? re how MM explains multiplying 2 negatives

[Momto6inIN]

DD is doing MM7. We're multiplying negatives today, and I'm confused about how the text explains it. When it talked about multiplying a negative by a positive, it explained that because 8x2 means 8 sets of 2s, then 8x-2 means 8 sets of -2s. Got it. But when it got to talking about multiplying -8x-2, it just said that mathematicians have been stumped by this for a long time and the way they solved the problem was just by deciding to say that multiplying 2 negatives gives a positive because it continues the pattern in a predictable way.

 

I get where she's coming from, and the subsequent exercises demonstrate this pattern, but wouldn't it be easier to explain that the negative in front of the 8 in this case means "opposite of" (this was discussed in a previous lesson, that the minus sign can mean opposite or negative or subtraction) and "the opposite of 8x-2" is positive? This makes total sense to me and I can't figure out why the text didn't just say it that way ... which makes me think there may be something I'm missing that would make it a bad idea to explain it to my DD this way.

 

Am I right in my thinking? Or can someone correct me please? Thanks!

[...................]

Wow, if you are correct that would really help my dd and I understand it. I'll be following. :)

[Elizabeth 2]

pretty much. It has to do with the distributive property. The negative sign stands for the inverse of the positive. So

 

-8 x -2 =

-(8 x -2) =

-(-16) =

16

 

If DC understand this, then algebra will be much easier.

 

I don't like her explanation as it dumbs it down more than necessary. If a further explanation is needed, then that requires the showing of proofs.

Edited September 23, 2016 by Elizabeth 2

[wendyroo]

For negative numbers I always talk piles of dirt and holes.

I've discussed before using them for addition/subtraction, but they work for multiplication as well.

piles are one positive amount of dirt
holes are one negative amount of dirt
+ means adding something
- means taking something away

 

Addition:

-7 + 1 = -6
7 holes + 1 pile = 6 holes

Subtraction:
3 - 5 = -2
3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

Subtracting a negative:
4 - -6 = 10
4 piles take away 6 holes
How do you take away a hole?  You fill it in.  But how can we take away holes that aren't there?  You haul in a hole's worth of dirt and you put it on the ground.  What does that leave?  A PILE!!
4 piles take away 6 holes leaves you with 10 piles of dirt

 

With multiplication, remember to always visualize flat ground to start with...no piles, no holes.

 

Multiplying two positives:

3 * 5 = 15

Adding 3 groups of 5 piles each means you have made 15 piles.

 

Multiplying a positive and a negative:

4 * -2 = -8

Adding 4 groups of 2 holes each means you have made 8 holes.

 

Or

-2 * 3 = -6

Taking away 2 groups of 3 piles each...well, since the ground is flat to start with, but you still need to "take away" 6 piles worth of dirt, you will end up with 6 holes.

 

Multiplying two negatives:

-10 * -3 = 30

Taking away 10 groups of 3 holes each...well, since the ground is flat to start with, but you still need to "take away" 30 holes, you will have to haul in 30 unit of dirt and you will find yourself with 30 piles.

 

Wendy

[MotherGoose]

For negative numbers I always talk piles of dirt and holes.

I've discussed before using them for addition/subtraction, but they work for multiplication as well.

piles are one positive amount of dirt

holes are one negative amount of dirt

+ means adding something

- means taking something away

 

Addition:

-7 + 1 = -6

7 holes + 1 pile = 6 holes

 

Subtraction:

3 - 5 = -2

3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

 

Subtracting a negative:

4 - -6 = 10

4 piles take away 6 holes

How do you take away a hole? You fill it in. But how can we take away holes that aren't there? You haul in a hole's worth of dirt and you put it on the ground. What does that leave? A PILE!!

4 piles take away 6 holes leaves you with 10 piles of dirt

 

With multiplication, remember to always visualize flat ground to start with...no piles, no holes.

 

Multiplying two positives:

3 * 5 = 15

Adding 3 groups of 5 piles each means you have made 15 piles.

 

Multiplying a positive and a negative:

4 * -2 = -8

Adding 4 groups of 2 holes each means you have made 8 holes.

 

Or

-2 * 3 = -6

Taking away 2 groups of 3 piles each...well, since the ground is flat to start with, but you still need to "take away" 6 piles worth of dirt, you will end up with 6 holes.

 

Multiplying two negatives:

-10 * -3 = 30

Taking away 10 groups of 3 holes each...well, since the ground is flat to start with, but you still need to "take away" 30 holes, you will have to haul in 30 unit of dirt and you will find yourself with 30 piles.

 

Wendy

Love love love this!!

