1 hr 1760 yd 1 mi 200 1760 yd 1 hr 352,000 yd 1 hr Cancel out common factors. Multiply. Write the ratio of yards per hour. The peregrine falcon can reach speeds up to 352,000 yards per hour. 1. Alberto Contador won the 2007 Tour de France with an average speed of about 39 kilometers per hour. What was Alberto s average speed in meters per hour? 39 km 1hr 1 km 39 1000 m 1 hr 1000 m 39,000 m 1 hr Alberto Contador s average speed was about 39,000 meters per hour. 2. Some elephants can eat up to 660 pounds of food per day. How much food can an elephant eat in tons per day? One ton is equal to 2000 pounds. 0.33 t/d 3. A sprinkler with a flow rate of 2 gallons per minute is watering a lawn. What is the flow rate of the sprinkler in gallons per hour? 120 gal/h Saxon. All rights reserved. 15 Saxon Algebra 1

Reteaching continued 8 Remember that area is measured in square units and volume is measured in cubic units. A covered patio measures 6.25 yards by 5 yards. What is the area of the patio in square feet? Step 1: Find the area of the patio. 6.25 yd 5 yd 31.25 yd 2 So, the conversion is 31.25 yd 2? ft 2. Step 2: Equate units. 1 yd 3 ft So, the unit ratio is 1 yd, or 3 ft 3 ft 1 yd. Step 3: Write the multiplication sentence. Then multiply. 31.25 yd yd 3 ft 3 ft 1 yd 1 yd 31.25 yd yd 3 ft 1 yd 31.25 3 ft 3 ft 1 3 ft 1 yd Cancel out common factors. Multiply. 281.25 ft 2 Write the area in square feet. The area of the outdoor patio is 281.25 square feet.. Mr. Greene s yard is 50 feet by 20 feet. He wants to buy sod to cover his yard. Each piece of sod is 1-yard square. What is the area of Mr. Greene s yard in square yards? 2 50 ft 20 ft 1000 ft 1000 ft ft 1 yd 3 ft 1 yd 3 ft 1000 1 yd 1yd 111 yd 2 9 The area of Mr. Greene s yard is about 111 square yards. 5. An interior room is 12 feet by 17 feet. Carpet pieces are 1-yard square. How many square yards of carpet must be purchased to cover the floor of the room? 22 2 3 yd 2 6. A hose with a flow rate of 15 cubic feet per hour is filling a large aquarium. What is the flow rate of the hose in cubic inches per hour? 25,920 in 3 /h Saxon. All rights reserved. 16 Saxon Algebra 1

Name Date Class Reteaching Evaluating and Comparing Algebraic Expressions 9 You have simplified expressions containing only numbers and operations. Now you will evaluate expressions that contain numbers and/or variables. These expressions are called algebraic expressions. Evaluate the expression for n 2 and p 1. 6p 3n np Step 1: Substitute 2 for n and 1 for p in the expression. 6 1 3 2 2 1 Step 2: Simplify using the order of operations. 6 1 3 2 2 1 6 6 2 6 Evaluate the expression for a 1 and b 3. 2 (b a) 3 5 b 2 Step 1: Substitute 1 for a and 3 for b in the expression. 2 (3 1) 3 5 3 2 Step 2: Simplify using the order of operations. 2 (3 1) 3 5 3 2 2 (2) 3 5 3 2 2 8 5 9 16 5 61 Complete the steps to evaluate each expression for the given values. 1. 8c a ac; a 3, c 2 2. 2 y 2 3 x 2 y; x 3, y 5 8(2) 3 3 (2) 2 ( 5 ) 2 3 (3) 2 5 16 12 6 2(25) 3(9) 20 22 50 27 20 57 Evaluate each expression for the given values. 3. 3b ab 2; a 5, b 1. 5(c d) 6(c 2d); c, d 1 10 61 5. 7x 2y 3xy; x 5, y 2 6. 2st t 2 s; s 3, t 2 61 28 7. m p 3 7p; m 10, p 2 8. q r 2 2( q); q 6, r 1 25 9. A cable company charges a $36 monthly fee and then $2.99 for each movie ordered. They use the expression 36 2.99m, where m is the number of movies ordered, to find the total amount to charge for each month. How much would the cable company charge for the month of June if three movies were ordered? $.97 Saxon. All rights reserved. 17 Saxon Algebra 1

