Unit 5 Practice Problems Lesson Problem Mai had $4.50. She spent $4.35 at the snack bar and $5.5 at the arcade. What is the exact amount of money Mai has left? A. $9.60 B. $0.60 C. $4.90 D. $5.90 C Problem A large cheese pizza costs $7.50. Diego has $40 to spend on pizzas. How many large cheese pizzas can he afford? Explain or show your reasoning. 5 pizzas. Sample reasoning: Each pizza costs about $8, and 8 5 = 40. Problem 3 Tickets to a show cost $5.50 for adults and $4.5 for students. A family is purchasing adult tickets and 3 student tickets.. Estimate the total cost.. What is the exact cost? 3. If the family pays $5, what is the exact amount of change they should receive?. $4 ( 6 + 6 + 4 + 4 + 4 = 4). $3.75 ( 5.50 + 5.50 + 4.5 + 4.5 + 4.5 = 3.75) 3. $.5 ( 5.00 3.75 =.5) Problem 4 Chicken costs $3.0 per pound, and beef costs $4.59 per pound. Answer each question and show your reasoning.. What is the exact cost of 3 pounds of chicken? 3. How much more does 3 pounds of beef cost than 3 pounds of
. What is the exact cost of 3 pound of beef? chicken?. $9.60 ( 3.0 3 = 9.60). $3.77 ( 4.59 3 = 3.77) 3. $4.7 ( 3.77 9.60 = 4.7) Problem 5 (from Unit 4, Lesson 6). How many -liter glasses can Lin fill with a -liter bottle of water? 5. How many -liter bottles of water does it take to fill a 6-liter jug?. 7 5 (or ). (She can fill 5 of the glasses with liter and then another half of that or with the other half liter, so that is 7 5 glasses. = or 5 7.). 0 3 (or ). (This can be obtained by computing 6, which is 3 3 6 3 or 0. This is correct as 0 bottles give 5 liters, and then more 3 liter is of the bottle.) 3 Problem 6 (from Unit 4, Lesson 4) Use the grid to complete this problem. Let the side length of each small square on the grid represent unit. Draw two different triangles, each with base 5 units and area 9. Why does each 4 units of your triangles have area 9? Explain or show your reasoning. 4 units Drawings vary but should show a height of 7 units. Sample reasoning: The base times the height is times the area of the triangle: (base) (height) = (9 ). Since, that means the 4 (9 ) (5 ) = 3 4 height should be (3 ) = 7. Problem 7 (from Unit 4, Lesson 0) Find each quotient. 5 0
5 a. 6. 5 6 b. 6 0 c. 6 4. 4 3. 40 Lesson Problem Use the given key to answer the questions.. What number does this diagram represent?. Draw a diagram that represents 0.6. 3. Draw a diagram that represents 0.304.. 0.05 3.. Problem Here are diagrams that represent 0.37 and 0.84.. Use the diagram to find the value of 0.37 + 0.84. Explain your reasoning.. Calculate the sum vertically. 3. How was your reasoning in the first two questions different? How was it similar or the same?.
. 3. Responses vary. Sample response: Using the diagrams, 0 thousandths can be bundled to make hundredth. Then 0 hundredths can be bundled to make tenth. These values can then be combined. Without diagrams, 0 of the thousandths can be converted into hundredth and 0 of the hundredths to tenth. The methods are similar. The diagrams show the bundling, but the method without a diagram is faster. Problem 3 For the first two problems, circle the vertical calculation where digits of the same kind are lined up. Then, finish the calculation and find the sum. For the last two problems, find the sum using vertical calculation.. 3.5 +. 0.5 +.5 3. 0.6 +.7 4. 3 + 0.. The second arrangement is correct. The sum is 4.5.. The first arrangement is correct. The sum is.65. 3..3 4. 3. Problem 4 (from Unit, Lesson 9) Andre has been practicing his math facts. He can now complete 35 multiplication facts in 90 seconds.. If Andre is answering questions at a constant rate, how many facts can he answer per second?. Noah also works at a constant rate, and he can complete 75 facts in minute. Who is working faster? Explain or show your reasoning.
