Middle School 6th grade Review Short Answer 1. Compare the fractions in each pair. Insert the correct sign: <, >, or =. Describe your reasoning. a. b. c. d. 2. Name three different fractions between and. 3. The diagram represents of a fifths fraction strip. Sketch the whole strip. 4. The drawing below shows part of an inch ruler. The picture has been enlarged. Each inch is broken into smaller pieces to measure more accurately than in whole inches. Label each mark on the ruler with a correct fraction. 5. You are invited to go out for pizza with several friends. When you get there, your friends are sitting in two separate groups. You can join either group. If you join the first group, there will be a total of 4 people in the group and you will be sharing 6 small pizzas. If you join the second group, there will be a total of 6 people in the group and you will be sharing 8 small pizzas. If pizza will be shared equally in each group, and you are very hungry, which would you rather join? Explain your choice. 6. Samuel is getting a snack for himself and his little brother. There are two muffins in the refrigerator. Samuel takes half of one muffin for himself and half of the other muffin for his little brother. His little brother complains that Samuel got more. Samuel says that he got and his brother got. What might be the problem? 7. Your best friend was absent when your class learned how to compare decimal numbers. Write a set of directions that would help your friend understand how to compare decimal numbers.
8. Decide whether each of the following statements is true or false. Explain. a. If you compare two fractions with the same denominator, the fraction with the greater numerator is greater. b. If you compare two fractions with the same numerator, the fraction with the greater denominator is greater. 9. In each figure, express the area shaded and the area not shaded as percents. a. b. shaded: not shaded: shaded: not shaded: c. shaded: not shaded: 10. In a recent survey of 600 people, 20% said chocolate chip cookie was their favorite ice cream. How many people in the survey favored chocolate chip cookie ice cream? Explain your answer. 11. Rename each of the decimal amounts as a fraction. a. 0.375 b. 0.6 c. 0.05
12. In each pair of pencils, the length of the new pencil is about what fraction of the length of the old pencil? a. b. 13. The average human body temperature is 98.6 (98 and ) Fahrenheit. For parts (a) (c), tell whether the thermometer shows this temperature. 14. Michael decided to make French toast for himself and his other four family members. He finds a recipe that calls for: 8 slices of bread 4 eggs 1 cup of milk 1 teaspoon of cinnamon a. If he makes a single recipe and shares the food equally with his family, how much French toast will each person receive? b. How much egg will each person have in their French toast? 15. The drawing shows the controls on a small, portable stereo system. Use the drawing to answer each of the following questions. Record all of your answers as fractions. a. What fraction of the total volume is the stereo playing? b. What fraction of the total bass output is the stereo playing?
c. What fraction of the total treble output is the stereo playing? d. If the volume of the stereo is turned down to half the current volume, what fraction of the total volume will be the new volume? Explain your reasoning. e. If the bass control on the stereo is adjusted up so that the stereo is playing at double the bass output it is playing at now, what fraction of the total bass output will be the new bass output? Explain your reasoning. 16. A bag contains 24 marbles (Note: You may want to use 24 cubes, chips, marbles, or other objects to help you solve this problem.) a. If 16 of the marbles are removed from the bag to play a game, what fraction of the marbles are left in the bag? b. Of the 16 marbles taken from the bag, one-fourth are put back in the bag. Now how many marbles are in the bag? Explain your reasoning. 17. Joey s father stops at the gas station to buy gas. The car has a 16-gallon tank, and the fuel gauge says there is of a tank of gas. a. How many gallons of gas are in the tank? b. If Joey s father buys 6 gallons of gas, what fraction of the tank will the car s fuel gauge read? c. What fraction of the gas tank is empty after Joey s father puts 6 gallons of gas in the tank? 18. Copy each number line below. In each case, two of the marks are labeled. Label the unlabeled marks with decimal numbers. a. b. c. d. 19. In one competition, the archery team had to shoot at targets from three different distances: 10 m, 20 m, and 30 m. The number of hits and the number of shots for each distance are given below. Write their score for each round as a fraction, a decimal, and a percent. a. at 10 m: 42 hits out of 50 shots b. at 20 m: 37 hits out of 50 shots c. at 30 m: 18 hits out of 50 shots 20. Blake was listening to his favorite radio station. He noticed that in one hour 15 songs were played and 3 of them were by the group from Michigan, RU Cold2. What percentage of the songs in that hour were by RU Cold2?
21. a. In stage A the middle one-third of a line segment is covered by a triangle. What fraction of the line is covered at stage A? What fraction is not covered? b. In stage B, the middle one-third of each of the two parts that were uncovered in stage A are covered. What fraction of the line is covered at stage B? What fraction is not covered? c. In stage C, the middle one-third of each of the parts that were not covered in stage B are covered. What fraction of the line is covered at stage C? What fraction is not covered? 22. For each part, state whether the sum of the fractions is less than, greater than, or equal to 1. Explain your thinking. a. b. c. d. e. f. g. h. i. 23. For each set of fractions, list all the possible pairs whose sum is between 1 and. Explain your thinking. a. b. c. d. e. f.
