Why is a negative times a negative a positive?

[wendyroo]

15 minutes ago, Not_a_Number said:

My point is that the best thing you can do for a student is to work with THEIR model and to make their specific model cohesive and linear in their heads. 

Maybe I am dooming my children by not doing what is best (🙄), but my goal is to offer them lots of models of a concept and give them lots of time to play with them, explore the similarities that allow them all to model one concept while looking superficially different, and figure out which are most helpful for them.

For example, at our house, 4 years old is the year of the fraction. My goal is for my four year olds to firmly understand that a fraction is a part of a whole, that you have to be careful to split your whole into equal parts, and that the parts can then be put together to form bigger fractions. I want them to be able to recognize and create the fractions: 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, 3/4, and 4/4.

If I stuck with one model, say a rectangle split into 2, 3 or 4 pieces, then I could have them rote cranking out those fractions in a hour. It is really not that hard for a bright kid to "split the box into as many pieces as the bottom number and color in as many pieces as the top number." That wasn't my goal at all. I also used almost no words or "explanations" when teaching fractions. With my kids, I always consider how I would teach a concept to someone with whom I did not share a language. My kids obviously do speak English, but from experience, I know that by the time my verbal explanations reach their brains, they sound much like the results of a game of telephone.

We do have a fraction motto: "Equal, equal, ___ equal parts, and I choose ___." But I don't really expect those words to help them until they have seen many, many examples. So they see half an apple, and half a cup, and half a length of string, and half of the M+Ms, and half as tall, and half the pie, and half way there (shown on a map), and half of the table, and half of the money, etc. We spend the year breaking almost everything into pieces, but almost entirely concretely.

My 4, 5 and 6 year olds have not been ready for "If I have 2 apples..." - nope, that requires translating words into a mental image. They were very much at the stage of "I have two apples right here...". Obviously, I could have halted all math education until their language skills caught up, but I chose a different path that worked for us...and seems to help some other kids as well.

[wendyroo]

20 hours ago, Not_a_Number said:

Getting a debt is pretty easy. You get a piece of paper that say IOU 5 dollars. You can see that this means you have to pay 5 dollars. Taking away debts is a little trickier.

As a side note, I have to say that these types of comments come off as very condescending.

I said, "Trying to think about "getting a debt" is incredibly unintuitive to me. I have to ponder over those words for a long time before I can figure out what situation they would model." I am clearly saying that that is a difficult model for me, so your response that I am wrong (about what is difficult for me?), and that it is in fact easy, is grating.

[Not_a_Number]

1 minute ago, wendyroo said:

If I stuck with one model, say a rectangle split into 2, 3 or 4 pieces, then I could have them rote cranking out those fractions in a hour. It is really not that hard for a bright kid to "split the box into as many pieces as the bottom number and color in as many pieces as the top number."

That's really not what I meant by "a single model." A model can be cohesive and widely applicable at the same time. But I am clearly not managing to communicate what I mean by that. I know it's been instrumental in how I teach. I also know that no one seems to know what the heck I mean by that. 

 

2 minutes ago, wendyroo said:

My kids obviously do speak English, but from experience, I know that by the time my verbal explanations reach their brains, they sound much like the results of a game of telephone.

As I said, I'm not a huge fan of explanations, even with kids who have an easier time with language than yours do. 

 

4 minutes ago, wendyroo said:

For example, at our house, 4 years old is the year of the fraction. My goal is for my four year olds to firmly understand that a fraction is a part of a whole, that you have to be careful to split your whole into equal parts, and that the parts can then be put together to form bigger fractions. I want them to be able to recognize and create the fractions: 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, 3/4, and 4/4.

Well, as you know, I like doing fractions after I do division, so I don't do them at age 4. I'd also never define it as part of a whole, because I don't want to exclude things like 5/3 in any way. 

I know you're a good teacher, wendyroo. I'm generally just excited to share things that worked better for me than what I think of as more standard methods. But I know I'm totally failing to communicate them.

[Not_a_Number]

Just now, wendyroo said:

As a side note, I have to say that these types of comments come off as very condescending.

I said, "Trying to think about "getting a debt" is incredibly unintuitive to me. I have to ponder over those words for a long time before I can figure out what situation they would model." I am clearly saying that that is a difficult model for me, so your response that I am wrong (about what is difficult for me?), and that it is in fact easy, is grating.

