Operations on negative integers

I was trying to teach my younger sister some math, and it drifted on to integers, and operations on negative integers. So questions like:

a) $-3+2 = ?$

b) $2- (-3)= ?$

c)$-3 -2 = ?$

had to be answered. So, I did not want to say that because minus of minus is plus, so the answer to b) is 5, and minus of plus is minus, so you can solve a) and c) likewise....and I explained in detail how we can simplify these and for b) particularly, I said:

Think of 2 as $5-3$ then $5-3-(-3)$ is the question. Now, I can say that is the same as $5+0-3-(-3)$ and by the definition of $-3$, $0-3=(-3)$, So now the expression becomes; $5 + (-3) - (-3)$ and since $(-3)$ is being added and subtracted, we will just cancel that and write 5. So, that is your answer!

But I think this is a little too long so I said that you should observe this pattern and then use the result $-(-x)=x$ and solve questions.

To see if she has really understood, I asked her to explain this to my mother, and the result was not satisfactory. So, my question is, How can I better explain it to her? Or rather, does this have any flaw which needs to be corrected?

Please suggest a method that will involve only basic algebra as she is in grade 4...

Aslo, if you find an easier way, do share it with me!

PS: I think the tags I added are not right, but thats all I could think of, so if I have gone wrong, do edit that.

I remember learning it as a child using something like a "digging a hole" metaphor.

You can think of "a box of dirt" as being a unit, and talk about boxes of dirt stacked on top of each other. If you dig a cubish hole in the ground and use it to fill one box, the empty space is supposed to represent $-1$. If there are no boxes full of dirt and no holes, then we are at 0. If you have a box of dirt and dump it into a 1-box-hole, it fills perfectly and you get $0$.

So how to deal with $-(-3)$? One can interpret this as "taking away 1-box vacancies". Taking away a one box vacancy is the same as filling it in, i.e. $-(-1)=+1$.

I guess what I have in mind is really thinking of $+1$ as a box of dirt, $-1$ as a vacancy (hole) of the exact same size as one box of dirt, and the positive numbers as stacks of filled unit boxes, and negative numbers as stacks of "vacancies" of the same size as a box of dirt.

There are obvious variations of this for children as they grow older. Obviously, if they understand how to use money, you can do the same picture but with owned dollars and owed dollars. If you "have" $-1$ dollars, you owe $1$ dollar. If you have $-3$ debt and you add $+2$, the combination would be that you still owe $-1$. The $-(-2)$ can be interpreted as "the removal of a debt of 2" which would be the same thing as gaining $+2$.

Electric charge is another model, but less down to earth than money and dirt, maybe.

Somehow I forgot about another obvious version, that of a thermometer. If you can imagine $+1$ as being a unit of "heat" and $-1$ as being a unit of cold, and $0$ as being room temperature, or freezing or whatever. (Of course this analogy breaks because of absolute zero, but you can get away with it for now.)

Now the idea of $-(-1)=1$ is "removing one cold is the same as adding a heat".

If you have $7$ hot and combine it with $-6$ (6 cold), then you are still warmer than 0 at $+1$.

If you are at 3 cold and add 4 more cold $-3-4$ then you have a total of seven colds $-7$.

Summary

Try to establish the idea of a "vacancy of one unit" as opposed to "one unit". We've mentioned that there are several models to do this (listed in the order that I like them):

1. Dirt
2. Temperature
3. Money
4. Electric charge

All of them rely on building the idea of a unit that "is present" and a unit that "is absent", and the idea that existence and absence cancel each other out.

• didnt get it really...how can I use this? Like the basic idea is to explain how $-1$ behaves when multiplied with another integer, so how do I do this? – Saurabh Raje Jul 10 '13 at 16:43
• basically, what is the metaphor for minus and what is the metaphor for plus? (when I say minus, I mean it as an operator, and not a sign) – Saurabh Raje Jul 10 '13 at 16:45
• @SaurabhRaje Sorry, still working on the post. I'll try to address these questions. – rschwieb Jul 10 '13 at 16:45
• @SaurabhRaje I just recalled temperature might be a good model to try for a child. See what I added about a "unit of heat" and "unit of cold" – rschwieb Jul 10 '13 at 17:00
• @SaurabhRaje Unfortunately, it's not perfect, and physics teachers will probably cringe at the metaphor. But then again, physics teachers are all constantly teaching us something and then telling us it is riddled with mistakes and then teaching us something better, so maybe they would understand :) – rschwieb Jul 11 '13 at 13:07

Basically addition and subtraction is just the same. Subtraction is just addition with negative numbers.

There are several ways to remember operations with signed integers:

First way (common way taught in school):

(I used to tutor a 5th grade student who is having a difficult time with math. This is the technique I thought of that time and I think she mastered solving operations with signed integers.)

There are three rules: (I ask her every time we solve a similar problem if what rule can we apply to solve it)

RULE 1. Addition with same signs:

RULE: When adding two integers with the same sign we just add the "absolute value" (or the numbers disregarding their signs) and copy the sign to the sum.

To make it easier for her to understand, it is like adding normal unsigned integers and the "sign" is just an identifier.

Example:

a. -2 **+** -4 = -6
b. 2 **+** 4 = 6

Imagine the plus sign being bold so as to remember that it is an operator. The sum depends on the addends on both sides of the operator. The two addends are independent and the plus sign will merge them depending on their signs.

RULE 2. Addition with different signs:

RULE: If both addends have opposing signs, subtract the "absolute value" of the larger number with the smaller number and copy the sign of the larger number.