24. A recipe for granola cookies calls for cup of butter and cup of chopped nuts. Because Jane likes moist cookies without too many nuts, she decides to increase the amount of butter by half and decrease the amount of chopped nuts by half. a. How much butter is required for Jane s new recipe? Explain how you got your answer. b. What amount of chopped nuts is required for Jane s new recipe? Explain your reasoning. c. Since Jane increased the butter by half and decreased the nuts by half, is the combined amount of butter and nuts the same as in the original recipe? Explain why or why not. 25. LiAnn works in the Olde Tyme Soda Shoppe. The shop sells chocolate shakes, double chocolate shakes, and triple chocolate shakes. A chocolate shake uses cup of chocolate syrup, a double chocolate shake uses cup of chocolate syrup, and a triple chocolate shake uses each kind could she make with 3 cups of chocolate syrup? cup of chocolate syrup. How many shakes of 26. Three groups of students are sharing leftover pizza (all the same size originally). In which group does each student get the most pizza? Explain your choice. a. Six students equally share of a pizza. b. Three students equally share of a pizza. c. Four students equally share of a pizza. 27. Max noticed a pattern in some fraction division problems that he computed. gives the same answer as 6 3. gives the same answer as 7 3. gives the same answer as 9 2. gives the same answer as 4 5. Describe the pattern that Max found. Explain why it works.
Find the value of N. Show your work. 28. N + = a. b. d. d. 29. Last Wednesday it snowed 2 1 in. On Sunday it snowed 3 2 in. On which day did it snow more? How much 7 3 more? 30. Insert decimal points into the two factors so that each of the following problems give the correct product. Explain how you made the problems. 31. Which difference is greater? Explain your thinking. a. 7.3 4.9 or 8.5 3.2 b. 25.041 8.3 or 31.241 14.5 c. 0.57 0.008 or 0.6 0.044 32. For each list, identify the greatest value. Explain your thinking. a. 35.7, 35.07, 35.007 b. 608.9, 609.8, 690.8 c. 75.0605, 75.6050, 75.6500 33. James used a calculator to complete each computation. But he forgot to write the decimal point in each answer. Write the correct answer for each computation. Problem 5.7 + 6.09 + 4.2 1599 3.007 2.9 + 35.054 35161 14.5 8.07 6.2 23 Answer without decimal point Correct answer
34.The diagram below shows a rectangular plot of land cut into squares of 2.65 acres each. a. What is the acreage of the shaded region? Explain your reasoning. b. What is the acreage of the unshaded region? Explain your reasoning. c. In this area, land sells for $2475 per acre. i. What would the price of the shaded region be? ii. What would the price of the unshaded region be? d. In this area, owners pay property taxes of $13.50 per thousand dollars of property value. What is the total annual property tax for the shaded and unshaded regions combined? Explain your reasoning. Use the number sentence to help you solve the following problems. 35. 123 4 = 492 a. 12.3 4 b. 1.23 4 c. 0.123 4 d. 0.123 40 e. 0.123 400 f. 0.123 4000 36. 492 4 = 123 a. 492 40 b. 492 400 c. 492 4000 d. 49.2 4 e. 4.92 4 f. 0.492 4 37. For each of the following problems, estimate the product. Describe your reasoning. a. 2.4 0.8 b. 5.21 1.1 c. 1.29 8 d. 12.2 e. 74.6 1.5 f. 3.04 100 38. The Midtown Middle School cheerleaders earned $175 at a car wash. If this amount is 25% of the cost of a new set of uniforms, what is the total cost for a set of uniforms? Explain your reasoning. 39. Mary and Ms. Miller are ordering merchandise to sell in the student store. Ms. Miller says that the cost of notebooks is 125% of last year s cost. a. Explain what Ms. Miller means. b. If a notebook cost $2.00 last year, what will a notebook cost this year? Explain how you found your answer. 40. 3.5 is what percent of 14? Explain your reasoning.
41. What is 60% of 115? Explain your reasoning. 42. Solve the following computations. Show your work. a. 11.46 + 32.917 b. 8.29 3.112 c. 12.3 4.2 d. 36.8 0.8 43. Write a complete fact family for each problem. a. 12.4 3.2 = 9.2 b. 3.2 4.1 = 13.12 44. Find the area and perimeter of this rectangle. Explain how you found your answers. 45. Because the winters are very windy and snowy for Sarah Fieldler, her mom decides to build a small snow shelter for her children to wait in before the school bus arrives in the morning. Mrs. Fieldler has only enough wood to build a shelter whose floor has a total perimeter of 20 feet. a. Make a table of all the whole number possibilities for the length and width of the shelter. b. What dimensions should Mrs. Fieldler choose to have the greatest floor area in her shelter? c. What dimensions should Mrs. Fieldler choose to have the least floor area in her shelter? d. Township building codes require 3 square feet for each child in a snow shelter. Which shelter from part a will fit the most children? How many children is this?
