Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency Specification Sheet Discuss Numerical Fluency Problems with your students by sharing their multiple correct strategies. This sharing will help students become flexible, efficient, and accurate with numerical reasoning while learning the TEKS deeply. NF Problem 6-1 6-2 6-3 Matrix # TEKS # TAKS Obj # 223 501 223 501 223 6.5 6.9A 6.5 6.9A 6.5 2 5 2 5 2 Answers 1 2 C B 1 2 C A 1 2 A There are 12 combinations of a pair of pants, a shirt, and a pair of shoes as listed below. 501 6.9A 5 Navy, red, white Navy, red, black Navy, green, white Navy, green, black Navy, orange, white Navy, orange, black 127 501 6.2B 6.9A 1 5 1 2 3 4 6-4 502 6.9B 5 5 6 127 501 6.2B 6.9A 1 5 1 2 3 4 6-5 502 6.9B 5 5 6 Austin ISD Secondary Mathematics Department Khaki, red, white Khaki, red, black Khaki, green, white Khaki, green, black Khaki, orange, white Khaki, orange, black 1, 2, 3, 4, 5, 6 All outcomes are equally likely as long as it is a fair (balanced) number cube. 1 P(6) = 6 1 1 1 3 1 P(2)+P(4)+P(6)= + + = or 6 6 6 6 2 1 1 6 2 4 2 1 – P(3)-P(6) = 1 - - = - = or 6 6 6 6 6 3 P(10)=0 1, 2, 3, 4, 5, 6, 7, 8 All outcomes are equally likely since each section of the circular spinner has equal area. 1 P(7)= 8 1 1 1 3 P(3)+P(4)+P(5)= + + = 8 8 8 8 1 1 1 1 1 4 P(1)+P(4)+P(6)+P(8)= + + + = or 8 8 8 8 8 2 P(12)=0 6th Six Weeks 2009-2010 Page 1 of 36 Grade 6 Mathematics NF Problem Numerical Fluency Problems Matrix # TEKS # TAKS Obj # 127 220 501 6.2B 6.3C 6.9A 1 2 5 Answers 1 2 3 6-6 4 502 6.9B 5 5 6 127 220 501 6.2B 6.3C 6.9A 1 2 5 1 2 3 4 6-7 502 6.9B 5 5 6 6-8 6-9 6-10 6-11 6-12 6-13 117 502 117 502 110 117 502 6.1B 6.9B 6.1B 6.9B 6.1A 6.1B 6.9B 1 5 1 5 1 1 5 1 2 117 6.1B 1 220 6.3C 2 502 6.9B 5 501 6.9A 5 502 6.9B 5 127 6.2B 1 135 6.2D 1 Austin ISD Secondary Mathematics Department Black ball, checkered ball, white ball Not all outcomes are equally likely because there are a different number of black, checkered, and white balls. 3 P(black)= 8 1 7 P(not white)=1- = 8 8 25 checkered balls because half of the balls are checkered. 5 50 3 5 P(not black)=1- = and = so 50 balls can be expected 8 80 8 8 not to be black. One, Two, Three, Four Not all outcomes are equally likely because “One” on the spinner has more area than the other outcomes. 1 P(Three)= 5 2 P(One)= 5 1 8 Since , it can be expected that “Three” can be landed on 5 40 the spinner for 8 of 40 spins. 2 20 Since , it can be expected the spinner was spun 50 times 5 50 if the spinner landed on “One” 20 times. B D 1 2 D D 1 2 B A 1 2 B B 1 There are 9 possible pairs of outcomes as listed below. Red, red Blue, red Yellow, red Red, yellow Blue, yellow Yellow, yellow Red, blue Blue, blue Yellow, blue 3 1 9 3 2 P(match)= 1 Estimate: a little more than 6; Exact: 6 2 3 20 Estimate: a little less than 6; Exact: 5.62 6th Six Weeks 2009-2010 Page 2 of 36 Grade 6 Mathematics Numerical Fluency Problems NF Problem Matrix # TEKS # TAKS Obj # 6-14 133 6.2C 1 127 135 502 418 127 135 127 6.2B 6.2D 6.9B 6.8D 6.2B 6.2D 6.2B 1 1 5 4 1 1 1 605 6.11A 6 127 6.2B 1 605 6.11A 6 127 6.2B 1 605 6.11A 6 127 6.2B 1 605 6.11A 6 131 605 L 6.11A 6 610 6.12A 6 131 605 610 112A 117 133 223 L 6.11A 6.12A 6.1F 6.1B 6.2C 6.5 6 6 1 1 1 2 6-15 6-16 6-17 6-18 6-19 6-20 6-21 6-22 6-23 6-24 6-25 1 $919 1 2 Estimate: a little more than 8; Exact: 8.18 Estimate: a little less than 4; Exact: 3.95 1 2 3/5 222 1 2 Estimate: about 18.25: Exact: 18.5 Estimate: about 14.5: Exact: 14.62 1 2 1 2 1 2 1 2 1 6-26 606 6.11B 6 6-27 133 220 127 6.2C 6.3C 6.2B 1 2 1 6-28 Answers Austin ISD Secondary Mathematics Department 2 1 2 1 of the cabbages 3 5 3 hours or 3 hours 25 minutes 12 1 2 inches 2 5 of Clyde’s allowance 8 5 4 pounds of apples 8 4 of the cookies in the cookie jar 9 0.52 meters or 52 cm $4.50 1155 miles 3 9 yards of blue string 5 1 of the pizza 8 12 boys 1. 120 2. 4.375 1. 5 hours 2. C Who Student Teacher Parent Seat position 1, 4, 7, 10, 13, 16, 19 2, 5, 8, 11, 14, 17, 20 3, 6, 9, 12, 15, 18, 21 H 1. 