Example:

a. -7 **+** 3 = -4 (7-3=4 and the sign of the bigger number which is 7 is negative)
b. 7 **+** -3 = 4 (7-3=4 and the sign of the bigger number which is 7 is positive)

RULE 3. Subtraction with same or opposite sign:

RULE: Change the sign of the subtrahend (second operand e.g. negative becomes positive) and proceed to addition (make the minus operation a plus operation.

All rules for addition can be applied here.

To sum up:

ADDITION: a. same sign - normal addition but copy the sign common to the addends b. opposite sign - normal subtraction of the same number but copy the sign of the bigger number

SUBTRACTION: -change the sign of the second operand and proceed to addition.

SECOND WAY/SHORTCUT:

As I stated earlier, addition and subtraction is just the same. It is just addition with integers where one or both of the addends can be negative.

Consider all subtraction as addition and refer below:

a. 3 - 7 (this is basically subtraction at first glance)

Consider it this way

3 + (-7)

there is an imaginary plus sign (addition between 3 and -7)

since 3 and -7 have opposite signs we use RULE 2.

3 - 7 = 3 + (-7) = -4

b. -4 - 8

can be written as

(-4) + (-8)

placing a parenthesis or thinking of the negative sign and the number itself as one entity really helps

-4 - 8 = (-4) + (-8) = - 12

from RULE 1

Hope this helps :)

EDIT As you noticed above, I focused on the operation addition since subtraction is based on addition. It is the most fundamental operation. We were taught from school that addition and subtraction is just moving to the right and left in a number line. I think that is the most intuitive way to do simple arithmetic on signed numbers for beginners ;)

Furthermore, I believe that doing Math is more of a gut feeling. You don't have to analyze every time you perform an operation the logic behind things. Plus, it's time consuming. Of course at first you have to but once you get a feel with what you are doing, solving problems will just come naturally. That's why I outlined these simple steps.

• Thanks a ton for such a long response, but I have a basic doubt here. Aren't you just giving the rules and asking the student to learn them? I wanted it to be intuitive and not memory based, so I think I need to give a "proof." Of these rules to the student. – Saurabh Raje Jul 11 '13 at 3:12
• I agree with Saurabh that this is basically relies on a student accurately applying rules established by fiat. This certainly is the tack taken by most American primary/secondary schools, but such an approach seems to have pedagogical problems. Students begin to expect math is all memorization and application of such rules. I think Saurabh is looking for some appealing physical metaphor to try to convey the idea rather than the rules. Once the idea is established, a student can figure out many of the rules themselves (a much better result :) ) – rschwieb Jul 11 '13 at 13:13
• @rschwieb, hit the nail on the head, my friend! – Saurabh Raje Jul 11 '13 at 14:20
• @SaurabhRaje you can utilize the number line. That is how I first learned to add signed numbers. Addition is done by moving through the number line to the right. Likewise, subtraction is done by going to the left of the number line ;) – chaine09 Sep 26 '13 at 19:28
• @SaurabhRaje you can first explain the concept of a number line. It is also useful to present these numbers in terms of money. It is easier to visualize arithmetic this way. I assumed that since you are working with addition and subtraction with signed numbers that your student already know the basics about addition and integers. – chaine09 Sep 26 '13 at 19:38

A positive integer can be represented as a length:

Create a line. Pick a point on it that you give the "nickname" $0$ (if the term "point" is confusing, use "dot" or something]. Now go any distance you like to the right, and mark off another point. The line segment between $0$ and this point, we decide to be the length $1$, or $1$ centimeter, if that's better. [It helps drawing this on graph paper].

To make the length corresponding to $19$, go $1$ length $19$ times to the right. To make the length corresponding to  $5+7$, go first $5$ lengths to the right, and from the point you landed on, proceed $7$ lengths to the right. The directions left and right is with respect to $0$.

To make the length $-19$, start from the point $0$ and go $1$ length $19$ times to the left. A negative number corresponds in other words to the "opposite" length. $-19$ is the result of moving in the opposite direction of how you move $19$.

To add $-3$ with $2$, go $1$ length three times to the left, and then go $1$ length two times to the right.

$-(-3)$ is how you move to get in the opposite direction of $-3$. What is the opposite of moving to the left? Moving to the right. $3$ times to the right from position $0$ is the length $3$.

$2-(-3)$ is what you get when you go two lengths of $1$ to the right, and then go three times of length $1$ in the opposite direction of the direction of $-3$ (left). Therefore, from the endpoint of $2$, we go three times the length of $1$ to the right.

$-3-2$ is just moving three unit lengths to the left, and then two unit lengths further left.

I would think that it would be easier to always evaluate the first-occurring integer first. That is, $-5 + 2$ is constructed first by looking at $-5$ and going $5$ times to the left, and then you look at $2$, and remember to construct it from the endpoint of $-5$.

This also paves a way to do multiplication. $5$ times $7$ is constructed by making the length corresponding to $7$. Then, you go $5$ times to the right with that length. Essentially you are treating $7$ as a unit.

$5$ times $-7$ is constructed by making the length $-7$ and then going $5$ times that length in the non-opposite direction of $-7$, that is, to the left.

$-5$ times $-7$ is constructed by first constructing $-7$ and from there you go five times in the opposite direction of $-7$ (starting from $0$). This is because, looking at $-7$ as the negative of $7$, and "pretending" that $7$ is the unit length, $(-7)$ times $(-5)$ is essentially $-(-5)$ times the distance of $7$ in the direction of $-(-5)$, that is, to the right.