46. Use the diagram below to answer the following questions. The squares on this grid are 1 centimeter long and 1 centimeter wide. a. What is the perimeter of the figure? Explain your reasoning and any strategies that you use. b. What is the area of the figure? Explain your reasoning and any strategies that you use. 47. Jason is planning to redecorate his bedroom. He measured the room and made this rough sketch. a. Jason is planning to buy paint for the walls and ceiling. Will he need to find the perimeter or area to figure out how much paint to buy? What unit of measure should he use? b. To determine how much new carpet to buy, will Jason need to find the perimeter or area? What unit of measure should he use? c. Jason also needs baseboard for around the bottom of the walls. Will he need to find the perimeter or area to figure out how much baseboard to buy? What unit of measure should he use? d. How much carpeting does Jason need? Show how you found your answer. e. How much baseboard does Jason need? Show how you found your answer. f. If a gallon of paint covers 350 square feet, how much paint does Jason need for the walls and ceiling? 48. A neighbor asks you to help her design a rectangular pen for her dog, Ruff. Your neighbor has 42 meters of fencing to use for the pen. a. What design would give Ruff the most space for playing? b. What design would give Ruff the best space for running?
After looking at your designs, your neighbor decided to use her house as one of the walls for the pen. Her house is 35 meters long. c. Using your neighbor's idea, now what design would give Ruff the most space for playing? d. What design would give Ruff the best space for running? 49. For each of the following, tell whether the given area is possible for a rectangle with a perimeter of 28 units and whole-number side lengths. Explain your reasoning. a. 24 sq. units b. 40 sq. units c. 42 sq. units d. 45 sq. units 50. A. 1. Find the area of each parallelogram below.
2. How are the heights of these parallelograms related to each other? 3. How are the areas of these parallelograms related to each other? B. 1. Find the area of each parallelogram below. b. How are the bases of these parallelograms related to each other? c. How are the areas of these parallelograms related to each other? C. 1. Find the area of each parallelogram below. 2. Based on the patterns in Parts A and B, sketch the third parallelogram. 3. How are the heights of these parallelograms related to each other? 4. How are the bases of these parallelograms related to each other? 5. How are the areas of these parallelograms related to each other? Find the area and perimeter of the figure. 51. 52. Suppose you built all the rectangles possible from 48 square tiles. a. Which rectangle would have the largest perimeter? Dimensions: Perimeter: b. Which rectangle would have the smallest perimeter? Dimensions: Perimeter: Make a line plot or a bar graph of a set of name-length data that fits the description. 53. 9 names with a median of 12 letters
54. Edna rolled a pair of six-sided number cubes several times and recorded the sums on the line plot. a. Which roll(s) occurred most often? Explain your reasoning. b. How many times did Edna roll the cubes? Explain how you found your answer. c. How do the sums on Edna s line plot vary? d. What is the median sum? Explain how you found your answer. e. Does Edna s line plot show categorical or numerical data? Explain your answer. f. If you rolled a pair of six-sided number cubes the same number of times as Edna did and recorded your results in a line plot, would you expect your line plot to look exactly like Edna s? Why or why not? Tell whether the answers to the question are numerical or categorical data. 55. How many hand spans are needed to measure the length of your teacher s desk? 56. The members of the drama club sold candy bars to help raise money for the school s next play. The stem-andleaf plot below shows how many candy bars each member of the drama club sold. a. How many students are in the drama club? Explain how you found your answer. b. How many students sold 25 or more candy bars? c. How do the numbers of candy bars sold by each student vary? d. What is the typical number of candy bars sold by each student? 57. Create two different groups of 5 students that have a mean of 3 children in their family.
58. A class investigated the question of how many paces it takes to travel from their class to the gym. They measured the distance by counting the number of paces each student walked. Every step made on the right foot counted as one pace. Here are their results: a. What is the median number of paces the students took to travel the distance? b. Make a bar graph that displays this information. Explain how the bar graph is similar to and different from a line plot. c. Who has the shorter pace: the student who traveled the distance in 17 paces or the student who traveled the distance in 25 paces? Explain your thinking. 59. Here are the results from the coin-toss experiment, Problem 1.1, for one class of 30 students for 10 days. They each toss a coin once. a. Which days seem to have the most unexpected results? b. Complete the table and make a graph to show how the accumulated percent of heads changes from day to day.
