Fraction Problems. Figure 1: Five Rectangular Plots of Land

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1 Fraction Problems 1. Anna says that the dark blocks pictured below can t represent 1 because there are 6 dark blocks and 6 is more than 1 but 1 is supposed to be less than 1. What must Anna learn about fractions in order to overcome her confusion?. The 5 rectangles in Figure 1 represent 5 rectangular plots of land. Two thirds of the total land in these 5 plots is to be used for a neighborhood garden. Shade a portion of the 5 rectangular plots that could be used for the neighborhood garden. Figure 1: Five Rectangular Plots of Land. According to our text, what does 5 mean?. Many people have difficulty understanding improper fractions, such as 5. Describe two distinctly different ways that you could help someone understand the meaning of What could you show someone to help them understand the meaning of an improper fraction such as 7, other than showing them pieces of pies (or the like)? 1

2 6. You showed Johnny the picture below to help Johnny understand the meaning of the fraction 5. (show a picture of 5/ of pie) But Johnny doesn t get it. He says the picture shows 5, not 5. 6 (a) How could you help Johnny interpret the picture as showing 5? (b) What is another way to show 5 to Johnny, other than with parts of objects? 7. (a) Give three different fractions that you can legitimately use to describe the shaded region in Figure. For each fraction, explain why you can use that fraction to describe the shaded region. (b) Write an unambiguous question about the shaded region in Figure that can be answered by naming a fraction. Explain why your question is not ambiguous. Figure : What Fraction is Shaded? 8. Discuss why it can be confusing to show an improper fraction such as 7 with pieces of pie or the like. What is another way to show the fraction 7? 9. If of a cup of a food gives you your daily value of potassium, then what fraction of your daily value of potassium is in 1 cup of the food? Draw a picture that helps you solve this problem. Use your picture to help you explain your solution. For each fraction in this problem, and in your solution, describe the whole associated with this fraction. In other words, describe what each fraction is of. 10. Use the meaning of fractions to explain clearly why = 7 7. (Do not use multiplication by 1 to explain this.)

3 11. People often say that 0 0 = because we can cancel the zeros. Explain the rationale for being able to cancel the zeros. Describe what the process of cancelling the zeros actually is in terms of the diagrams in Figure, which show 0 0 of a pie shaded and of a pie shaded. Figure : Pies 1. Susie wants to know why it s true that = but it s not true that = + +. Susie says that in both cases you are doing the same thing to the top and the bottom of, so she wants to know why one way is correct but the other is not. Explain this to Susie, using pictures or diagrams to support your explanation. Susie does not know about multiplying fractions. 1. Susie says that when you do the same thing to the top and bottom of a fraction you get an equivalent fraction. Is Susie right, or is it possible to do the same thing to the top and the bottom of a fraction and not get an equivalent fraction?

4 1. Using the example 8 1 describe how to put a fraction in simplest form and explain why this procedure makes sense. In terms of pictures, what are you doing when you put a fraction in simplest form? 15. Using the fractions 1 and, describe how to give two fractions common denominators. In terms of pictures, what are you doing when you give fractions common denominators? 16. Ken ordered of a ton of sand. Ken wants to receive 1 of his order 5 now (and of his order later). What fraction of a ton of sand should Ken receive now? (a) Draw a picture to help you solve the problem. Explain how your picture helps you to solve the problem. (b) What are the different wholes that the fractions in this problem refer to? For each fraction in the problem and in the solution to the problem, describe the corresponding whole. 17. Tyrone is making a recipe that calls for of a cup of ketchup. Tyrone only has 1 of a cup of ketchup. Assuming that Tyrone has enough of the other ingredients, what fraction of the recipe can Tyrone make? (a) Draw a picture to help you solve the problem. Explain how your picture helps you to solve the problem. (b) What are the different wholes that the fractions in this problem refer to? For each fraction in the problem and in the solution to the problem, describe the corresponding whole. 18. Plot 5, 5, and on the number line below in such a way that each 6 number falls on a tick mark. Lengthen the tick marks of whole numbers.