..5 facts per second ( 35 9 =.5). Andre is faster, because Noah can only answer.5 facts per second. ( 75 60 =.5) Lesson 3 Problem Here is a base-ten diagram that represents.3. Use the diagram to find.3 0.46. Explain how you found the difference, or label your diagram to show your steps. 0.67. Sample response: First, unbundle tenth into 0 hundredths and then take away 6 hundredths from the 3 hundredths, leaving 7 hundredths. Next, unbundle the one as 0 tenths. After taking away 4 tenths, 6 tenths are left. So the answer is 0.67. Problem Compute the following sums. If you get stuck, you can draw base-ten diagrams.. 0.07 + 0.004. 0.03 + 0. 3.. + 0.45. 0.03. 0.3 3..345 (Diagrams for b and c shown here.)
Problem 3 A student said we cannot subtract.97 from 0 because.97 has two decimal digits and 0 has none. Do you agree with his statement? Explain or show your reasoning. Disagree. Sample explanation: The number.97 is equal to 97 hundredths. 0 can be written as 0.00 or,000 hundredths. We can subtract 97 from,000 to get,803 hundredths, so 0.97 = 8.03. Problem 4 Decide which calculation shows the correct way to find 0.3 0.006 and explain your reasoning. D. Sample reasoning: It is the only one that shows the decimal points correctly lined up so that the same base-ten units are aligned vertically. Problem 5 Complete the calculations so that each shows the correct difference. Problem 6 (from Unit 5, Lesson )
The school store sells pencils for $0.30 each, hats for $4.50 each, and binders for $3.0 each. Elena would like to buy 3 pencils, a hat, and binders. She estimated that the cost will be less than $0.. Do you agree with her estimate? Explain your reasoning.. Estimate the number of pencils could she buy with $5. Explain or show your reasoning.. Disagree. Sample reasoning: The hat costs more than $4, and two binders cost more than $6. Even without the pencils the cost is already more than $0.. Answers vary, but should be around 5 or 6. Sample reasoning: She could buy 3 pencils for every dollar, so for $5, she could buy around 5 pencils. Problem 7 (from Unit 4, Lesson 5) A rectangular prism measures 7 cm by cm by 5 cm.. Calculate the number of cubes with edge length cm that fit in this prism.. What is the volume of the prism in cm? Show your reasoning. If you are stuck, think about how many cubes with -cm edge lengths fit into cm 3..,60 cubes. 395 cm 3. Sample reasoning: Eight cm cubes fit in a cm cube and,60 of these cm cubes fit in the prism. So,,60 8 of the cm cubes fit in the prism. That means the volume of the prism in cm 3 is,60 8 = 395. Problem 8 (from Unit, Lesson ) At a constant speed, a car travels 75 miles in 60 minutes. How far does the car travel in 8 minutes? If you get stuck, consider using the table. minutes distance in miles 60 75 6 8.5 miles (or equivalent). Possible strategy: minutes distance in miles 60 75 6 7.5 8.5 Lesson 4 Problem For each subtraction problem, circle the correct calculation.. 7. 3.67
. 6.4 Problem Explain how you could find the difference of and 0.978. Answers vary. Sample responses: can be unbundled into 0,000 ten-thousandths. 0.978 is,978 tenthousandths. To find the difference, we subtract: 0,000,978 = 8,0. The difference is 8,0 ten-thousandths or 0.80. can be written as.0000. In a vertical calculation, we can show the being unbundled into 0 tenths, of those tenths being unbundled into 0 hundredths, of those hundredths being unbundled into 0 thousandths, and of the thousandths being unbundled into 0 ten-thousandths. Subtracting 0.978 from those digits gives us 0.80. Problem 3 A bag of chocolates is labeled to contain 0.384 pound of chocolates. The actual weight of the chocolates is 0.3798 pound.. Are the chocolates heavier or lighter than the weight stated on the label? Explain how you know.. How much heavier or lighter are the chocolates than stated on the label? Show your reasoning.. Lighter. Reasoning varies. Sample reasoning: 0.3798 is 3,798 tenthousandths. 0.384 is 384 thousandths, which is equal to 3,840 tenthousandths, so 0.384 is greater than 0.3798.. 0.004 ounce lighter. Reasoning varies. Sample reasoning: 3,840 ten-thousandths subtracted by 3,798 ten-thousandths is tenthousandths, because 3,840 3,798 = 4. 0.3798 is 0.000 away from 0.3800, and 0.3800 is 0.004 away from 0.384, so 0.3798 is ( 0.000 + 0.004) or 0.004 away from 0.384. 0.384 0.3798 = 0.004.
