digit delight: problem-solving activities using 0 through 9

Digit Delight: Problem-solving Activities Using 0 through 9Author(s): Don S. BalkaSource: The Arithmetic Teacher, Vol. 36, No. 3 (November 1988), pp. 42-45Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193500 .

Accessed: 12/06/2014 22:51

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 Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 62.122.79.21 on Thu, 12 Jun 2014 22:51:47 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctm

http://www.jstor.org/stable/41193500?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Digit Delight: Problem-solving Activities

Using 0 through 9 By Don S. Balka

Number puzzles of all kinds offer intriguing challenges to elementary and junior high school students. As the title suggests, this article presents several problem-solving activities in- volving only the digits 0 through 9. For classroom use, sets often ceramic tiles (each 1-inch square) can be inex- pensively made using marking pens, or plastic sets called "Try-A-Tiles" can be purchased. Each set can be stored in a small plastic bag for stu- dent use. Students will find the direc- tions easy to follow. The teacher's guidance is helpful to direct students' interaction about their strategies and later to discuss maximum and mini- mum solutions. Problems can be shown on an overhead projector, as bulletin-board puzzles, or on individ- ual sheets in learning centers.

The problems take two forms: those in which tiles are used only in the problem itself, not in the answer; and those in which tiles are used for the problem and the answer. An example of each is illustrated in figure 1 .

Additionally, other well-known puzzles are solvable using some or all of the ten digits. Two such puzzles are shown in figure 2 (magic squares and magic triangles). Previous issues of the Arithmetic Teacher (October 1976; March 1977; October 1978) and the Mathematics Teacher (October 1974; May 1983) have published dis- cussions of these puzzles.

Most puzzles, including those sug-

Don Balka is an associate professor in the mathematics department at Saint Mary's Col- lege, Notre Dame, IN 46556, where he teaches undergraduate mathematics and mathematics education courses.

Fig. 1

(a) Using tiles in the problem

Use six tiles to create two three-digit addends whose sum is the greatest possible.

d an "DDP

One possible solution: ПЛ Fj] ПП

~1 8 3 9~

(b) Using tiles in the problem and answer

Use nine tiles to create two three-digit addends and a three-digit sum that is the greatest possible.

DD D + D D D

One possible solution: Г7] Г4П ПП

Fig. 2

(a) Magic square Fill in the square using nine tiles so that the sum of the numbers in every row, column, and diagonal is the same.

□DD DDD DDD

ИШШ ШЕИ

Magic sum: 15

(b) Magic triangle Fill in the squares using nine tiles so that the sum of the numbers on each side of the triangle is the same.

D D D

D D D D D D

Magic sum: 15

42 Arithmetic Teacher



gested in figures 2(a) and 2(b), have more than one solution. For example, the magic square and magic triangle could be completed to give magic sums of twelve and twenty-three as shown in figure 3. Thus, students are given ample opportunity for problem solving: 1. What is the largest sum you can

find? 2. What is the smallest sum you can

find? 3. Is a particular sum unique, or can

you rearrange the tiles to get the same solution?

4. How many different sums can you find?

5. Did you discover a pattern in the solutions?

Consider these questions in relation to the puzzle in figure 4. Notice a pattern in each solution: 0 + 5 = 2 + 3 or 0 + 4 = 1 + 3 in figure 4(a); 4 + 6 = 7 + 3 or 4+ 1 = 2 + 3 in figure 4(b);

8 + 7 = 6 + 9 or 8 + 4 = 3 + 9 in figure 4(c); 8 + 6 = 9 + 5 or 8+1 = 4 + 5 in figure 4(d); and 9 + 4 = 7 + 6 or 9 + 5 = 8 + 6 in figure 4(e). Thus, we should be able to generate a triangle having a magic sum of, say, 8, or 10, or 17. First, find two pairs of two addends whose sum is the same. Then, simply add the necessary digits to each side of the triangle as shown in figure 5. Note that certain pairs can- not be used to generate particular sums, since duplicate digits would be needed. For example, 4 + 3 and 5 + 2 could not be used to generate a sum of

9, since an additional 2 would be needed (5 + 2 + 2).

