problem solving || solving problem-solving

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Solving problem-solving Author(s): Larry Lesser Source: The Mathematics Teacher, Vol. 96, No. 8, PROBLEM SOLVING (NOVEMBER 2003), p. 533 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/20871426 . Accessed: 04/05/2014 06:19 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org This content downloaded from 185.50.4.105 on Sun, 4 May 2014 06:19:57 AM All use subject to JSTOR Terms and Conditions

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Page 1: PROBLEM SOLVING || Solving problem-solving

Solving problem-solvingAuthor(s): Larry LesserSource: The Mathematics Teacher, Vol. 96, No. 8, PROBLEM SOLVING (NOVEMBER 2003), p.533Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871426 .

Accessed: 04/05/2014 06:19

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 185.50.4.105 on Sun, 4 May 2014 06:19:57 AMAll use subject to JSTOR Terms and Conditions

Page 2: PROBLEM SOLVING || Solving problem-solving

ffj READER REFLECTIONS Wm wmk \m*

' ^e ?PPrec,0*e the interest and value the views of those who write. Readers commenting on articles are encouraged to send copies W? III .;?/. of their correspondence to the authors. For publication: All letters for publication are acknowledged, but because of the large

f|||^ ^?s number submitted, we do not send letters of acceptance or rejection. Please TYPE AND DOUBLE-SPACE all letters to be con

gPBgi, . sidered for publication. Letters should not exceed 250 words and are subject to abridgement. At the end of the letter, include your - - *-* name and affiliation, if any, including zip or postal code and e-mail address, in the style of the section.

Solving problem-solving In honor of this focus issue, I want to share two ways that I help students recall the steps of problem solving.

First, I have written a heuristic that "sticks"-DECAL:

Describe the problem

Explore its structure

Create a model

Apply the model

Link to new situations

Second, I paraphrased P?lya's four steps in the chorus of a lyric, which can be sung to the tune of Paul Simon's "Fifty Ways to Leave Your Lover."

Chorus: Just say what's known, Joan, and what's to find, Caroline.

What rings a bell, Nell, to what we've done?

Make a plan, Stan, follow it through, Sue.

Try to extend, Ken, and we'll be done!

Larry Lesser

[email protected] Armstrong Atlantic State University Savannah, GA 31419

Lesser's lyrics appeared previously in "Sum of

Songs: Making Mathematics Less Monotone" in the May 2000 issue of the Mathematics

Teacher.?Ed.

Square investigations In "Reader Reflections: Finding the square of any given number x" on page 566 of the November 2002 issue of the Mathematics Teacher, Ben Khazan notices a pattern and finds an equation for obtaining the value for a square of a number he did not know from the value of a square that he did know. Certainly, such investigations are to be encouraged.

Khazan notices that the difference between the squares can be found by taking twice the difference {y

- x) times the larger number (y) and subtracting the square of the difference. Mentioning trial and error, he

presents the equation

y2 = x2 + 2y(y-x)-(y-x)2. Readers might be interested in seeing a

related pattern. Note that

x2 + 2y(y-x)-(y-x)2 = 2 + 2( y

- + x)(y -

)-(y- )2 = x2 + 2x{y-x) + (y-x)2.

Thus, the difference in the squares can be found by taking twice the difference times the smaller number and adding the square of the difference.

These equations can be arrived at more

directly. Note that

x2 = (y + x-y)2 = y2 + 2y(x-y) + (x-y)2,

from which Khazan's equation follows; or

y2 = (x+y-x)2 = x2 + 2x(y-x) + (x-y)2,

the alternative pattern. John F. Goehl Jr.

jgoehl@mail. barry.edu Barry University Miami Shores, FL 33161

Surprising result I found the following observation and results

interesting and think it would be a nice prob lem for students.

Let AB be a diameter of a semicircle, and let C be any point on the diameter. Further, let D and E be two points on the semicircle, such that triangles ADC and CEB are isosce les. If the angles of the triangles are (a, a, and 180 - 2a) and (b, b, and 180 - 26), then show that either a = ? or that cos 2a + cos 2b = -1. See figure 1 (Deshpande).

Proofs are left to the reader.

M. N. Deshpande dpratap@nagpur. dot net. in Institute of Science

Nagpur 440001 India

Think about using Deshpande's "Surprising result" to encourage students to think about

proof and justification as part of problem solving. To let them investigate the problem, pose the following prompts:

Are these four cases distinct? Are they the

only four cases?

For each of these cases, justify that a = 6 or that cos 2a + cos 26 = -1.

Create a model of this problem using a

dynamic construction environment.

Explain how your model can be used to demonstrate the four cases and support the writer's conclusions. -Ed.

Solution: Four possible cases exist:

In cases I and IV, cos 2a + cos 26 = -1; and in cases II and III, a = 6.

Fig. 1 (Deshpande)

In other IUCTM journals Readers of the Mathematics Teacher might enjoy the following articles in the November 2003 issue of Mathematics Teaching in the

Middle School:

"Learning Geometric Concepts through Ceramic Tile Design," by Claire V. Bell

"Using Friday Puzzlers to Discover Arith metic Sequences," by Arlene Yolles

For a complete listing of the contents of all the journals, see the NCTM Web site at

www.nctm.org. Mr

Vol. 96, No. 8 ? November 2003 533

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