[email protected]

This looks long because I painstakingly wrote out everything - didn't leave
any gaps. So you have to read each line - don't skim. Please try it - I
really would appreciate it MOST if some of you who don't think they
understand why a negative times a negative is a positive would give this
example a real go. I made it up and I want to know if it helps people "get
it." I'm not claiming you can "get it" without an effort - I'm asking you to
do me the favor of making an effort - I don't think you'll find it terribly
painful, the whole point is that I think this is a pretty clear example,
bettter than others I've come across or made up myself. (I HOPE I don't have
any typos - that would be really discouraging - I've proofread - but if
someone catches one, please let me know FAST so I can fix quickly.)

If you have kids who are at the right stage to maybe try to follow this, that
would be GREAT too, if they'd "review it" for me either by reading it
themselves or by you presenting the examples to them.

--pam

Question: Why do you get a positive answer when you multiply a negative
number times a negative number?

Answer:
The reason positive and negative multiplying and dividing are so hard to
intuitively grasp is that we have to give SOME meaning to the ideas of
positive and negative for them to make sense. You don't usually have to
consciously do this for just positives because you do it automatically in
your head. If I ask you, "What is 2 times 30?" You can say it is 60. You
don't have to THINK about what the positives mean -- you don't stop and think
about it as +2 times +30 equals +60, because you are comfortable with
positive numbers meaning - it could mean that you received two payments of 30
dollars each so you got 60 dollars or they could mean things like a car
drives 2 hours at 30 miles an hour so it has gone 60 miles. These are
meanings that just make sense - you very likely had some experience with such
meanings before you ever learned to multiply. You KNEW that if 3 people gave
you 2 cookies each that you'd end up with 6 cookies, probably before you
learned to multiply. So you had some "hooks" to use when you learned to
multiply positive numbers together. Not so with negative numbers - you very
possibly didn't have any automatic hooks for those.

When you work, in REAL life, with positive AND negative numbers, the meanings
will be clear. But when you're just doing pure "math" problems, there is a
tendency to try to do it without any context - with no meaning attached - and
that is too hard, at first. You can do it later, when you're CONVINCED
already that there are ways for it to mean something real.

So - you have to give meaning to positive and negative.

For example - imagine a car sitting on a long east/west road. Draw a
horizontal line on a piece of paper. Put a mark in the middle of the line -
that's where you're standing, watching this car. The east is to your right
and the west is to your left. You might want to use a toy car for this. Sit
it on that marked spot in the center of your line.

We're going to multiple two signed numbers.

The sign of the first number indicates which direction the car is facing. If
it is facing east, we will give the first number a "+" sign and if the car is
facing west we'll give the first number a "-" sign. So - the meaning of the
sign on the first number is whether the car itself is facing east or west.

The value of the first number is going to tell us what the speed of the car
is when it takes off (we'll pretend it is at a constant speed - not worry
about the time it takes to accelerate to that speed).

So if the first number is +30, then it means the car is facing east and that
its speed is 30 miles per hour. If the first number is -30, it means the car
is facing west and that its speed is 30 miles per hour.

The sign of the second number indicates whether the car is in drive or in
reverse. If the sign is positive, it means the car is in "drive" and if the
sign is negative, it means the car is in "reverse."

The value of the second number is how many hours the car goes (at the speed
given by the first number).

So if the second number is -2, then it means that the car is in reverse and
it travels for 2 hours. If the second number is, +2, then the car is in drive
and it travels for 2 hours.

Suppose we want to multiply +30 times +2.

The "+" in the first number means the car is facing east (to the right). The
value means that the car is traveling 30 mph.

The + in the second number means the car is in "drive". And the value of 2
means that it travels for 2 hours.

Okay - make your toy car face east (to the right). It is in "drive" so when
it starts moving it is going to also MOVE to the east - the way it is facing.
And it is going to go 30 mph for 2 hours so it will end up 60 miles to the
east.

So our answer is going to be +60. The "+" in the answer tells us that the car
ends up to the east of its starting point. The value 60 tells us how many
miles from the starting point.

*************
Suppose we want to multiply -30 times +2.

The "-" in the first number means the car is facing west (to the left). The
value means that the car is traveling 30 mph.

The + in the second number means the car is in "drive". And the value of 2
means that it travels for 2 hours.

Okay - make your toy car face west (to the left). It is in "drive" so when it
starts moving it is going to also MOVE to the west- the way it is facing. And
it is going to go 30 mph for 2 hours so it will end up 60 miles to the west.

So our answer is going to be -60. The "-" in the answer tells us that the car
ends up to the west (left) of its starting point. The value 60 tells us how
many miles from the starting point.

***************
Suppose we want to multiply +30 times -2.

The "+" in the first number means the car is facing east (to the right). The
value means that the car is traveling 30 mph.