5 19. Plot,.1, and.15 on the number line below in such a way that 7 each number falls on a tick mark. Lengthen the tick marks of whole numbers (if any). 0. The line segment below units long. Show a line segment that is 5 units long. Explain how you know your segment is the correct length. unit 1. Plot 1 5 and at least two other nearby integers on a number line and explain why you have plotted these numbers appropriately.. If your full daily value of potassium is 600 milligrams, then how many milligrams is 5% of your daily value of potassium? (a) Show how to solve the problem with the aid of a picture. Explain how your picture helps you solve the problem. (b) Explain how to solve the problem numerically.. If the normal rainfall for August is.5 inches, but only 1.75 inches of rain fell in August, then what percent of the normal rainfall fell in August? (a) Show how to solve the problem with the aid of a picture. Explain how your picture helps you solve the problem. (b) Explain how to solve the problem numerically.. If the full capacity of a tank is 5 liters and the tank is filled with only 15 liters, then what percent full is the tank? (a) Show how to solve the problem with the aid of a picture and common fractions. Explain how your picture helps you solve the problem. (b) Explain how to solve the problem numerically. 5. If $85,000 is 0% of the budget, then what is the full budget? 5

6 (a) Show how to solve the problem with the aid of a picture. Explain how your picture helps you solve the problem. (b) Explain how to solve the problem numerically. 6. Explaining your reasoning clearly, use pictures to help you answer the following: a. What percent of 1 5 is 5? b. What percent of 5 is 1 5? 7. If of a cup of juice gives you 100% of your daily value of vitamin C, then what percent of your daily value of vitamin C will you get in 1 full cup of juice? Solve this problem by drawing a diagram. Explain how your diagram helps you to solve the problem. 8. If Company A sells 0% as many cars as Company B, then what are Company B s car sales, when they are calculated as a percentage of Company A s sales? Draw a picture or diagram to help you solve the problem. Use your picture to help you explain your answer. 9. Describe three general methods for determining which of two fractions is greater. Illustrate the three methods with the fractions 5 and Explain clearly and in detail why we can determine which of two fractions is greater by giving the two fractions common denominators. What is the rationale behind this method? What are we really doing when we give the fractions common denominators? 1. Show how to use the cross-multiplying method to determine which of 5 8 is greater. and 7 1. Explain clearly and in detail why we can determine which of two fractions is greater by using the cross-multiplying method. What is the rationale behind this method? What are we really doing when we cross-multiply in order to compare fractions?. Give two different methods for solving the following problem. Find three different fractions in between and whose numerators and 5 denominators are all whole numbers. 6

7 . Conrad says that > because > and 8 > 7. Regardless of 8 7 whether or not Conrad s conclusion is correct, discuss whether or not Conrad s reasoning is valid. 5. Minju says that fractions that use bigger numbers are greater than fractions that use smaller numbers. Make up two problems for Minju to help her reconsider her ideas. For each problem, explain how to solve it, and explain why you chose that problem for Minju. 6. Use reasoning other than converting to decimals, using common denominators, or cross-multiplying to determine which of and is greater. Explain your reasoning clearly and in detail. 7. Use reasoning other than converting to decimals, using common denominators, or cross-multiplying to determine which of 1 1 and is greater. Explain your reasoning clearly and in detail. 8. Use reasoning other than converting to decimals, using common denominators, or cross-multiplying to determine which of 8 5 and is 9 6 greater. Explain your reasoning clearly and in detail. 9. Use reasoning other than converting to decimals, using common denominators, or cross-multiplying to determine which of 1 1 and is 81 greater. Explain your reasoning clearly and in detail. Addition and Subtraction 0. Using the example +, explain why we add fractions the way we do. What are we really doing when we carry out the procedure of adding fractions? What is the logic behind this procedure? Explain your answer clearly and in detail. 1. Frank says that + = and uses the picture in Figure as evidence. 6 Explain why Frank s method is not a valid way to add fractions. Be specific. (Do not explain how to do the problem correctly, explain where the flaw is in Frank s reasoning.). Explain how to convert a mixed number, such as 5, to an improper fraction, and explain why this procedure makes sense, in other words, explain the rationale behind the procedure. 7

