Math folks-help me explain the difference between pos/neg and addition/subtraction...

[Zoo Keeper]

We have approached this from *multiple* ways, and it doesn't seem to sink in.  It should have sunk in by now, right? :confused1:

 

Yes, this is for my oldest (freshman), who keeps on stumbling over this in his algebra. The equations are getting more complex, and he is getting twisted around and very frustrated.  He thinks it is all "so random."  I know we can't really move forward until he gets this nailed down.

 

Help a sleep-deprived, overextended mama out, please. It's like we need some of those U.N headphones that will translate my words to something that finally sinks in for him. ;)

 

 

[SparklyUnicorn]

There is essentially no difference.  What specifically trips him up? 

 

Can you give an example of a problem he finds confusing?

 

 

[Zoo Keeper]

Sparkly,  he thinks *very* left to right.  So when he simplifies expressions, instead of combining like terms he tries to add or subtract each term in order from left to right.   He is very, very, procedural in math, and does not transfer concepts well.  He can do very well on page after page of lessons on adding/subtracting/mult/dividing integers (one concept at a time), but then be so balled up when it comes simplifying an expression and solving an equation.  He gets the idea of balancing an equation, he gets using inverse operations, but it's like there is a block with pos/neg terms.  "What do you mean that's a negative 6?! That's a subtraction sign!  You can't just call it a negative 6 and move things around! You have to take away the 6!…"

[Mike in SA]

Try this from the AL forum:

 

http://forums.welltrainedmind.com/topic/624199-help-with-explaining-a-maths-concept/

 

 

Does it help?  If not, happy to clarify...

 

Edited November 4, 2016 by Mike in SA

[Zoo Keeper]

Try this from the AL forum:

 

http://forums.welltrainedmind.com/topic/624199-help-with-explaining-a-maths-concept/

 

 

Does it help?  If not, happy to clarify...

 

 

Thanks, Mike.

 

Processing…   :)

[SparklyUnicorn]

Could you post an example of a problem that generally confuses him?

 

 

[Daria]

Dragonbox has been really helpful for me with that concept.

[ondreeuh]

Maybe pulling out counters to show it conceptually? Like this:

[skimomma]

My dd has struggled with this in the past and what helped her was to change all subtraction into addition using parenthesis.  

 

2-3x+2y+7x-4-9y

 

becomes

 

2+(-3x)+2y+7x+(-4)+(-9y)

 

I am very mathy so did not and still do not understand why this helps her think it through, but it does.  She will still do this occasionally with complex equations.

[SparklyUnicorn]

My dd has struggled with this in the past and what helped her was to change all subtraction into addition using parenthesis.  

 

2-3x+2y+7x-4-9y

 

becomes

 

2+(-3x)+2y+7x+(-4)+(-9y)

 

I am very mathy so did not and still do not understand why this helps her think it through, but it does.  She will still do this occasionally with complex equations.

 

The only thing I can imagine is maybe it makes the negatives stand out more so she doesn't forget that they are negative/subtractions.

[SparklyUnicorn]

The sign follows the number is what I tell one of mine.  You can reorder addition and subtractions no problem so long as you bring the sign with it. 

 

So 5-6 is -6+5.  If it helps to order stuff together then order stuff together.

 

If I were to organize this:

 

2-3x+2y+7x-4-9y

 

I'd do -3x+7x+2y-9y+2-4.

 

I'd go left to right looking for what I'd add next to my ordered problem and lightly cross out the stuff I accounted for.  Again, the sign follows the number wherever you move it to.  Then you can calculate left to right. 

 

But I'm not entirely sure if THAT is the confusion. 

 

When I see something like - -5 I put a parenthesis around -5 so -(-5) and think ok negative times negative is positive.  Some people change the two negatives into positives.  Sometimes I do that, depends on the problem.  But I find adding the parenthesis often helpful there.

 

 

[daijobu]

<pulling out AoPS prealgebra book, turning to page 17...>

 

So, -X is defined to be the negation of the number X, which is the number we add to X to get zero.  

 

Definition:  -X + X = 0

 

It is also called the "additive inverse of X" or "minus X" or "negative X."

 

the negation of 4 is -4

the negation of 3 is -3

the negation of 2 is -2

the negation of 1 is -1

the negation of 0 is ???

the negation of -1 is ???

the negation of -2 is ???

 

In each case, ask yourself what do you add to 0 to get zero?  What do you add to the negation of 1 to get zero?  