60. Suppose that Aisha, Billie, and Caitlin play a game in which a nickel and a dime are tossed. If neither coin shows heads up, Aisha wins; if two coins show heads up, Billie wins; if one coin shows heads up, Caitlin wins. Caitlin says that this is a fair game because each player has a chance to win. a. How would you convince her otherwise? b. What would you say to Caitlin if each player has one win in the first three plays and she says, See, the game is fair? 61. Use this circle to draw a spinner with six sections. Make the spinner so that it is equally likely that the spinner will land in each of the six sections. What fraction of the circle is each section? 62. Use this circle to draw a spinner with six sections. Make this spinner so that it is not equally likely that the spinner will land in each of the six sections. What fraction of the circle is each section? 63. An ordinary six-sided number cube has the numbers from 1 through 6 on its faces. a. If you roll a six-sided number cube, what are the possible outcomes? b. Suppose you roll a six-sided number cube 18 times. How many times would you expect to roll a 5? What are you assuming about the possible outcomes? c. Takashi and Glen are playing a game. For each turn, a number cube is rolled. If the roll is an even number, Takashi gets a point. If the roll is odd, Glen gets a point. Is this a fair game? Explain your reasoning. 64. A bucket contains 24 blocks. Some are blue, some are green, some are red, and some are yellow. The theoretical probabilities of drawing a blue, green, or red block are: P(blue), P(green), P(red). a. How many blue blocks are in the bucket? b. How many green blocks are in the bucket? c. How many red blocks are in the bucket? d. How many yellow blocks are in the bucket? e. What is the probability of drawing a yellow block? f. What is the probability of not drawing a yellow block?
65. Use the spinner to answer the following questions. a. When you spin the pointer are the three possible outcomes landing on a one, two or three equally likely? Explain. b. If you were to spin the pointer 120 times, how many times would you expect to land on two? 66. List all the proper factors of 35. 67. Which of these numbers are square numbers? Explain. 25 36 48 68. Evonne and Dolphus found a new Product Game board. Three of the factors and one of the products were not filled in. a. What are the other three factors you would need in order to play the game using this board? b. What product is missing? 69. Terrapin Crafts wants to rent between 35 and 40 square yards of space for a big crafts show. The space must be rectangular, and the side lengths must be whole numbers. Find the number(s) between 35 and 40 with the most factor pairs that gives the greatest number of rectangular arrangements to choose from. 70. What number has the prime factorization? Show how you found the number.
71. Carlos is packing sacks for treats at Halloween. Each sack has to have exactly the same stuff in it or the neighborhood kids complain. He has on hand 96 small candy bars and 64 small popcorn balls. 72. 3 a. What is the greatest number of treat sacks he can make? b. How many of each kind of treat is in one sack? Describe how you can tell whether a given number is a multiple of the number shown. 73. Use concepts you have learned in this unit to make a mystery number question. Each clue must contain at least one word from your vocabulary list. 74. a. List the first ten square numbers. b. Give all the factors for each number you listed in part (a). c. Which of the square numbers you listed have only three factors? d. If you continued your list, what would be the next square number with only three factors? 75. A mystery number is greater than 50 and less than 100. You can make exactly five different rectangles with the mystery number of tiles. Its prime factorization consists of only one prime number. What is the number? 76. The numbers 10, 20, and 30 on the 30-board in the Factor Game all have 10 as a factor. Does any number that has 10 as a factor also have 5 as a factor? Explain your reasoning. 77. The numbers 14, 28, and 42 on the 49-board in the Factor Game all have 7 as a factor and also have 2 as a factor. Does any number that has 7 as a factor also have 2 as a factor? Explain your reasoning. 78. In each of the rectangles shown below, only the tiles along the length and width are shown. For each rectangle, explain how many square tiles it would take to make each rectangle. a. b. c. 79. Kyong has built two rectangles. Each has a width of 7 tiles.
a. If each rectangle is made with an even number of tiles that is greater than 40 but less than 60, how many tiles does it take to make each rectangle? Explain your reasoning. b. What is the length of each of Kyong s rectangles? Explain your reasoning. c. Without changing the number of tiles used to make either rectangle, Kyong rearranges the tiles of each rectangle into different rectangles. What is a possibility for the length and width of each of Kyong s new rectangles? Explain your reasoning. 80. An odd number that is less than 160 has exactly three different prime factors. What is the number? Explain your reasoning. 81. A number sequence is an ordered series of numbers that follow a pattern or rule. Jason has developed a secret rule for generating his own number sequence. Here are the first five terms in the sequence: 3, 15, 45, 225, 675,... and so on. Use Jason s sequence to answer the following questions. a. What is Jason s rule for finding the numbers in his number sequence? Explain how you found your answer. b. What are the next two terms in Jason s number sequence? c. What is the greatest common factor of all the terms in Jason s sequence, no matter how many new numbers he adds to the sequence? Explain your reasoning. 82. In the 1,000-locker problem, what was the last locker touched by the students indicated? a. both students 20 and 25 b. both students 13 and 19 c. all three students 3, 4, and 5 d. all three students 30, 40, and 50 83. a. Use an angle ruler to draw a 100 angle. b. Use an angle ruler to measure the following angle.
