Four cards
(Integers in grades 1 to 8)
Task 1.
From index cards or poster board make four cards of the size of a normal playing card. With a marker write 1 on one side of the first card, and -1 on the other side. Similarly write 3 and -3 on the second card, 9 and -9 on the third, and 27 and -27 on the fourth card. The numbers should be large and readable. (In the first and second grades children may need help in making their cards.)
Task 2.
Put some cards on the table (one, two, three, or four cards from just one set), with either side facing up, and add all the numbers that are visible (you may use a calculator). Record the numbers and their sums.
Question 1.
What numbers can you get as sums?
Question 2.
Can you get the same number in more than one way?
In first and second grade you may start with two or three cards only, and not use the 27-card until children become familiar with the whole procedure.
About recording.
• You record what you did as an addition problem, and not as a subtraction problem.
27 + -9 + 1 = 19 is fine, but 27 - 9 + 1 = 19 is not acceptable.
• You may write the sum on either side of the equal sign. So both -9 + -3 + 1 = -11, and
-11 = -9 + -3 + 1 are acceptable.
• You may add the numbers in any order, but record them from the biggest absolute value to the smallest. 1 + -3 + -9 is not acceptable.
About the use of calculators.
• In early grades students should use a four-operation calculator such as the TI-108. They should be told that they may use either the [+] or the [M+] key, together with the [+/-] key, but they cannot use the [-] and [M-] key in this addition problem. They can be shown that the same result can be obtained by using subtraction, but they should not use it in this problem.
• In higher grades students may use a scientific calculator such as the TI-34 II. But they also should not use the [-] key, but only the [+] key and the [(-)] key.
Answers.
• You can form any number between -40 and 40, except 0. With only three cards you get the numbers between -13 and 13. It is acceptable to say that you get 0 by not putting any cards on the table, but it is not necessary.
• Each number can be formed in only one way.
Remark.
In early grades the results can be displayed as a long table.
-40 = -27 + -9 + -3 + -1
-39 = -27 + -9 + -3
-38 = -27 + -9 + -3 + 1
-37 = -27 + -9 + -1
.......................
-1 = -1
0
1 = 1
2 = 3 + -1
.......................
40 = 27 + 9 + 3 + 1
The table can be made as follows. Each child writes a few lines of the table, each one on a narrow strip of paper, and then the strips are attached to a display board.
Follow-up problems in higher grades.
(1) Generalization from 27 to 3n.
Which numbers can be formed as the sum of the numbers -1, 1, ... , -3n, 3n? Answer: Numbers between -(3n+1 -1)/2 and (3n+1-1)/2.
This problem can be treated as an informal investigation or as an introduction to proofs by mathematical induction.
(2) Numbers in base 3 and other bases.
The above problem is really the question of representing numbers in base 3, when the digits are -1, 0, and 1, instead of the more common 0, 1, and 2. A discussion of other bases, and choice of digits, may follow.