8 add them together: Figure : Frank s Idea for Adding +. Which of the following problems can be solved by adding 1 + 1? For those problems that can t be solved by adding 1 + 1, solve the problem in another way if there is enough information to do so, or explain why the problem cannot be solved. (a) One third of the boys in Mrs. Scott s class want to have a peanut butter sandwich for lunch. One fourth of the girls in Mrs. Scott s class want to have a peanut butter sandwich for lunch. What fraction of the children in Mrs. Scott s class want to have a peanut butter sandwich for lunch? (b) One third of the pizzas served at a party have pepperoni on them. One fourth of the pizzas served at the party have mushrooms on them. What fraction of the pizzas served at the party have either pepperoni or mushrooms on them? (c) The pizzas served at a party all have only one topping. One third of the pizzas served at the party have pepperoni on them. One fourth of the pizzas served at the party have mushrooms on them. What fraction of the pizzas served at the party have either pepperoni or mushrooms on them?. Can the following story problem be solved by subtracting 1 1? If not, explain why not, and solve the problem in another way if there is enough information to do so. Story problem: There is 1 of a pie left over from yesterday. Julie eats of the leftover pie. Now how much pie is left? 1 5. Denise says that 1 = 1 and gives the reasoning indicated in Figure 5 to support her answer. Is Denise right? If not, what is wrong with her reasoning and how could you help her understand her mistake and fix it? Don t just explain how to solve the problem correctly, explain where Denise s reasoning is flawed. 8

9 First I showed /. Then when you take away half of that you have 1/ left. Figure 5: Denise s Idea For 1 6. Write a story problem for Show how to solve + 1 in two different ways. In each case, express your answer as a mixed number. Explain why why both of your 5 methods are legitimate. 8. Show how to solve 1 in two different ways. In each case, express your answer as a mixed number. Explain why why both of your 5 methods are legitimate. 9. A farmer has two plots of land. Twenty percent of the first plot is planted with cotton and 0% of the second plot is planted with cotton. Can we calculate the percentage of the farmer s (total) land that is planted with cotton by adding 0% + 0%? Explain your answer. 50. A farmer has two plots of land, both of the same size. Twenty percent of the first plot is planted with cotton and 0% of the second plot is planted with cotton. What percentage of the farmer s (total) land is planted with cotton? Explain your answer. Percent Increase and Percent Decrease 51. The price of a gizmo went from $.50 down to $1.50. Find the percent decrease in the price of the gizmo in two ways: (a) by calculating (show your work), and (b) by using pictures and words. tons of waste per month to pro- 5. A community goes from producing 1 ducing 1 tons of waste per month. 9

10 (a) Show how to use a picture to help you calculate the percent increase in a community s monthly waste production. (b) Show how to calculate the percent increase in the community s monthly waste production numerically. 5. Fill in the blanks so as to make the statements below correct: (a) 600 is % of 500. (b) 100 is % more than 0. (c) 68 is % less than If the price of a barrel of oil goes up by 0%, and then goes back down by 0% (of the new, raised price), will the final price of oil (after raising and lowering) be equal to the original price of oil? If not, which will be greater, the original price or the final price? Determine the answers to these questions without doing any calculations. Explain your reasoning clearly. Multiplication 55. Here is how Nya solved the problem 7: Half of 7 is 6. Half of 6 is 18. Then to add 6 and 18 I did 0 plus 1, which is 5. Write a string of equations that incorporate Nya s ideas. Which properties of arithmetic did Nya use (knowingly or not) and where? Be thorough and be specific. Write your equations in the following format: 7 = some expression = some expression. = Which of the following are story problems for 1 Explain briefly in each case. and which are not? 10

