DEVELOPING PROBLEM-SOLVING SKILLS
Author(s): STEPHEN KRULIK and JESSE A. RUDNICK
Source: The Mathematics Teacher, Vol. 78, No. 9 (DECEMBER 1985), pp. 685-692, 697-698 Published by: National Council of Teachers of Mathematics
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activities
Edited by CHRISTIAN R. HIRSCH, Western Michigan University, Kalamazoo, Ml 49008
DEVELOPING PROBLEM-SOLVING SKILLS
By STEPHEN KRULIK and JESSE A. RUDNICK, College of Education, Temple University, Philadelphia, PA 19122
Teacher's Guide Introduction: An Agenda for Action states
that "problem solving must be the focus of school mathematics in the 1980s." It further suggests that "the mathematics curriculum should be organized around problem solv ing," that "mathematics teachers should create classroom environments in which problem solving can flourish," and that
"appropriate curricular materials to teach problem solving should be developed for all grade levels" (NCTM 1980, 2, 4). The devel opment of materials that are suitable for direct use in the mathematics classroom is, of course, a most important recommen dation, since without proper instructional materials, teachers have little direction in what content to emphasize and how it might best be presented. Curricular materi
als previously offered in this series of Agenda-related activities focused on the problem-solving skills of make a drawing and work backward (Schaaf 1984) and guess and test and simplification (Laing 1985). The activities in this article are intended to pro vide suitable materials for introducing and reinforcing the learning of two additional problem-solving skills, namely, make an or ganized list and search for a pattern.
Grade levels: 7-10
Materials: Copies of the worksheets for each student and a set of transparencies for demonstration purposes
Objectives: (1) To develop the problem solving skills of making and reading an or ganized list and searching for a pattern; (2) to provide practice in using the general heuristics of the problem-solving process
Directions: This activity consists of two subactivities : solving problems by making an organized list (worksheets 1 and 2) and solving problems by searching for a pattern (worksheets 3 and 4). Each subactivity will require approximately one instructional period and out-of-class work by the stu dents. Allow ample class time for discussion of students' solutions to the problems. Al though the second subactivity builds on the first, it is not necessary that the two subac
tivities be used in consecutive class periods.
Begin by distributing a copy of the first worksheet to each student. This sheet is de
signed to be used by the entire class in a teacher-centered activity. Using a transpar ency of sheet 1, present the first problem
Editor's note: In An Agenda for Action: Recommendations for School Mathematics of the 1980s, the National Council of Teachers of Mathematics set directions for mathematics programs for the decade of the eighties. We are now halfway through that decade. Whereas significant progress has been reported in several areas of the rec ommendations, continued and expanded efforts are necessary if the goals envisioned in the Agenda are to be fully realized. During the 1985-86 school year, this section is again devoted to mathematical activities in worksheet form that can contribute to effective implementation of actions recommended in the Agenda. This material may be photoreproduced by classroom teachers for use in their own classes without requesting permission from the Council.
Reachers who have developed other mathematical activities related to themes from the Agenda are encouraged to submit manuscripts, in a format similar to the "Activities "
already published, to the managing editor for review.
December 1985
685
and explain how the list was constructed.
Assume that the first tag chosen is a 3 and the second tag is a 6. This leaves 2, 5, and 1 to be distributed as shown in the first three vertical entries on the list. Now we have exhausted the 6s. The next choice should be a 3 first and a 2 second, with 5 and 1 distrib uted as shown in the next two vertical en
tries. Instruct the class to proceed in a simi lar manner to complete the list and then answer parts (b)-(d). After discussing the solution to this problem, review with the students how the use of an organized list enabled them to keep track of all pos sibilities as they worked. Note that the list
is the answer.
Assign the students to complete the solution to problem 2 by using the list that has been started for them. Discuss their solutions. Then distribute the second work sheet and direct the students to complete problem 3. When discussing their solutions to this problem, note that Chen does not need to purchase exactly the 17 pounds of grass seed; it is more economical to buy 18 pounds. In this problem, the list is not the answer, but it leads directly to the answer.
