Subtraction
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Introduction This is the last unit in which we use only two-column boards. We describe subtraction using two new concepts, zero and negative numbers. This approach requires some explanation, because negative numbers currently are not introduced until children enter late elementary or middle school, so using this concept in the first grade may seem premature. 
Above is an image of the two-column board used in the lesson.
When a quantity such as distance, speed, weight, volume, age, or cost remains constant, we can describe it by one number, using an appropriate unit. But when a quantity changes, we need at least two numbers. The first number gives the value of the quantity before the change and the other its value after the change. The size of the change is the difference of these two values. But there are three cases to consider:
The value before plus its change equals the value after. Thus the change can be a positive number, zero, or a negative number. Remark This introduction of negative numbers, using appropriate examples, has been successfully introduced in first grade classrooms. (And in our opinion this explanation is also better than the currently used concept of "directed numbers" that was borrowed from the concept of vector space, and the "old fashioned" concept of "negative quantity" such as debt.) The Operation of Subtraction on a Counting Board Introducing the Concept Before learning the concept of negative numbers and using it, children need to gather some data that can be used to observe a change of some variable magnitudes. For example, with help from the teacher, children can measure the temperatures at noon outside of school for two weeks and display the results in their classroom. For example: 76, 78, 91, 91, 84, 83, 68, 65, 77, ... After learning how to compare two numbers using both white and red counters as shown in the unit about Comparing Numbers, children are given pairs of consecutive numbers from the list representing temperature, and they are asked to answer the following questions, "Did the temperature increase, decrease, or remain the same? What was the change?" They are asked to use red counters to represent the first number (temperature before the change) and white counters to represent the second number (temperature after the change) when they compute the difference on the counting board. When no counter remains on the board, then the temperature remains the same and the change is recorded as 0. Numbers 91, 91 
In the leftmost image, the first number, 91, is represented with red counters and the second number, 91 is represented with white
counters. The rightmost image shows the board after regrouping. The rightmost board shows that the temperature did not change; the difference is 0 degrees.
Numbers 91, 84 
In the leftmost image, the first number, 91, is represented with red counters and the second number, 84 is represented with white
counters. The rightmost image shows the board after regrouping. The rightmost board shows that the temperature decreased; the difference is -7 degrees.
Numbers 65, 77 
In the leftmost image, the first number, 65, is represented with red counters and the second number, 77 is represented with white
counters. The rightmost image shows the board after regrouping. The rightmost board shows that the temperature increased; the difference is 12 degrees.
Notice that -7 degrees did not mean that the temperature was below zero. It only means that it decreased by 7 degrees. This example is from a school in the southern part of New Mexico where most children have never seen temperatures below zero. We suggest that in most other places, different lists of data should be used to introduce negative numbers. Negative numbers are used for many different purposes. But showing them at the same time can easily lead to confusion. General Method of Addition and Subtraction We consider two kinds of numbers: positive, represented by white counters, and negative, represented by red counters. A positive number represents the amount of increase of a quantity, and a negative number represents the decrease of a quantity. You can add positive and negative numbers together by using all regrouping rules and removing red and white counters that are put on the same location. There are three possible outcomes of addition: a positive number, a negative number, and zero. To subtract a number means to add its opposite. Important Exercise The difference between adding only positive numbers and adding both positive and negative numbers is very small, so it does not require any special practice. But the results are often unexpected. This is illustrated by the following task: Task Children are given a written list of positive numbers within the range 1 to 100.
2. Next, they have to add all the differences to the first number, 41. 
The child begins by representing 41, the first number, on the board.
The child adds the difference, -4, to the number currently on the board. In the leftmost image,
a red token representing -4 is added to the current board. The rightmost board shows the result
after regrouping.
The child adds the next difference, -26, to the number currently on the board. In the leftmost image,
red tokens representing -26 are added to the current board. The rightmost board shows the result
after regrouping.
The child adds the next difference, 54, to the number currently on the board. In the leftmost image,
white tokens representing 54 are added to the current board. The rightmost board shows the result
after regrouping.
The child adds the next difference, -15, to the number currently on the board. In the leftmost image,
red tokens representing -15 are added to the current board. The rightmost board shows the result
after regrouping.
The child adds the next difference, -20, to the number currently on the board. In the leftmost image,
red tokens representing -20 are added to the current board. The rightmost board shows the result
after regrouping. Notice that the result, 30, is the last number of the original list.
Number Board index |