Problem 4 Complete the calculations so that each shows the correct sum...964. 0.6 3. 0.4863 Problem 5 (from Unit 4, Lesson 4) A shipping company is loading cube-shaped crates into a larger cube-shaped container. The smaller cubes have side lengths of feet, and the larger shipping container has side lengths of 0 feet. How many crates will fit in the large shipping container? Explain your reasoning. 64 crates. Reasoning varies. Sample reasoning: Four crates can fit in a length of 0 feet because 4 = 0. So the container can fit 4 4 4 or 64 crates. The volume of the larger container is 000 cubic feet because 0 0 0 = 000. The volume of a crate is 5 5, since 8 = 5. Then 64 crates fit inside the container because 5 8 5 000 5 = 64. 8 Problem 6 (from Unit, Lesson 3) For every 9 customers, the chef prepares loaves of bread. Here is double number line showing varying numbers of customers and the loaves prepared.. Complete the missing information.. The same information is shown on a table. Complete the missing information. 3. Use either representation to answer these questions. How many loaves are needed for 63 customers?
customers loaves 9 How many customers are there if the chef prepares 0 loaves? 7 4 4 How much of a loaf is prepared for each customer?. See double number line.. customers loaves 9 8 4 7 6 63 5 or equivalent 9 3. 4 loaves 90 customers 9 of a loaf Lesson 5 Problem. Find the product of each number and. 00..8 350..704. What happens to the decimal point of the original number when you multiply it by? Why do you think that is? Explain your reasoning. 00... 0.8 3. 3.50 4. 0.0704 5. Answers vary. Sample response: The decimal point moves places to the left. Multiplying a decimal number by means dividing by 00, which 00 moves the decimal point places to the left. Problem Which expression has the same value as A. 6 54 00,000 B. 6 54 00,000 C. 6 (0.) 54 (0.0) 6 54 (0.0000) (0.06) (0.54)? Select all that apply.
D. 6 54 (0.0000) E. 0.0094 A, B, E Problem 3 Calculate the value of each expression by writing the decimal factors as fractions, then writing their product as a decimal. Show your reasoning... 3. 4. 5. (0.0) (0.0) (0.3) (0.) (.) 5 (0.9) (.) (.5). 0.000 because 0.0 = and 0.0 =, so the product is 00 00 0,000 3. 0.06 because 0.3 = and 0. =, so the product is 0 0 3. 6 because 5 = 60 and. is one tenth of 9 4. 0.99 because 0.9 = and. =, so the product is 0 0 3 5. 3 because.5 = and twice this is 3 6 00 99 00 Problem 4 Write three numerical expressions that are equivalent to (0.0004) (0.005). Answers vary. Possible responses: 4 (0.000) 5 (0.00) 4 5 (0.000) (0.00) 4 5 0,000,000 4 5 0,000,000 Problem 5 (from Unit 5, Lesson 3) Calculate each sum.. 33. +.95..075 + 7.05 3. 0.40 + 9.8. 35.05. 8.8 3. 9.68
Problem 6 (from Unit 5, Lesson 4) Calculate each difference. Show your reasoning.. 3..78. 3. 0.376 3. 0.9 0.45..4..734 3. 0.655 Sample reasoning: Problem 7 (from Unit, Lesson 3) On the grid, draw a quadrilateral that is not a rectangle that has an area of 8 square units. Show how you know the area is 8 square units. Answers vary. Sample responses: Lesson 6 Problem Find each product. Show your reasoning... 3. (.) (0.) (0.34) (0.0) 0 (0.00). 0.3. Sample reasoning:. is a tenth of and 0. is a hundredth of, so the product of. and 0. is a thousandth of or 3,,000 which is 0.3. 34 68 =