Reteaching continued 9 Compare the expressions when a 2 and b 1. Use,, or. b 2 ab 3 b 3 2 a 3 b Step 1: Substitute 2 for a and 1 for b in the expressions. (1) 2 (2)(1) 3 (1) 3 2 (2) 3 (1) Step 2: Simplify using the order of operations. (1) 2 (2)(1) 3 (1) 3 2 (2) 3 (1) 1 (2)(1) 3(1) 2 (8)(1) 1 (8)(1) 3 16 (1) 1 (8) 3 16 Step 3: Compare using,, or. Since 16, b 2 ab 3 b 3 2 a 3 b when a = 2 and b = 1. Complete the steps to compare the expressions when x 7 and y 2. Use,, or. 10. 2(x y) 3x 0.5y 12 x xy 2(x y) 3x 0.5y 12x xy 2(7 2) 3(7) 0.5(2) 2(9) 3(7) 0.5(2) 18 21 1 38 12 (7) (7)(2) 8 1 98 2(x y) 3x 0.5y 12 x xy when x 7 and y 2. Compare the expressions for the given values. Use <, >, or. 11. 3 a 2 b 2 b 2 a 2 b 5b; a 1, b 3 12. 2 h (2h k 2 ) h 2 (2h k ); h 3, k 3 13. 5 (x y) 2 2y (x y ) 2 ; x 5, y 2 1. k 2 2j j 3 2k; j 3, k 7 15. Cell phone company A charges a $30 monthly fee and 15 cents per minute. They use the expression 30 0.15m to find the total amount to charge for each month. Cell phone company B charges a $25 monthly fee and 17 cents per minute. They use the expression 25 0.17m to find the total amount to charge for each month. Which cell phone company charges less for 300 minutes during a month? A Saxon. All rights reserved. 18 Saxon Algebra 1

Name Date Class Reteaching Adding and Subtracting Real Numbers 10 You have solved problems by adding and subtracting pairs of numbers. Now you will solve problems by adding and subtracting three or more rational numbers. 7. Simplify 2 6 1 7 7 7 3 Step 1: Write the problem as addition. 7 3 2 6 7 7 1 7 Step 2: Group the terms with like signs. 2 7 1 Step 3: Add. 7 6 7 3 9 6 7 7 7 3 7 Simplify 2.1 0.22 5.25 (3.81). Step 1: Write the problem as addition. 2.1 (0.22) 5.25 (3.81) Step 2: Group the terms with like signs. 2.1 5.25 (0.22) (3.81) Step 3: Add. 7.39.03 3.36 Complete the steps to simplify each expression. 1. 1 9 9 5 2 9 5 9 9 9 1 1 5 9 9 2 9 9 9 5 9 7 9 2 9 2 9 2. 7.2 (2.78) 3. 5.12 7.2 (2.78) ( 3. ) 5.12 7.2 5.12 (2.78) ( 3. ) 12.36 6.18 6.18 Simplify. 3. 1 5 5 3 5 0. 3 11 11 5 11 2 11 5. 2 5 7 7 7 6 13 7 7 or 1 6 6. 7 3 13 13 5 13 3 13 5 13 7. 8.3 5.16 (7.22) 10.9 8. 22.15 1.56 29.0 5.33 9. 1.3 (2.7) 3.1 1.25 6.02 10. 3.19 (21.7) 3.79 15.2.8 11. Mrs. Lewis has $156 in her checking account. She writes two checks for $31.19 and $15.76 and makes one deposit for $119. What is her new balance? $228.05 Saxon. All rights reserved. 19 Saxon Algebra 1

Reteaching continued 10 Order the numbers from least to greatest. 7, 0.6, 2 10 5, 1 Step 1: Place each number on a number line. 2 7-0.6 5 10 1-2 -1 0 1 Step 2: To order the numbers, read the numbers on the number line from left to right. 0.6, 2 5, 7 10, 1 Complete the steps to order the numbers from least to greatest. 12. 1.75, 0.3, 1, 5-2 -1.75-5 - 1-1 0 The numbers in order from least to greatest 0.3 1 13. 3, 1.25, 5 5 10, 0. -2-1.25-5 10 0. 3 5-1 0 The numbers in order from least to are 1.75, 5, 1, 0.3. greatest are 1.25, 5 10, 0., 3 5. 1 Order from least to greatest. 1. 1 3,, 1., 2 15. 0.9, 9 3, 1, 0.05 10 2, 1., 3, 1 3 16. 5.8, 7, 3 5, 3.025 17. 3 8 1, 0.9, 0.05, 9 10 2, 5, 1.2, 0.8 6 7, 5.8, 3 5 8, 3.025 3 2, 1.2, 0.8, 5 6 18. 2 3, 7, 1.5, 2.7 19. 1.75, 8 5, 1.5, 0.3 10 2 3, 2.7, 7, 1.5 1.75, 1.5, 0.3, 8 5 10 20. Terra is deep-sea diving. She descends 20 feet below sea level, ascends 7 feet, descends 15 feet and ascends feet. What is her current position in relation to sea level? 2 ft below sea level or 2 ft Saxon. All rights reserved. 20 Saxon Algebra 1

Name Date Class Reteaching Determining the Probability of an Event INV 1 The probability of an event is the likelihood that the event will occur. You can estimate the probability of an event by performing an experiment. An experiment is an activity involving chance. The more trials you perform, the more accurate your estimate will be. experimental probability number of times an event occurs number of trials An experiment consists of randomly selecting marbles from a bag. Use the results in the table to find the experimental probability of each event. a. selecting a green marble number of times an event occurs 8 number of trials 12 8 15 5 8 0 b. not selecting a white marble number of times an event occurs number of trials 12 8 5 12 8 15 5 1 5 25 0 5 8 Outcome Frequency Red 12 Green 8 White 15 Blue 5 An experiment consists of randomly selecting pens from a bag. Use the results in the table and complete the steps to find the experimental probability of each event. 1. selecting a blue pen number of times an event occurs 9 number of trials 2. not selecting a red or a blue pen number of times an event occurs number of trials 15 5 9 7 9 36 1 15 7 15 5 9 7 22 36 11 18 Outcome Frequency Black 15 Red 5 Blue 9 Green 7 An experiment consists of selecting letters from a bag. Use the results in the table to find the experimental probability of each event. 8 3. Selecting the letter M 5. Selecting a vowel 5 2 15 Outcome M A R B L E Frequency 8 7 10 1 2. Not selecting the letter B 6. Not selecting L or E 7 9 29 5 Saxon. All rights reserved. 21 Saxon Algebra 1

