Problem Solving: Tips For TeachersAuthor(s): Phares G. O'Daffer, Stephen Krulik and Jesse A. RudnickSource: The Arithmetic Teacher, Vol. 33, No. 4 (December 1985), pp. 38-39Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194113 .

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Problem botoing Tip> For Tecichar}

Phares G. O'Daffer,

Stephen Krulik Jesse A. Rudnick,

Edited by I

By and

Illinois State University, Normal, IL 61761

, Temple University, Philadelphia, PA 19122

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Fig. 1

38 Arithmetic Teacher

Ъ i Strategy Spotlight I

Reduction Чшшешшшешешшшшшшешш^ш

The strategy of reduction permits students to solve problems that might involve a large number of cases or a very complex figure. To use the strategy, stu- dents should reduce the number of cases to a simple few and gradually build up the situation. Usually a list or a table must be made and examined for a pattern. Problem: A piece of string art is made by connecting nails that are evenly spaced on the vertical axis to nails that are evenly spaced on the horizontal axis with straight lines made of colored strings. The same number of nails is used on each axis. Connect the nail farthest from the origin on one axis to the nail nearest the origin on the other axis. Continue in this manner until all the nails are connected. How many segments of string are used and how many intersec- tions do they make if you connect 8 nails on each axis?

• To solve the problem, we reduce the number of nails to the simplest case, namely, one nail on each axis. This situation requires 1 piece of string and yields 0 intersections. (See fig. 1 .)

• Now increase the number of nails to 2 on each axis. We use 2 pieces of string and have 1 inter- section. (See fig. 2.)

• Increase the number of nails on each axis to 3. We have 3 pieces of string and 3 intersections (fig. 3).

• If we increase the number of nails on each axis to 4, we obtain 4 pieces of string and 6 intersections (fig. 4).

Fig. 3 Fig. 4

Fig. 5 Let's summarize what we have done so far with a

table:

Number of Nails on Number of Number of Each Axis Strings Intersections

1 1 0 2 2 1 3 3 3 4 4 6

• Some students may find this information sufficient to recognize a pattern. That is, if the number of nails on each axis is n, then the number of strings

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December 1985 39

is also n. The number of intersections seems to nails are on each axis. This method will usually ■ follow the pattern 0, 1,3, 6, 10, 15, 21, 28, 36, lead to a rather large, messy configuration of lines I .... Students should then be encouraged to try that will be difficult, if not impossible, to count. You I one additional "string" to see if their conjecture should encourage students to reduce the case to I does hold true (fig. 5). It does! a simpler one in attempting to solve this problem. I

Thus if 8 nails are on each axis, we use 8 seg- #

f s f t0

™ow-up activity, formula

students for

might be asked to I ments of string and have a total of 28 intersections.

seg- ХгУ t0 EeveloP tf]e formula for the nth (or 9eneral> I

term, Tn, in the final column of the table: ■ • Notice that as their initial attempt, some students n(n- 1) I

may begin by drawing the situation in which 8 'n= 2 I

/ / f^^^^^^^^Hf '^^b H Developing Problem-solving Skills L· ■

MĚĚĚ ^^^^^^^^^Шт ^^R # Students may not be familiar with the follow- D Щ ]^^Щ ф^ЩЯ^ШШ^Ш nlirtir in9 strategies that are often used in problem ■ I Ata^/ Τ ^^ШмР^ Ι

w*' solving. Prepare a poster that includes these В I ^Л^У I Ш 1 w*'

Γ, λ strategies and encourage your students to I I ^^У^^^т ' ̂ ' 1 ̂-^У^ N. refer t0 the list as they work· The months I I

^/P3^ ρ ^ Ϊ ^ rt 4/ Д that these strategies were discussed in the I I

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I Problem Corner y' Arithmetic Teacher are included in parenthe- ■ I ji j ^ s^ Jř' ses ^m H Щ/^ЛА

j ТгУ this Problem with students in ( È'' Draw a Picture (October 1 984) ■ I

ГС /17 9rades4-8- Courage them to re- g'l ч Experimentation-Simulation (November 1985) II ^^J/ duce the problem to a simpler case Г$

ч F/nd a Pařřem (January 1 985) || when they start.

g Guess and Check (September 1984) ■ I ■

The 10 members of the Wacky I Logical Reasoning (February 1985) ■ I Comedy Basketball Team always I Make a Table (November 1984) ■ ■ have a warm-up drill in which each ■ Organize a List (December 1984) ■ ■ player tosses the ball to each of I Solve a Simpler Problem (March 1985) H I the other players. How many times I И/ог/с Backward (April 1985) ■ I is the ball tossed during this drill? ■ Write an Equation (May 1985) ■ ■

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. I

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