500 2. x = 45.51 6th Six Weeks 2009-2010 Page 3 of 36 Grade 6 Mathematics NF Problem Matrix # TEKS # Numerical Fluency Problems TAKS Obj # 6-29 117 6.1B 1 6-30 122 6.2E 1 127 502 218 220 6.2B 6.9B 6.3A 6.3C 1 5 2 2 6-31 6-32 Austin ISD Secondary Mathematics Department Answers 1. 2. 3. 4. 1. 2. 3. 4. 1. 2. 0.6 repeating 0.875 3.20 2.5625 59 59 5 5 12 75% or 0.75 1. 5:2 2. 4 6th Six Weeks 2009-2010 Page 4 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-1 1 2 Jasmine bought 3 CDs that were originally priced at $12.98 each. Each CD was on sale for $4.50 off the original price. Which equation can be used to find t, the total price of the 3 CDs? A t = 12.98 – 4.50 B t = 3(12.98) – 4.50 C t = 3(12.98) – 3(4.50) D t = 3(4.50) – 3(12.98) Daniel takes one sandwich and one piece of fruit to school for lunch. He has a choice of a ham or turkey sandwich. His choices for fruit are an apple, a pear, or an orange. Which table lists all possible combinations for Daniel’s lunch? A B Fruit Apple Apple Pear Pear Sandwich Ham Ham C Ham Turkey Turkey Turkey Fruit Apple Apple Apple Orange Orange Orange Fruit Apple Apple Pear Pear Orange Orange Sandwich Ham Ham Ham Turkey Turkey Turkey Fruit Apple Apple Orange Orange Pear Pear Sandwich Ham Turkey Ham Turkey Sandwich Ham Turkey Ham Turkey Ham Turkey Austin ISD Secondary Mathematics Department D 6th Six Weeks 2009-2010 Page 5 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-2 1 2 Doris and 3 friends ordered an extra large pizza for $13.25, 4 large drinks for $1.50 each, and one order of chicken wings for $5.75. If they split these costs evenly, which equation can be used to find a, the amount in dollars and cents each person should pay, not including tax or tip? A a = 13.25 + 1.50 + 5.75 4 B a = 13.25 + 4 1.50 + (5.75 4) C a = (13.25 + 4 1.50 + 5.75) 4 D a = (13.25 + 3 1.50 + 5.75) 3 A penny and a nickel are tossed. Which tree diagram shows all the possible outcomes? Penny Heads Tails Nickel Heads Tails Heads Tails Penny Heads Tails Nickel Heads Tails Penny Heads Tails Nickel Tails Heads Penny Heads Tails Nickel Heads Tails Heads A B C D Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 6 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-3 1 2 If the cost of renting a bicycle is a basic fee of $8 plus an additional $2 per hour for each hour the bicycle is rented, which equation can be used to find c, the cost in dollars of the rental for h hours? A c = 2h + 8 B c = 8h + 2 C c = 2(h + 8) D c = 8(h + 2) The table below shows the colors of all of Juan’s pants, shirts, and shoes. Pants Navy Khaki Shirts Red Green Orange Shoes White Black If Juan wears one pair of pants, one shirt and one pair of shoes, what are all the possible combinations of clothing he can wear? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 7 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-4 A normal six-sided number cube as shown below is tossed. 1 What are all the possible outcomes? 2 Are all outcomes equally likely? Be prepared to explain your reasoning. 3 What is the probability that a 6 will be rolled? 4 What is the probability that an even number will be rolled? 5 What is the probability that a multiple of 3 will NOT be rolled? 6 What is the probability that a 10 will be rolled? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 8 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-5 The spinner shown below is spun once. 1 2 8 3 4 7 6 5 1 What are all the possible outcomes? 2 Are all outcomes equally likely? Be prepared to explain your reasoning. 3 What is the probability that a 7 will be spun? 4 What is the probability that a number greater than 2 but less than 6 will be spun? 5 What is the probability that a prime number will NOT be spun? 6 What is the probability that a 12 will be spun? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 9 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-6 One ball is randomly drawn from the container shown below. 1 What are all the possible outcomes? 2 Are all outcomes equally likely? Be prepared to explain your reasoning. 3 What is the probability of drawing a black ball? 4 What is the probability of NOT drawing a white ball? 5 If one ball was randomly drawn from the container 50 times, how many would you expect to be checkered balls? 