Instruct students to solve problems 4, 5, and 6 in a similar manner. Depending on the previous problem-solving experiences of the students, you may need to suggest that for problem 6, they begin by making a drawing.
To introduce the second subactivity, dis tribute sheet 3 to each student and use a transparency of this sheet to discuss the patterning rule for each exercise. Allow
students as wide a selection of possible rules as you can, provided that their pat terning rule fits all the examples given in the series.
Next distribute sheet 4. Allow time for the students to complete problem 1. In the discussion that follows, point out the value of the organized list in looking for the pat tern and emphasize the usefulness of the discovered pattern in completing the solu tion. Assign problems 2 and 3 to be solved using the same skill.
Follow-up activities: The problem-solv ing skills of make an organized list and search for a pattern should be reinforced throughout the school year. Additional problems whose solutions are amenable to the use of these skills can be found in Dolan and Williamson (1983), Krulik and Rudnick (1980, 1984), and in the Lane
County Mathematics Project (1983). Final ly, note that although a problem can often be solved by algebra, an algebraic solution may not always be possible, and thus facili ty in making and reading an organized list is an important skill.
Solutions:
Sheet 1: l.a.
First tag 3333336662 Second tag 6662252255 Third tag 2515115111
Score 11 14 10 10 6 9 13 9 12 8
l.b. Every possible triple of numbers selected from {3, 6, 2, 5, 1} appears in the table. If, for example, an eleventh column was added consisting of the entries 5, 1, and 3, it would duplicate the entries in column six.
I.e. The completed list shows eight different scores, l.d. The possible scores are 6, 8, 9, 10, 11, 12, 13, and 14.
686
Mathematics Teacher2.
Clock 1 2:00 2:06
2:12 2:18 2:24Clock 2
2:00 2:08 2:16 2:24The cuckoos come out together at 2:24.
Sheet 2:1.
Number of 5-Pound
Boxes
Cost at
$6.58 a Box
Number of 3-Pound
Boxes
Cost at
$4.50 a Box
Total Pounds
Total Cost
$26.32 0 -0 20 $26.32
19.74 $ 4.50 18 24.24
13.16 13.50 19 26.66
6.58 18.00 17 24.58
27.00 18 27.00
Chen should buy three 5-pound boxes and one 3-pound box; this combination is cheaper than buying exactly the 17 pounds of seed required.
2. Number of
Packages of 4
Number of Packages of 3
1
Number of Singleton Packages
0
11
12 15
The order can be filled in fifteen different ways.
3.
Number of Checks
10 11 12 13 14 15 16 17 18 19 20 21
Cost for Checks,
@ $0.10
$0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10
Total with $2.00
Service Charge
$2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10
Cost for Checks,
@ $0.05
$0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05
Total with $3.00
Service Charge
$3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05
At least twenty-one checks a month must be written if the new plan is to save you money.
(Solutions continued on next page) REFERENCES
Dolan, Daniel T., and James Williamson. Teaching Problem-solving Strategies. Menlo Park, Calif.:
Addison-Wesley Publishing Co., 1983.
Krulik, Stephen, and Jesse A. Rudnick. Problem Solv ing: A Handbook for Teachers. Newton, Mass.:
Allyn & Bacon, 1980.
-. A Sourcebook for Teaching Problem Solving.
Newton, Mass. : Allyn & Bacon, 1984.
Laing, Robert A. "Extending Problem-solving Skills."
Mathematics Teacher 78 (January 1985):36-44.
Lane County Mathematics Project. Problem Solving in Mathematics. Palo Alto, Calif.: Dale Seymour Pub
lications, 1983,
National Council qf Teachers of Mathematics. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, Va. : The Council,
1980.
Schaaf, Oscar F. "Teaching Problem-solving Skills."
