Problem Solving: Tips For TeachersAuthor(s): John A. Van de Walle, Barbara Wilroot and David BarnesSource: The Arithmetic Teacher, Vol. 35, No. 7 (March 1988), pp. 32-33Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193366 .

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Problem >ol'>ing Tip3 For Teacher} 9 M

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Gifted Problem Solvers I

Teachers often search for ways to differentiate learn- ing experiences for their students with high abilities. Problem solving with appropriate extensions offers a very viable tool for meeting needs at various ability levels. With practice, creating extensions becomes easy and interesting. In fact, students will soon help by creating their own extensions through brainstorm- ing in cooperative learning groups.

Extensions seem to fall into four mayor catego- ries: (1) reaching for a pattern or generalization, (2) introducing new concepts or vocabulary from other areas of mathematics, (3) extending divergent think- ing (creativity), and (4) initiating discussions that pre- sent opportunities for value judgments. The grade level suggestions for using the following problems should be interpreted with flexibility. Wide variations of interest, ability, and experience occur at any grade level.

Pattern and Generalization Problem: If you use pattern blocks, how many trian-

gles are required to make a larger, similar triangle (fig. D?

Extensions (K-2) Can you make an even larger triangle? If nine triangles were in the bottom row, how many triangles

Edited by John A. Van de Walle Virgìnia commonwealth university Richmond, VA 23284-0001 Prepared by Barbara Wilmot and

David Barnes Illinois State University Normal, IL 61761

would you have used? (81) I

(2-4) How many smaller squares are needed to I make a larger square? If twelve were in the bottom ■ row, how many squares would you need? (144) How ■ many hexagons are required to make a larger hexa- I gon? (Can you try to explain why it cannot be done?) I

(4-8) If you use only two colors (shapes) of blocks, I | can you make a hexagon? A regular hexagon? What I combinations of blocks will work? How many of each I shape will you need? Can you make a hexagon with I three shapes? Four? Five? I

New Concepts and Vocabulary I Problem: If a book that has 167 pages is open and I the sum of the page numbers is 137, what pages I are showing? (68, 69) I

32 Arithmetic Teacher

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Extensions (1-3) If page 1 is on the right side of the book and page 2 is on the left, will page 39 be on the right or the left? (right) Will page 86 be on the right or the left? (left) How many sheets of paper are in the book? (84) (4-8) If the product of the pages that are showing is 16 002, what are the page numbers? (126, 127) What is the probability that when you open the book, one of the page numbers is a multiple of three? (55/84) Or that on exactly one of the pages the sum of the digits is greater than ten? (7/84) Or that the tens digit of the page number on the right is greater than the ones digit? (29/79; 79 right-hand pages have a tens digit.)

Creativity Problem: A farmer has a hen, a cat, and a bag of

seed to carry across the river. His boat can only carry him and one other item. He cannot leave the cat alone with the hen or the hen alone with the seed. How can he get them all across the river?

Extensions (1-8) Add another animal or object to the farmer's load. Now how many trips will it take to get every- thing across safely? What kinds of things (or animals) can the farmer add? What kinds of things will not work? (Note: Any additional animal that can cause harm to one of the original animals or corn provides an experience with a problem with no solu- tion. Students can prove that no solution exists!)

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'^^^^^^^^^^^^^^^^^^^^^^ Ц To Enhance Creativity ■

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Ward> and '°9'Са1 thinking' I / form ofProbler^s that in ÊÊ * Have students hold onto their creative ■ / food с Personal or socV°IVe

in letal S0l7)e Ш nonmathematical solutions until a solution ■

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anirnal letal dllemma Щ has been found usin9 the actual situation and I В I • After Po^er anrfXPer/rTienra" Ш restrictions in the problem. (Example: Taping В

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Of Various Ш t0 create a new situation- Decide if the new I ^^^^^jf^^ted.

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Щ^^^^^^^^^^^^^^ Ш tions, or prove the impossibility of a solution. H

Part of the Tip Board is reserved for techniques that you've found useful in teaching problem solving in your class. Send your ideas to the editor of the section. W

March 1988 33

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