The "-" in the second number means the car's transmission is in "reverse".
And the value of 2 means that it travels for 2 hours.

Okay - make your toy car face east (to the right). It is in "reverse" so when
it starts moving it is going to MOVE the opposite of the way it is facing -
to the west (left). And it is going to go 30 mph for 2 hours so it will end
up 60 miles to the west.

So our answer is going to be -60. The "-" in the answer tells us that the car
ends up to the west (left) of its starting point. The value 60 tells us how
many miles from the starting point.

*******************

And now the moment you've all been waiting for <G>.

Suppose we want to multiply -30 times -2.

The "-" in the first number means the car is facing west (to the left). The
value means that the car is traveling 30 mph.

The "-" in the second number means the car's transmission is in "reverse".
And the value of 2 means that it travels for 2 hours.

Okay - make your toy car face west (to the left). It is in "reverse" so when
it starts moving it is going to MOVE the opposite of the way it is facing.
Since it is facing west and it is reverse, it'll MOVE to the east. And it is
going to go 30 mph for 2 hours so it will end up 60 miles to the east.

So our answer is going to be +60. The "+" in the answer tells us that the car
ends up to the east (right) of its starting point. The value 60 tells us how
many miles from the starting point.

*******************

So here is ONE example where the signs of the numbers have some meaning and
so the sign of the answer has a meaning too. There are other similar kinds of
examples. If you FOOLED with enough of them BEFORE you had to actually learn
the paper-and-pencil computational skill - then it wouldn't be that hard - it
would make sense and slip into place in your brain. It is harder to do it
after you've got it into your head that positives and negatives are
confusing.

Fetteroll

on 5/6/02 3:14 PM, PSoroosh@... at PSoroosh@... wrote:

I thought that was *very* good! :-)

> But when you're just doing pure "math" problems, there is a
> tendency to try to do it without any context - with no meaning attached - and
> that is too hard, at first.

Maybe: But when you're just doing textbook math with no context, the meaning
is hard to figure out.

> we'll pretend it is at a constant speed - not worry
> about the time it takes to accelerate to that speed

constant speed and not worry about the time

or

constant speed don't worry about the time

> The sign of the second number indicates whether the car is in drive or in
> reverse. If the sign is positive, it means the car is in "drive" and if the
> sign is negative, it means the car is in "reverse."
>
> The value of the second number is how many hours the car goes (at the speed
> given by the first number).

Why would the sign for drive or reverse be attached to the hours it travels?

Joyce

[email protected]

In a message dated 5/6/2002 2:42:06 PM Pacific Daylight Time,
fetteroll@... writes:


> I thought that was *very* good! :-)
>
> > But when you're just doing pure "math" problems, there is a
> > tendency to try to do it without any context - with no meaning attached -
> and
> > that is too hard, at first.
>
> Maybe: But when you're just doing textbook math with no context, the
> meaning
> is hard to figure out.
>

Impossible. Unless you done so much stuff with positives and negatives
BEFOREHAND, that it just happens that you can think of a meaning for a
problem. I doubt there are many of us who got that opportunity.

> > we'll pretend it is at a constant speed - not worry
> > about the time it takes to accelerate to that speed
>
> constant speed and not worry about the time
>
> or
>
> constant speed don't worry about the time
>

I'm not understanding. What I meant was that if the car is starting out, if
it isn't moving and then is moving, it'll take a few seconds to get up to 30
mph so, in reality it won't be going 30 mph for the whole 2 hours. It'll be
for 2 hours minus a few seconds. So I was saying that we should just ignore
those few seconds and pretend the car is going the full 30 mph right from the
get-go.

> > The sign of the second number indicates whether the car is in drive or in
> > reverse. If the sign is positive, it means the car is in "drive" and if
> the
> > sign is negative, it means the car is in "reverse."
> >
> > The value of the second number is how many hours the car goes (at the
> speed
> > given by the first number).
>
> Why would the sign for drive or reverse be attached to the hours it
> travels?

It is arbitrary. I could say that it is because they go "in reverse" or "in
drive" for that many hours.

The sign of the first number is which way the car is facing. Again, you could
ask why that would be attached to how fast it moves? It is arbitrary. I could
say that it is how fast it is going while facing in that direction, though.

BUT - I could switch the meanings and it would still make just as much sense.
For those with the patience - I'll do it:

********************


So - you have to give meaning to positive and negative.

For example - imagine a car sitting on a long east/west road. Draw a
horizontal line on a piece of paper. Put a mark in the middle of the line -
that's where you're standing, watching this car. The east is to your right
and the west is to your left. You might want to use a toy car for this. Sit
it on that marked spot in the center of your line.

We're going to multiple two signed numbers.

The sign of the first number indicates whether the car is in drive or in
reverse. If the sign is positive, it means the car is in "drive" and if the
sign is negative, it means the car is in "reverse."

