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February 6th, 2015, 02:39 AM   #1
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question about Subtraction as a concept

Ok so I was thinking about how I do subtraction in my head and wanted to understand why both methods work.
Lets say I have the problem 27-2.
I will think to go back to 25 (since 25+2=27) and kind of remove the last 2 "objects" in my mind to see that there are 25 left. Almost like a right to left removal of 2.
But if the problem is 64-57.
I will think that 64 is 57+7 and sort of remove the first 57 objects in my mind and get 7 left over.
So why do both of these strategies work and is this how most people think about subtraction? Like why does my mind kind of arrange the number into groupings of 10 from left to right or down to up (why am I thinking about first and last objects?)

Hope my questions make some kind of sense, just wanted to think more about how I understand this concept. Thanks for all the help everyone!
lastly a question about the interpretation of "the difference":

To me it seems weird that when I am doing to the removal of the last couple of object (like in the 27-2 example) that I get the difference between 27 and 2 which is of course 25. But I guess no matter how you arrange 27 you will always get 25 and 2 so the difference is 2. Is that a correct interpretation?
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February 6th, 2015, 03:35 AM   #2
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When you are learning maths in school, you can (and should) be assigned problems like this:

10 - 2 = ?

But also like this:

2 + ? = 10

So you learn to solve these two different problems.

"If I remove 2 from ten, what is left over?"
"What do I have to add to 2 to get 10?"

So why do they give the same result? The two equations above can be shown to be equivalent.

Replace question mark with an algebraic unknown, x:
10 - 2 = x

Add 2 to both sides of the equation in order to keep it balanced:

10 - 2 + 2 = x+2
10 = x + 2

Therefore, the two ways of solving problems achieve the same solution. This means that either method can be used to solve either problem.


Just to make it really clear, it is totally acceptable to do the following:

Evaluate 10 - 4.

Ask yourself "What do I have to add to 4 to make 10? It's 6!"

10 - 4 = 6.



Final point:
When teaching subtraction to kids, especially those with learning difficulties, ask the kid to count up from the lower number to reach the upper number on their fingers, saying each number out loud:

10 - 4...
Start on 4...
"5", "6", "7", "8", "9", "10"
(looks at fingers, sees 6 up)
10-4 = 6

They will generally find this much easier to execute and understand than counting backwards from 10:

10-4...
Start on 10...
"9", "8" ,"7" ,"6" ,"5" ,"4"
(looks at fingers, sees 6 up)
10-4 = 6

Although once the kid gets comfortable with this sort of maths and can routinely achieve the correct answer, they should be introduced to the equivalence of the two ways of doing it.

Last edited by Benit13; February 6th, 2015 at 03:37 AM.
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February 6th, 2015, 05:15 AM   #3
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Quote:
Lets say I have the problem 27-2.
I will think to go back to 25 (since 25+2=27) and kind of remove the last 2 "objects" in my mind to see that there are 25 left. Almost like a right to left removal of 2.
But if the problem is 64-57.
I will think that 64 is 57+7 and sort of remove the first 57 objects in my mind and get 7 left over.
I don't see how these are "two different methods". It looks like you are doing exactly the same thing in both. But how do you get the "7" so that you can "think that 64 is 57+7" in the first place?
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February 6th, 2015, 05:41 AM   #4
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Quote:
Originally Posted by Country Boy View Post
I don't see how these are "two different methods". It looks like you are doing exactly the same thing in both. But how do you get the "7" so that you can "think that 64 is 57+7" in the first place?
The OP stated that he is solving problem 1 by removing 2 from 27 (i.e. counting backwards from 27 to 25) and problem 2 by knowing that adding 7 to 57 gives 64. I explained each method in my previous post.

The 7 is obtained either by counting up from 57 to 64 or by memorising that 7+7 gives 14, so similarly 57+7 = 64.

I don't think too many people would solve 64 - 57 by counting backwards down from 64 to 57 (method 1) and I don't think anybody at all would actually subtract 57 from 64 by counting down all the way to 7 (which I've seen some kids try before)!
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February 11th, 2015, 05:04 PM   #5
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I don't know what, if anything, that has to do with the subject at hand.
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February 11th, 2015, 10:38 PM   #6
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Quote:
Originally Posted by Benit13 View Post


Final point:
When teaching subtraction to kids, especially those with learning difficulties, ask the kid to count up from the lower number to reach the upper number on their fingers, saying each number out loud:

10 - 4...
Start on 4...
"5", "6", "7", "8", "9", "10"
(looks at fingers, sees 6 up)
10-4 = 6
This makes sense to do (for me) if the number being subtracted (the 4 in this case) is above 5, since that "distance" is small to count up from.

My interpretation is that when doing it like this, you see 4 fingers and 6 more (to the right of the original 4) and since removing the first 4 leaves the 6 you know the 10-4=6


Quote:
Originally Posted by Benit13 View Post
They will generally find this much easier to execute and understand than counting backwards from 10:

10-4...
Start on 10...
"9", "8" ,"7" ,"6" ,"5" ,"4"
(looks at fingers, sees 6 up)
10-4 = 6

Although once the kid gets comfortable with this sort of maths and can routinely achieve the correct answer, they should be introduced to the equivalence of the two ways of doing it.
I've never actually thought of doing subtraction this way (and didn't mean to imply that I did in my first post). What I would do with 10-4 is to think of 10 things lined up from left to right and remove the last 4 starting from the 10th and ending at the 6th since I know the 6 + 4 is 10 and kind of remove those last 4 and know that 6 has to remain.

Does that all make sense from an interpretation standpoint?

I guess my question was really why do we either start from the end of the number and remove items from right to left (in the case of a small take away) or start with the items being removed and add up left to right to the number started with. Is that just the easiest way to keep track of things and a way to exploit our number system because we come to realize the removing any 4 items from 10 will always leave 6 regardless of which 4? Because if it didn't then we could exploit which "items" to remove and end up with more than 6?

Hope that makes sense, I know I'm blabbering a little but it's hard to get my thought process down into words sometimes.

Thanks for your help Benit!
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February 11th, 2015, 11:14 PM   #7
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And to add:

I feel like the notion of A=B+C and A=C+B plays into this in that both are equivalent. So we come to use either way of removing B like I talked about in the post above and come to realize that the method we are using represents a more abstract concept that whatever way remove B from A we get C. So we use a mental method that is easiest to keep track based on the fact that addition is commutative so we can almost think of whatever we are subtracting as the last thing added or first thing added in the original sum to make it easier to follow.

I think anybody doing the subtraction problem 12-3 probably thinks about it by removing the two units and then one more from the 10 to get 9, so basically they will go back 3 units from 12 on the number line (kind of removing the last 3 and seeing the 9 underlying).

What has seemed a little strange to me is when we are finding the distance between 3 and 12 on a number line or graph we probably remove those last 3 units in our mind and see 9 underlying but if we were adding up the 3 and 9 from left to right (as they would be oriented on a number line) it's strange that we are not going to remove those first 3 to see the 9 distance (above or to the right of the original 3) but instead the last 3 (from the 9 part to the right of the 3 part). But I guess it doesn't really matter what 3 you remove from the collection because you can always reorient by the commutative property of addition to fit the same distance of 12. That make sense?

Last edited by mathz; February 12th, 2015 at 12:00 AM.
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