Reteaching continued INV 1 You can use experimental probability to make predictions. A prediction is an estimate or guess about something that has not yet happened. Kiro is baking cookies to sell at the track team bake sale. He wants to sell only the cookies that weigh at least 30 grams. Out of the first 2 cookies, 3 cookies weigh less than 30 grams. a. What is the probability a cookie will weight less than 30 grams? Express the probability as a percent. number of times an event occurs number of trials 3 2 1 12.5% 8 b. If Kiro bakes 10 dozen cookies, about how many of the cookies are likely to weigh less than 30 grams? 12.5% (10 12) 0.125 120 15 15 cookies are likely to weigh less than 30 grams. Inspectors test 500 cars for air pollution emissions. Thirteen of them fail the test. Complete the steps to answer each question. 7. What is the probability that a car chosen at random will fail the test? Express your answer as a fraction and a percent. number of times an event occurs 2.6% number of trials 500 8. If 9000 cars are scheduled for the smog emissions test, about how many will likely fail? 13 2.6% 9000 0.026 9000 23 23 cars are likely to fail the test. A machine assembles 600 boxes. An inspector determines that 59 of the boxes have no defects. 9. What is the probability that a box chosen at random will have no defects? Express your answer as a fraction in lowest terms and as a percent. Fraction: 99 100; Percent: 99% 10. The machine assembles 800 boxes. About how many will have no defects? 792 boxes Saxon. All rights reserved. 22 Saxon Algebra 1

Name Date Class Reteaching Multiplying and Dividing Real Numbers 11 You have used multiplication and division to solve problems with whole numbers. Now you will use multiplication and division to solve problems with signed numbers. Multiplying and Dividing Signed Numbers The product or quotient of two numbers with the same sign is a positive number. The product or quotient of two numbers with opposite signs is a negative number. Simplify the expression 6(3). Justify your answer. Step 1: Determine the sign ( or ) for the product. The product of two numbers with opposite signs is negative. 6(3) Step 2: Multiply. 6(3) 18 18 Step 3: Justify your answer. Multiplying two numbers with opposite signs results in a negative product. Simplify the expression 10 (2). Justify your answer. Step 1: Determine the sign ( or ) for the quotient. The quotient of two numbers with the same sign is positive. 10 (2) Step 2: Divide. Step 3: Justify your answer. Dividing two numbers with the same sign results in a positive quotient. Complete the steps to simplify each expression. Justify your answer. 1. (5) 2. 16 (2) The product of two numbers with the The quotient of two numbers with same sign is positive. opposite signs is negative. (5) 20 16 (2) 8 Multiplying two numbers with the same Dividing two numbers with opposite sign results in a positive product. signs results in a negative quotient. Simplify each expression. 3. 6(7) 2. 3(1.6).8 5. (12)(5) 60 6. 21 7 3 7. 22 (11) 2 8. 2 (7) 6 9. Mr. Young s monthly bank withdrawals can be represented by $810.00. How can you represent his average withdrawal if the month had 30 days? $27 Saxon. All rights reserved. 23 Saxon Algebra 1

Reteaching continued 11 Two numbers whose product is 1 are called reciprocals. Dividing by a fraction is the same as multiplying by the reciprocal of the divisor. Evaluate the expression 9 1 2. Step 1: Determine the sign ( or ) for the quotient. The quotient of two numbers with the same sign is positive. 9 1 2 Step 2: Divide. 9 2 1 Multiply by the reciprocal of 1 2. 9 2 1 8 9 8 9 Complete the steps to evaluate each expression. 10. 9 16 3 8 The quotient of two numbers with opposite 11. 5 6 7 8 The quotient of two numbers with signs is negative. the same signs is positive. 9 16 3 9 8 16 8 72 3 8 3 2 5 6 7 8 5 6 8 7 0 2 20 21 Evaluate each expression. 12. 3 1 3 2 1 1. 1 7 9 12 21 16. 2 7 1 3 6 7 18. 2 1 9 2 9 13. 7 10 1 2 1 2 5 15. 1 11 12 3 11 17. 8 9 5 10 1 7 19. 7 7 8 32 9 9 20. The average annual minimum temperature in Pinecreek, Minnesota is 5 F. In Northwood, Iowa, the average annual minimum temperature is half as cold. What is the average annual minimum temperature in Northwood, Iowa? 22.5 F Saxon. All rights reserved. 2 Saxon Algebra 1