11 (a) There is of a cake left. One half of the children in Mrs. Brown s class want cake. How much of the cake will the children get? (b) A brownie recipe used of a cup of butter for a batch of brownies. You ate 1 of a batch. How much butter did you consume when you ate those brownies? (c) Three quarters of a pan of brownies is left. Johnny eats 1 of a pan of brownies. Now what fraction of a pan of brownies is left? (d) Three quarters of a pan of brownies is left. Johnny eats 1 of what is left. How many brownies did Johnny eat? (e) Three quarters of a pan of brownies is left. Johnny eats 1 of what is left. What fraction of a pan of brownies did Johnny eat? 57. Write a story problem for Write one story problem for and one story problem for. Say which is which. 59. Ken ordered of a ton of sand. Ken wants to receive 1 of his order 5 now (and of his order later). What fraction of a ton of sand should Ken receive now? (a) Solve the problem numerically and explain why you can set the problem up numerically the way you do. (b) Draw a picture to help you solve the problem. Explain how your picture helps you to solve the problem. Discuss how the way you set the problem up numerically is related to your picture. 60. Write a simple story problem for 5. Use your story problem and use pictures to explain clearly why it makes sense that the answer to the fraction multiplication problem is 5. In particular, explain why the numerators are multiplied and why the denominators are multiplied. 11

12 61. Write a simple story problem for and use your story problem and pictures to help you solve the problem. Explain why it would be easy to interpret the picture incorrectly as showing that = 8. Explain 1 how to interpret the picture correctly. 6. Explain why it would be easy to interpret the picture in Figure 6 incorrectly as showing that = 9. Explain how to interpret the picture 1 correctly and explain why your interpretation fits with the meaning of. Figure 6: A Picture for 6. In order to understand fraction multiplication thoroughly, we must be able to work simultaneously with different wholes. Using the example, explain why this is so. What are the different wholes that are 5 associated to the fractions in the problem (including the answer 5 to the problem)? 6. In oder to understand fraction multiplication thoroughly, we must understand how to divide a whole number of objects into equal parts, such as dividing 8 cookies equally among people. Where do we need to understand how to divide a whole number of objects into equal parts in order to understand thoroughly? Be specific In addition to understanding the meaning of fractions, list two other important ideas that a person must develop in order to understand fraction multiplication thoroughly. Explain why the ideas you list are required for understanding fraction multiplication thoroughly. 66. Discuss why we must develop an understanding of multiplication that goes beyond seeing it as repeated addition. Division 1

13 67. When Mary converted a recipe from metric measurements to U.S. Customary measurements, she discover that she needed 8.6 cups of flour. Mary has a 1 cup measure, a 1 cup measure, a 1 cup measure, and a measuring tablespoon, which is 1 of a cup. How should Mary use her 16 measuring implements to measure the 8.6 cups of flour as accurately as possible? Explain your reasoning. 68. Use the meaning of fractions and the meaning of division to explain why =. Your explanation should be general, in the sense that you could see why the equation = would still be true if other numbers were to replace and. 69. Use the large, subdivided square in Figure 7 to help you explain why the decimal representation of 1 is Figure 7: Show Show how to use pennies and dimes to help you explain why 1 5 =., 1 =.5, and 1 8 = Which of the following problems can be solved by the division problem 1? For those that can be, which interpretation of division is used? For those that cannot be solved by 1, determine how to solve the problem another way, if it can be solved. 1

