Subtracting integers

[Amom123]

I am teaching my son how to add and subtract positive and negative integers. I have taught him how to change the subtraction problems to addition by "adding the opposite". He wants to know why you do this? It is at the point where he is so frustrated because I have no answer to justify why it is done this way. Does anyone have a concrete explanation I can give him?

 

[Amom123]

Thankyou for the website. We will take a look at it tomorrow with fresh eyes.

With the thermometer/ temperature comparison it gets tricky when you have a problem like 6-(-4)= 10. It is hard for him to visualize this.
I know the lesson was using the inverse property of addition a-b=a+(-b) as well as the identity property of addition.

I appreciate you taking the time to help us with this question.

 

[Amom123]

The banking analogy solves all the questions of plus and minus as signs and operators, but it may not be meaningful to a child. I shall try to think of one that is more physically intuitive. (Even most adults find banking to be a bit mysterious. In fact, in recent years it has become apparent that even many bankers find it mysterious.)[/QUOTE]



So very true!
Thanks so much for your help.

 

[Mrspi]

I was not able to dream up a plausible physical analogy for the 6 - (-2) situation, but I came up with a logical explanation.

This is not directly relevant to a child's actual language, but it is a straight forward application of logic, and children can be painfully logical.

What is a not-cat. Well, a dog is a not-cat, a mouse is a not-cat, a house is a not-cat. In fact, everything that is not a cat is an example of a not-cat.

So what is an example of a not-not-cat. Well it cannot be a dog because a dog is a not-cat. It cannot be a mouse because a mouse is a not-cat. It cannot be a house because a house is a not-cat. Everything that is not a cat is a not-cat so the only thing can be a not-not-cat is a cat.

The negation of a negation is a positive. Stop not doing your homework means to start doing it.

So 6 - (-2) has a double negative and must be replaced by a positive 6 + 2. Similarly (-6) * (-2) = -(-(6 * 2)) = + 6 * 2 = 12.
Best I can do. Sorry.

One winter morning when I woke up, the temperature was -2 degrees F. By afternoon, the temperature was 6 degrees F. How much did the temperature change?

To find the amount of change, we would reason this way:

amount of change = ending temperature - beginning temperature

So,

amount of change = 6 - (2)
or, using the rule for subtracting, we can change that to "adding the opposite" and get

amount of change = 6 + 2
amount of change = 8
The temperature ended up 8 degrees higher than it was to begin with.

Now, does that make sense at the 4th grade level?

The temperature started at -2 degrees F, which means it was 2 degrees BELOW zero. To get to 0, it had to go UP 2 degrees, right? And to get from 0 to 6, it had to go UP 6 more degrees. UP 2 degrees and then UP 6 more degrees is UP 8 degrees.

And that's the best I can do!

 

[daon2]

Ask him if he would rather have -$3 dollars or +$2 dollars added to his allowance. Follow through with his request. He will either make the right choice, justifying your lessons, or a terrible mistake, something he will never again forget.

When mathematics is formalized it is often hard to understand. In reality subtraction is defined in terms of addition. It is "something to get used to" rather than to completely understand. The answer to his question is a rather unsatisfying "because that is what it means."