Mathematics Teacher 77 (December 1984) :694-99.
-~~- 688 -Mathematics Teacher
4.
Total Area
Area of Square 1
Area of Square 2
Side of
Square 1
Side of
Square 2
Perimeter of Square 1
Perimeter of Square 2
130 1 129
130 126
130 121 11 12 44
130 16 114
130 25 105
130 36 94
130 49 81 28 36
130 64 66
130 81 49 36 28
130 100 30 10
130 121 11 44 12
*These values will be nonintegral.
Notice that two possible answers satisfy the condition that the sum of the areas is 130 but that only one satisfies the condition that the sum of the perimeters is 64. Thus the answer is that the sides differ by 2 centimeters.
Sheet 3:1.
2. Answers will vary.
3. 216
4. Answers will vary.
5. 21
6. Answers will vary.
7.
ooo 8.
9. 3750
10. [
Patterning rule:
Patterning rule:
Patterning rule : Patterning rule : Patterning rule : Patterning rule : Patterning rule:
Patterning rule:
Patterning rule : Patterning rule :
Each term is one-half of the previous term.
Names beginning with the letter " J"
Each term is the cube of the number of the term.
Names of various rock groups
These are Fibonacci numbers. Each term is the sum of the two terms that precede it.
Names in alphabetical order
Alternating sequence of squares and circles.
The number of figures of a given shape in creases by one each time the shape occurs.
Each fraction has the denominator of the pre vious fraction as its numerator, and the denomi nator is the next integer.
Each term is five times the previous term.
The sequence of squares repeats with the upper portion of each odd-numbered term shaded and
the lower portion of each even-numbered term shaded.
December 1985
689
Sheet 4: l.a. 4; 5
l.b. Minutes Forward
Years Gained
11 13
15 18
19 23
23 28
27 33
By setting the timer ahead 27 minutes you will gain 33 years. The year will be 1985 + 33 = 2018.
2. Week
20
Weekly Salary
$1.00 2.00 4.00 8.00 16.00 32.00 64.00 128.00
Total Earnings
524 288.00
$1.00 3.00 7.00 15.00 31.00 63.00 127.00 255.00
1 048 575.00
An analysis of this list suggests two patterns: the weekly salary can be expressed as 2"-1, where is the number of the week ; the total earnings can be expressed as 2" ? 1, where is
again the number of the week. From the latter pattern, it follows that the total earnings after twenty weeks of employment would be 220 ? 1 = $1 048 575.00.
3.
Stop #
Number
Picked Up
Number
Dropped Off
Number
Aboard 11 14 12 17
10
15
16
24
The organized list enables us to see the pattern: 24 passengers will be aboard at the end of the sixteenth stop. Observe that the sequence of numbers of passengers actually consists of two subsequences: 5, 8, 11, 14,17 and 3, 6, 9,12,15.
690
Mathematics TeacherMAKE AN ORGANIZED LIST SHEET 1
1. The five tags shown are placed in a box and mixed. Three tags are then drawn out at one time. If your score is the sum of the numbers on the three tags drawn, how many different scores are possible? What are the possible
scores
To keep track of the different scores, it is helpful to prepare an organized list. Let's begin by assuming that the first of the three tags drawn is the "3"
and then listing the possibilities for the other two tags.
First Tag Second Tag Third Tag
Score 11 14 10 10
a. Complete the chart.
b. Explain why this list accounts for all possible drawings of three tags where the order of selection is unimportant._
c. How many different scores are possible?_
d. What are the possible scores? _
2. Jim's Repair-Your-Clock Shop has two cuckoo clocks that were brought in
for repairs. One clock has the cuckoo coming out every six minutes, whereas
the other has the cuckoo coming out every eight minutes. Both cuckoos come out at exactly 2:00. When will they both come out together again?
Let's start to make an organized list of the times that the two cuckoos come
out:
Clock 1 2:00 I 2:06 2:12
Clock 2 2:00 ! 2:08
Finish the list and solve the problem.
