help answer this

[Guest Barb B]

Now, get me I am not asking how to do this but why. We are doing algebraic addition and dd asks why. Anyone answer why we say 3 + (-4) and not 3 - 4. My brain is dead and I found the why of this hard.

Barb

[ereks mom]

Now, get me I am not asking how to do this but why. We are doing algebraic addition and dd asks why. Anyone answer why we say 3 + (-4) and not 3 - 4. My brain is dead and I found the why of this hard.

Barb

 

3 + (-4) = 3 - 4

The two expressions mean the same thing. Some people find it easier to manipulate the numbers if it's written 3 + (-4), and some find it easier if it's written 3 - 4.

[Jamauk]

3 + (-4) = 3 - 4

The two expressions mean the same thing. Some people find it easier to manipulate the numbers if it's written 3 + (-4), and some find it easier if it's written 3 - 4.

 

:iagree:

[usetoschool]

I am absolutely sure someone (anyone :lol:) will come up with a more accurate and mathematical answer but I would say to my kids "because all addition/subtraction is really addition with a list of numbers - don't think of the + sign to mean Do Something, but as a marker telling you whether it is a positive or negative number."

 

So the problem you gave is really (+3), (-4) and it is all addition and this is just a way of introducing the concept.

[Teachin'Mine]

Because the concept is being introduced and later on there will be more complicated expressions inside those parentheses. So they're taught to think in terms of adding a negative and then later simplifying it into a subtraction problem. Then when they get 3 - (-4) they will be able to convert that into an addition problem more easily since they're used to having the number, positive or negative, separate from the operation performed.

[kiana]

To emphasize that addition and subtraction are really the same operation -- that subtraction is just the addition of the negative. It's obvious here but when it's something like 3 - (3 - 4) or maybe 3 - (x^2 - 15x - 22) they'll need to be able to simplify that easily and naturally.

[emubird]

Use a number line. Put your pencil on 3 and count back 4.

 

Why it works becomes a little more obvious with a visual demonstration. It may also make it more clear that it's just the same thing as 3-4.

 

Kids often wonder why, when they've finally gotten one concept down, the books have to switch things around on them. I've been using the old line: "you'll understand later" a lot. Now that I've said it so many times and been proved right, they listen to me. (Along with the line: "just do it that way and you'll start to understand why it works, even though it makes no sense at first.")

 

Also, you might want to teach her to simplify the expression into something that makes sense to her. There's no reason why she can't do that in this situation. It may even help her understand better what's going on.

[creekland]

My guys always did better when it was the traditional 3 - 4, so I let them skip the 3 + (-4) step. About half the kids at school would do better if allowed to skip the step too. They find it incredibly confusing to add a negative when one should subtract that number UNLESS the first number is also negative.

 

The most understood way I've found to explain it IME is to tell kids if the signs are the same (e.g. -3 -4 = -7 or 3 +4 = 7) then add the numbers and keep the sign. If the signs differ (-3 +4 = 1 or 3 -4 = -1) then subtract the numbers (always highest absolute value - lowest absolute value) and keep the sign of the greater term. I agree it also helps to show students how/why this works on a number line so they can see it visually.

 

Of course kids also need to know that two negatives = a positive. That's separate.

[Lawana]

ETA: Probably not helpful.

Edited September 3, 2011 by Lawana
clarification

[songbirdie]

My understanding of it is this: In algebra, we aren't really adding or subtracting numbers in concept. We are actually always *combining* positive or negative numbers. It simply represents a different way to think of manipulating the numbers, even if 4 - 3 yields the same answer as +4 + (-3). I like to think of it in terms of points given in a game. . .if a team had 4 points but then I realized I'd given the 3 points for a goal to the wrong team, I would have to give the 4 point team a "minus 3" (negative 3) to bring their total back to 1. You still end up with the total of 1, but your thought process was changed from "subtracting 3" to "giving/adding a negative 3".

[songbirdie]

http://www.mathsisfun.com/positive-negative-integers.html

 

Go down about 1/3 of the way to where it says something about having negative numbers. The stories they tell with the numbers totally helped us to "get it".

