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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 1 Teaching Mathematics through Problem Solving in Lower- and Upper-Secondary School Frank K. Lester, Jr. Indiana University

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 2 Themes for this session: The teacher’s role Developing habits of mind toward problem solving

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 3 The cylinders problem LAUNCH: Do cylinders with the same surface area have the same volume?

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Teaching Math through Problem Solving Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet 4 SUMMARIZE Have students report about their findings. Have students report about their findings. Encourage student-to-student questions. Encourage student-to-student questions. Look back: How is this problem related to problems we have done before? Look back: How is this problem related to problems we have done before? What have we learned about the relationship between circumference and volume? What have we learned about the relationship between circumference and volume? Examine the formulas for surface area and volume (Big math ideas) Examine the formulas for surface area and volume (Big math ideas) SA = (2π)R*H; V = πR 2 *H

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Teaching Math through Problem Solving Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet 5 Extending the Activity Have students conjecture about what is happening to the volume as the cylinder continues to be cut, getting shorter and shorter (and wider and wider). Have students conjecture about what is happening to the volume as the cylinder continues to be cut, getting shorter and shorter (and wider and wider). Some students may become interested in exploring the limit of the process of continuing to cut the cylinders in half and forming new ones. Some students may become interested in exploring the limit of the process of continuing to cut the cylinders in half and forming new ones. What if the cylinders have a top and bottom? What if the cylinders have a top and bottom?

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Teaching Math through Problem Solving Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet 6 Qualities of the Lesson A question is posed about an important mathematics concept.A question is posed about an important mathematics concept. Students make conjectures about the problem.Students make conjectures about the problem. Students investigate and use mathematics to make sense of the problem.Students investigate and use mathematics to make sense of the problem. The teacher guides the investigation through questions, discussions, and instruction.The teacher guides the investigation through questions, discussions, and instruction. Students expect to make sense of the problem.Students expect to make sense of the problem. Students apply their understanding to another problem or task involving these concepts.Students apply their understanding to another problem or task involving these concepts.

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 7 12 345 678 987 654 321

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 8 12 345 678 987 654 321

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 9 HoM 1: Mathematics is the study of patterns and structures, so always look for patterns. 11 x 11 = 121 111 x 111 = 12 321 1 111 x 1 111 = 1 234 321 11 111 x 11 111 = 123 454 321... 111 111 111 x 111 111 111 = 12 345 678 987 654 321

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 10 Which triangles can be divided into 2 isosceles triangles?

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 11 Find the sum of the interior angles of the star in at least 5 different ways.

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 12 HoM2: Develop a willingness to receive help from others and provide help to others. HoM3: Learn how to make reasoned (and reasonable) guesses. HoM4: Become flexible in the use of a variety of heuristics and strategies.

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 13 For what values of n does the following system of equations have 0, 1, 2, 3, 4, or 5 solutions? x 2 - y 2 = 0 (x - n) 2 + y 2 = 1

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 14 Determine the sum of the series: 1/12 + 1/23 + 1/34 + 1/45 + 1/n(n+1)

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 15 HoM 5: Draw a picture or diagram that focuses on the relevant information in the problem statement. HoM 6: If there is an integer parameter, n, in the problem statement, calculate a few special cases for n = 1, 2, 3, 4, 5. A pattern may become evident. If so, you can then verify it by induction.

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 16 (A 2 + 1)(B 2 + 1)(C 2 + 1)(D 2 + 1) ABCD If A, B, C, and D are given positive numbers, prove or disprove that ≥ 16

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 17 If x, y, z, and w lie between 0 and 1, prove or disprove that (1 - x)(1 - y)(1 - z)(1 - w) > 1 - x - y - z - w

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 18 HoM 7: If there are a large number of variables in a problem, all of which play the same role, look at the analogous 1- or 2- variable problem. This may allow you to build a solution from there. HoM 8: If a problem in its original form is too difficult, relax one of the conditions. That is, ask for a little less than the current problem does, while making sure that the problem you consider is of the same nature.

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 19 Suppose n distinct points are chosen on a circle. If each point is connected to each other point, what is the maximum number of regions formed in the interior of the circle?

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Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet Problem Solving in School Mathematics 20 HoM 9: Be skeptical of your solutions. HoM 10: Do not do anything difficult or complicated until you have made certain that no easy solution is available.

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