34. 0.0068. Sample reasoning: = or 0.0068. 00 68 00,000 3. 0.4. Sample reasoning: 0.00 is thousandths or, so the product,000 40 of 0 and 0.00 is 0, which equals or 0.4.,000,000 Problem You can use a rectangle to represent (0.3) (0.5).. What must the side length of each square represent for the rectangle to correctly represent (0.3) (0.5)?. What area is represented by each square? 3. What is (0.3) (0.5)? Show your reasoning.. 0.. 0.0 square units 3. The area is 0.5 because there are 5 squares, and 5 (0.0) = 0.5. Problem 3 One gallon of gasoline in Buffalo, New York costs $.9. In Toronto, Canada, one liter of gasoline costs $0.9. There are 3.8 liters in one gallon.. How much does one gallon of gas cost in Toronto? Round your answer to the nearest cent.. Is the cost of gas greater in Buffalo or in Toronto? How much greater?. $3.46. (3.8) (0.9) = 3.458, and this is closer to 3.46 than to 3.45.. The cost of one gallon of gas is $.7 more in Toronto. Problem 4 (from Unit 5, Lesson ) Calculate each sum or difference.. 0.3 + 3.7. 0.99 4.97 3. 5.99 + 3.5 4..893 0.353. 4. 6.0 3. 39.5 4..54 Problem 5 (from Unit 4, Lesson ) 49 7 Find the value of using any method. 50 6
5 (or equivalent) Problem 6 (from Unit, Lesson ) Find the area of the shaded region. All angles are right angles. Show your reasoning.,400 square units. Reasoning varies. Sample reasoning: The region can be enclosed with a 60-by-30 rectangle, which has an area of,800 square units. Three of the corners of that rectangle have a rectangular region removed. The removed areas are 00 square units (upper left), 50 square units (lower left), and 50 square units (upper right). The area of the shaded region, in square units, is, 800 (00 + 50 + 50) or, 800 400, which is,400. Problem 7. Priya finds (.05) (.8) by calculating 05 8, then moving the decimal point three places to the left. Why does Priya s method make sense?. Use Priya s method to calculate (.05) (.8). You can use the fact that 05 8 =,940. 3. Use Priya s method to calculate (0.005) (0.04)...05 = 05 and, so 00.8 = 8 0 (.05) (.8) = (05 8). This is the same as finding and then moving the decimal point three places to the left.. Since 05 8 =,940, (.05) (.8) =.940 because the decimal point in,940 moved three places to the left. 3. 5 4 = 360. The decimal needs to be moved 7 places to the left because the decimal point of 0.005 was moved four places to the right to get 5, and the decimal point of 0.04 was moved three places to the right to get 4. So the answer is 0.0000360. Lesson 7 Problem Here is a rectangle that has been partitioned into four smaller rectangles. For each expression, choose a sub-rectangle whose area, in square units, matches the expression.,000 05 8. 3 (0.6). (0.4) 3. (0.4) (0.6) 4. 3
. B. C 3. A 4. D Problem Here is an area diagram that represents (3.) (.4).. Find the areas of subrectangles A and B.. What is the area of the 3. by.4 rectangle?. Rectangle A: 3. square units, Rectangle B:.4 square units. 4.34 square units ( 3. +.4 = 4.34) Problem 3 Draw an area diagram to find (0.36) (0.53). Label and organize your work so that it can be followed by others. 0.908. Sample diagram and reasoning: Area of A is (0.5)(0.3) = 0.5. Area of B is (0.03)(0.3) = 0.009. Area of C is (0.5)(0.06) = 0.03. Area of D is (0.03)(0.06) = 0.008. The area of the rectangle, in square units, is 0.5 + 0.009 + 0.03 + 0.008 = 0.908. Problem 4 Find each product. Show your reasoning.. (.5) (.4). (0.64) (0.8). 3.5. Sample reasoning: (.4) =.8 and (0.5) (.4) = 0.7. The product is.8 + 0.7 or 3.5.. 0.584. Sample reasoning:
Area of A is (0.8)(0.6) = 0.48. Area of B is (0.0)(0.6) = 0.006. Area of C is (0.8)(0.04) = 0.03. Area of D is (0.0)(0.04) = 0.0004. The area of the rectangle, in square units, is 0.48 + 0.006 + 0.03 + 0.0004 = 0.584. Problem 5 (from Unit 5, Lesson 3) Complete the calculations so that each shows the correct sum or difference. Problem 6 (from Unit, Lesson ) Diego bought mini muffins for $4.0.. At this rate, how much would Diego pay for 4 mini muffins? number of mini muffins price in dollars. How many mini muffins could Diego buy with $3.00? Explain or show your reasoning. If you get stuck, consider using the table. 4.0