84. One of the most common places we see angles is on the faces of clocks. At the start of each hour, the minute hand is pointed straight up, at the 12. Without using an angle ruler, find the measure of the following angle. 85. Naomi wants measurements of 3, 6, 6, and 12 for the side lengths of a quadrilateral. Marcelo says he cannot make a quadrilateral with these lengths. Is he right? Explain. Decide whether the given statement is true or false. Give explanations or sketches to support your answers. 86. Any two quadrilaterals that have sides of the same lengths will be identical in size and shape. SAMPLE Two quadrilaterals with side lengths measuring 5, 7, 9, and 11 will be the same size and shape. Use a coordinate grid like the one shown below. 87. If (2, 0) and (5, 5) are two vertices of a triangle that does not have a right angle, what might be the coordinates of the other vertex of the triangle? 88. Use an protractor to measure each angle.
89. 90. Draw a 90º angle. Use the given data and what you know about relations among sides and angles to find the lengths and angle measurements of all sides and angles in the figure. 91. parallelogram 92. Jack has made a tiling with quadrilateral shapes. He can pick up a shape from his tiling, turn it 90º, and it will fit back where it was. Kenesha has used a different quadrilateral to make a tiling. Kenesha's quadrilateral will not fit back into the pattern when she turns it 90º. a. What might Jack's quadrilateral look like? Draw or describe it, and explain why it fits. b. What might Kenesha's quadrilateral look like? Draw or describe it, and explain why it doesn't fit back into the tiling pattern when it is turned 90º. 93. a. Is the triangle below a regular polygon? Explain why or why not. b. Could this triangle be used to tile a surface? Explain why or why not. 94. A square has a perimeter of 16.4 centimeters. What is the length of each side? Explain. 95. Answer parts (a) and (b) for each polygon. a. Is the shape a regular polygon? Explain why or why not. b. Could the shape be used to tile a surface? Make a sketch to demonstrate your answer.
96. 97. An isosceles triangle has two 50º angles. What is the measure of the third angle? Explain how you found your answer. 98. One angle of a parallelogram measures 40º and another angle measures 140º. What are the measures of the other two angles? Explain how you found your answer. 99. A quadrilateral has two sides of length 6. The sum of the lengths of the other two sides is 15. Use this information to answer the following questions. a. Suppose the two sides of length 6 are right next to each other. What might the lengths of the other two sides be? Explain your reasoning. b. Suppose the quadrilateral is a rectangle and the two sides of length 6 are opposite each other. What would the lengths of the other two sides have to be? Explain how you found your answer. c. Could the quadrilateral have two sides of length 6, one side of length 13.5, and one side of length 1.5? Explain why or why not. 100. A triangle has side lengths measuring 3 and 7. The measurement of the longest side is missing. Ted says that one possibility for the unknown side length is 11. Do you agree with Ted? Explain why or why not.
Middle School 6th and 7th grade Review Answer Section SHORT ANSWER 1. a. ; Possible strategy: Using as a benchmark: and is larger than half because you would need 6 pieces out of 12 to be equal to half. b. ; Possible strategies: Common Denominators: I found common denominators and looked to see which numerator was larger. I knew that. ; Common Numerators: I made the numerators the same so I could compare the size of the pieces. and. 48ths would be bigger pieces, then has to be the bigger fraction. ; One piece over Half: Each fraction is one piece more than a half. Since is smaller than, the is closer to a half or the smaller fraction. c. ; Possible strategies: Use common denominators or rewrite each fraction in lowest terms to see they are equivalent. They may remember that the fractions are equivalent from work in the problems. d. ; Possible strategy: Since the numerators are equal, compare the size of the pieces. Eighths are cut into fewer pieces, so the piece size will be bigger, making it the larger fraction. 2. Possible answers: 3. 4. from left to right: or or or or 1, or 5. If you want the most pizza possible, join the first group. In the first group, you would share 6 pizzas among 4 people, so you would receive pizzas. In the second group, you would share 8 pizzas among 6 people, so you would receive pizzas. 6. The muffins may not have been the same size to start with.
