Title: Teaching Mathematics through Problem Solving in Lower- and Upper-Secondary School
1Teaching Mathematics through Problem Solving in
Lower- and Upper-Secondary School
- Frank K. Lester, Jr.
- Indiana University
2Themes for this session
- The teachers role
- Developing habits of mind toward problem solving
3The cylinders problem
- LAUNCH Do cylinders with the same surface area
have the same volume?
4SUMMARIZE
- 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
5Extending 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?
6Qualities 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.
712 345 678 987 654 321
812 345 678 987 654 321
9HoM 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
10Which triangles can be divided into 2 isosceles
triangles?
11Find 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.
13For 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
14Determine 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
17If 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.
19Suppose 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.