Name Date Class Reteaching Using the Properties of Real Numbers to Simplify Expressions 12 You have simplified expressions. Now you will use the properties of addition and multiplication to simplify expressions. The six properties below are true for all real numbers. Identity Property of Addition Identity Property of Multiplication Adding a and zero equals a. Multiplying a by 1 equals a. a 0 a 12 0 12 a 1 a 1 Commutative Property of Addition Order does not affect the sum of a and b. a b b a 2 8 8 2 Associative Property of Addition Grouping does not affect the sum of numbers. a b c a b c 2 3 2 3 Commutative Property of Multiplication Order does not affect the product of a and b. a b b a 6 3 3 6 Associative Property of Multiplication Grouping does not affect the product of numbers. a b c a b c 2 3 2 3 Complete the steps to identify each property illustrated. 1. (a b) c a (b c) 2. a 1 a (2 7) 5 2 (7 5) 8 1 a Grouping does not affect the product Multiplying by 1 equals a. of the numbers. Associative Property of Multiplication Identity Property of Multiplication Identify each property illustrated. 3. 9 9. (5 ) 6 5 ( 6) Commutative Property of Multiplication Associative Property of Addition 5. 1 6. (8 9) 3 8 (9 3) Identity Property of Associative Property of Multiplication Multiplication Saxon. All rights reserved. 25 Saxon Algebra 1

Reteaching continued 12 Simplify the expression 12x 1. Justify each step. 12 Step 1: 12x 1 12 Step 2: 1 12x Commutative Property of Multiplication 12 Step 3: 1 12 12 x Associative Property of Multiplication Step : 1 x x Identity Property of Multiplication Complete the steps to simplify the expression. Justify each step. 7. 12 x 16 8. 1 3 x 3 12 x 16 12 16 x Commutative Property of Addition 1 3 x 3 3 1 x Commutative Property 3 28 x Simplify. of Multiplication x Simplify. Simplify each expression. Justify each step. 9. 23 5x 7 10. 1 12 x 23 5x 7 1 12 x 23 7 5x Commutative Prop. of Add. 1 12 x Commutative Prop. of Mult. 30 5x Simplify. 11. 12 x 6 3x 12. 3 x 21 7 12 6 x 3x Commutative Prop. of Add. 1 3 x Simplify. 21 3 7 x Commutative Prop. of Mult. 18 7x Simplify. 9x Simplify. Saxon. All rights reserved. 26 Saxon Algebra 1

Name Date Class Reteaching Calculating and Comparing Square Roots 13 You have used exponents to find the square of a number. Now you will estimate the square root of a number and compare the values of square roots. Estimate the value 15 to the nearest integer. Step 1: 15 is not a perfect square. Step 2: 15 is between the perfect squares 9 and 16. Step 3: 15 is between 3 and because 9 3 and 16 Step : 15 is closer to the number because 15 is closer to 16 than 9. Estimate the value 0 to the nearest integer. Step 1: 0 is not a perfect square. Step 2: 0 is between the perfect squares 36 and 9. Step 3: 0 is between 6 and 7 because 36 6 and 9 7. Step : 0 is closer to the number 6 because 0 is closer to 36 than 9. Complete the steps to estimate the square root of each number. 1. Estimate 7 to the nearest integer. 2. Estimate 79 to the nearest integer. 7 is not a perfect square. 79 is not a perfect square. 7 is between the perfect squares and 9. 79 is between the perfect squares 6 7 is between 2 and 3 because and 81. 2 and 9 3. 79 is between 8 and 9 because 7 is closer to the number 3 because 7 6 8 and 81 9. is closer to 9 than. 79 is closer to the number 9 Estimate the square root of each number. 3. 5. 8 2 3 5. 3 6. 105 6 10 7. 8 8. 77 9 9 9. 2 10. 5 5 7 11. 119 12. 228 11 15 because 79 is closer to 81 than 6. Saxon. All rights reserved. 27 Saxon Algebra 1

Reteaching continued 13 Compare the expressions. Use,, or. 9 25 16 16 Step 1: Simplify the expression. 3 5 Step 2: Add. 8 8 Step 3: Compare. 8 8 Compare the expressions. Use,, or. 81 36 100 1 Step 1: Simplify the expression. 9 6 10 1 Step 2: Add. 15 11 Step 3: Compare. 15 11 Complete the steps to compare the expressions. Use,, or. 13. 9 25 36 1. 00 121 289 196 Step 1: Simplify the expression. Step 1: Simplify the expression. 7 2 5 6 20 11 17 1 Step 2: Add. Step 2: Add. 9 11 31 31 Step 3: Compare. Step 3: Compare. 9 11 31 31 Compare the expressions. Use,, or. 15. 25 6 9 9 16. 625 25 00 9 13 10 30 23 17. 9 81 1 16 18. 169 1 81 289 16 16 25 26 19. 1 6 121 81 20. 100 361 256 32 20 20 29 3 21. The area of a square classroom is 576 square feet. The area of the square classroom is four times the area of the square teacher s room. What are the dimensions of the teacher s room? 12 ft 12 ft Saxon. All rights reserved. 28 Saxon Algebra 1

