Tick-tack-fourAuthor(s): DIANA JORDANSource: The Arithmetic Teacher, Vol. 15, No. 5 (MAY 1968), pp. 454-455Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185811 .

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base-four numeral of 1230321 and the same procedure as in converting binary numerals to base four. I arrived at the base-sixteen numeral 1£39. I converted both nondecimal numerals to base ten, as a check, and arrived at the base-ten numeral 6,969.

I next tried out my discovery in the realm of the powers of three; however, I did not go very far. Knowing that 32 = 9, I decided to see if base-three numerals could be converted to base-nine numerals, using the same procedure as in converting base two to base four. Using Table 4 and the base-three numeral 2012, I arrived at the base-nine numeral 65. Checking 2012 in base three and 65 in base nine by con- version to base ten, both are equal to 59 in that base.

Table 4

Base nine 012345678

Base three 00 01 02 10 11 12 20 21 22

It is one of my pet sayings to my pupils in arithmetic that as soon as you arrive at a rule in mathematics, you try to see how

far you can stretch it. So I tried the con- version method on converting numerals in base four to base eight and converting numerals in base three to base six. The rule, or method, did not work in the last two cases. So, apparently, the conversion method will work only in cases where the larger-base numeration system is a whole power of the smaller-base numeration sys- tem, the exponent of the base being the cue as to how many digits should be in each group when dividing the smaller base numeral into groups for conversion to the larger base.

Editor's Note. - Repeatedly, teachers and chil- dren inquire how one may change base-two numerals to base eight without first translating both into base ten, or how base-four numerals may be expressed in base twelve, etc. Another article in the journal has given one approach. The editor has suggested parallel number lines as a simple technique for young children. Now Mr. Barrett makes yet another suggestion. Able learners will enjoy experimenting with the ideas in this article and in testing out the hypothesis advanced. I wonder if there is some easy way to change base-three numerals to base six with- out translating both to base ten? - Charlotte W. Junge.

Tick -tack -four DIANA JORDAN

College of Education, Wayne State University, Detroit, Michigan Miss Jordan is a student at the College of Education. The activity she discusses was developed while teaching a Grade 3 class in the Aymer Jay Hamilton Laboratory School at Chico State College, Chico, California, during the 1965/66 school year.

Vjhildren are intrigued by numbers and their relationships to one another. If we, as teachers, use this curiosity, it is possible to develop many games and activities in arithmetic which can serve as replacement for traditional drill.

As an example: put the number square shown in Figure 1 on the blackboard and

3 4

5 6

Figure 1

454 The Arithmetic Teacher

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ask the children to help you find the sums of each vertical, horizontal, and diagonal row.

Then, together, look at the finished dia- gram, Figure 2, and ask the children what patterns they see in these numbers and their sums.

9 8 10 9

7 ~3 4 7~~

11 5 _6_ 11

9 8 10 9

Figure 2

Here are a few of the patterns which the children may notice:

1. The beginning numbers are in order: 3, 4, 5, 6.

2. The first vertical sums are the same: 3 + 5 = 8, 5 + 3 =• 8.

3. The second vertical sums are also the same: 4 + 6 = 10, 6 + 4 =• 10.

4. The diagonal sums follow the same pattern: 3 + 6 = 9, 6 + 3 = 9, and 5 + 4 = 9, 4 + 5 = 9.

5. The horizontal sums also follow a similar pattern: 3 + 4 = 7, 4 + 3 = 7, and 5 + 6 = 11, 6 + 5 = 11. (Some children may explain that 2, 3, 4, and 5 as given above are examples of the com- mutative law.)

6. The sum of the corner sums of the top row, 9 + 9, is equal to the sum of the two inside sums of the top row, 8 + 10. This pattern is also true for the rela- tionship between the other corner and interior sums. See Figure 3.

[9] ®j@ {Щ 3 4 5 6

Figure 3

Next, challenge the children to see if they always need to use four consecutive

numbers in order for these relationships to remain true. (In this way you are asking them to do inductive thinking as they verify their basic principles.)

Each child will usually select, as an example to try, one that reflects his own level of ability. (Competent third graders have tried such squares as the one shown in Figure 4, while children working at a

3459 4241

7165 1997

Figure 4

lower level of ability in the same class- room have tried such squares as the one shown in Figure 5. In each case the indi-

3 7

1 8

Figure 5

vidual differences of the children met the criteria of the challenge.)

Children in one class developed, from the sixth pattern enumerated above, a rule which they called the "Law of Tick-Tack- Four": The sum of the outside corners is equal to the sum of the inside rows. The class continued to show interest in this law for several weeks. Each day, at least one child attempted to test this new gener- alization with a special example of his own. Everyone participated, and no one complained about all the drill and practice in addition facts that he was doing.

Practice can be fun and exciting, if we provide the opportunity for our classes.

Editor's Note. - It is not uncommon to have elementary school teachers of mathematics ex- press a need for new and effective ways of pro- viding practice on basic skills and understandings. Miss Jordan's procedure provides practice on basic facts, on laws of operation, and on habits of mathematical inquiry. - Charlotte W. Junge.

May 1968 455

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