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Problem Solving Non-standard Problems

Problem Solving Non-standard Problems. Sara Hershkovitz Center for Educational Technology Israel. Imagine a standard classroom of fifth-grade students, receiving the following nonstandard problems to solve:. Problem No. 1.

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Problem Solving Non-standard Problems

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  1. Problem SolvingNon-standard Problems Sara Hershkovitz Center for Educational Technology Israel

  2. Imagine a standard classroomof fifth-grade students, receiving the following nonstandard problems to solve: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  3. Problem No. 1 How many two-digit numbers, up to one hundred, have a tens digit that is larger than the units digit? Student’s answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  4. What if not… Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  5. Problem No. 2: How many two-digit numbers, up to one hundred, have a units digit that is larger than the tens digit? Student’s answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  6. Problem No. 3: Look at the following numbers: 23, 20, 15, 25,which number does not belong? Why? Students' answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  7. Problem No. 4: 100 nuts are divided among 25 children, unnecessarily in equal portions. Each child receives an odd number of nuts. How many nuts does each child receive? Students' answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  8. Problem No. 5:(Elaborated after Paige, 1962) We made change from one Shekel into smaller coins: 5, 10, and 50 Agorot (or cents). Make change so that you hold three times as many 10-Agorot coins as 5-Agorot coins. Students' answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  9. Problem No. 6: A witch wants to prepare a frog drink. She can buy dried frogs only in packages of five or eight. Note that the witch has to buy the exact number of dried frogs she needs. How many frogs can she buy? What is the largest number of frogs she cannot buy? Students' answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  10. What if not…. In a similar witch problem,the dried frogs come in packages of four or eight. What is the largest number of frogs that the witch cannot buy? back Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  11. Problem No. 7: Insert different numbers in the blank spaces, so that the four-digit number received divides by 3. __ 1 4 __ Students' answers: A teacher’s answer: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  12. Problem No 8: I have a magic handbag. If I leave some money in it overnight, I find twice as much money in the morning, plus one unit. Once, I forgot some money in the handbag for two nights, and on the third day I found 51 dollars. How much money was in the handbag before the first night? Students' answers: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  13. Discussion Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  14. A.The types of problems: How do the problems differ? What are the different types of problems? Can the types be characterized? Write a few novel problems for each problem type. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  15. B. Promoting creativity: Creativity, as defined by some researchers (Guilford 1962, Haylock 1987, Silver 1994), contains the following components: Fluency: measured by the total number of replies. Flexibility:measured by the variety of categories given. Originality: measured by the uniqueness of a reply within a given sample of replies. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  16. Do such problems promote creativity? Which additional types of problems can be employed for promoting creativity? Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  17. C. Using these problems in the classroom: We routinely use these problems in the classroom, approximately once a week, without connecting them to ongoing topics studied in class. Our goal is to allow students to experience the following: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  18. Extending and applying previous mathematical knowledge. • Constructing relationships among topics by integrating various topics within one problem. • Encouraging the use of different strategies: working systematically, while using naïve strategies. • Stimulating reflections on student experiences. Articulating the solution in various ways: words, or diagrams. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  19. We emphasize the following merits of class discussion: • An opportunity for students to explain how they think about problems; different problems provide different ways of analysis. • Individual ways of presentation may bring up disparate relations or different mathematical points of view. • Students are encouraged to understand and assimilate someone else’s strategies. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  20. A discussion emphasizes that mathematics entails active processes, such as investigating, looking for patterns, framing, testing, and generalizing, rather than just reaching a correct answer. • The discussion demonstrates that mathematical thinking involves more questions than answers. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  21. Thank you Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  22. Problem No. 1 - Avi T(Teacher): How many ……. (posing the question)? A(Avi): Mmmm…. (thinking). T: Do you understand the problem? A: Yes. T: Can you find an example for such a number? A: 52. T: Ok; so how many two-digit numbers are there? A: A lot. . Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  23. T: How many? A: A lot. T: How many? Give a number. A: 8. T: Tell me what they are. A: 53, 42, 85, 31, 64, 97, 75, 61, 98; that’s all Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  24. Problem No. 1 - Ben:I’m going to find them in the “First-hundred table”: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  25. Now I will count them: 1,2,3,…..45. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  26. Problem No. 1 - Galit 21, 30, 31, 32, Mmmm. I see 10, 20, 21, 30, 31, 32 (thinking a bit before writing down each number.) Then Galit began writing faster: 40, 41, 42, 43, 44, and deleted the 44. T: Can you explain what you are doing? Galit didn’t answer, and began writing quickly: 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74. 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92,93, 94, 95, 96, 97, 98 Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  27. Galit raised her head and said: Now I have to count them: 1,2,3,4 ... T: I saw that you began writing quickly. What happened at that point? G: I saw that for each tens number in the list, I could write the numbers less one 1,2,3,4,5,.. T: Do you know how to sum the numbers by a shorter method? G: 6,7,8,9,10,11,12,………43,44.45. There are 45 numbers. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  28. Problem No. 1 - Dana D: The smallest number is 20, Mmmm; no, sorry, 10,…. and the largest is …98 98 – 10 = 88; there are 88 numbers. T: Mmmm. D: Sorry, between the numbers there are also 55, and 34 and 28; I have to think. 20, 21 ……..30, 31, 32 T: You forgot 10. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  29. D: Yes, 10, 20, 21, 30, 31, 32 40, 41, 42, 43 I see; In ten I have one number,in twenty there are two numbers, in thirty there are three, in forty there are four; I can continue up to ninety where there are nine.. So, 1+2+3+4+5+6+7+8+9= 1+2 is 3, plus 3 is 6, 10, 15, 21, 28, 36,….45. There are 45 numbers. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  30. Problem No. 1 - Hadar Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  31. Now I see that there are ten (while pointing to the arrows), twenty, thirty, forty,forty five …numbers. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  32. Problem No. 1 - Vered Vered wrote in the columns: Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  33. Now I can sum them;8+7+6+5+4+3+2+1are 15, 21, 26, 30, 33, 35, 36. Actually, I see the same unit in each row. back Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  34. Problem No. 2 - Ziva This problem is symmetrical to the previous one, so the answer is the same: 45 numbers. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  35. Problem No. 2 - Chedva If I think of the “First-hundred table”, 100 has three digits, so 99 numbers are left; subtract the previous 45, so there are 55 numbers. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  36. Problem No. 2 - Mick Beginning with 10: 12, 13, 14. 15, 16, 17, 18, 19 Beginning with 20: 23, 24, 25, 26, 27, 28, 29 With 30: 34, 35, 36, 37, 38, 39 ………….. ………….. 89 With 80: --- with 90: it’s 36= 8+7+6+5+4+3+2+1 Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  37. Problem No. 2 - Yoni I’m thinking of the “First-hundred table”. 100 has three digits, and 99 numbers are left. 1-9 are one-digit numbers, so there are only 90 numbers left. 11-99 do not fit, as well, because they have the same digits; we have to remove 9 additional numbers, and then we are left with 81 numbers. If we remove 45 numbers, from the previous problem, we end up with 36 numbers. back Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  38. Problem No. 3 - Adi 20 – It is the only even number. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  39. Problem No. 3 - Bill 15 - The only number that divides by 3. 20 - The units digit is 0; It doesn’t have units. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  40. Problem No. 3 - Gur 15 - It is in the 2nd ten and the rest are in the 3rd ten. 20 - The only round number. This number has more factors 23 - Not a multiple of 5. 25 - The sum of its digits is the largest. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  41. Reasons for choosing a certain number as exceptional 15 • It is under 20 • Its tens digit is 1, and the rest have the digit 2 • It is in the 2nd ten and the rest are in the 3rd ten • It is the smallest number • The only number that divides by 3 Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  42. 20 • The only even number • A multiple of 2; divides by 2 • The units digit is 0; it doesn’t have units • The sum of its digits is doesn’t fit the series • The number divides by 10 • The only round number • The only number divides by 4 • The number that has more factors Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  43. 23 • Not a multiple of 5 • Doesn’t divide by 5 • Not in the series that adds 5 to each number • The only prime number • Doesn’t appear in the multiplication table • The only number that has the digit 3 in it Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  44. 25 • A square number • It is the largest number • The sum of its digits is the largest • Is 30 in approximation Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  45. Distribution of Number Property Categories Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  46. Unique Replies Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  47. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  48. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  49. back Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

  50. Problem No. 4 - Aluma It's impossible; 100 : 25 = 4That means that each child gets 4 nuts, but it is an even number. Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

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