Name Date Class You have multiplied and divided fractions. Now you will use fractions to determine the likelihood of an event. A bag contains 2 red marbles, 3 blue marbles, and 5 yellow marbles. What is the probability of randomly choosing a yellow marble? 5 yellow marbles Step 1: P(yellow) 10 marbles in all Step 2: P(yellow) 5 10 Step 3: 5 0.5 50% 10 Reteaching Determining the Theoretical Probability of an Event 1 A bag contains 2 red marbles, 3 blue marbles, and 5 yellow marbles. What is the probability of randomly not choosing a yellow marble? Step 1: P(not yellow) 2 red 3 blue 10 marbles in all Step 2: P(not yellow) 5 10 Step 3: 5 0.5 50% 10 Complete the steps to find the probability. Use the information from the examples above. 1. P(red) 2. P(red or yellow) P(red) 2 red marbles 10 marbles in all 2 P(red) 2 P(red or yellow) 2 red 5 yellow 10 marbles in all P(red or yellow) 7 10 10 10 0.2 20 % 7 0.7 70 % 10 Refer to the spinner at the right. Find the probability of each event. A H G B C F D E 3. P(A) 5. P(D or E) 7. P(A, B, or C) 1 8. P(B) 1 6. P(not H) 3 8 8. P(not G or not F) 1 8 7 8 3 Saxon. All rights reserved. 29 Saxon Algebra 1

Reteaching continued 1 You roll a number cube one time. What is the probability that you roll a number less than 3? Step 1: 1 and 2 are the numbers less than 3 on a number cube. Step 2: P(1 or 2) 2 1 6 3 You roll a number cube one time. Do you have a greater chance of rolling a 3 or a 5? Step 1: P(3) 1 6 Step 2: P(5) 1 6 Step 3: 1 1, so the chance of rolling a 3 is 6 6 the same as the chance of rolling a 5. A bag contains red marbles, 5 blue marbles, 3 white marbles, and 3 green marbles. Complete the steps to find the probability. 9. P(red or blue) 10. Do you have a greater chance of picking a red marble or a white marble? There are red marbles and 5 blue marbles. 9 P(red or blue) red or blue marbles 15 marbles in all 9 P(red or blue) 15 3 5 P(red) P(white) 15 3 15 3 15, so you have greater chance 15 of picking a red marble than picking a white marble. A bag contains red marbles, 5 blue marbles, 3 white marbles, and 3 green marbles. 11. P(blue or white) 8 15 12. P(not green) 12 15 5 13. Do you have a greater chance of picking a red marble or a blue marble? blue 1. Do you have a greater chance of picking a white marble or a blue marble? blue 15. Do you have a greater chance of picking a green marble or a white marble? equal Saxon. All rights reserved. 30 Saxon Algebra 1

Name Date Class Reteaching Using the Distributive Property to Simplify Expressions 15 You have used the Identity, Commutative, and Associative Properties of real numbers to simplify expressions. Now you will also use the Distributive Property to simplify numeric expressions. The Distributive Property For all real numbers a, b, c, a(b c) ab ac and a(b c) ab ac Simplify 2(5 7). 2(5 7) 2(5) 2(7) Distribute the 2. 10 1 Multiply. 2 Add. Simplify 3(6 ). 3(6 ) 3(6) 3() Distribute the 3. 18 12 Multiply. 6 Subtract. Simplify. (2 6). Rewrite the expression as 1(2 6), then distribute. 1(2 6) (1)(2) (1)(6) Distribute the 1. 2 6 Multiply. 8 Subtract. Simplify. (3 7). (3 7) ()(3) ()(7) 12 28 0 Distribute the. Multiply. Add. Complete the steps to simplify each expression. 1. 5(3 ) 2. 6(7 3) 5(3 ) 6(7 3) 5 (3) 5 () Distribute the 5. 6 (7) 6 (3) Distribute the 6 15 20 Multiply. 2 18 Multiply. 35 Add. 60 Subtract. Simplify each expression. 3. (8 3) 20. 3(7 2) 27 5. 3(9 5) 12 6. 12(3 2) 60 7. 10(6 2) 80 8. 15(5 2) 5 Saxon. All rights reserved. 31 Saxon Algebra 1

Reteaching continued 15 The Distributive Property can also be used to simplify algebraic expressions. When multiplying algebraic expressions, remember to add the exponents of powers with the same base. Simplify 3(z 8). 3(z 8) (3)(z) (3)(8) 3 Distribute the 3. 3z 2 Multiply. Simplify 2t(st t s). Multiply 2t by each of the terms in the parentheses. 2t(st t s) (2t)(st) (2t)(t ) (2t)(s) 2st 2 2t 2 8st Simplify (p 7)3. (p 7) 3 3(p) 3(7) Distribute the 3. 3(p) 21 Multiply. Simplify ab(ac 2 b 2 ). Multiply ab by each of the terms in the parentheses. ab(ac 2 b 2 ) (ab)(ac 2 ) (ab)(b 2 ) a 2 bc 2 ab 3 Complete the steps to simplify each expression. 9. (9 y)8 10. 3p 2 q(pq 2q 2 ) (9 y)8 3p 2 q(pq 2q 2 ) 8 (9) 8 (y) Distribute the 8. Multiply 3p 2 q by each of the terms in 72 8y Multiply. the parentheses. Simplify each expression. 3p 2 q (pq) 3p 2 q (2q 2 ) 3p 3 q 2 6p 2 q 3 11. (y 15)5 5y 75 12. 9(x 2) 9x 18 13. 7b(b 2 ac) 7b 3 28abc 1. 8z(2z 6) 16z 2 8z 15. (3p pq 3 )2pq 6p 2 q 2p 2 q 16. 5cw 2 (3c 2 cw 2 ) 15c 3 w 2 5c 2 w 17. Last spring members of the drama club sold tickets to the school play. They sold 300 tickets ahead of time and 250 tickets at the door. One ticket cost $6. Write an expression to show the total amount of money collected. Simplify the expression using the Distributive Property. 6(300 250); $3300 Saxon. All rights reserved. 32 Saxon Algebra 1

