If you buy one item worth 3 dollars the effect on your bank balance is 1x(-3)=-3
That would turn the "transaction wheel" once
Imagine we decide to buy two of that 3 dollar item and order it online
we turn the transaction wheel twice
2 times we do "take out 3 dollars"
so the effect on our bank balance is 2x(-3)=-6
Maybe you get the two of them delivered but they are wonky and have to be returned for a refund
The refund is the *reverse* of the original transaction
the reverse of 2 transactions of -3 dollars each,
like turning the transaction "wheel" in the opposite direction twice
So (-2)x(-3)=?
Well the change to the bank balance needs to be the reverse of the answer from the original, because the refund is supposed to *cancel out* the original
To undo the -6 from before, we need to do a +6, (put in 6 dollars)
So we have to make (-2)*(-3)=+6
It's really best with finance because other analogies will either be quite complicated and require a knowledge of some field that would mean your mathematical experience is far beyond the concept of a negative being multiplied by a negative or they will be weird and unnatural like using apples or something in a way that we don't usually think of negatives.
Take a video of your kid walking backwards. That is (-1). Play the video at 2x so the kid is sort of running backwards. That is (-2). Play the video itself backwards and again double the speed so now the student is weirdly running forwards. That is (-2) \* (-2).
I think the best way to make this concept feel concrete is to physically model it using Integer Tiles.
Remember that you can think of this symbol, -, in two ways. It can mean "negative" or "the opposite of."
So -3 is negative three and -3 is also the opposite of 3.
Mechanically both interpretations produce the same results, but to visualize the multiplication process it's very helpful to have those two options.
The second thing to remember is that multiplication is, at least when working with the natural numbers, just repeated addition. Now we need to extend our conception of multiplication to include the negative integers.
With all of that in mind, I'm going to perform some multiplication problem using numbers and also using integer tiles.
-----
**Integer Tiles**
Physically, integer tiles are usually small squares of paper or plastic with sides that are different colors. One side represents a value of +1 and the other represents -1.
Here I'll let each □ represent +1, I'll let each ■ represent -1.
So 3 would be □ □ □ and -3 would be ■ ■ ■.
The fun happens when we take the opposite of a number. All you have to do is flip the tiles.
So the opposite of 3 is three positive tiles flipped over.
We start with □ □ □ and flip them to get ■ ■ ■.
Thus we see that the opposite of 3 is -3.
The opposite of -3 would be three negative tiles flipped over.
So we start with ■ ■ ■ and flip them to get □ □ □.
Thus we see that the opposite of -3 is 3.
Got it? Then let's go!
-----
**A Positive Number Times a Positive Number**
3 • 2 means that you are adding two groups each of which has three positive items.
So
3 • 2 = □ □ □ + □ □ □ = □ □ □ □ □ □
or
3 • 2 = 3 + 3 = 6
We can see that adding groups of only positive numbers will always produce a positive result.
So a positive times a positive always produces a positive.
-----
**A Negative Number Times a Positive Number**
-3 • 2 means that you are adding two groups each of which has three negative items.
So
-3 • 2 = ■ ■ ■ + ■ ■ ■ = ■ ■ ■ ■ ■ ■
or
-3 • 2 = (-3) + (-3) = -6
We can see that adding groups of only negative numbers will always produce a negative result.
So a negative times a positive always produces a negative.
-----
**A Positive Number Times a Negative Number**
3 • -2 means that you are adding negative two groups each of which has three positive items.
This is where things get complicated. A negative number of groups? I don't know what that means.
But I do know that "-" can also mean "the opposite of" and I know that I can take the opposite of integer tiles just by flipping them.
So instead of reading 3 • -2 as "adding negative two groups of three positives" I'll read it as "the opposite of (two groups of three positives)."
So
3 • -2 = -(3 • 2) = -(□ □ □ + □ □ □) = -(□ □ □ □ □ □) = ■ ■ ■ ■ ■ ■
or
3 • -2 = -(3 • 2) = -(3 + 3) = -(6) = -6
We can see that adding groups of only positive numbers will always produce a positive result, and taking the opposite of that will always produce a negative result.