. $.40.. 8 mini muffins, which would cost $.80. He does not have enough money for 9 mini muffins, because that would cost $3.5. number of mini muffins price in dollars 4.0 0.35 8.80 9 3.5 Lesson 8 Problem Here are an unfinished calculation of (0.54) (3.8) and a 0.54-by-3.8 rectangle.. Which part of the rectangle has an area of 0.43? Which part of the rectangle has an area of.6? Show your reasoning.. What is (0.54) (3.8)?. 0.43 is the area of the 0.8 by 0.54 rectangle because (0.8) (0.54) = 0.43..6 is the area of the 3 by 0.54 rectangle because 3 (0.54) =.6...6 ( 0.54 +.6 =.6) Problem Explain how the product of 3 and 65 could be used to find (0.03) (0.65). Answers vary. Sample response: We can use vertical calculation to find 3 times 65, which equals 95. Because 0.03 is 3 hundredths and 0.65 is 65 hundredths, 95 will need to be multiplied by (0.0) (0.0) or 0.000. Multiplying by 0.000 moves the decimal point 4 places to the left, so the product is 0.095. Problem 3 Use vertical calculation to find each product.. (5.4) (.4). (.67) (3.5)..96. 5.835
Problem 4 A pound of blueberries costs $3.98 and a pound of clementines costs $.49. What is the combined cost of 0.6 pound of blueberries and.8 pounds of clementines? Round your answer to the nearest cent. $6.87. Sample reasoning: (3.98) (0.6) =.388, or about $.39. (.49) (.8) = 4.48, or about $4.48. The combined cost is.39 + 4.48 or 6.87. Problem 5 (from Unit 5, Lesson 3) Complete the calculations so that each shows the correct sum or difference. Problem 6 (from Unit 5, Lesson 4) Which has a greater value: 7.4 0.00 or 7.39 0.00? Show your reasoning. 7.4 0.00 has a greater value. 7.4 0.00 = 7.3978 and 7.39 0.00 = 7.3888. Problem 7 (from Unit, Lesson ) Andre is planting saplings (baby trees). It takes him 30 minutes to plant 3 saplings. If each sapling takes the same amount of time to plant, how long will it take Andre to plant 4 saplings? If you get stuck, consider using the table. number of saplings time in minutes 3 30 4 40 minutes (or equivalent). Possible strategy:
number of saplings time in minutes 3 30 0 4 40 Lesson 9 Problem Here is one way to find,05 5 using partial quotients. Show a different way of using partial quotients to divide,05 by 5. Responses vary. Sample response: Problem Andre and Jada both found 657 3 using the partial quotients method, but they did the calculations differently, as shown here.. How is Jada's work similar to and different from Andre s work?. Explain why they have the same answer.. Similarities: Andre and Jada both subtracted multiples of 3 several times. They both added the numbers being multiplied by 3 to find the quotient, and both ended up with 9. Differences: Jada subtracted multiples of 3 more times than Andre and the multiples of 3 that she subtracted were
different.. Andre and Jada have the same answer, since they both calculated 657 3 by subtracting multiples of 3 until there was no remainder and both gave the number of multiples of 3 subtracted as the answer. Problem 3 Which might be a better way to evaluate,50 46: drawing base-ten diagrams or using the partial quotients method? Explain your reasoning. Answers vary. Sample response: The partial quotient method works better. Dividing,50 into 46 equal groups by drawing will take too long. With the partial quotient method, the groups don t need to be drawn. Problem 4 Here is an incomplete calculation of 534 6. Write the missing numbers (marked with? ) that would make the calculation complete. Problem 5 Use the partial quotients method to find,03 43. Responses vary. Sample response: Problem 6 (from Unit 5, Lesson 8) Which of the polygons has the greatest area? A. A rectangle that is 3.5 inches wide and 6. inches long. B. A square with side length of 4.6 inches. C. A parallelogram with a base of 5.875 inches and a height of 3.5 inches. D. A triangle with a base of 7.8 inches and a height of 5.4 inches.