7. When comparing two decimal numbers that are both less than 1, you need to compare place-value amounts. For example, 0.37 is less than 0.6. This is because 0.37 means 37 hundredths or 3 tenths and 7 hundredths, and 0.6 means 6 tenths. Tenths are greater than hundredths, so 0.6 is more than 0.37. 8. a. True; possible explanation: If the denominators are the same, each piece is the same size. The fraction with the larger numerator is the larger fraction because numerators tell how many pieces you have. b. False; possible explanation: If the numerators are the same, the number of pieces is the same. As the denominator increases, the size of the pieces decreases, so the fraction with the smaller denominator is the larger fraction. 9. a. percent shaded: 12%; percent not shaded: 88% b. percent shaded: 8%; percent not shaded: 92% c. percent shaded: 30%; percent not shaded: 70% 10. 120 people; possible explanations: 20% means 20 out of every 100. You have 6 hundreds, so. 20% percent can be written as, and a fifth of 600 is 120. 11. a. or 12. a. b. or c. or b. 13. a. yes b. no c. yes 14. a. slices 15. a. b. of an egg b. c. d. ; half of is e. ; twice is 16. a. or b. 12 17. a. 6 b.
c. 18. a. b. c. d. 19. a. ; 0.84; 84% b. ; 0.74; 74% c. ; 0.36; 36% 20. 20% 21. a. At stage A, of the line is covered and of the line is not covered. b. At stage B, of the line is covered and of the line is not covered. c. At stage C, of the line is covered and of the line is not covered. 22. a. b. c. d. e. f. g. h. i. 23. a. and ; and b. and c. None; all sums are too great. d. and e. and f. and ; and ; and 24. a. cup
b. cup c. No; Jane is increasing and decreasing different numbers by half ( and ), so the new sum will be different. 25. milkshake: 3 = 24 shakes double milkshake: 3 triple milkshake: 3 = 12 double shakes = 8 triple shakes 26. C; the four students who share of a pizza ( 4) will each get of the pizza. If you divide the amount of leftover pizza by the number of students in the other groups, you get: Group A: 6 = of the pizza for each student. Group B: 3 = of the pizza for each student. 27. When you divide fractions with common denominators, you only have to divide the numerators. The denominator is the size of the fraction you are working with. The numerator is the number of those size parts. For example, with divided by, eights are the size of the fractional parts that you have. You are trying to find out how many times three-eighths will go into six-eighths. This is the same as asking how many times 3 goes into 6. 28. C; and 29. Sunday; 1 11 in. 21 30. Possible answers: and Possible explanation: In the first problem there had to be a total of 2 decimal places in the two factors. In the second problem there had to be a total of 1 decimal place in the two factors. 31. a. 8.5 3.2; If you use benchmarks to estimate, 8.5 3 is a greater difference than 7 5. b. They are the same. c. 0.57 0.008; Students probably have to compute the two differences. 32. a. 35.7; The whole number parts are the same and 0.7 is the greatest decimal part. 33. b. 690.8; This has the greatest whole number part. Since it has the greatest whole number, the size of the decimal part is irrelevant. c. 75.6500; The whole number parts in each number are the same. The decimal 0.6500 (which is 0.65) in 75.6500 is the greatest because it has a 6 in the tenths place. The number 75.0605 does not have any tenths. The number 75.6050 has a 6 in the tenths place but it does not have any thousandths and 75.6500 has 5 thousandths. Problem Answer without the decimal point 5.7 + 6.09 + 4.2 1599 15.99 3.007 2.9 + 35.054 35161 35.161 Correct answer
14.5 8.07 6.2 23 0.23 34. a. 23.85 b. 29.15 c. i. $59,028.75 ii. $72,146.25 d. $1770.86 (2.65 20 2475 131175, 131.175 13.5 1770.86) 35. a. 49.2 b. 4.92 c. 0.492 d. 4.92 e. 49.2 f. 492. 36. a. 12.3 b. 1.23 c. 0.123 d. 12.3 e. 1.23 f. 0.123 37. a. Possible estimate: 2; Using benchmarks, 2 1 = 2. One number was rounded up and the other was rounded down. b. Possible estimate: 5.21; 5.21 1 5.21 c. Possible estimate: 10; 8 10 d. Possible estimate: 6; 12 6 e. Possible estimate: about 112.5; (75 1) + (75 ) 75 + 37.5 112.5 f. Possible estimate: 300; 3 100 300 38. They need to raise. 39. a. The cost is 1.25 times what the cost was last year. b. 40. 25% 41. 69 42. a. 44.377 b. 5.178 c. 51.66 d. 46 43. a. 12.4 3.2 = 9.2; 12.4 9.2 = 3.2; 3.2 + 9.2 = 12.4; 9.2 + 3.2 = 12.4 b. 3.2 4.1 = 13.12; 4.1 3.2 = 13.12; 13.12 4.1 = 3.2; 13.12 3.2 = 4.1 44. Area: 20 sq. cm, Perimeter: 18 cm; Possible explanation: I multiplied 5 by 4 to find the area. To find the perimeter I added the length and the width, then multiplied this answer by 2, in order to account for both lengths and both widths. 45. a. The problem does not specify to list the areas of each shelter. Nonetheless, they are included in the table below. Some students may list the 4 by 6 rectangle separately from the 6 by 4 rectangle, while some will list only one of these. Either way is acceptable.