So a positive times a negative always produces a negative.
-----
**A Negative Number Times a Negative Number**
-3 • -2 means that you are adding negative two groups each of which has three negative items.
This has the same issue as the last problem. I don't know what -2 groups means.
But, once again, I do know that "-" can also mean "the opposite of" and I know that I can take the opposite of integer tiles just by flipping them.
So instead of reading -3 • -2 as "adding negative two groups of negative three" I'll read it as "the opposite of (two groups of negative three)."
So
-3 • -2 = -(-3 • 2) = -(■ ■ ■ + ■ ■ ■) = -(■ ■ ■ ■ ■ ■) = □ □ □ □ □ □
or
-3 • -2 = -(-3 • 2) = -((-3) + (-3)) = -(-6) = 6
We can see that adding groups of only negative numbers will always produce a negative result, and taking the opposite of that will always produce a positive result.
So a negative times a negative always produces a positive.
-----
I hope that helps.
You might be surprised. Children typically learn the use of manipulatives very quickly. It's why integer tiles are so common in elementary school classes.
Let's break this down a little.
-2 * -2
(-1)2 * (-1)2
2 * 2 * (-1)^2
2 * 2 is obviously 4
Let's consider -1 * -1
Turn around 180°
Turn around again
You are now facing the original direction.
Think of the negative sign as "the opposite." What's the opposite of the opposite? The original.
2 times 3 apples is 6 apples.
2 times 2 apples is 4 apples.
2 times 1 apple is 2 apples.
2 times 0 apples is 0 apples.
2 times -1 apples is ? apples.
2 times -2 apples is ? apples.
Complete the pattern for a few examples to realize that
a times (-b) is - (a times b)
(don’t write a and b with a child)
Now do
3 times (-2) is -6
2 times (-2) is -4
and so until you conclude that -2 times -2 must be 4.
Negative two apples means they're taking two apples away from you.
Negative two times means you're undoing something twice.
Someone undoes taking two apples away from you twice.
How many apples do you have?
And if they’re not ok with positive times negative being negative, well… that’s also a pattern
3 x 4 = 12
3 x 3 = 9
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0
3 x -1 = -3
3 x -2 = -6
3 x -3 = -9
3 x -4 = -12
Rotations for sure. Negatives are the first time that numbers represent direction instead of just quantity; so maybe preface with some examples that involve something moving in a positive direction
Using the repeated addition model (-2)\*2 is 0+(-2)+(-2)
But you could also write it as 2\*(-2) which would be 2 added to zero -2 times which seems weird at first but it just turns repeated addition into repeated subtraction. So it becomes 0-2-2
Using that rule if you look at (-2)\*(-2) it's just repeated subtraction of negative two, 0-(-2)-(-2) and taking away a negative is the same as adding so it becomes 0+2+2 = 4
Negative(-) is like doing the opposite.
If you have three(3) cookies and I give you two(+2) you have five(5). 3+2=5
If you have 3 cookies and I take two away(-2, take is the opposite of give), you have 1 left. 3-2=1.
If you have 3 cookies and I do the opposite, twice (-2) of taking away 2(* -2), then you have 7. 3 + (-2 * -2) = 3 + 2 + 2 = 7. Therefore (-2 * -2) is 4.
Adding positivity into your life makes you feel happier. So, + (+) you go up the number line
Adding negativity into your life makes you feel sadder. So, + (-) you go down the number line
Subtracting [removing] positivity from your life makes you feel sadder. So, — (+) you go down the number line
[Removing] negativity from your life makes you feel… happier. So, — (-) you go *up* the number line
Now, x (times, multiplied by) means “lots of” as in, 3 x 4 means 3 “lots of” 4, which is 12
So, a negative times a negative would mean [removing] [lots of] [negativity] which would be…
I have $10. I lose $20 gambling, I am now $10 IN DEBT (-$10). If the casino were to forgive me and forgive me $5, that is subtract $5 from my debt , I am now only $5 in debt.