The value of the first number is going to tell us what the speed of the car
is when it takes off (we'll pretend it is at a constant speed - not worry
about the time it takes to accelerate to that speed).

So if the first number is +30, then it means the car is in drive and that
its speed is 30 miles per hour. If the first number is -30, it means the car
is in reverse and that its speed is 30 miles per hour.

The sign of the second number indicates which direction the car is facing. If

it is facing east, we will give the second number a "+" sign and if the car
is
facing west we'll give the second number a "-" sign. So - the meaning of the
sign on the second number is whether the car itself is facing east or west.

The value of the second number is how many hours the car goes (at the speed
given by the first number).

So if the second number is -2, then it means that the car is facing west and
it travels for 2 hours. If the second number is, +2, then the car is facing
east
and it travels for 2 hours.

Suppose we want to multiply +30 times +2.

The "+" in the first number means the car is in drive. The
value means that the car is traveling 30 mph.

The + in the second number means the car is facing east. And the value of 2
means that it travels for 2 hours.

Okay - make your toy car face east (to the right). It is in "drive" so when
it starts moving it is going to also MOVE to the east - the way it is facing.

And it is going to go 30 mph for 2 hours so it will end up 60 miles to the
east.

So our answer is going to be +60. The "+" in the answer tells us that the car

ends up to the east of its starting point. The value 60 tells us how many
miles from the starting point.

*************

and so on --- I switched the meanings of the "signs" of the two numbers --
but it still works out sensibly.

--pamS


[Non-text portions of this message have been removed]

Fetteroll

on 5/6/02 6:28 PM, PSoroosh@... at PSoroosh@... wrote:

>>> we'll pretend it is at a constant speed - not worry
>>> about the time it takes to accelerate to that speed
>>
>> constant speed and not worry about the time
>>
>> or
>>
>> constant speed don't worry about the time
>>
>
> I'm not understanding.

I was getting lazy and didn't write the whole thing out. And those weren't
as understandable as:

we'll pretend it is at a constant speed - we won't worry
about the time it takes to accelerate to that speed

The implied "will not" after the dash seemed to need to be explicit.

>> Why would the sign for drive or reverse be attached to the hours it
>> travels?
>
> It is arbitrary. I could say that it is because they go "in reverse" or "in
> drive" for that many hours.

Well, to *me* it makes perfect sense. I can "see" the math from your example
and why multiplying two negatives makes a positive. Perhaps because I've
done vector math and it reminded me of it?

But it seemed like the type of question that a nonmathie would wonder. The
direction the car faces seems clearly related to the car. But forward and
reverse also seem clearly related to the car. But attaching the negative to
time makes it seem like going back in time. Why would one direction be given
to the car and the other given to time? Why not both to the car?

But don't worry about it. Wait to see if a nonmathie actually does ask the
question!

Joyce

[email protected]

In a message dated 5/6/2002 4:21:31 PM Pacific Daylight Time,
fetteroll@... writes:


> But attaching the negative to
> time makes it seem like going back in time. Why would one direction be
> given
> to the car and the other given to time? Why not both to the car?

Aha -- I have another example that does that instead -- makes one of the
numbers going forward or back in time. Maybe that would be more clear. I'll
write that one up later, too, and see what you guys think. I didn't think it
was as good since going back in time is pretend <G>.

--pam


[Non-text portions of this message have been removed]

joanna514

--- In Unschooling-dotcom@y..., PSoroosh@a... wrote:
> This looks long because I painstakingly wrote out everything -
didn't leave
> any gaps. So you have to read each line - don't skim. Please try
it - I
> really would appreciate it MOST if some of you who don't think they
> understand why a negative times a negative is a positive would give
this
> example a real go. I made it up and I want to know if it helps
people "get
> it." I'm not claiming you can "get it" without an effort - I'm
asking you to
> do me the favor of making an effort - I don't think you'll find it
terribly
> painful, the whole point is that I think this is a pretty clear
example,
> bettter than others I've come across or made up myself. (I HOPE I
don't have
> any typos - that would be really discouraging - I've proofread -
but if
> someone catches one, please let me know FAST so I can fix quickly.)

Very clear.
I didn't even need the toy car!
When I first (lamely) tried it, I tried to think of the negative
numbers you get when playing cards(like rummy when you have them left
in your hand) and I couldn't put multiplying them into any context.



>
> We're going to multiple two signed numbers.>>>
>

This was the only typo I saw. Shouldn't it be "multiplY"

Thanks for the math lesson!
Joanna

[email protected]

In a message dated 5/7/2002 6:16:09 AM Pacific Daylight Time,
Wilkinson6@... writes:


> >
> > We're going to multiple two signed numbers.>>>
> >
>
> This was the only typo I saw. Shouldn't it be "multiplY"
>
> Thanks for the math lesson!

Woops - thank you!!

--PamS


[Non-text portions of this message have been removed]