b. The 5 by 5 shelter has the largest area: 25 sq. ft. c. The 1 by 9 shelter has the least area: 9 sq. ft. d. The 5 by 5 shelter and the 4 by 6 shelter each hold 8 children under the code. The 5 by 5 shelter would have a little bit of extra space for these 8 children, but they would each hold the same number. 46. a. 34 centimeters. Students will need to measure the diagonal sides, which are each 5 centimeters. b. 42 square centimeters. Possible explanation: We cut the shape into a 5 by 6 rectangle and two triangles, each with base 4 centimeters and height 3 centimetrs. The area of the figure is the sum of the areas of these smaller shapes: 6 + 6 + 30 = 42. 47. a. area, square feet b. area, square yards (or square feet) c. perimeter, feet d. The room is 3 yards (9 feet) by 4 yards (12 feet), so 3 4 = 12 square yards of carpet are needed (108 square feet is also correct). e. (9 + 12) 2 = 42 feet of baseboard (Some students may argue for less than 42 feet-say, 39 feet-because of the door opening not needing baseboard. This is a reasonable answer as well.) f. Possible answer: Two of the walls need 12 8 = 96 square feet of paint, two of the walls need 9 8 = 72 square feet of paint, and the ceiling needs 12 9 = 108 square feet of paint, so there is (96 2) + (72 2) + 108 = 444 square feet to cover. This would require 444 350 = about 1.27 gallons of paint, so you would need or 2 gallons (if the paint came only in full gallons). 48. a. The idea here is to find the pen with the largest area. If students use whole units, the pen with the largest area is a 10 11 rectangle. A more sophisticated answer would take into account the fact that whole units were not a restriction; thus the pen with the largest area would be a 10.5 10.5 square. b. The idea here is to find the pen with the longest running area. The longest and thinnest design using whole-number units is a 20 1 rectangular pen. Some students might argue for other sizes, such as a 19 2 pen, to give the dog more room to turn around. c. If students use whole units, the pen could have fence side lengths of 10, 22, and 10 meters or 11, 20, and 11 meters which, with the house as the fourth side of the pen, results in an area of 220 square meters. A more sophisticated answer would involve considering rational numbers; the pen could have side lengths of 10.5, 21, and 10.5 and an area of 220.5 square meters. d. Students might argue for a 3.5 35 meter pen because the house (which is 35 meters long) is to be one side on the pen and that leaves 7 meters for the two short ends. Others might argue that you could use the fencing to extend the house wall and suggest a 1.5 37 meter pen or a 1 37.5 meter pen. 49. a. Yes, a rectangle b. Yes, a rectangle c. Not possible
d. Yes, a rectangle 50. A. 1. 2 cm 2, 6 cm 2 and 18 cm 2 2. The height of each parallelogram is 3 times the height of the next smaller parallelogram. 3. The area of each parallelogram is 3 times the area of the next smaller parallelogram. B. 1. 2 cm 2, 4 cm 2 and 8 cm 2 2. The base of each parallelogram is twice the base of the next smaller parallelogram. 3. The area of each parallelogram is twice the area of the next smaller parallelogram. C. 1. 2 cm 2, 12 cm 2 (and 72 cm 2 for the third one drawn in part (b).) 2. The next parallelogram should have a base of 8 cm and a height of 9 cm. 3. The height of each parallelogram is three times the height of the next smaller parallelogram. 4. The base of each parallelogram is twice the base of the next smaller parallelogram. 5. The area of each parallelogram is six times the area of the next smaller parallelogram. 51. Area: 24 sq. in., Perimeter: 22 in. 52. a. The 1 by 48 rectangle would have the largest perimeter: 98 units. b. The 6 by 8 rectangle would have the smallest perimeter: 28 units. 53. Answers will vary, but because there is an odd number of values, at least one value (the median) must be located at 12. 54. a. 6, 7, 8, and 9 b. 26; found by counting the data points c. 4 to 12 d. The median must lie between the two middle data points, which are both 8. e. Numerical, since the data show numerical outcomes (i.e., the sum of the dice). f. The distribution would probably not look exactly the same. The measures of center might be very close, but the data points themselves would likely vary. 55. numerical data 56. a. 27 b. 13 c. 10 to 58 d. Possible answer: 24 (median) 57. Possible answers: group 1: 3, 1, 1, 2, 8; group 2: 5, 2, 2, 3, 3 58. a. 22 paces b. The line plot and the bar graph show representations of the same data. The bar graph requires a vertical scale to read the numbers of data in each group, while the numbers on the line plot can be found by counting the X s in each column.
c. The student who traveled the distance in 25 paces has the shorter pace. It took that student more paces to travel the same distance as the student who traveled the distance in only 17 paces. 59. a. June 3 and June 8 results are furthest from the expected 15 heads. b. Students might make a bar graph to show how the percents change. They may need help setting up the graph. 60. a. Make a list of all possible outcomes and find the theoretical probabilities. This would show that HH or TT can occur only once while HT or TH can occur twice. b. Tell her that we need to play more rounds to see the patterns that emerge over the long run. 61. Each section is 60 or 62. Possible answer: of the circle.