[Momto6inIN]

pretty much. It has to do with the distributive property. The negative sign stands for the inverse of the positive. So

 

-8 x -2 =

-(8 x -2) =

-(-16) =

16

 

If DC understand this, then algebra will be much easier.

 

I don't like her explanation as it dumbs it down more than necessary. If a further explanation is needed, then that requires the showing of proofs.

Thank you for verifying!

DD does understand this and it made perfect sense to her ... which made me think I must be explaining it wrong because pre-algebra has been her nemesis so far lol

[Momto6inIN]

For negative numbers I always talk piles of dirt and holes.

I've discussed before using them for addition/subtraction, but they work for multiplication as well.

piles are one positive amount of dirt

holes are one negative amount of dirt

+ means adding something

- means taking something away

 

Addition:

-7 + 1 = -6

7 holes + 1 pile = 6 holes

 

Subtraction:

3 - 5 = -2

3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

 

Subtracting a negative:

4 - -6 = 10

4 piles take away 6 holes

How do you take away a hole? You fill it in. But how can we take away holes that aren't there? You haul in a hole's worth of dirt and you put it on the ground. What does that leave? A PILE!!

4 piles take away 6 holes leaves you with 10 piles of dirt

 

With multiplication, remember to always visualize flat ground to start with...no piles, no holes.

 

Multiplying two positives:

3 * 5 = 15

Adding 3 groups of 5 piles each means you have made 15 piles.

 

Multiplying a positive and a negative:

4 * -2 = -8

Adding 4 groups of 2 holes each means you have made 8 holes.

 

Or

-2 * 3 = -6

Taking away 2 groups of 3 piles each...well, since the ground is flat to start with, but you still need to "take away" 6 piles worth of dirt, you will end up with 6 holes.

 

Multiplying two negatives:

-10 * -3 = 30

Taking away 10 groups of 3 holes each...well, since the ground is flat to start with, but you still need to "take away" 30 holes, you will have to haul in 30 unit of dirt and you will find yourself with 30 piles.

 

Wendy

Seriously, this is really great!

[kiwik]

There was a good video using Lego the other day too if you search.

[almondbutterandjelly]

pretty much. It has to do with the distributive property. The negative sign stands for the inverse of the positive. So

 

-8 x -2 =

-(8 x -2) =

-(-16) =

16

 

If DC understand this, then algebra will be much easier.

 

I don't like her explanation as it dumbs it down more than necessary. If a further explanation is needed, then that requires the showing of proofs.

Umm, that's not the distributive property, which has addition or subtraction in it.  a(b+c) = ab + ac

 

However, you are basically using the associative property.  

-8 x -2=

(-1)(8) x (-2)=

(-1)(8 x -2)=

-(-16)=

16

 

Your math is fine.  Just make sure you call it the right property or that could have unfortunate ramifications.  :)

[Catalytic]

For negative numbers I always talk piles of dirt and holes.

I've discussed before using them for addition/subtraction, but they work for multiplication as well.

piles are one positive amount of dirt

holes are one negative amount of dirt

+ means adding something

- means taking something away

 

Addition:

-7 + 1 = -6

7 holes + 1 pile = 6 holes

 

Subtraction:

3 - 5 = -2

3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

 

Subtracting a negative:

4 - -6 = 10

4 piles take away 6 holes

How do you take away a hole?  You fill it in.  But how can we take away holes that aren't there?  You haul in a hole's worth of dirt and you put it on the ground.  What does that leave?  A PILE!!

4 piles take away 6 holes leaves you with 10 piles of dirt

 

With multiplication, remember to always visualize flat ground to start with...no piles, no holes.

 

Multiplying two positives:

3 * 5 = 15

Adding 3 groups of 5 piles each means you have made 15 piles.

 

Multiplying a positive and a negative:

4 * -2 = -8

Adding 4 groups of 2 holes each means you have made 8 holes.

 

Or

-2 * 3 = -6

Taking away 2 groups of 3 piles each...well, since the ground is flat to start with, but you still need to "take away" 6 piles worth of dirt, you will end up with 6 holes.

 

Multiplying two negatives:

-10 * -3 = 30

Taking away 10 groups of 3 holes each...well, since the ground is flat to start with, but you still need to "take away" 30 holes, you will have to haul in 30 unit of dirt and you will find yourself with 30 piles.

 

Wendy

 

Could you come tutor me through college algebra, please?  That's such an awesome way to explain it!

[Julie Smith]

For negative numbers I always talk piles of dirt and holes.

