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

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


1
Teaching Mathematics through Problem Solving in
Lower- and Upper-Secondary School
  • Frank K. Lester, Jr.
  • Indiana University

2
Themes for this session
  • The teachers role
  • Developing habits of mind toward problem solving

3
The cylinders problem
  • LAUNCH Do cylinders with the same surface area
    have the same volume?

4
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 (2p)RH V pR2H

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).
  • 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?

6
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.

7
12 345 678 987 654 321
8
12 345 678 987 654 321
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

10
Which triangles can be divided into 2 isosceles
triangles?
11
Find the sum of the interior angles of the star
in at least 5 different ways.
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.

13
For what values of n does the following system of
equations have 0, 1, 2, 3, 4, or 5
solutions? x2 - y2 0 (x - n)2 y2
1
14
Determine the sum of the series
  • 1/12 1/23 1/34 1/45 1/n(n1)

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.

16
(A2 1)(B2 1)(C2 1)(D2 1)
ABCD
  • If A, B, C, and D are given positive numbers,
    prove or disprove that

16
17
If x, y, z, and w lie between 0 and 1, prove or
disprove that
  • (1 - x)(1 - y)(1 - z)(1 - w) gt 1 - x - y - z - w

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.

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?
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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