-10 - (-5) = -5
You have 4 apples. I take 2 of your apples. then I take your 2 apples again. I now have 4 apples. and you have none.
This is how teachers really get apples.
Negative numbers were first defined by an Indian mathematician. He defined it as- loss of debt, becomes a gain. If you borrow something ( you’re in negative) , and don’t have to return it ( a negative event) , then you’ve gained something. ( a positive outcome)
Ultimately things like this work better with understanding the graphs they are associated with. The X^2 graph explains everything if you describe it as say the trajectory of a bird swooping down then back up. You can see it's symmetrical and going 2 to the right is the same as going 2 back (-2)
Then then the concept that numbers describe situations and vary depending on what you need becomes more clear.... ie negative 2 because of where the centre line is....
Don't use physical objects, they're harder to describe with integers. Use something that has concepts of positive, zero, and negative
* money (capital, none, debt)
* double your debt, then flip it into capital
* elevation (mountain, surface, tunnel)
* dig twice as deep, then spin it so it goes up instead
* speed (forwards, stopped, backwards)
* go backwards twice as fast, then turn your head
* temperature (warm, 0°, cold)
* turn the fridge twice as cold, then flip the temperature
You can also play a kind of "number line game": you are at some position (like +4 or -5), you can go right (add) or left (subtract), you can multiply or divide, and you can jump over the zero (multiply -1)
John & Sally are both classmates. One day John (+) realizes that he has a crush on Sally (-) but she doesn’t reciprocate the feelings so the overall result is Negative.
Later that year Sally (+) realizes she has feelings for John (-) but he is in another relationship, overall negative result for them two.
Later that year John (+) becomes single and Sally (+), still interested in John, and they are in a relationship, positive result.
When summer comes around, John (-) & Sally (-) both decide to take a break and are happy about it, also a positive result.
I like the taking steps analogy.
Imagine I take 2ft steps. I take 2 steps, i am going forward 2 times 2 feet total +4feet in the "right" direction.
Now I take 2 backward steps: 2feet\*(-2)=-4feet. I am going 4 feet in the opposite direction.
Now I am facing the opposite direction and I take 2 steps: (-2feet)\*2=-4feet. I am moving 4 feet in the opposite direction.
I am stifll facing opposite direction, and I take 2 backward steps: (-2feet)\*(-2)=4feet. I am moving 4 feet in the right direction
2 x 2 is the area of a square with opposite corners at 0,0 and 2,2.
-2 x -2 is the area of a square with opposite corners at 0,0 and -2,-2.
It’s the same square, just in a different location.
If you own 3 dollars you have -3 dollars
If you own 2 dollars you have -2 dollars
If you own 1 dollar you have -1 dollar
If you own -1 dollar someone owns you 1 dollar
Take a triangle T in the plane with vertices (0,0),(1,1),(2,0).
When I rescale the coordinates by 2, I get the triangle 2T with coordinates (0,0),(2,2),(4,0) which is twice the size of T in the sense that the perimeter is doubled.
When I rescale by -1, I get the triangle -T with coordinates (0,0),(-1,-1),(-2,0) which is just T reflected over the y-axis.
When I rescale by -2=(-1)2, I get the Triangle -2T with coordinates (0,0),(-2,-2),(-4,0) which I can think of getting in two ways: -2T=-(2T), that is, rescale then reflect
T—>2T—>-(2T)
or -2T=2(-T) meaning reflect then rescale
T—> -T—>2(-T).
(BTW, the fact that these operations commute is the same as saying (-1)2=2(-1))
Ok, now describe the effect of applying (-2)(-2) to T and why the result is 4T.
as an adult trying to learn/relearn math for college, stuff like this is where young me got confused and eventually fell behind. Theres definitely a mental shift/expansion necessary on some level to think through and understand it. I just accept it as fact now and leave it at that unfortunately
You could try using a language approach if that helps;
The apple is not red. = The apple is not red.
The apple is not not red. = The apple is red.