From the Mathematics Teacher, December 1985
MAKE AN ORGANIZED LIST SHEET 2
1. Helen Chen wants to seed her front lawn.
Grass seed can be bought in 3-pound boxes that cost $4.50, or in 5-pound boxes that cost $6.58. She needs exactly 17 pounds of seed. How many boxes of each size should she purchase to get the best buy?
The use of an organized list can also help in solving this problem.
Number of
5-Pound Boxes
Cost at
$6.58 a
Box
Number of
3-Pound Boxes
Cost at
$4.50 a
Box
Total Pounds
Total Cost
$26.32 0
-0 20$26.32
19.74 $4.50
0 6
Complete the list and solve the problem.
Make an organized list to solve each of the following problems.
2. A customer ordered 15 blueberry muffins. If the muffins are packaged singly or in sets of 3 or 4, in how many different ways can the order be filled?
3. A bank has been charging a monthly service fee of $2.00 plus $0.10 a check
for a personal checking account. To attract more customers it is advertising
a new "reduced cost" plan with a monthly service charge of $3.00 plus only
$0.05 a check. How many checks must you write each month for the new plan to save you money?
4. A piece of wire 64 centimeters in length is cut into two parts. The parts are then each bent to form a square. The total area of the two squares is 130 square centimeters. How much longer is a side of the larger square than a side of the smaller square? (Consider only whole-number solutions.)
From the Mathematics Teacher, December 1985
SEARCH FOR A PATTERN SHEET 3
For each of the following sets, give another element of the set. State in your own words what you think the patterning rule is.
1. 80, 40, 20, 10,_
Patterning rule :_?
2. James, Jill, Joan, John,
Patterning rule :_-_
3. 1, 8, 27, 64, 125,_
Patterning rule :_
4. Styx, Beatles, Who, Kansas,_
Patterning rule :_
5. 1, 1, 2, 3, 5, 8, 13,_
Patterning rule :_
6. Alvin, Barbara, Carla, Dennis, _ Patterning rule :_-?
7.
Patterning rule :
9. 6, 30, 150, 750, _ Patterning rule :
10.
h, y, H, U, h,
Patterning rule :
From the Mathematics Teacher, December 1985
SEARCH FOR A PATTERN SHEET 4
1. Scientists have invented a time machine.
By setting the dial, you can move forward in time. Set it forward 3 minutes and you will be in the year 1988. Set it forward 7 minutes and you will be in the year 1993 ;
set it forward 11 minutes and you will be in the year 1998; set it forward 15 min utes and you will be in the year 2003. If
the machine continues in this manner, in what year will you be if you set the timer
ahead 27 minutes? (This year is 1985.)
What do we know? Forward 3 minutes, gain of 3 years (1988 ?
1985) Forward 7 minutes, gain of 8 years (1993 ?
1985) Forward 11 minutes, gain of 13 years (1998 ? 1985) Forward 15 minutes, gain of 18 years (2003 ?
1985) What do we want? Forward 27 minutes, gain of how many years?
Plan: Make an organized list and search for a pattern.
Minutes Forward
Years Gained 8
11 13
15 18
27
a. Complete the patterning rule : Every time we move the timer ahead
_ minutes, we gain an additional
_years.
b. Now complete the chart and solve the problem..
Solve the following problems by first making an organized list and then
looking for a pattern.
2. Carlos was offered a part-time job that included on-the-job training. Because of the training, he was to be paid $1.00 the first week, $2.00 the second, $4.00
the third, $8.00 the fourth, and so on. How much money would Carlos have earned after twenty weeks of employment?
3. An empty streetcar picks up five passengers at the first stop, drops off two
passengers at the second stop, picks up five passengers at the third stop, drops off two passengers at the fourth stop, and so on. If it continues in this manner, how many passengers will be on the streetcar after the sixteenth
stop?
From the Mathematics Teacher, December 1985