C Problem 7 (from Unit 5, Lesson 4) One micrometer is a millionth of a meter. A certain spider web is 4 micrometers thick. A fiber in a shirt is hundred-thousandth of a meter thick.. Which is wider, the spider web or the fiber? Explain your reasoning.. How many meters wider?. The fiber is wider. hundred-thousandth is 0 millionths, and 0 millionths is more than 4 millionths (it s 6 hundred thousandths more).. 6 hundred-thousandths Lesson 0 Problem Kiran is using long division to find 63 7. A. Hundreds B. Tens C. Ones D. Tenths B He starts by dividing 6 by 7. In which decimal place should Kiran place the first digit of the quotient (8)? Problem Here is a long-division calculation of 97 7.. There is a 7 under the 9 of 97. What does this 7 represent?. What does the subtraction of 7 from 9 mean? 3. Why is a written next to the from 9 7?. Answers vary. Sample response: The 7 under the 9 represents 700 (because it is written directly under the hundreds place of 97).. Answers vary. Sample response: It means a subtraction of 7 groups of hundred from 9 hundreds. 3. Answers vary. Sample response: To represent the 0 in 97. There is hundreds left after 7 hundreds are subtracted from 9 hundreds. The hundreds is combined with the ten from 97, which makes tens. Problem 3 Han's calculation of 97 9 is shown here.
. Find 80 9.. Use your calculation of 80 9 to explain how you know Han has made a mistake. 3. Identify and correct Han s mistake.. 80 9 = 60. Han is mistaken because the product of 97 9 and 9 should be 97, not,60. 3. Answers vary. Sample response: Han s mistake is that when he brought down the 7 from 97 and saw that 7 tens could not be divided into 9 groups (or 7 is not a multiple of 9), he did not write 0 above the 7 before bringing down the ones. Here is the correct long division calculation: Problem 4 Find each quotient.. 93. 77 3. 369 Problem 5 (from Unit 5, Lesson 7) One ounce of a yogurt contains of. grams of sugar. How many grams of sugar are in 4.5 ounces of yogurt? A. 0.7 grams B..7 grams C. 7. grams D. 7 grams
C Problem 6 (from Unit 5, Lesson 4) The mass of one coin is 6.78 grams. The mass of a second coin is 7. grams. How much greater is the mass of the second coin than the first? Show your reasoning. 0.50 grams, because 7. 6.78 = 0.50 Lesson Problem Use long division to show that the fraction and decimal in each pair are equal. 3. and 0.75 4 3. and 0.06 50 7 3. and 0.8 5 Problem Mai walked of a 30-mile walking trail. How many miles did Mai walk? Explain 8 or show your reasoning. 3.75 miles. Reasoning varies. Sample reasoning: of 30 is 30 8 = 3.75 8 Problem 3 Use long division to find each quotient. Write your answer as a decimal.. 99. 6 5 3.,988 8 Problem 4 9 9 8 36 To find the decimal of, Tyler reasoned: is equivalent to and to, so 5 5 50 00 9 the decimal of is 0.36. 5. Use long division to show. Is the decimal of also 0.36? Use 8 50
that Tyler is correct. 50 long division to support your answer. 8 Yes, the decimal of is also 0.36. 50 Problem 5 (from Unit 5, Lesson 4) Complete the calculations so that each shows the correct difference.. 0.67. 0.985 3. 0.37 Problem 6 (from Unit 5, Lesson 6) Use the equation 4 5 =,860 and what you know about fractions, decimals, and place value to explain how to place the decimal point when you compute (.4) (0.5)..4 is 4 (0.0) and 0.5 is 5 (0.0). So (.4) (0.5) can be written as 4 5 (0.0) (0.0), which is (,860) (0.000) or 0.86. Lesson Problem Here is a diagram representing a base-ten number. The large rectangle represents a unit that is 0 times the value of the square. The square represents a unit that is 0 times the value of the small rectangle. Here is a diagram showing the number being divided into 5 equal groups.