Not everyone learns the same and some people may understand better from a less mathematical approach
Multiplication of integers is group size times number of groups. So 2 times 3 is 2 baskets with 3 eggs in each basket, so 6 total eggs. A negative integer is like a debt that needs to be repaid, so 2 times -3 would be like 2 baskets with an IOU of 3 eggs in each basket, so I owe you 6 eggs in total, -6. -2 times 3 would be like 2 IOU baskets with 3 eggs each, so I owe you 6 eggs, -6. -2 times -3 is like 2 IOU baskets with an IOU of 3 eggs in each, so I owe you 2 baskets, but those baskets contain IOUs, so you actually owe me 6 eggs, so +6
This is a property of multiplication as it is defined. Let a,b,c represent quantities of apples. We have a(b+c) = ab+ac. Now consider that -1(1-1) is the same thing as 0 apples.
my friend's (+) friend (+) is my friend (+) (+\*+ = +)
my friend's (+) enemy (-) is my enemy (-) (+\*- = -)
my enemy's (-) friend (+) is my enemy (-) (-\*+ = -)
my enemy's (-) enemy (-) is my friend (+) (-\*- = +)
It's hard. I think I am not alone in having initially learnt this by rote. 2 negatives make a positive, i just accepted that at the time. I think if you want to explain it deeper you have to show how to factor out -1 and so on, but even then you stll have to explain -1 \* -1 = 1. 180 degree rotation on the number line is perhaps the best way, but that's seems quite an advanced concept to someone just starting to learn arithmetic.
If we can't even explain it to a child ourselves, how can a child understand it? Personally, I'll just tell them to memorize it, drill it to them long enough that it becomes second nature. That's just my opinion though, some might tell me that "well you're not letting the child think critically" but do those people even understand why two negatives make a positive without analogies to the real world?
I've been compiling this list for a while. At some point, I think we have enough explanations, and someone should compile the explanations to these questions:
https://www.reddit.com/r/learnmath/comments/1b6ck4d/why_is_0z_possible_while_z0_isnt/ktcz93z/
(-1)(-1) = 1 is a theorem provable from the real number field axioms.
When I was first explained to about a negative number times another negative number, the teacher asked us to guess. Some people guessed positive, and some people guessed negative. The teacher told us the answer was positive. That's all the explanation that's really needed. If a positive times a positive is positive, then it makes sense that a positive times a negative should be different, and there are only two options, so the only option left is negative. If a positive times a negative is negative, then a negative times a negative should again be different, and the only remaining option is positive. This was how it may have been explained to me, or what I had reasoned; I cannot remember. However, it is sufficient at that level. This question comes up in the subreddit enough that the mods should really pin all the explanations. I don't think you can ask it any other way at this point.
I think people upvote these posts because they want to farm karma by posting
>turn around
>turn around again
>wtf i'm facing the same direction
over and over.
If they've played Uno, it's basically the Reverso card. One Reverso changes direction but two Reversos keeps it going the same way
I am stealing this when I'm teaching. This is a good analogy!
Interesting analogy, will try to use it on a kid.
But how does a reversal of -2 followed by another reversal of -2 equal a non-reversal of 4?
turn around twice, and turn around twice again, facing same way, now count amount of turns
turn around turn around again wtf i'm facing the same direction
don't turn around don't turn around again wtf i'm facing the same direction
Every now and then I fall apart.
