This is a continuation of the chapter on negative numbers for seventh grade. The next installment will be on multiplying and dividing negative numbers.
Sub-Chapter 4: Subtracting Integers
This lesson provides a clarification and, for some, a revelation, that subtraction of two numbers is the addition of an additive inverse. In formal mathematical terms, x – y is defined as x+(-y) where -y is the additive inverse of y. They have seen this already with problems such as 4+(-10) on the number line, which they have learned is the same as 4-10. There is one important case that hasn’t yet been explored, which is subtracting a negative integer; e.g., 5-(-10). This particular case will be the new procedure that they learn; everything up to that point is a clarification of what has come before.
At least one of the warm-ups I’ve used for this lesson provides a segue to the subtraction of a negative number. Specifically:
The temperature yesterday was 4 below zero. Today it is zero. By how much did the temperature increase?
Adding the opposite. I start this lesson by asking the class to find 7 + (-4). After this is done, I think ask them to tell me what 7 -4 equals. I ask why we obtained the same answer and if I don’t hear something along the lines of “It’s the same problem” (which sometimes happens despite all my intentions) then I’m not afraid to say “Do you suppose that 7 +(-4) is the same problem as 7 – 4?”
At this point I disclose that whenever they have been adding negative integers, they are subtracting. Subtraction is really the addition of an additive inverse. Stated more simply, subtracting a number is the same as adding the opposite of this number. I quickly give an example: The problem 7 – 3 is the same as adding the additive inverse of 3, which is 7 + (-3). I ask them to write it without brackets as we did in the previous lesson: 7 – 3.
Addition and subtraction are what are called inverse operations. The students have been using this fact for years, having been told that 5-2 is a number which when added to 2 equals 3. That is 5 = 2 + x. They have worked with number bonds or number families, so they are familiar with 2+3 = 5, 5-3 = 2, and 5-2 = 3. When we have a problem like 4 – 10 (warm up problem 3), we are finding what number added to 10 equals 4. They may have solved it using the “good day, bad day” technique, seeing first that there is a loss and it’s a loss of 6, or -6. I then write 10 + (-6) and ask what it is equal to. They can see the answer is 4.
I remind them that this is how they were taught to check if their answer to a subtraction problem is correct. This may seem like new information even with reminders that this is what they have been doing the past few days starting with the “gains and losses; good day, bad day” technique. Therefore, more examples are necessary to reinforce the procedures so they are comfortable with doing such problems. I then give them three or four problems to do and have them check the answers. Thus, to check that 3- 6 equals -3, the student would add 6 + (-3) to obtain 3, which checks.
What Subtraction Represents. A problem like 10 – 4 can represent the loss of 4 things—those things being many different items such as money, weight, length and so on. For example, the question “If the temperature was 10 degrees and it decreased by 4 degrees, what is the resulting temperature?” is answered by subtracting 4 from 10. If, however, the question were “The temperature was 4 degrees and now it is 10 degrees. By how much did it change?”, the answer is still six, but the numbers represent different things.
In the first instance, six represents the new temperature after a decrease of four degrees. In the second it is the amount of increase in temperature from four to ten degrees.
Subtracting a Negative Integer. Now we come to subtraction of a negative integer such as 10 – (-4). The second model is what we use when we present subtraction of a negative integer in order to keep things straightforward for students, I limit the examples to finding the amount of change rather than what a loss of -4 represents. (An example of a problem that asks what’s left after a loss of -4 would be: A person has $10 in his bank account after $4 has been deducted in error. The bank corrects this error by removing the debit of $4. This is done by subtracting the loss of $4: 10-(-4), which then becomes 10 + 4 or $14. Even adults may find it confusing that ancelling a debt can be represented by the subtraction of a negative number. It is therefore highly likely that seventh graders will find the concept difficult. Since the goal is for students to subtract negative numbers, it is far easier to explain the procedure using the “find the difference” model discussed above, rather than by the “find what’s left” model.)
I first point out how they answered the question of “If it were 4 degrees yesterday and 10 degrees today, what is the change in temperature?” I want students to see the form they used: Today’s temperature minus yesterday’s temperature.