. If a large rectangle represents,000, what division problem did the second diagram show? What is its answer?. If a large rectangle represents 00, what division problem did the second diagram show? What is its answer? 3. If a large rectangle represents 0, what division problem did the second diagram show? What is its answer?.,30 5. The answer is 64.. 3 5. The answer is 6.4. 3. 3. 5. The answer is.64. Problem. Explain why all of these expressions have the same value.. What is the common value?. Answers vary. Sample response: The expressions all have the same value because the numerator and denominator are both being multiplied by 0 to get from one expression to the one above it. This does not affect the quotient (because dividing the 0 in the numerator by the 0 in the denominator results in ).. 50 4500 90 450 9 45 0.9 4.5 0.09 Problem 3 Use long division to find each quotient.. 7.89. 39.54 3 3. 0.76 5. 3.945. 3.8 3. 0.035 Possible calculations:
Problem 4 Four students set up a lemonade stand. At the end of the day, their profit is $7.5. How much money do they each have when the profit is split equally? Show or explain your reasoning. $4.38. Answers vary. Sample explanation: Four people are sharing $7.5 equally, so each person gets $7.5 4. Each person can be given $4, and then $.5 remains. Each person can be given $0.30, and then $0.3 remains. So they each get $0.08 more. That means each person gets a total of 4 + 0.30 + 0.08 or $4.38. Problem 5 (from Unit 5, Lesson 8). A standard sheet of paper in the United States is inches long and 8.5 inches wide. Each inch is.54 centimeters. How long and wide is a standard sheet of paper in centimeters?. A standard sheet of paper in Europe is.0 cm wide and 9.7 cm long. Which has the greater area, the standard sheet of paper in the United States or the standard sheet of paper in Europe? Explain your reasoning.. 7.94 cm by.59 cm. The European paper. Reasoning varies. Sample reasoning: The difference in the length is substantially larger than the difference in width, so the European paper probably has a larger area. Calculating shows the standard European paper is 63.7 sq cm while the standard United States paper is 603.46 sq cm. Lesson 3 Problem A student said, To find the value of 09. 6, I can divide,09 by 60.. Do you agree with this statement? Explain your reasoning.. Calculate the quotient of 09. 6 using any method of your choice.. Yes. Reasoning varies. Sample reasoning: As long as both dividend and divisor are multiplied by the same power of 0 (or just the same non-zero number), the quotient has the same value.. 8.. Methods vary. Sample response: 09. (dividend) and 6 (divisor) can be multiplied by 0 to get,09 60. The value of this quotient is 8.. Problem
Here is how Han found 3.59 3:. At the second step, Han subtracts 5 from 55. How do you know that these numbers represent tenths?. At the third step, Han subtracts 39 from 39. How do you know that these numbers represent hundredths? 3. Check that Han s answer is correct by calculating the product of.43 and 3.. Explanations vary. Sample explanation: The second 5 of the 55 is written in the tenths column (directly under the tenths place of 3.59), so it represents 5 tenths. The first 5 of 55 is written in the ones column (directly under the ones place of 3.59), so it represents 5 ones, which is 50 tenths. So, the 55 represents 55 tenths. The of 5 is written in the tenths column and the 5 is in the ones column, so the 5 represents 5 tenths.. Explanations vary. Sample explanation: The 9 of 39 is written in the hundredths column (directly under the hundredths place of 3.59), so it represents 9 hundredths. The 3 of 39 is written in the tenths column (directly under the tenths place of 3.59), so it represents 3 tenths, which is 30 hundredths. So, the 39 represents 39 hundredths. 3. (.43) 3 = 3.59, so Han is correct. Calculations vary. Sample calculation: (.43) 3 = (.43) (0 + 3) = 4.3 + 6. +.9 = 3.59. Problem 3. Write two division expressions that have the same value as 6. 3... Find the value of 6. 3.. Show your reasoning.. Answers vary. Sample responses: 6. 3 and 6. 0.3.. 9. Reasoning varies. Sample reasoning: Problem 4 A bag of pennies weighs 5. kilogram. Each penny weighs.5 grams. About how many pennies are in the bag? A. 0 B. 00 C.,000 D. 0,000 B
Problem 5 (from Unit 5, Lesson 3) Find each difference. If you get stuck, consider drawing a diagram...5.6. 0.7 0.4 3..3.75 4. 73.3..5.6 = 0.9. 0.7 0.4 = 0.3 3..3.75 = 9.55 4. 73.3 = 7.7 Problem 6 (from Unit 4, Lesson ) Plant B is 6 inches tall. Plant C is 4 4 inches tall. Complete the sentences and 3 5 show your reasoning.. Plant C is times as tall as Plant B.. Plant C is inches (taller or shorter) than Plant B. 6. Plant C is times as tall as Plant B. Plant B is 6 inches and Plant C is 5 3 4 4 inches tall. Reasoning varies. Sample reasoning: 5 4 64 0 64 4 6 = = = 5 3 5 3 5 3 6 0 5. Plant C is inches shorter than Plant B. ( 5 4 0 4 6 6 4 = 6 4 = = ) 3 5 5 5 5 5 Problem 7 (from Unit 3, Lesson 5) At a school, 460 of the students walk to school.. The number of students who take public transit is 0% of the number of students who walk. How many students take public transit?