I've discussed before using them for addition/subtraction, but they work for multiplication as well.

piles are one positive amount of dirt

holes are one negative amount of dirt

+ means adding something

- means taking something away

 

Addition:

-7 + 1 = -6

7 holes + 1 pile = 6 holes

 

Subtraction:

3 - 5 = -2

3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

 

Subtracting a negative:

4 - -6 = 10

4 piles take away 6 holes

How do you take away a hole? You fill it in. But how can we take away holes that aren't there? You haul in a hole's worth of dirt and you put it on the ground. What does that leave? A PILE!!

4 piles take away 6 holes leaves you with 10 piles of dirt

 

With multiplication, remember to always visualize flat ground to start with...no piles, no holes.

 

Multiplying two positives:

3 * 5 = 15

Adding 3 groups of 5 piles each means you have made 15 piles.

 

Multiplying a positive and a negative:

4 * -2 = -8

Adding 4 groups of 2 holes each means you have made 8 holes.

 

Or

-2 * 3 = -6

Taking away 2 groups of 3 piles each...well, since the ground is flat to start with, but you still need to "take away" 6 piles worth of dirt, you will end up with 6 holes.

 

Multiplying two negatives:

-10 * -3 = 30

Taking away 10 groups of 3 holes each...well, since the ground is flat to start with, but you still need to "take away" 30 holes, you will have to haul in 30 unit of dirt and you will find yourself with 30 piles.

 

Wendy

Thank you Wendy. I can see pikes and holes being very useful when I get to this concept with Youngest.

[Janeway]

DD is doing MM7. We're multiplying negatives today, and I'm confused about how the text explains it. When it talked about multiplying a negative by a positive, it explained that because 8x2 means 8 sets of 2s, then 8x-2 means 8 sets of -2s. Got it. But when it got to talking about multiplying -8x-2, it just said that mathematicians have been stumped by this for a long time and the way they solved the problem was just by deciding to say that multiplying 2 negatives gives a positive because it continues the pattern in a predictable way.

 

I get where she's coming from, and the subsequent exercises demonstrate this pattern, but wouldn't it be easier to explain that the negative in front of the 8 in this case means "opposite of" (this was discussed in a previous lesson, that the minus sign can mean opposite or negative or subtraction) and "the opposite of 8x-2" is positive? This makes total sense to me and I can't figure out why the text didn't just say it that way ... which makes me think there may be something I'm missing that would make it a bad idea to explain it to my DD this way.

 

Am I right in my thinking? Or can someone correct me please? Thanks!

I explain the negatives as a debt.

[Momto6inIN]

I explain the negatives as a debt.

Yes, but how do you multiply 2 debts and come out with a positive amount? Lol

That's what I was having trouble explaining to her.

[wendyroo]

Yes, but how do you multiply 2 debts and come out with a positive amount? Lol

That's what I was having trouble explaining to her.

 

You don't because you don't typically multiply debt x debt.

 

Think about inches.  If you multiple 3 inches x 2 inches you end up with 6 SQUARE inches.  That is very different than multiplying 3 (unit-less) pieces of ribbon that are each 2 inches long and ending up with 6 inches of ribbon.

 

So instead of thinking about multiplying two debts, think about multiplying a negative number of groups (unit-less) by the amount of debt in each group.

 

So, let's start with positives.  Your friend gives you 3 bills each of which are worth $5.  You are now $15 dollars richer.

3 (positive because the thing is coming to you) * $5 = +$15 for you

 

Now, you give your friend 2 bills each worth $4.  You are now $8 poorer.

-2 (negative because the thing is leaving your possession) * $4 = -$8 for you

 

But now, instead of giving bills back and forth, you are passing debt back and forth.  I always picture this debt as slips of paper that say, "I owe friend $X".

 

So, let's say friend lets you eat 3 of her cupcakes.  Each cupcake is worth $1, so you now have three new slips of paper that say "I owe friend $1".  You are $3 in debt to friend...you theoretically have $3 less to your name than you did before you ate the cupcakes.

3 (positive because the thing is coming to you) * -$1 (negative because these are debts) = -$3 for you

 

Now you let friend eat 2 of your slices of pizza which are each worth $4.  That means 2 new $4 debts have LEFT your possession.  Friend now has 2 pieces of paper that say she owes you $4.

-2 (negative because the thing is leaving your possession) * -$4 (negative because these are debts) = +$8 for you.  

Your friend is $8 in debt to you...you theoretically have $8 more to your name than you did before friend ate the pizza.

 

Wendy

[eternalsummer]

I like the holes explanation.  You must have posted it before because I used it last week to explain negatives to DD11.  We are both visual people so it made sense to her that way.  With DH, who is more abstract/conceptual, the debts way of seeing it makes more sense.

 

Possibly this is why I can't understand economics :)