Next, I ask: “If it were 4 degrees yesterday but 10 degrees today what is the change in temperature?” Using the form defined above, we obtain 10-4 which represents an increase of 6 degrees.
We are now ready to present the problem of 10 – (-4). The question becomes: “If it were 10 degrees today, but -4 degrees yesterday, what was the change in temperature.” It can also be stated, “What was the increase in temperature from -4 to 10 degrees? The problem can now be written as 10 – (-4), which I leave up on the board.
Drawing a number line (either vertical or horizontal), I plot -4 and 10. As mentioned earlier, one of the warm-up problems for this lesson asked how much must the temperature increase from -4 degrees to reach zero. Although one would hope that students remember the warm-up problem, I usually have to remind them of it, as well as the answer: the temperature must increase 4 degrees to get to zero degrees from -4. That is, -4 + 4 equals zero.
Transferring this to the number line, it is apparent that zero to 10 degrees is an increase of 10 degrees, so we add 4 to 10—an increase of 14 degrees.
I then explain that rather than using a number line to calculate it, we can use the “add the opposite rule. The problem 10 – (-4) becomes 10 added to the opposite of -4 or 10 + 4, which is 14.
Another example using depth under water illustrates subtracting a negative number as well. I scaffold the problem by first starting with all positive numbers: A bird was 5 feet in the air and flew up to 10 feet. What was the distance upward that it flew? Students will easily see that it is “new height minus original height”, or 10 – 5.
The problem is now changed so that a bird dives underwater 5 feet to catch a fish, and then flies upward to a height of 10 feet above the water. What was the upward distance that it flew? I provide prompts, such as “How do we represent 5 feet underwater?” (-5) and “How is the problem written?” 10 – (-5).
I also want to show problems where there is a decrease. For example: “The temperature today is -5 degrees; yesterday it was 6 degrees. What is the change in temperature?” The problem is written as -5 – 6, which equals -11, or a decrease of 11 degrees.
After these worked examples, students are now given four or five problems that require subtracting a negative. One or two problems will be word problems but the others are strictly numerical.
Vertical vs Horizontal Number Lines. Some teachers I have spoken with recommend using vertical rather than horizontal number lines. They have observed that when the number line is placed vertically, the students more easily grasped the idea of subtracting a negative. It appeared that up = positive, down = negative was a “natural” visualization as opposed to left = negative, right = positive. This is something to keep in mind if students struggle with this and other concepts—teachers can move back and forth between vertical and horizontal number lines as necessary.
Common error (again). Students will continue making the mistake of seeing problems like -4 -4 as zero. Now there are three ways to adjust their thinking.
1) Gain/loss method: “I lost $4 and then I lost $4 more.”
2) Number line method: -4 -4 on the number line is (-4) + (-4).
3) The problem -4 – 4 is not the same as -4 – (-4). The latter equals zero because when one adds the opposite it becomes -4+4, and adding a number’s opposite will always equal zero.
Giving it a rest. Mastering the operations with negative numbers will be confusing for some students at first. Although these lessons continually refer to, build upon and reinforce the “gain/loss” procedure, some students may become overwhelmed by new information as well as the mathematical way of stating things. In particular, they are now learning to think of subtraction as the addition of an opposite number.
It is advisable to give students time to work with the newly learned procedures to ensure that they are comfortable with them and are achieving automaticity. Many textbooks, however, provide an additional topic such as showing pictorially how addition and subtraction work using circles where each circle represents a positive or negative unit integer. The purpose is to spotlight the conceptual underpinning of adding and subtracting negative integers. An example of how it works is shown in Figure 3.
In my experience, when students are still trying to master the procedures for adding and subtracting negative integers, additional approaches may confuse more than enlighten. Some students may not be ready to absorb the yet another pictorial approach, particularly when they have adjusted to the pictorial approach using the number line.
The pictorial approach can be given later when, in the estimation of the teacher, students appear proficient and comfortable working with negative integers. I tend to not present it, particularly if I see that students are successful in working with negative numbers. Ultimately it is a judgment call, based on a teacher’s experience with teaching it, as well as their understanding of how it works and how and when to present it.