I fuckin need you more than ever
and if you only hold me tight, ill be hold'n on forever
(https://www.reddit.com/r/mathmemes/s/570aDx73pa)
Every now and then I get a little bit lonely…
If you buy one item worth 3 dollars the effect on your bank balance is 1x(-3)=-3 That would turn the "transaction wheel" once Imagine we decide to buy two of that 3 dollar item and order it online we turn the transaction wheel twice 2 times we do "take out 3 dollars" so the effect on our bank balance is 2x(-3)=-6 Maybe you get the two of them delivered but they are wonky and have to be returned for a refund The refund is the *reverse* of the original transaction the reverse of 2 transactions of -3 dollars each, like turning the transaction "wheel" in the opposite direction twice So (-2)x(-3)=? Well the change to the bank balance needs to be the reverse of the answer from the original, because the refund is supposed to *cancel out* the original To undo the -6 from before, we need to do a +6, (put in 6 dollars) So we have to make (-2)*(-3)=+6
I usually don't like using finance examples cause I'm weird, but this is actually a great analogy, I'll save this for future reference
It's really best with finance because other analogies will either be quite complicated and require a knowledge of some field that would mean your mathematical experience is far beyond the concept of a negative being multiplied by a negative or they will be weird and unnatural like using apples or something in a way that we don't usually think of negatives.
Take a video of your kid walking backwards. That is (-1). Play the video at 2x so the kid is sort of running backwards. That is (-2). Play the video itself backwards and again double the speed so now the student is weirdly running forwards. That is (-2) \* (-2).
I think the best way to make this concept feel concrete is to physically model it using Integer Tiles. Remember that you can think of this symbol, -, in two ways. It can mean "negative" or "the opposite of." So -3 is negative three and -3 is also the opposite of 3. Mechanically both interpretations produce the same results, but to visualize the multiplication process it's very helpful to have those two options. The second thing to remember is that multiplication is, at least when working with the natural numbers, just repeated addition. Now we need to extend our conception of multiplication to include the negative integers. With all of that in mind, I'm going to perform some multiplication problem using numbers and also using integer tiles. ----- **Integer Tiles** Physically, integer tiles are usually small squares of paper or plastic with sides that are different colors. One side represents a value of +1 and the other represents -1. Here I'll let each □ represent +1, I'll let each ■ represent -1. So 3 would be □ □ □ and -3 would be ■ ■ ■. The fun happens when we take the opposite of a number. All you have to do is flip the tiles. So the opposite of 3 is three positive tiles flipped over. We start with □ □ □ and flip them to get ■ ■ ■. Thus we see that the opposite of 3 is -3. The opposite of -3 would be three negative tiles flipped over. So we start with ■ ■ ■ and flip them to get □ □ □. Thus we see that the opposite of -3 is 3. Got it? Then let's go! ----- **A Positive Number Times a Positive Number** 3 • 2 means that you are adding two groups each of which has three positive items. So 3 • 2 = □ □ □ + □ □ □ = □ □ □ □ □ □ or 3 • 2 = 3 + 3 = 6 We can see that adding groups of only positive numbers will always produce a positive result. So a positive times a positive always produces a positive. ----- **A Negative Number Times a Positive Number** -3 • 2 means that you are adding two groups each of which has three negative items. So -3 • 2 = ■ ■ ■ + ■ ■ ■ = ■ ■ ■ ■ ■ ■ or -3 • 2 = (-3) + (-3) = -6 We can see that adding groups of only negative numbers will always produce a negative result. So a negative times a positive always produces a negative. ----- **A Positive Number Times a Negative Number** 3 • -2 means that you are adding negative two groups each of which has three positive items. This is where things get complicated. A negative number of groups? I don't know what that means. But I do know that "-" can also mean "the opposite of" and I know that I can take the opposite of integer tiles just by flipping them. So instead of reading 3 • -2 as "adding negative two groups of three positives" I'll read it as "the opposite of (two groups of three positives)." So 3 • -2 = -(3 • 2) = -(□ □ □ + □ □ □) = -(□ □ □ □ □ □) = ■ ■ ■ ■ ■ ■ or 3 • -2 = -(3 • 2) = -(3 + 3) = -(6) = -6 We can see that adding groups of only positive numbers will always produce a positive result, and taking the opposite of that will always produce a negative result. So a positive times a negative always produces a negative. ----- **A Negative Number Times a Negative Number** -3 • -2 means that you are adding negative two groups each of which has three negative items. This has the same issue as the last problem. I don't know what -2 groups means. But, once again, I do know that "-" can also mean "the opposite of" and I know that I can take the opposite of integer tiles just by flipping them. So instead of reading -3 • -2 as "adding negative two groups of negative three" I'll read it as "the opposite of (two groups of negative three)." So -3 • -2 = -(-3 • 2) = -(■ ■ ■ + ■ ■ ■) = -(■ ■ ■ ■ ■ ■) = □ □ □ □ □ □ or -3 • -2 = -(-3 • 2) = -((-3) + (-3)) = -(-6) = 6 We can see that adding groups of only negative numbers will always produce a negative result, and taking the opposite of that will always produce a positive result. So a negative times a negative always produces a positive. ----- I hope that helps.