14 (a) of a bag of gummi worms make 1 a cup. How many bags of gummi worms does it take to make one cup? (b) of a bag of gummi worms make 1 a cup. How many cups of gummi worms are in one bag? (c) You have of a container of chocolate syrup and a recipe that calls for 1 of a cup of chocolate syrup. How many batches of your recipe can you make (assuming you have enough of the other ingredients)? (d) You have of a cup of chocolate syrup and a recipe that calls for 1 of a cup of chocolate syrup. How many batches of your recipe can you make (assuming you have enough of the other ingredients)? (e) If of a pound of candy costs 1 of a dollar, then how many pounds of candy should you be able to buy for 1 dollar? (f) If you have of a pound of candy and you divide the candy in 1, then how much candy will you have in each portion? (g) If 1 of a pound of candy costs $1, then how much should you expect to pay for of a pound of candy? 7. Grandma s favorite muffin recipe uses 1 cups of flour for one batch of 1 muffins. Answer each of the following questions, explaining clearly which meaning of division you are using each time. In each case, calculate the answer without the use of a calculator. (a) How many cups of flour are in one muffin? (b) How many muffins does 1 cup of flour make? (c) If you have cups of flour, then how many batches of muffins can you make? 7. Write a story problem for 1 and draw pictures to help you solve the problem. Explain clearly how to interpret the pictures. 7. Sally is working on the division problem 1 1 by using the story problem, how many 1 cups of water are in 1 cups of water? Sally draws a picture like the one in Figure 8 and gives the answer 1. When Sally checks her work by inverting and multiplying, she realizes that she has made a mistake. Her answer should be 1, not 1. How should Sally interpret her picture so as to get the correct answer, 1? 1

15 Figure 8: A Picture for Write one story problem for 1 and another story problem for 1, making clear which problem is which. In each case, draw pictures to help you solve the problem. Explain clearly how to interpret your pictures. 76. Which of the following division problems can be interpreted as asking: How many groups of are in? =? =? =? 17 8 =? =? =? =? 8 17 =? Circle all that apply and explain your reasoning. 77. Bruce is working on the division problem longhand, but he can t remember what to do about decimal points. Instead, Bruce solves the division problem longhand and gets the answer.51. Bruce knows that the answer to will be.51, except that the decimal point in.51 must be shifted over somehow. (a) Explain to Bruce how he can reason about the sizes of the numbers to determine where the decimal point is supposed to go in the 15

16 answer to already done. Make use of the work that Bruce has (b) Show Bruce how to write 75. and 16.9 as improper fractions and use fraction division to determine where the decimal point should go in.51. Make use of the work that Bruce has already done. 78. Alice and Jose are planning to mix red and yellow paint. They are considering which of the two following paint mixtures will make a more yellow paint: a mixture of parts red to 7 parts yellow a mixture of parts red to 8 parts yellow Alice says that both paints will look the same because to make the second mixture you just add one part of each color to the first mixture. Because you add the same amount of each color, the second mixture should look the same as the first mixture. Jose says that the second mixture should be more yellow than the first because it uses 8 parts yellow and the first mixture only uses 7 parts yellow. Discuss the children s ideas. Is their reasoning valid or not? Which paint will be more yellow and why? 79. A dough recipe calls for cups of flour and 1 1 cups of water. You want to use the same ratio of flour to water to make a dough with 10 cups of flour. How much water should you use? (a) Set up a proportion to solve the problem. Explain why you can set up the proportion as you do. What is the logic behind the procedure of setting up a proportion? What do the two fractions that you set up mean and why do you set those two fractions equal to each other? (b) Solve the proportion by cross-multiplying. Explain why it makes sense to cross-multiply. What is the logic behind the procedure of cross-multiplying? (c) Now solve the problem of how much water to use for 10 cups of flour in a different way, by using the most elementary reasoning you can. Explain your reasoning clearly. 16

17 The problems (later: many/some of the problems) in this list are from the Instructor s Manual for Arithmetic for Elementary Teachers, first edition, by Sybilla Beckmann, Addison-Wesley, 00 (expected). 17

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