36 Sample Problems for Arithmetic and Problem Solving | MATH 5001, Assignments of Mathematics

Download 36 Sample Problems for Arithmetic and Problem Solving | MATH 5001 and more Assignments Mathematics in PDF only on Docsity! Sample Problems for MATH 5001, University of Georgia 1. Give three different decimals that the bundled toothpicks in Figure 1 could repre- sent. In each case, explain why the bundled toothpicks can represent that decimal. Figure 1: Which Decimals Can These Bundled Toothpicks Represent? 2. Label the tick marks on the three number lines in Figure 2 in three different ways. In each case, your labeling should fit with the fact that the tick marks at the ends of the number lines are longer than the other tick marks. You may further lengthen the tick marks at either end as needed. 7.1 7.1 7.1 Figure 2: Label These Number Lines 3. Anna says that the dark blocks pictured in Figure 3 can’t represent 1 4 because there are 6 dark blocks and 6 is more than 1 but 1 4 is supposed to be less than 1. What must Anna learn about fractions in order to overcome her confusion? 4. (a) Give three different fractions that you can legitimately use to describe the shaded region in Figure 4. 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 4 that can be answered by naming a fraction. Explain why your question is not ambiguous. 1 Figure 3: Representing the Fraction 1 4 Figure 4: What Fraction is Shaded? 5. If 3 4 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. 6. Use the meaning of fractions to explain clearly why 2 3 = 2 · 4 3 · 4 . (Do not use multiplication by 1 to explain this.) 7. Plot 5 6 , 5 4 , and 4 3 on the number line in Figure 5 in such a way that each number falls on a tick mark. Lengthen the tick marks of whole numbers. Figure 5: A Number Line 8. If the normal rainfall for August is 2.5 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. 9. Find a number between 7.8651 and 7.8652 and plot all three numbers visibly and distinctly on a copy of the number line in Figure 6. Label all the longer tick marks on your number line. 10. Which of the following could the pictures in Figure 7 be used to illustrate? Circle all that apply. 120 > 45 12 > 4.5 12 > .45 .12 > .45 .12 > .045 2 21. To solve 341 − 176, a student writes the following: 341 176 −5 −30 200 200 −30 170 −5 165 341 − 176 = 165 Describe the student’s solution strategy and discuss why the strategy makes sense. Expanded forms may be helpful to your discussion. 22. A rug is 4 feet wide and 5 feet long. Use the meaning of multiplication to explain why we can calculate the area of the rug by multiplying. 23. A box is 2 feet deep, 3 feet wide, and 4 feet tall. Use the meaning of multiplication to explain why we can calculate the volume of the box by multiplying. 24. Which property or properties of arithmetic do you use when you calculate 3× 70 by first calculating 3 × 7 = 21 and then putting a zero on the end of 21 to make 210? Write equations to show which properties are used and where. 25. Write at least two different expressions for the total number of triangles in Fig- ure 8. Each expression is only allowed to use the numbers 3, 4, and 5, the multiplication symbol, and parentheses. In each case, use the meaning of multi- plication to explain why your expression represents the total number of triangles in Figure 8. Figure 8: How Many Triangles? 26. Keisha says that it’s easy to multiply even numbers by 5 because you just take half of the number and put a zero on the end. Write equations that incorporate Keisha’s method and that demonstrate why her method is valid. Use the case 5 × 8 for the sake of concreteness. Write your equations in the following form: 5 × 8 = some expression = some expression ... ... = 40. 27. Ashley knows her 1×, 2×, 3×, 4×, and 5× multiplication tables well. 5 (a) Describe how the three pictures in Figure 9 provide Ashley with three differ- ent ways to determine 6×8 from multiplication facts that she already knows well. In each case, write an equation that corresponds to the picture and that shows how 6 × 8 is related to other multiplication facts. Figure 9: Different Ways to Think of 6 × 8 (b) Draw pictures showing two different ways that Ashley could use the mul- tiplication facts she knows well to determine 6 × 7. In each case, write an equation that corresponds to the picture and that shows how 6×7 is related to other multiplication facts. 28. Halley calculates 45% of 280 in the following way: Half of 280 is 140. I know 10% is 28, so 5% is half of that, which is 14. So I get 140 minus 14, which is 126. (a) Explain briefly why it makes sense for Halley to solve the problem the way she does. What is the idea behind her strategy? (b) Write a string of equations that incorporate Halley’s ideas. Which properties of arithmetic did Halley use (knowingly or not) and where? Be thorough and be specific. Write your equations in the following format: 45% × 280 = some expression = some expression ... ... = 126. 29. (a) Use the partial products algorithm to calculate 34 ×27 (b) Use the meaning of multiplication and a picture to give a clear and thorough explanation for why the partial products algorithm gives the correct answer to the multiplication problem in part (a). (Use graph paper for your picture.) (c) Show why the partial products algorithm calculates the correct answer to the multiplication problem in part (a) by writing equations that use properties of arithmetic and that incorporate the calculations of the partial products algorithm. (FOIL is not a property of arithmetic.) Write your equations in 6 the following format: 34 × 27 = some expression = some expression ... Identify the properties of arithmetic that you used and show where you used them. (d) Relate your equations for part (c) to your picture for part (b). 30. Which of the following are story problems for 1 2 × 3 4 and which are not? Explain briefly in each case. (a) There is 3 4 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 3 4 of a cup of butter for a batch of brownies. You ate 1 2 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 2 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 2 of what is left. How many brownies did Johnny eat? (e) Three quarters of a pan of brownies is left. Johnny eats 1 2 of what is left. What fraction of a pan of brownies did Johnny eat? 31. Write a simple story problem for 3 4 · 3 5 . Use your story problem and use pictures to explain clearly why it makes sense that the answer to the fraction multiplication problem is 3 · 3 4 · 5 . In particular, explain why the numerators are multiplied and why the denomina- tors are multiplied. 32. For each of the following story problems, write the corresponding division problem, state which interpretation of division is involved (the how many groups? or the how many in each group?, with or without remainder), and solve the problem. (a) Given that 1 quart is 4 cups, how many quarts of water is 35 cups of water? (b) If your car used 15 gallons of gasoline to drive 330 miles, then how many miles per gallon did your car get? (c) If you drove 240 miles at a constant speed and if it took you 3 1 2 hours, then how fast were you going? 7