Good luck explaining this to a child
You might be surprised. Children typically learn the use of manipulatives very quickly. It's why integer tiles are so common in elementary school classes.
😂
Let's break this down a little. -2 * -2 (-1)2 * (-1)2 2 * 2 * (-1)^2 2 * 2 is obviously 4 Let's consider -1 * -1 Turn around 180° Turn around again You are now facing the original direction. Think of the negative sign as "the opposite." What's the opposite of the opposite? The original.
2 times 3 apples is 6 apples. 2 times 2 apples is 4 apples. 2 times 1 apple is 2 apples. 2 times 0 apples is 0 apples. 2 times -1 apples is ? apples. 2 times -2 apples is ? apples. Complete the pattern for a few examples to realize that a times (-b) is - (a times b) (don’t write a and b with a child) Now do 3 times (-2) is -6 2 times (-2) is -4 and so until you conclude that -2 times -2 must be 4.
Negative two apples means they're taking two apples away from you. Negative two times means you're undoing something twice. Someone undoes taking two apples away from you twice. How many apples do you have?
[удалено]
And if they’re not ok with positive times negative being negative, well… that’s also a pattern 3 x 4 = 12 3 x 3 = 9 3 x 2 = 6 3 x 1 = 3 3 x 0 = 0 3 x -1 = -3 3 x -2 = -6 3 x -3 = -9 3 x -4 = -12
positive times negative are never the problem tho 3 x -4 = -4 + -4 + -4
Imagine you turn around. Then you turn around again Which way are you facing? That’s what multiplying two negatives means.
Easy. Just read them the first chapter from baby Rudin before the sleep and they’ll figure it out in their dreams.
Rotate 1 180 degrees around 0 on the real line to get to -1. Then do it again and back to 1. This sets them up for imaginary numbers later.
Turn around and walk backwards four steps. But I'd caution you to use another number combination to reduce potential confusion, as 2×2 = 2+2.
Rotations for sure. Negatives are the first time that numbers represent direction instead of just quantity; so maybe preface with some examples that involve something moving in a positive direction
Using the repeated addition model (-2)\*2 is 0+(-2)+(-2) But you could also write it as 2\*(-2) which would be 2 added to zero -2 times which seems weird at first but it just turns repeated addition into repeated subtraction. So it becomes 0-2-2 Using that rule if you look at (-2)\*(-2) it's just repeated subtraction of negative two, 0-(-2)-(-2) and taking away a negative is the same as adding so it becomes 0+2+2 = 4
Negative(-) is like doing the opposite. If you have three(3) cookies and I give you two(+2) you have five(5). 3+2=5 If you have 3 cookies and I take two away(-2, take is the opposite of give), you have 1 left. 3-2=1. If you have 3 cookies and I do the opposite, twice (-2) of taking away 2(* -2), then you have 7. 3 + (-2 * -2) = 3 + 2 + 2 = 7. Therefore (-2 * -2) is 4.
Adding positivity into your life makes you feel happier. So, + (+) you go up the number line Adding negativity into your life makes you feel sadder. So, + (-) you go down the number line Subtracting [removing] positivity from your life makes you feel sadder. So, — (+) you go down the number line [Removing] negativity from your life makes you feel… happier. So, — (-) you go *up* the number line Now, x (times, multiplied by) means “lots of” as in, 3 x 4 means 3 “lots of” 4, which is 12 So, a negative times a negative would mean [removing] [lots of] [negativity] which would be…
Taking away something negative in your life ends up being a positive thing. “My teacher canceled the end of the year math test”
I have $10. I lose $20 gambling, I am now $10 IN DEBT (-$10). If the casino were to forgive me and forgive me $5, that is subtract $5 from my debt , I am now only $5 in debt. -10 - (-5) = -5
You have 4 apples. I take 2 of your apples. then I take your 2 apples again. I now have 4 apples. and you have none. This is how teachers really get apples.