[MyThreeSons]

I am currently teaching my co-op Physics students to work with vectors. When doing so graphically, thinking of subtraction as the addition of the opposite makes it SOOO much easier.

 

I tutor a couple of high school students in math. Way back when they first learned math, their father taught them to think of subtraction as the addition of the opposite. That's how they've always done it, so they took to it right away in Algebra.

[creekland]

I am currently teaching my co-op Physics students to work with vectors. When doing so graphically, thinking of subtraction as the addition of the opposite makes it SOOO much easier.

 

I tutor a couple of high school students in math. Way back when they first learned math, their father taught them to think of subtraction as the addition of the opposite. That's how they've always done it, so they took to it right away in Algebra.

 

Now see, this too evidently depends on the person. I do vectors a ton in school when teaching and I can't say I see it easier to add the opposite. In fact, even now when I think about it, it's confusing to me to think of it that way.

 

I tell students to think of signs as direction, + means one direction, - means the other. It's far easier for me to see it that way and many of them get their "aha" moments thinking of it that way too.

 

The way I see it, you simply aren't adding numbers if the signs differ. You're subtracting them. With vectors, you go from one point to the next, but absolute value-wise, I'm still subtracting if the signs change and adding them if they don't.

 

I suppose this "proves" that different brains are wired differently!

[MyThreeSons]

Now see, this too evidently depends on the person. I do vectors a ton in school when teaching and I can't say I see it easier to add the opposite. In fact, even now when I think about it, it's confusing to me to think of it that way.

 

I tell students to think of signs as direction, + means one direction, - means the other. It's far easier for me to see it that way and many of them get their "aha" moments thinking of it that way too.

 

 

 

uhhhhh ..... I'm not sure how you say it, but it seems to me that "adding the opposite" when it comes to vectors is the same as going in the opposite direction.

[creekland]

uhhhhh ..... I'm not sure how you say it, but it seems to me that "adding the opposite" when it comes to vectors is the same as going in the opposite direction.

 

 

It's the terminology that is different. "Adding" to me means just that, adding the absolute values of the numbers. My brain doesn't connect adding the opposite with subtracting actual numbers. I know how they teach it. I know the concept, but it doesn't come naturally to me, therefore, unless needed for something, I skip it and allow my boys to do the same. When I come across others at school who also fail to have their neurons connect with that terminology (and there are many), I tell them my "sign" method of determining adding or subtracting and it clicks almost immediately. With vectors, when I tell people to think of signs as directions (or directions as signs pending the actual problem), then it also works well.

 

Adding the opposite is just a confusing paradox for some of our brains (and evidently not for others).

[regentrude]

It's the terminology that is different. "Adding" to me means just that, adding the absolute values of the numbers. My brain doesn't connect adding the opposite with subtracting actual numbers. I know how they teach it. I know the concept, but it doesn't come naturally to me, therefore, unless needed for something, I skip it and allow my boys to do the same. When I come across others at school who also fail to have their neurons connect with that terminology (and there are many), I tell them my "sign" method of determining adding or subtracting and it clicks almost immediately. With vectors, when I tell people to think of signs as directions (or directions as signs pending the actual problem), then it also works well.

.

 

I frequently observe that students have trouble subtracting algebraic quantities that are, in itself, negative: they will add extra minus signs and mess up the signs. The same happens when adding vector components: frequently, when they add components of a vector that happen to be negative, many will add an extra minus sign, thus rendering the algebraic expression incorrect.

 

Do you think the root to this problem is how they are taught in high school?

It is an ongoing problem we observe every single semester - to get them to understand that a summation means PLUS, irrespective of the signs of the quantities that are being summed, is like pulling teeth. If it says: SUM of A and B, that is A+B, whether A and B are positive or negative. But boy, do they find that difficult.

[wapiti]

It just so happens that our next two lessons are Negation and Subtraction (for about 15 pages) in the first unit of AoPS Prealgebra. The book distinguishes them as follows:

 

Negation and subtraction look the same, but are different operations. Negation takes one number and returns its opposite. Subtraction takes two numbers and returns their difference.