Name Date Class Reteaching Simplifying and Evaluating Variable Expressions 16 You have simplified expressions that contain variables. Now you will evaluate expressions that contain variables. To evaluate an expression that contains variables Substitute each variable in the expression with a given numeric value. Find the value of the expression. Evaluate the expression for the given values of the variables. m[n(m n)] for m 5 and n 2 5 2 m[n(m n)] m[n(m n)] 5[2(5 2)] 5[2(7)] 5[1] 70 Substitute each variable in the given value. Add. Multiply inside the brackets. Multiply. Complete the steps to evaluate the expression for the given values of the variables. 1. p(5qr) pq for p 2, q 3 and r p(5qr) pq 2 (5)(3)() (2)(3) 120 6 20 Substitute each variable in the given value. Multiply. Divide. Evaluate each expression for the given values of the variables. 2. 5c[z(c 2z)] for c 2 and z 1 0 3. 5j(2k j ) for j 1 2 and k 3 2.5. w [x z(w + z) w] for x 2, w 1, and z 3 3 5. (3d )(def )(2f ) for d 3, e 2, and f 1 3 12 Saxon. All rights reserved. 33 Saxon Algebra 1

Reteaching continued 16 A variable expression can often be simplified before it is evaluated. First simplify s(2 t ) s. Then evaluate it for s and t 2. Justify each step. s(2 t ) s 2s st s 3s st 3() ()(2) 12 8 Distributive Property Combine like terms. Substitute. Multiply. Add. First simplify x(2y x) 3xy. Then evaluate it for x 5 and y 3. Justify each step. x(2y x) 3xy 2xy x 2 3xy Distributive Property xy x 2 Combine like terms. (5)(3) (5) 2 Substitute. (5)(3) 25 Evaluate the exponent. 15 25 Multiply. 0 Add. Complete the steps to simplify each expression. Then evaluate for p 3 and q 2. Justify each step. 6. pq 3q(3 2p) q 7. 2q(p q) 3pq Distribute the 3q pq 9q 6pq q Combine 2pq 2 q 2 3pq 5pq 8q Like terms. 5pq 2 q 2 Substitute 5(3 ) (2) 8(2 ) 5( 3 ) (2) 2(2 ) 2 30 16 6 Multiply. Add. 5(3 ) (2) 2( ) 30 8 22 Simplify the expression. Then evaluate for x and y 2. Justify each step Distribute the 2q Combine Like terms. Substitute. Evaluate. the exponent. Multiply. Add. 8. x(3 y) 2x xy Sample: x (3 y) 2x xy 12x xy 2x xy 10x 3xy 10() 3()(2) 0 2 16 Distribute the x. Combine like terms. Substitute. Multiply. Subtract. Saxon. All rights reserved. 3 Saxon Algebra 1

Name Date Class Reteaching Translating Between Words and Algebraic Expressions 17 You have simplified and evaluated variable expressions. Now you will translate words into algebraic expressions. An algebraic expression, or variable expression, is an expression that contains at least one variable. Translating Words into Operations Words sum, total, more than, added, increased by, plus less, minus, decreased by, difference, less than product, times, multiplied quotient, divided by, divided into Operation Addition Subtraction Multiplication Division Write an algebraic expression for the phrase x plus 3. Find the word increased by in the table. It means addition. x 3 Write an algebraic expression for the phrase the quotient of r and 11. Find the word quotient in the table. It means division. r 11 Complete the steps to write an algebraic expression for each phrase. 1. the product of 7 and t 2. z decreased by 9 Find the word product in the table. Find the words decreased by in It means multiplication. the table. They mean subtraction. 7 t z 9 Write an algebraic expression for each phrase. 3. the total of m and 10. 16 divided by w m 10 16 w 5. the difference of q and 12 6. 17 multiplied by s q 12 17 s 7. Krista owns 3 more books than Laurie, who owns b books. Write the expression that shows the number of books Krista owns. b 3 Saxon. All rights reserved. 35 Saxon Algebra 1