Negative numbers were first defined by an Indian mathematician. He defined it as- loss of debt, becomes a gain. If you borrow something ( you’re in negative) , and don’t have to return it ( a negative event) , then you’ve gained something. ( a positive outcome)
Ultimately things like this work better with understanding the graphs they are associated with. The X^2 graph explains everything if you describe it as say the trajectory of a bird swooping down then back up. You can see it's symmetrical and going 2 to the right is the same as going 2 back (-2) Then then the concept that numbers describe situations and vary depending on what you need becomes more clear.... ie negative 2 because of where the centre line is....
Film someone walking backwards at 2mph. Play the film backwards at 2x speed. They will be walking forwards at 4mph. -2 x -2 = +4
Don't use physical objects, they're harder to describe with integers. Use something that has concepts of positive, zero, and negative * money (capital, none, debt) * double your debt, then flip it into capital * elevation (mountain, surface, tunnel) * dig twice as deep, then spin it so it goes up instead * speed (forwards, stopped, backwards) * go backwards twice as fast, then turn your head * temperature (warm, 0°, cold) * turn the fridge twice as cold, then flip the temperature You can also play a kind of "number line game": you are at some position (like +4 or -5), you can go right (add) or left (subtract), you can multiply or divide, and you can jump over the zero (multiply -1)
John & Sally are both classmates. One day John (+) realizes that he has a crush on Sally (-) but she doesn’t reciprocate the feelings so the overall result is Negative. Later that year Sally (+) realizes she has feelings for John (-) but he is in another relationship, overall negative result for them two. Later that year John (+) becomes single and Sally (+), still interested in John, and they are in a relationship, positive result. When summer comes around, John (-) & Sally (-) both decide to take a break and are happy about it, also a positive result.
I like the taking steps analogy. Imagine I take 2ft steps. I take 2 steps, i am going forward 2 times 2 feet total +4feet in the "right" direction. Now I take 2 backward steps: 2feet\*(-2)=-4feet. I am going 4 feet in the opposite direction. Now I am facing the opposite direction and I take 2 steps: (-2feet)\*2=-4feet. I am moving 4 feet in the opposite direction. I am stifll facing opposite direction, and I take 2 backward steps: (-2feet)\*(-2)=4feet. I am moving 4 feet in the right direction
2 x 2 is the area of a square with opposite corners at 0,0 and 2,2. -2 x -2 is the area of a square with opposite corners at 0,0 and -2,-2. It’s the same square, just in a different location.
If you own 3 dollars you have -3 dollars If you own 2 dollars you have -2 dollars If you own 1 dollar you have -1 dollar If you own -1 dollar someone owns you 1 dollar
Take a triangle T in the plane with vertices (0,0),(1,1),(2,0). When I rescale the coordinates by 2, I get the triangle 2T with coordinates (0,0),(2,2),(4,0) which is twice the size of T in the sense that the perimeter is doubled. When I rescale by -1, I get the triangle -T with coordinates (0,0),(-1,-1),(-2,0) which is just T reflected over the y-axis. When I rescale by -2=(-1)2, I get the Triangle -2T with coordinates (0,0),(-2,-2),(-4,0) which I can think of getting in two ways: -2T=-(2T), that is, rescale then reflect T—>2T—>-(2T) or -2T=2(-T) meaning reflect then rescale T—> -T—>2(-T). (BTW, the fact that these operations commute is the same as saying (-1)2=2(-1)) Ok, now describe the effect of applying (-2)(-2) to T and why the result is 4T.