Other triangles with sums between six and twenty-one can easily be con- structed following the procedures sug- gested previously. Thus, we have found answers to all five questions posed:

1. The largest sum is twenty-one. 2. The smallest sum is six. 3. Figures 4(c) and 4(d) show two

triangles, both with a magic sum of eighteen; therefore, a particular sum is not unique.

Fig. 3

Fig. 4

Fig. 5

November 1988 43

ШИШ Ш Ш

Magic sum: 12 Magic sum: 23

Fill in the squares using six tiles so that the sum of | | | | each side of the triangle is the same. i - i i - i i - i

Solutions:

Smallest sum: Same sum: Largest sum: 6 12 18 21 (a) (b) (c) (d) (e)

| О | | О | | / |

Magic sum: 8 Magic sum: 10 Magic sum: 17



Fig. 6 Fig. 7

(a) Use nine tiles to create three (b) Fill in the squares so that each two-digit addends and a three- line has the same sum. digit sum that is the greatest . - , , - , .- , , - , possible.

greatest ЙИИШ . - , , - , .- , , - ,

(с) Use seven tiles to create two (d) Fill in the squares so that each two-digit factors and a three- line has the same sum. digit product that is the greatest possible. HÍLHIE

Fig. 8

Largest sum Smallest difference

I - II - II -

I I - II - II - I

44 Arithmetic Teacher



4. Each of the sums six through twenty-one is possible.

5. A pattern was illustrated in figure 4. Other patterns can be found.

We can also generate a new puzzle by adding to the triangle a fourth row with four additional tiles (see fig. 6). Can you find a different magic sum?

A variety of challenging computa- tional puzzles exist, especially when multiplication is involved, or when nine digits or all ten digits are required for the solution (see fig. 7). Some problems, in fact, may not be solvable with the tiles, and discovering this fact is a good problem-solving activity for students.

For multiplication puzzles, such as

the one illustrated in figure 7(c), students must often focus simultaneously on place- value concepts, basic com- putational facts, and number theory. After students have tried a guess-and- test strategy, teachers may want to suggest that students use their calcu- lators to assist in solving the puzzles and that they also keep a record of upper and lower bounds for their trials. Quick checks of the trials may eliminate many unnecessary calcula- tions.

Other digit puzzles and a solution to each are given in figure 8. In addition, many letters of the alphabet can be transformed into "digit delights."

The number of problem-solving activities with the digits is only as great as your imagination. The puzzles are

challenging and open-ended, and they provide opportunities for applying many strategies. Try them with your students. Delight with the digits!

Bibliography

Ajose, Sunday A. "Subtractive Magic Trian- gles." Mathematics Teacher 76 (May 1983):346-47.

Bernard, John E. "Constructing Magic Square Number Games." Arithmetic Teacher 26 (October 1978):36-38.

Ewbank, William A. "A Vest-Pocket Game." Illinois Mathematics Teacher 28 (September 1977):3-6.

Lott, Johnny W. "Behold! A Magic Square." Arithmetic Teacher 24 (March 1977):228-29.

О 'Sullivan, Ellen P. "Magic Squares for Aver- age Learners, Too." Arithmetic Teacher 23 (October 1976):427-28.

Williams, Horace E. "A Note on Magic Squares." Mathematics Teacher 61 (October 1974):511-13. W

From tha Fila

| money | ^^MONEY BAGS ^

The teaching of concepts related to money requires the use of manipulatives, preferably the coins themselves. One way to give hands-on practice while preventing coins from rolling off the desks is to use resealable plastic bags. On the outside of each bag draw a large circle. In the bag place the coins necessary for the particular lesson and seal the bag. Each child receives his or her own bag.

Instruct children to shake the bags to force all coins to the bottom. From this point questions can be asked, such as, "Show me a penny," and students move the coins with their fingers to the area inside the circle. The teacher can quickly scan the bags to ensure that all students have responded correctly. Other questions: "Show me one way to make six cents." "You have five cents to spend. Put five pennies inside the circle. You spend two cents. Move two cents out of the circle. How much do you have left?" Later, two bags can be used to compare quantities and values.

From the file of Donald E. Van Ostrand, Howard County Public Schools, Ellicott City, MD 21043

- Readers are encouraged to send in two copies of their classroom-tested ideas for "From the File" to the Arithmetic Teacher for review. -

November 1988 45



digit delight: problem-solving activities using 0 through 9

Documents