6 If one ball was randomly drawn from the container 80 times, how many would you expect NOT to be black balls? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 10 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-7 The spinner shown below is spun once. 1 What are all the possible outcomes? 2 Are all outcomes equally likely? Be prepared to explain your reasoning. 3 What is the probability that the spinner will land on “Three”? 4 What is the probability that the spinner will land on “One”? 5 If the spinner was spun 40 times, how many times would you expect the spinner to land on “Three”? 6 If the spinner landed on “One” 20 times, how many times would you expect the spinner to have been spun? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 11 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-8 Use the following information to answer problems 1 and 2 below. In a mathematics game, students created the following vocabulary word with their cards. P 1 2 R O P O R T I O N If one of the cards from the word “PROPORTION” was selected at random, what is the probability the card showed a vowel? A 3 10 B 2 5 C 1 10 D 3 5 If one of the cards from the word “PROPORTION” was selected at random, what is the probability the card did NOT show a P? A 20% B 40% C 60% D 80% Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 12 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-9 Use the following information to answer problems 1 and 2 below. Brianna has a bag with 10 blue marbles, 4 green marbles, 8 purple marbles, and 2 red marbles. 1 2 If Brianna draws one marble at random from the bag, what is the probability the marble will be purple? A 8.24% B 2 66 % 3 C 25% D 1 33 % 3 If Brianna wants the probability of picking a blue marble to 1 be , what could she do? 2 A Remove 1 green, 1 purple, and 1 red marble B Add 2 blue marbles C Add 2 green, 2 purple, and 2 red marbles D Add 4 blue marbles Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 13 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-10 Use the following information to answer problems 1 and 2 below. Each box pictured below holds the number of marbles stated below the box. In each box, half of the marbles are black. Also, in each box, there is only one red marble. 10 Marbles 1 2 100 Marbles 1000 Marbles If one marble is selected at random from the middle-sized box, what is the probability that the marble will NOT be black or red? A 48% B 49% C 50% D 51% Which box would give you the greatest chance of picking a red marble? A The box with the 10 marbles B The box with the 100 marbles C The box with the 1000 marbles D All three boxes have the same chance of picking a red marble Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 14 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-11 Use the following information to answer problems 1 and 2 below. At Mendez Middle School, in a survey of 100 students about their favorite pizza topping, 25 students chose pineapple. 1 2 According to the results of the survey, if a student is picked at random from Mendez Middle School, what is the chance that the student prefers pineapple as a pizza topping? A 1 out of 5 B 1 out of 4 C 1 out of 2 D 1 out of 6 Which proportion can be used to predict s, the number of students who like pineapple as a favorite topping, if 220 Mendez Middle School students are asked about their favorite topping? A 25 220 100 s B 1 s 4 220 C 25 100 220 s D 1 220 4 s Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 15 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-12 Use the following information to answer problems 1 and 2 below. Two bags each contain one red, one yellow, and one blue block as shown below. 1 If a pair of blocks, one block from each bag, is randomly selected, what are all the possible outcomes of the pairs of blocks that can be selected? 2 If a pair of blocks, one block from each bag, is randomly selected, what is the probability that the pair of blocks will have matching colors? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 16 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-13 First, state an estimated answer for each problem. Then find the exact answer. Write each answer in lowest terms. 1 2 1 9 3 5 4 Estimated Answer: _________________ Exact Answer: _____________________ Don’t Forget to Estimate!!! 2 14 8.38 Estimated Answer: _________________ Exact Answer: _____________________ Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 17 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-14 1. Alex earns $16 dollars per hour for his work at a factory. He works 32 hours per week and gets paid every 2 weeks. If the factory takes $105 out of each check for taxes how much money will Alex be paid? Be prepared to explain your reasoning. Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 18 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-15 First, state an estimated answer for each problem. Then find the exact answer. Write each answer in lowest terms. 1 1 3 4.68 2 Estimated Answer: _________________ Exact Answer: _____________________ Don’t Forget to Estimate!!! 2 5.7 1 3 4 Estimated Answer: _________________ Exact Answer: _____________________ Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 19 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-16 1. The National Weather Service states that there is a 60% chance of a thunderstorm. What is this probability written as fraction is simplest form? 2. The movie Gone with the Wind is 3 hours and 42 minutes long. How many minutes is this? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 20 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-17 First, state an estimated answer for each problem. Then find the exact answer. Write each answer in lowest terms. 1 1 19 2 7 9 4 12 3 Estimated Answer: _________________ Exact Answer: _____________________ Don’t Forget to Estimate!!! 2 4 13 14.12 1 5 10 Estimated Answer: _________________ Exact Answer: _____________________ Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 21 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-18 Write each answer in lowest terms. 1 of the cabbages 2 planted this spring on the community farm. If Mrs. Florio 1 picked of the cabbages, what fraction of the cabbages did 6 Mrs. Lee pick? 1 Mrs. Florio and Mrs. Lee together picked 2 Jill studied math for 2 2 3 hours and Spanish for 1 hours. 3 4 What was her total study time for math and Spanish? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 22 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-19 Write each answer in lowest terms. 5 inches on one side. 6 What is the perimeter of this triangle? 1 An equilateral triangle measures 2 Clyde expected to spend 3 of his allowance on candy but 4 1 of his allowance on candy. What 8 fraction of his allowance did Clyde save? actually spent only Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 23 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-20 Use the following information to answer problems 1 and 2 below. Write each answer in lowest terms. Mr. and Mrs. Garcia are preparing food for a family reunion. 3 3 pounds of apples to bake pies and 5 8 4 pounds of apples to make applesauce. How many more pounds of apples does he use for the pies than for the applesauce? 1 Mr. Garcia uses 10 2 Mrs. Garcia baked cookies and placed them in a cookie jar. 2 of the cookies in the cookie jar have chocolate chips, 9 1 while of the cookies have oatmeal and raisins. What 3 fraction of the cookies in the cookie jar does not have chocolate chips, oatmeal, or raisins? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 24 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-21 1 During track practice, Trent jumped 4.2 meters and Will jumped 3.68 meters. How much further did Trent jump than Will? 2 Clarissa went to a sandwich shop to buy her lunch. She bought a turkey sandwich for $3.10, a bag of pretzels for $0.88 and lemonade for $1.10. The tax was $0.42. How much change did Clarissa received back from a $10 bill? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 25 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-22 1 An airplane flew at an average speed of 462 miles for 2 1 2 hours. How far did the plane fly? 2 Simone is going to make shoelaces in school colors for 2 everyone on her basketball team. She will use of a yard 5 of blue string for one shoelace. There are 12 players on the basketball team. How many yards of blue string does Simone need to buy? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 26 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-23 1 Armando found 3 4 of a pizza leftover in the refrigerator. He ate 1 6 of the pizza in the refrigerator. How much of the pizza did he eat? 2 There are 32 students in Mrs. Hall’s class. 3 8 of the students are boys. How many boys are in Mrs. Hall’s class? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 27 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-24 1. What is the least common multiple of 5, 6 and 8? 2. Write 4 3/8 as a decimal. Be prepared to explain how you converted. Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 28 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-25 1. Michelle’ family is planning a trip from Dallas to San Antonio. Dallas is about 272 miles from San Antonio. If Brandon’s dad drives an average of 60 miles per hour, about how long will the trip take? 2. Let d represent the number of dogs that Mark walks in 1 day. Which expression shows the number of dogs Mark walks in 7 days? Be prepared to explain your reasoning. A. 7+d B. d-7 C. 7d D. d/7 Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 29 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-26 A parent group is planning an awards dinner for students, teachers, and parents. The parent group plans to seat the guests around a circular table that has seating for 30. The guests will be seated in the order of student, teacher, and parent, in a repeating pattern. Will the 20th guest be a student, a teacher or a parent? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 30 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-27 1. When Javier sits down today to read, he notices that he is on page 20 of a 200 page book. He decides to read 4 pages of this book every day until he is finished. If this pattern continues, what page of the book will Javier be on in 10 more days? Be prepared to explain your reasoning. F. 24 G. 44 H. 60 J. 130 Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 31 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency #6-28 1. You are making gift baskets that each contain 6 bananas, 4 oranges, and 5 apples. If you need to make 100 gift baskets, how many apples do you need? 2. What is the solution to the equation? 97.56 + X = 143.07 Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 32 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-29 Write each fraction or mixed number as a decimal. Show your work and be prepared to explain your strategies. 1. 2/3 2. 7/8 3. 3 1/5 4. 2 9/16 Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 33 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency #6-30 Evaluate each expression and be prepared to explain the Order of Operations. 1. 9 X 8 – 13 2. (9 X 8) - 13 3. 30 / 2 – 5 X 2 4. (30 / 2) - (5 X 2) Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 34 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-31 1. Paula will use 15 ft of ribbon to decorate gift baskets for her friends. If each basket requires 1.2 feet of ribbon, how many baskets can she decorate? 2. There is a 25% chance of snow on Friday. What is the probability that it will NOT snow on Friday? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 35 of 36 Grade 6 Mathematics Numerical Fluency Problems Numerical Fluency 6-32 1. There were 18 teachers and 45 students registered to participate in a 5K walk-a-thon. Which ratio accurately compares the number of students to the number of teachers? A. 1 to 5 B. 5:2 C. 3 to 15 D. 18 to 45 2. Katie drives at an average rate of 60 miles per hour. How many hours will it take her to drive 240 miles? Austin ISD Secondary Mathematics Department 6th Six Weeks 2009-2010 Page 36 of 36