The two large sections are of the circle. The four small sections are of the circle. 63. a. 1, 2, 3, 4, 5, 6 b. About three 5s assuming that all possible outcomes are equally likely. c. Yes, there is the same number of even and odd outcomes. 64. a. 2 b. 3 c. 8 d. 11 e. f. 65. a. No, they are not equally likely. P or P P or b. 50 times. If spun 120 times, you would expect to land on two or 50 times. 66. 1, 5, 7, 35 67. 25 and 36; Each of these numbers can be expressed as a number times itself. 5 5 = 25 and 6 6 = 36. If you made a tile model you could arrange each number of tiles into a square. 68. a. 3, 4, 6 b. 81 69. 36; If we consider an rectangle to be different from a rectangle, of the numbers 36, 37, 38, and 39, the number 36 gives nine choices of rectangles. 70. The number is 2,100. = = 2,100 71. a. Carlos can make 32 sacks. b. Each sack would contain 3 small candy bars and 2 small popcorn balls. 72. Possible answer: It can be divided by 3 without a remainder. 73. Answers will vary. 74. a. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 b.
c. 4, 9, 25, 49 (the squares of primes) d. 121 75. 81 76. Yes; since 2 and 5 are factors of 10, any number that has 10 as a factor must also have 5 as a factor. 77. No; for example, the number 35 has 7 as a factor, but since it is an odd number it does not have 2 as a factor. 2 and 7 are both prime factors of 14; for a number to be a factor of 2 and 7, it would also be a factor of 14. 78. In order to determine the number of tiles in each of the rectangles, multiply the tiles along the length by the tiles on the width. a. 60 b. 40 c. 40 79. a. One rectangle is made with 42 tiles, and the other is made with 56 tiles. These are the only two even multiples of 7 between 40 and 60. b. The rectangle with 42 tiles has a length of 6, and the rectangle with 56 tiles has a length of 8. These answers are found by finding the other number in the factor pair with 7 for each rectangle. c. Students answers will vary. For 42:,, or. For 56:,, or. 80. 105; an odd number cannot have a factor of 2, and 3, 5, and 7 are the only three primes with a product less than 160. 81. a. Alternate multiplying terms by 5 and then 3. In other words, multiply the first term by 5 to get the second term, multiply the second term by 3 to get the third, multiply the third term by 5 to get the fourth, multiply the fourth term by 3 to get the fifth, and so on. b. 3,375 and 10,125. c. The greatest common factor is 3, since it is the first term in the sequence and a prime number. 82. a. Locker 1,000 b. Locker 988 c. Locker 960 d. Locker 600
83. a. b. about 55 84. 60 85. Marcelo is not right. You can make a quadrilateral with these four line segments since the sum of the three smallest sides (3 + 6 + 6) is greater than the length of the longest side (12). 86. False; it is possible to sequence the four line segments in more than one arrangement, such as 5-7-9-11, 5-9-7-11 or 5-9-11-7. Also, any quadrilateral can be "squished" into different-shaped quadrilaterals. 87. Possible answer: (0, 5). Note: (5, 0) and (2, 5) will give right triangles and thus are incorrect. Make sure that two sides are greater than the third side in order for the figure to be a triangle. 88. about 20º 89. about 125º 90. 91. Opposite sides must be of equal length, so the short side is 1 cm, and the long side is 3 cm. Opposite angles must be equal, so the other acute angle is 45º. The two obtuse angles are equal, and each must be = 135º. 92. a. Jack's shape must be a square. Because its sides and angles are the same measure, the square is the only quadrilateral that can be turned 90º and fit back into the same spot. b. Kenesha's shape could be any quadrilateral other than a square. For example, students may choose a rectangle or a parallelogram and show how the adjacent angles, being different sizes, make it impossible to fit back into the original space. 93. a. No, because all sides and angles of the triangle are not equal. b. Yes; any triangle can be used to tile the plane. 94. Each side would have length = 4.1. 95. a. The shape is not a regular polygon because it does not contain sides and angles that are all the same size. b. The shape can be used to tile a surface. Drawings will vary. 96. a. The shape is not a regular polygon because it does not contain sides and angles that are all the same size. b. The shape can be used to tile a surface. Drawings will vary. 97. Since the sum of the angles must be 180º, the third angle must have a measure of 80º. 98. Since parallelograms have two pairs of equal angles, the other two angles must measure 140º and 40º. 99. a. Possible answers: 5 and 10, 3 and 12, 7 and 8
b. Each side would have a length of 7.5. c. No; the sum of any three sides must be greater than the length of the fourth side. In this case, the sum of the three shortest sides, 1.5 + 6 + 6 = 13.5, is not greater than the fourth side of 13.5. It is equal to it. 100. Ted s estimate is too large. The third side must have a length less than 7 + 3 = 10.