 

Point out the the negation of a negation is:  - ( -X) = X

 

Even though -X is sometimes called "negative X" it is not necessarily a negative number.  

 

We can replace -X with (-1)*X, that is "the negation of 1 times X"  So: (-1)*X = -X

 

I recommend getting a copy of the AoPS prealgebra book and continuing with their exercises as the book slowly and clearly explains every step and gradually brings you to something like:

 

a - b = a + (-b)

 

and so on.  It is really important that he get this, and this is a good opportunity to review this in detail.  

 

ETA: cleaned up typo.

Edited November 4, 2016 by daijobu

[Zoo Keeper]

My dd has struggled with this in the past and what helped her was to change all subtraction into addition using parenthesis.  

 

2-3x+2y+7x-4-9y

 

becomes

 

2+(-3x)+2y+7x+(-4)+(-9y)

 

I am very mathy so did not and still do not understand why this helps her think it through, but it does.  She will still do this occasionally with complex equations.

 

 

The only thing I can imagine is maybe it makes the negatives stand out more so she doesn't forget that they are negative/subtractions.

 

 

The sign follows the number is what I tell one of mine.  You can reorder addition and subtractions no problem so long as you bring the sign with it. 

 

So 5-6 is -6+5.  If it helps to order stuff together then order stuff together.

 

If I were to organize this:

 

2-3x+2y+7x-4-9y

 

I'd do -3x+7x+2y-9y+2-4.

 

I'd go left to right looking for what I'd add next to my ordered problem and lightly cross out the stuff I accounted for.  Again, the sign follows the number wherever you move it to.  Then you can calculate left to right. 

 

But I'm not entirely sure if THAT is the confusion. 

 

When I see something like - -5 I put a parenthesis around -5 so -(-5) and think ok negative times negative is positive.  Some people change the two negatives into positives.  Sometimes I do that, depends on the problem.  But I find adding the parenthesis often helpful there.

 

I think some of this will help!  I will read and think some more after I convince the stragglers to go to bed (and stay in bed!) so I *can*  think. 

[daijobu]

My dd has struggled with this in the past and what helped her was to change all subtraction into addition using parenthesis.  

 

2-3x+2y+7x-4-9y

 

becomes

 

2+(-3x)+2y+7x+(-4)+(-9y)

 

I am very mathy so did not and still do not understand why this helps her think it through, but it does.  She will still do this occasionally with complex equations.

 

This helps her think it through because what we are really doing is adding some numbers to the negation of other numbers.  

 

Subtraction is defined:  a - b = a + (-b)

 

I wrote my equations like this ad nauseum until I don't know how long.  

 

When I was in high school.  

[daijobu]

Sparkly,  he thinks *very* left to right.  So when he simplifies expressions, instead of combining like terms he tries to add or subtract each term in order from left to right.   

 

<Turning to page 4 in AoPS PreAlgebra...>

 

Does he remember the commutative property of addition?  Addition is commutative.  a+b = b+a.  

 

Addition is also associative.  (a+b) + c = a + (b+c).  

 

You can add things in any order, and the answer is the same.

 

Which would you rather add?

 

(i)  72 + (19 + 28)

or

(ii) (72+28) + 19

 

This comes up in Math Olympiad a lot.  <pulling old math olympiad exam book from shelf...>

 

What is the value of the following?

9+91+18+82+27+73+36+64+45+55

You have 5 seconds.  

(9+91)+(18+82)+(27+73)+(36+64)+(45+55) = 100 + 100 + 100 + 100 + 100 = 500.  

 

I think your best bet is to review using the AoPS prealgebra book the relevant sections.  The books takes you through slowly and deliberately each of these properties.  

[wapiti]

You could also try these AoPS prealgebra videos
 

1.5 Subtraction with Negatives Part 1

1.5 Subtraction with Negatives Part 2

1.5 Subtraction Problems

1.5 Subtraction and the Distributive Property

 

[daijobu]

You could also try these AoPS prealgebra videos

 

 

I'm enjoying them now.  I love how he writes 7 + (-2) = 5 + 2 + (-2) = 5 + 0 = 5.

 

Brilliant.  

[lisabees]

Agree with using AoPS, but even after the concept is understood, your ds needs to see enough of these problems to really cement it.  I would give him 5 problems a day on a whiteboard, even after he has moved on to other chapters.

[Zoo Keeper]

I'm still reading and thinking through all your posts; thanks so much to all of you who took the time to help. :)