. The number of students who bike to school is 5% of the number of students who walk. How many students bike to school? 3. The number of students who ride the school bus is 0% of the number of students who walk. How many students ride the school bus?. 9 students ( 460 0. = 9). 3 students ( 460 0.05 = 3) 3. 506 students ( 460.0 = 506) Lesson 4 Problem A roll of ribbon was meters long. Diego cut 9 pieces of ribbon that were 0.4 meter each to tie some presents. He then used the remaining ribbon to make some wreaths. Each wreath required 0.6 meter. For each question, explain your reasoning.. How many meters of ribbon were available for making wreaths?. How many wreaths could Diego make with the available ribbon?. 8.4 meters. Reasoning varies. Sample reasoning: Diego used 9 (0.4) or 3.6 meters for the presents, which leaves 8.4 meters in the roll, because 3.6 = 8.4.. 4 wreaths. ( 8.4 0.6 = 4) Problem The Amazon rainforest covered 6.4 million square kilometers in 994. In 04, it 50 covered only as much. Which is closest to the area of the Amazon forest in 59 04? Explain how you know without calculating the exact area. A. 6.4 million km B. 5.4 million km C. 4.4 million km D. 3.4 million km E..4 million km B. 5.4 million km is the closest. Sample reasoning: The rain forest covered about 6.5 million km 50 5 5 5, and is about. Since of 6 million is 5 million, of 59 6 6 6 6.5 million is about 5.4 million. Problem 3 To get an A in her math class, Jada needs to have at least 90% of the total number of points possible. The table shows Jada s results before the final test in the class.
Jada s points total points possible Homework 4 50 Test 87 00 Test 8 00 Test 3 9 00. Does Jada have 90% of the total possible points before the final test? Explain how you know.. Jada thinks that if she gets at least 9 out of 00 on the final test, she will get an A in the class. Do you agree? Explain.. No, before the final exam, Jada has 40 points out of 450. But 90% of 450 is (0.9) 450 = 405. So she is 4 points short of 90%.. Answers vary. Sample responses: No, Jada is 4 points short of 90% before the last test, so her score on the last test has to be at least 4 points more than 90% to make up for this. Maybe. This would give Jada 493 points out of 550. The value of 493 550 is a little more than 89.6. If the teacher rounds up, Jada will get an A. Problem 4 (from Unit 5, Lesson 4) Find the following differences. Show your reasoning.. 0.5 0.08. 0.06 0.035 3. 3.57.604. 0.3. Sample reasoning: 0.5 is 5 thousandths and 0.08 is 8 thousandths. 5 8 = 3, so the difference is 3 thousandths.. 0.0745. Sample reasoning: 0.06 can be written as 0.060 and the subtraction can be done using vertical calculation (as shown). 3. 0.9706. Sample reasoning: 3.57.604 can be thought of as.604 +? = 3.57, and we can use vertical calculation to see what number, when added to.604, makes 3.57 (the missing number is shown in boxes in the calculation). Problem 5 (from Unit 5, Lesson 3) Find these quotients. Show your reasoning.. 4... 3.5 0.4 3. 70.8 0.08.. Reasoning varies. Sample reasoning (decomposing into sums of multiples of ): 4 = (0 + ) = 0 + = 0 + =.. 33.5. Reasoning varies. Sample reasoning (decomposing into sums of
multiples of 4 plus remainder): 3.5 4 = (00 + 3 +.5) 4 = 5 + 8 +.5 = 33.5. 3.,8.5. Reasoning varies. Sample reasoning (using long division):