Teaching mathematics through problem solving in lower and upper secondary school
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Teaching Mathematics through Problem Solving in Lower- and Upper-Secondary School. Frank K. Lester, Jr. Indiana University. Themes for this session:. The teacher’s role Developing habits of mind toward problem solving. The cylinders problem.

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Teaching mathematics through problem solving in lower and upper secondary school l.jpg

Teaching Mathematics through Problem Solving in Lower- and Upper-Secondary School

Frank K. Lester, Jr.

Indiana University


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Themes for this session:

The teacher’s role

Developing habits of mind toward problem solving


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The cylinders problem

LAUNCH: Do cylinders with the same surface area have the same volume?


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SUMMARIZE

  • Have students report about their findings.

  • Encourage student-to-student questions.

  • Look back: How is this problem related to problems we have done before?

  • What have we learned about the relationship between circumference and volume?

  • Examine the formulas for surface area and volume (Big math ideas)

    SA = (2π)R*H; V = πR2*H


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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).

  • 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?


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Qualities of the Lesson

  • A question is posed about an important mathematics concept.

  • Students make conjectures about the problem.

  • Students investigate and use mathematics to make sense of the problem.

  • The teacher guides the investigation through questions, discussions, and instruction.

  • Students expect to make sense of the problem.

  • Students apply their understanding to another problem or task involving these concepts.




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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

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.

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111 111 111 x 111 111 111 = 12 345 678 987 654 321




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  • HoM2: 5 different ways. 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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For what values of 5 different ways.n does the following system of equations have 0, 1, 2, 3, 4, or 5 solutions? x2 - y2 = 0 (x - n)2 + y2 = 1


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Determine the sum of the series: 5 different ways.

1/1•2 + 1/2•3 + 1/3•4 + 1/4•5 + 1/n•(n+1)


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  • HoM 5: 5 different ways. 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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(A 5 different ways.2 + 1)(B2 + 1)(C2 + 1)(D2+ 1)A•B•C•D

≥ 16

If A, B, C, and D are given positive numbers, prove or disprove that


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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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  • HoM 7: that 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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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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  • HoM 9: point is connected to each other point, what is the maximum number of regions formed in the interior of the circle? 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.