Fair enough 🙂 . I'm sorry that I said it's easy. All I meant was that it's less tricky than other situations with debt, which I find REALLY hard to model. But you're right -- obviously if this model isn't intuitive, it's not going to be easy. 

[Not_a_Number]

Maybe I'm weird, but I actually really like finding new models of operations and new ways of teaching them! I was delighted when I found a new way to teach division in Liping Ma's book 😄 . I taught it to DD8 as her primary model, even though it's not my own. 

I can see that other people don't feel that way, but I swear I'm just trying to share things I've learned through arduous experimentation 🙂 . 

[Carol in Cal.]

To the OP:  

When I teach this kind of thing, I also always advise that the kiddos remember one equation that illustrates it and is easy to follow, and then they can apply it to more complex situations correctly.

For why this is so I use the number line idea.  When you multiply a positive by a positive, you go in the positive direction, toward the right, because you are adding, and that is positive.  When you multiply a positive by a negative, you turn in the negative direction, because your negative sign changes this to a subtraction type process.  Minus is subtraction.  When you multiply a negative by a negative, you change directions twice, because each minus sign changes the direction, so you end up going to the right (positive direction.). 

The thing I suggest that they remember is 2x2 is 4, 2x-2 is -4, -2x-2 is 4.  If they just write those three equations or a little number line sketch before they start to work, then they have a model to follow.

That’s the how.  The why is that multiplication is repeated addition or repeated subtraction, depending on which sign dominates.  

[Ausmumof3]

On 2/24/2021 at 3:28 AM, sweet2ndchance said:

I know you asked NotaNumber but I just wanted to add how I think of it.

Take -3 x -3.

Think of it as money. You owe 3 dollars to 3 people.

If those 3 people forgive your debt, you now have -3 debts to people. Those debts have essentially been taken away.

So your -$3 x -3 = $9. You would be in the black and not in the red.

I probably didn't explain it the best but it makes sense to me and my kids.

This is how I explained it to my kids.  It’s still sinking in for DD.

[Ausmumof3]

One other explanation I’m trying with dd this week is the scuba diver one which is similar to debt anyway but I like to give several different ways.  So scuba diva is 6m below the surface I.e -6,  he then moves so that he shortens that distance so he takes away -2 m from the -6 m and he’s now at -4.  He does it again and is now at -2.  He does it again and is at 0.  It’s still using multiplication as repeat addition (-3x-2) just using a different real life illustration.  Hoping i can make it less foggy by then.  I really like thinking of negatives and positives as directions (left right or up down) it makes it all clearer for me personally. And fits in neatly to graphing.

[The Governess]

I like thinking of negatives as opposites. So -a is the opposite of a. If that is true, then -(-a) (the opposite of negative a) would be a.
 

Going further, -a is the same as (-1)(a).
 

So (-a)(-a) would be the same as (-1)(a)(-1)(a). Using the commutative property, we could rearrange that to (-1)(-1)(a)(a), which could be rewritten as -(-a)(a). Since the opposite of -a is a, we are left with (a)(a), which will have a positive value.

Edited March 7, 2021 by lovelearnandlive
Spacing

[Roadrunner]

15 minutes ago, lovelearnandlive said:

I like thinking of negatives as opposites. So -a is the opposite of a. If that is true, then -(-a) (the opposite of negative a) would be a.
 

Going further, -a is the same as (-1)(a).
 

So (-a)(-a) would be the same as (-1)(a)(-1)(a). Using the commutative property, we could rearrange that to (-1)(-1)(a)(a), which could be rewritten as -(-a)(a). Since the opposite of -a is a, we are left with (a)(a), which will have a positive value.

This is the way AoPS did it in prealgebra, if I remember correctly. I like this as well. 

[Not_a_Number]

6 hours ago, Roadrunner said:

This is the way AoPS did it in prealgebra, if I remember correctly. I like this as well. 

AoPS does it as the inverse of addition, I believe: as the number you add to x to get 0. Then it uses a variety of properties to derive things. 

I didn’t find that this approach made kids solid in the operation in my class, though. Maybe if we came back to it enough, it would have, I don’t know.

Edited March 7, 2021 by Not_a_Number

[Not_a_Number]

8 hours ago, Ausmumof3 said:

This is how I explained it to my kids.  It’s still sinking in for DD.

It took us a loooong time for it to sink in. We had to model lots of scenarios to get the hang of it.