 

and yet:

Subtraction is defined as addition of a negation

 

but unlike addition, subtraction is neither communative nor associative, so, we change subtractions to additions (of negations?) so that we can use the cummutative and associative properties. This will be a fun week. I hope dd enjoys it as much as I will. This thread is motivating :)

[creekland]

I frequently observe that students have trouble subtracting algebraic quantities that are, in itself, negative: they will add extra minus signs and mess up the signs. The same happens when adding vector components: frequently, when they add components of a vector that happen to be negative, many will add an extra minus sign, thus rendering the algebraic expression incorrect.

 

Do you think the root to this problem is how they are taught in high school?

It is an ongoing problem we observe every single semester - to get them to understand that a summation means PLUS, irrespective of the signs of the quantities that are being summed, is like pulling teeth. If it says: SUM of A and B, that is A+B, whether A and B are positive or negative. But boy, do they find that difficult.

 

 

Actually, I think the "root" of the problem is too much calculator use at a young age. We have the same problem in high school. Of course, the calculator would often do the problem correctly IF they used parentheses, but they don't. Otherwise, they just don't know what to do with negative signs - period. They have a vague idea most of the time, but that's it. Since calculators are allowed, the best a teacher will do is remind them to always use parentheses. With vectors they also need to fully understand coordinate pairs. Some do, most don't.

 

Forgetting to distribute negative signs is also a major issue.

 

Then, of course, there are some who never memorize terminology (one of those math things that needs to be memorized), so we also have to repeat over and over again what sum and difference mean to a fair number of students. I think that may be a bi-product of this technology dependent generation. They are so used to looking up anything they need (as opposed to memorizing it) that it carries over to math.

 

Speaking specifically of our school (because I've no idea how common it is), block scheduling also hurts. Technically, we're supposed to cover 2 lessons in 1 day. This would work IF the students would go home and put time into learning the lessons, but they don't. They'll do between 5 - 10 problems of homework - IF they understand how to do it. They'll scribble down something and leave it if they don't, because we can only grade on completion, not correctness. I agree with not penalizing for incorrect homework, but it means kids won't try to get something they don't understand. That would require mental work. In my earlier teaching days (I've been there 12 years now) kids would TRY to do problems they didn't just "get." Now they won't.

 

So, we're trying to speed along to complete a course and the students are only willing to work in class (and sometimes not even there). In the end, we don't complete the course anyway (generally we get to 7 of 12 or 14 chapters with skipping parts along the way) and several don't understand what we covered anyway. Tests are REALLY dumbed down. And there still aren't many As. But they get a full credit.

 

We only get to vectors in the last week or two of the class and they aren't on a final for those who give one (many teachers don't). There aren't many motivated students at that point especially when we "change" things on them to work off north.

 

So, IMO it's a third degree equation with roots at:

 

Too much calculator dependence. (Many can't do 6x7. Some can't/won't do 4x2.)

Too little willingness to memorize even the basics. (Terminology can't be worked out.)

Too little effort to keep up in class. (Once one gets lost in math...)

[creekland]

I should also add that in our school kids are mixed grades in class. This means one can get a more math talented 10th grader in Alg 2 with a "needs the course to pass, but could care less about it" 12th grader. If we had our talented or "willing to work" math students together, a teacher could do tons more with them. Those who "need the credit" could be slowed down to see if they could learn some of the basics reasonably well instead of trying to do the whole course. The "need the credit" students also have a decent chance of being class disruptors...

 

So, I was thrilled this past year when they did keep the talented 9th graders together for Geometry (my son was one of them). BUT, the teachers NEVER did anything more with them. They just moved more quickly. They still had the same dumbed down tests and they didn't cover any additional topics. Our school has a goal that all students in a course must do the exact same thing as any other class. Otherwise, it wouldn't be "fair." They have different levels of courses in other subjects. (Level 3 = 4 year college bound, Level 2 = regular course, perhaps cc bound, Level 1 is usually middle school level material - or lower.) But in math, they don't differentiate. A student takes Geometry or Alg 2. They're all Level 2. They could make one or two sections Level 3 and then tailor the course to do more with those students, but they won't.