Reteaching continued 17 You can also translate algebraic expressions into words. Use the table on the previous page to find the operation represented by the symbol. Then replace the symbol with the appropriate word(s). Use words to write the algebraic expression r 5 in two different ways. Find the subtraction operation in the table. Replace the subtraction symbol with the appropriate word(s). r minus 5 r decreased by 5 Use words to write the algebraic expression 9 d in two different ways. Find the multiplication operation in the table. Replace the multiplication symbol with the appropriate word(s). the product of 9 and d 9 times d Complete the steps to use words to write each algebraic expression in two different ways. 8. x 15 9. 21 z Find the addition operation in the table. Replace the addition symbol. Find the division operation in the table. Replace the division symbol. x plus 15 21 divided by z x increased by 15 the quotient of 21 and z Use words to write each algebraic expression in two different ways. 10. 21 11. 2v 1 k Sample: the quotient of Sample: 1 more than 2 times 21 and k, 21 divided by k v, the sum of 2 times v and 1 12. 2v 1 13. 1 3 w 8 Sample: 1 more than Sample: one-third w 2 times v, the sum of minus 8, the difference 2 times v and 1 of one-third w and 8 Write a sentence that could be described by each expression. 1. 2p 3 15. 1 3 a Sample: Erin has 3 more Sample: Ryan s sister than twice as many borrowed one-third of his paperclips as Janet. allowance plus $. Saxon. All rights reserved. 36 Saxon Algebra 1

Name Date Class Reteaching Combining Like Terms 18 You have learned that algebraic expressions are made up of terms. Now you will learn about like and unlike terms in algebraic expressions. Like terms are two or more terms that have the same variable or variables raised to the same power. Unlike terms are two or more terms with different variables, or with the same variable or variables raised to a different power. Example Type 3x, 8x, 99x Like Terms 5pqr, 2pqr, 955pqr Like Terms 2bc, 2cd, 2de Unlike Terms 3fg, 3fh, 3fj Unlike Terms Simplify the expression 16a 9a. 16a 9a (16 9)a Distribution Property 7a Simplify. Simplify the expression 7z (2z) (3z). 7z (2z) (3z) (7 2 3)z Distribution Property 2z Simplify. Complete the steps to simplify each expression. 1. 7d 3d (2d) 2. 3pq p 2qp 7d 3d (2d) 3pq p 2qp (7 3 2 )d Distributive Property 3pq 2qp p Rearrange the terms. 2d Simplify. 3pq 2pq p Rearrange the factors. (3 2 )pq p Distributive Property 5pq p Simplify. Simplify each expression. 3. 9a (3a) 2a. pq 3pq 8pq a 10pq 5. 3gh 2fgh 11gh 6. 5wx 7k 2xw 8gh 2fgh 7wx 7k Saxon. All rights reserved. 37 Saxon Algebra 1

Reteaching continued 18 You can also combine like terms with exponents. Example Type 5 xy 2, 12 xy 2, x y 2 Like Terms 5 y, 2 y, 955 y Like Terms 2x, 2 x 2, 2 x 3 Unlike Terms 3 a 2, 3 b 2, 3 c 2 Unlike Terms Simplify the expression 2 m 3 n n 2 n m 3 5 n 2. 2 m 3 n n 2 n m 3 5 n 2 2 m 3 n nm 3 n 2 5 n 2 Rearrange the terms. 2 m 3 n m 3 n n 2 5 n 2 Rearrange the factors. (2 1) m 3 n (1 5) n 2 Use the Distributive Property. 3 m 3 n 6 n 2 Simplify. Complete the steps to simplify the expression. 7. a 3 b 2 ba 3 b 2 a 3 5ab a 3 b 2 ba 3 b 2 a 3 5ab a 3 b 2 3 b 2 a 3 ba 5ab Rearrange the terms. a 3 b 2 3 a 3 b 2 ab 5ab Rearrange the factors. (1 3) a 3 b 2 ( 5) ab Use the Distributive Property. 2 a 3 b 2 ab Simplify. Simplify each expression. 8. jk jk 2 jk 2j k 2 9. 6 c 2 d cd 12 dc 2 2 3jk 3j k 6 c 2 d cd 10. 2 x 3 x 2 7 x qx 2 x 11. ab ab 2 ba 16 b 2 a x 3 x 2 q x 2 2 5ab 15a b 12. Emily is installing a rectangular vegetable garden in her yard. She wants the length of the garden to be 5 feet longer than 2 times the width. The diagram represents the measurements of her garden. Find the perimeter of the garden as a simplified variable expression. Then evaluate the expression for w 15 feet. P 6x 10; P 100 ft 2w + 5 w Saxon. All rights reserved. 38 Saxon Algebra 1

Name Date Class Reteaching Solving One-Step Equations by Adding or Subtracting 19 You have used addition and subtraction to combine real numbers. Now you will use addition and subtraction to solve one-step equations. An equation is a statement containing an equal sign. The two quantities on either side of the equal sign are equal. A solution of an equation that contains one variable is a value of the variable that makes the equation true. After you have solved an equation, substitute the solution for the variable to make sure you are correct. State whether the value of the variable is a solution of the equation. y 7 12 for y 5 y 7 12 (5) 7 12 Substitute 5 for y. 12 12 State whether the value of the variable is a solution of the equation. c 11 for c 7 c 11 (7) 11 Substitute 7 for c. 3 11 Complete the steps to determine if the value of the variable is a solution of the equation. 1. 1 z 8 for z 6 2. 9 m 13 for m 1 z 8 9 m 13 1 (6) 8 Substitute 6 for z. 9 m 13 1 6 8 9 ( ) 13 Substitute for m. 20 8 13 13 State whether the value of the variable is a solution of the equation. 3. a 5 for a 9. 10 q 3 for q 6 solution, 9 5 not a solution, 10 6 3 5. 15 20 h for h 5 6. x 2 for x 6 not a solution, 15 20 5 solution, 6 2 7. t 2 for t 2 8. 5 p 5, for p 10 not a solution, 2 2 solution, 5 10 5 9. g 3 5 for g 2 10. 8 w 3 for w 12 solution, 2 3 5 not a solution, 8 12 3 Saxon. All rights reserved. 39 Saxon Algebra 1