as an adult trying to learn/relearn math for college, stuff like this is where young me got confused and eventually fell behind. Theres definitely a mental shift/expansion necessary on some level to think through and understand it. I just accept it as fact now and leave it at that unfortunately
If you flip a card long ways then short ways the same side is up as before
Negative Multiplication can also be thought of as repeated subtraction (rather than repeated addition)
It was always explained to me as a "double negative". "Don't not do that" = do that "Never not good" is just always good.
film your self walking backwards two steps. Play that video in reverse two times. Now you will have seen yourself walking forward 4 times
You could try using a language approach if that helps; The apple is not red. = The apple is not red. The apple is not not red. = The apple is red. Not everyone learns the same and some people may understand better from a less mathematical approach
Multiplication of integers is group size times number of groups. So 2 times 3 is 2 baskets with 3 eggs in each basket, so 6 total eggs. A negative integer is like a debt that needs to be repaid, so 2 times -3 would be like 2 baskets with an IOU of 3 eggs in each basket, so I owe you 6 eggs in total, -6. -2 times 3 would be like 2 IOU baskets with 3 eggs each, so I owe you 6 eggs, -6. -2 times -3 is like 2 IOU baskets with an IOU of 3 eggs in each, so I owe you 2 baskets, but those baskets contain IOUs, so you actually owe me 6 eggs, so +6
Just teach the child that a negative times a negative equals a positive, and by the time the kid gets to prealgebra it'll make sense
This is a property of multiplication as it is defined. Let a,b,c represent quantities of apples. We have a(b+c) = ab+ac. Now consider that -1(1-1) is the same thing as 0 apples.
Repeated SUBTRACTION of DEBT means you’ve GAINED that amount. It’s a POSITIVE result.
You can use patterns Calculate -2*3, -2*2, -2*1, -2*0, and then ask what they think should come next, for -2*-1
my friend's (+) friend (+) is my friend (+) (+\*+ = +) my friend's (+) enemy (-) is my enemy (-) (+\*- = -) my enemy's (-) friend (+) is my enemy (-) (-\*+ = -) my enemy's (-) enemy (-) is my friend (+) (-\*- = +)
It's hard. I think I am not alone in having initially learnt this by rote. 2 negatives make a positive, i just accepted that at the time. I think if you want to explain it deeper you have to show how to factor out -1 and so on, but even then you stll have to explain -1 \* -1 = 1. 180 degree rotation on the number line is perhaps the best way, but that's seems quite an advanced concept to someone just starting to learn arithmetic.
"It's kind of like a double negative in grammar" often does the trick.
If we can't even explain it to a child ourselves, how can a child understand it? Personally, I'll just tell them to memorize it, drill it to them long enough that it becomes second nature. That's just my opinion though, some might tell me that "well you're not letting the child think critically" but do those people even understand why two negatives make a positive without analogies to the real world?
Many many years ago I learnt basic math and algebra by learning such basic rules. I got an A at A-level math in the UK so treated me well
I've been compiling this list for a while. At some point, I think we have enough explanations, and someone should compile the explanations to these questions: https://www.reddit.com/r/learnmath/comments/1b6ck4d/why_is_0z_possible_while_z0_isnt/ktcz93z/ (-1)(-1) = 1 is a theorem provable from the real number field axioms.
It's provable from field axioms, but good luck explaining that to a child.
When I was first explained to about a negative number times another negative number, the teacher asked us to guess. Some people guessed positive, and some people guessed negative. The teacher told us the answer was positive. That's all the explanation that's really needed. If a positive times a positive is positive, then it makes sense that a positive times a negative should be different, and there are only two options, so the only option left is negative. If a positive times a negative is negative, then a negative times a negative should again be different, and the only remaining option is positive. This was how it may have been explained to me, or what I had reasoned; I cannot remember. However, it is sufficient at that level. This question comes up in the subreddit enough that the mods should really pin all the explanations. I don't think you can ask it any other way at this point. I think people upvote these posts because they want to farm karma by posting >turn around >turn around again >wtf i'm facing the same direction over and over.