Reteaching continued 19 You can use properties of equality and inverse operations to find the solution of an equation. Properties of Equality Addition Property of Equality If you add the same number to both sides of an equation, the equation will still be true. Subtraction Property of Equality If you subtract the same number from both sides of an equation, the equation will still be true. Inverse operations are operations that undo each other. To solve an equation, use inverse operations to isolate the variable on one side of the equal sign. You must use the same operation on each side. Inverse Operations Add Subtract Multiply Divide Solve w 2 9. Then check the solution. w 2 9 _ 2 _ 2 w 11 Add 2 to both sides to undo the subtraction. Check Substitute 11 for w. w 2 9 (11) 2 9 9 9 Solve f 12. Then check the solution. f 12 f 8 Subtract from both sides to undo the addition. Check Substitute 8 for f. f 12 (8) 12 12 12 Complete the steps to solve the equation. Then check the solution. 11. z 8 6 Check Substitute 2 for z. z 8 6 z 8 6 8 8 Subtract 8 from both sides (2) 8 6 z 2 to undo the addition. 6 6 Solve. Then check the solution. 12. v 11 3 13. d 13 5 v 1; 1 11 3 d 8;8 13 5 1. 12 q 7 15. a 1 1 3 6 a 1 q 5;12 5 7 2 ; 1 2 1 3 1 6 16. A video game is on sale for $5, which is $1 off its regular price. What is the regular price of the video game? $59 Saxon. All rights reserved. 0 Saxon Algebra 1

Name Date Class You have used number lines to plot real numbers. Now you will graph real numbers on a coordinate plane. An ordered pair, or two numbers in parentheses, identifies a point on the plane. It is written as (x, y). The first number is the x-coordinate. It is the distance of the point left or right of the origin. If negative, it is to the left of the origin. The second number is the y-coordinate. It is the distance of the point above or below the origin. If negative, it is below the origin. Graph (2, 3) on a coordinate plane. Label the point. Start at the origin. Move 2 to the right. Move 3 up. - y (2,3) O x - Reteaching Graphing on a Coordinate Plane 20 II Quadrant - Graph (, 2) on a coordinate plane. Label the point. Start at the origin. Move to the left. Move 2 up. -2 2 III -2 Quadrant - y O (0,0) x 2 (-,2) - I Quadrant IV Quadrant O - y x Complete the steps to graph each ordered pair on a coordinate plane. 1. (2, ). 2. (3, 2) Start at the origin. Move 2 to the right. y (2,) O x - - Start at the origin. Move 3 to the left. Move up. Move 2 down O - (-3,-2) - y x Graph each ordered pair on a coordinate plane. Label each point. 3. (3, ). (3, 5) y y 6 6 (-3,) O x -6-6 - O x -6-6 (-3,-5) - -6-6 5. (0, 2) 6. (3, 0) 6 y 6 y O x -6-6 (0,-2) - -6 O (3,0) x -6-6 - -6 Saxon. All rights reserved. 1 Saxon Algebra 1

Reteaching continued 20 When there is a relationship between two variable quantities, the dependent variable always depends on the value chosen for the independent variable. Variables Independent variable: The variable whose value can be chosen. (Input variable) Dependent variable: The variable whose value is determined by the input value of another variable. (Output variable) Complete the table for the equation y 2x 1. Since the values of x are given, x is the independent variable. The dependent variable is y. Substitute 3 for x in the equation. x y y 2x 1 Write the equation. 3 7 1 3 y 2(3) 1 Substitute 3 for x. 0 1 y 6 1 Evalute. 1 0 y 7 2 Substitute the other values to complete the table. Complete the steps to fill in the table for the equation. 7. Complete the table for the equation y 3x 6 Substitute 1 for x in the equation. y 3x 6 Write the equation. y 3(1) 6 Substitute 1 for x. y 3 6 Evalute. y 9 Substitute the other values to complete the table. x y 1 9 0 6 1 3 2 0 Complete the table for each equation. 8. y 5x 2 x 2 1 1 1 9. y x 9 5 y 8 3 3 7 x 1 0 3 y 13 9 3 7 10. Anita knits scarves. The equation y 20x 80 represents her earnings, where x is the number of scarves sold and y is the money earned. Find the amount of money Anita earns when 5, 10, 15, and 20 scarves are sold. Make a graph to represent the equation y 20x 80. 20, 120, 220, 320 00 300 200 100 O y 5 10 15 20 x Saxon. All rights reserved. 2 Saxon Algebra 1