A query was posted by Madiraju on Mathematics24x7
when i was teaching in algebra class one of my students got the doubt how the product of two negatives will be positive?
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Senad said...
You could use basic reasoning and explain this :
For example, you are probably familiar with this one:
" The enemy(-) of my enemy(-) is my friend(+) "
" The enemy(-) of my friend(+) is my enemy (-)"
" A friend(+) of my enemy(-) is also my enemy(-)"
etc ...
But we all know it is a convention ( -1 ) * (-1) = 1 ...
Whit Ford said...
I prefer an expanded version of the (brief) number line explanation that was given:
- Multiplication represents repeated addition, so (3) x (5) represents three fives added together
- To represent this on the number line, I start at zero then move the the right by five units three times. I will land at (+15).
- If a negative sign is involved, it turns addition into subtraction... which would cause me to move in the other direction on the number line.
- To represent (3) x (-5) on the number line, I need to "add negative five" three times which is the same thing as "subtracting five three times", so I will start at zero, and move to the left by five units three times. I will land at (-15)
- To represent (-3) x (-5) on the number line, I need to "subtract negative five" three times. This reverses the previous example. I still start at zero facing left to move by negative five, but the (-3) means that instead of moving the way I am facing three times, I move backwards... to the right by five units three times, landing at (+15).
- The essence of this explanation could be turned into a physical demonstration in the classroom: Have someone stand up facing "forward". Ask them to walk forward for three steps = (3) x (steps). Ask them to, still facing forward, walk backwards for three steps = (3) x (- steps). Now ask them to face "backwards" and walk "backwards" for three steps = (-3) x (- steps). They end up having moved in the forward direction, and the rule about double negatives logically cancelling each other out in an English sentence should come to mind: they did not not move forward.
Senad said...
Well even though I'm not a teacher, and I even agree with what you said, I need to disapprove the second method (walking).
The best way I think is to go with my example, it is a basic principle that even a 5 year old would understand. It requires no previos knowlage, and you don't have to walk a mile to understand it. Certanlly, as a student- I appreciate correct methods that will produce best possible results in small amount of time.
For example, if (-1) * (-1 ) would be ( -1 ), if that was a convention, ( " The enemy of my enemy is my enemy " ) would be in contradiction to all things known to man, even politics doesn't work that way :). And try explaining distribution law, using your mentioned " walk " method... Or you coud try to explaing in what cases, (-1)^n would result in 1, or in (-1).
Whit Ford said...
I cited the walking example as a way to try help someone understand *why* a double negative is a positive. The rules you cite certainly are often used in life, but if a young student is questioning why the rules are true in the first place... well, that can be a tough question to answer to their satisfaction. I cited the walking example in an attempt to connect very concrete initial steps to an abstraction in the later steps: backwards. But hopefully students have enough personal life experience with "backwards" to have a decent and growing intiutitive sense of what it means and can stand for.
Once a student accepts that the product of two negatives is a positive, and has been working successfully with it for a while with simple multiplication problems, I would abandon attempting to connect increasingly complex operations, like exponentiation, with the number line or physical examples - and base explanations on the rules of multiplication that the students have acquired a comfort with from personal experience as you advocate.
I explain the sign of (-2)^n to students studying the laws of exponents by telling them to count the number of negative signs that will be in the final product (n), and recall that a double negative makes a positive. Therefore, I can use the associative property of multiplication to group the repeated multiplication problem into pairs of negative signs which cancel one another out. Since 15 or 16 year old students should be very comfortable with the concepts of odd and even by now, they are usually quite comfortable with visualizing pairs of negative signs cancelling each other out, therefore if n is even, the answer is positive... and if n is odd, the answer will be negative.
I find it helpful to explain multiplication as repeated addition, and exponentiation as repeated multiplication. This implies that exponentiation is repeated-repeated addition... the contemplation of which gives most people a headache! While this explanation is true, I agree that trying to take a framework too far in a completely concrete way can cause more difficulties than it solves. Does that mean the initial explanation should not be used because it will not extend readily (or perfectly) to all arithmetic operations? I think it has value nevertheless... because it helps a student to gain a strong conceptual understanding of the nature of a fundamental operation, one which they will depend on as they learn about increasingly complex functions which depend on the operations they first learned in grade school.
Carolyn said...
One of the visual examples I use for algebra students is on the two dimensional coordinate plane. If they label quadrants in counter-clockwise order of I, II, III and IV, then quadrant I and quadrant III are your positive quadrants.
Since I always show multiplication as an area of length times width, then if you multiply on the coordinate plane you multiply an area of X axis times Y axis. Therefore, by way of example, (-3) x (-4) means go left 3 and then down 4. If you do that you have just created a rectangle in quadrant III. Since quadrant III is positive, you have created a positive area of 12 square untis.
You can then reverse this into divison. If you draw a rectangle 3 x 4 untis, in either quadrant I or III you have a positive area of 12 square units. You then have two choices as to what the sides will be: (3) x (4) or (-3) x (-4). Therefore the area (12) divided by the side (3) is the top (4), or the area (12) divided by the side (-3) is the top (-4).
I don't know if it's the best explanation, but I try to use as much visual as I can. And this is the only way that I can think of to actually visualize the rules. It makes more sense to them than just saying "pos x pos = pos", "neg x neg = pos", etc.
The same principals then work in quadrants II and IV which are negative quadrants.... (-3) x (4) means left 3 and up 4. You now have a rectangle of 12 square units which has a value of (-12) in that quadrant. Again you can draw this in either quadrant II or IV. Your area of (-12) will have a length of (-3) and a width of (4) in quadrant II and your area of (-12) will have a length of (3) and a width of (-4) in quadrant IV. So again, I can reverse the multiplication into division.
I do all my multiplcation/division visually with area - even when it is multiplication and division of fractions. I like to be able to see as much as possible. I dont know if this breaks down as the math get to higher levels, but it works well from very, very easy concepts on through a lot of algebra. If I am dealing with problems that have all real numbers, I use if for completing the square and quadratics, etc. Beyond that I don't know... (A friend of mine who used to work as an engineer, but now teaches math, told me that when working with imaginary numbers I need to use the "a by bi" coordinate plane instead of the "x by y". However, I don't remember ever learning anything about a plane that used a and bi to visualize what to do with your imaginary numbers. In fact, I don't remember ever being shown an "a + bi" plane at all. So I am at a loss at to how to visually demonstrate anything with with imaginary numbers - except that the parabola doesn't have any x intercepts on the (x,y) coordinate plane.)
I was not ever taught math that way when I learned it. But once I realized that I could visualize things that way, (and that was as an adult learner, years after I had a math degree) I took it as far as I could in my mind. Since it has been many, many years since I took any kind of Calculus classes, etc., and since I now teach/tutor only up through College Algebra level, I honestly don't remeber enough of what you learn in those higher level classes to know if this would work well for an explanation at higher levels and finding areas under curves, etc. But for what I show students, it seems to help them visualize how the + and - multiplication/division rules work. And they can also use it for completing the square and they can use it for multiplying binomials.
Hopefully it does not prove to be problematic for them in levels higher than Algebra.... For those of you who teach Calc, etc. let me know if you think that visualizing that way on the coordinate plane would cause any problems later.
Bradford said...
The only true way two negatives will ever make a positive is for us to make friends with our enemies. That said, let's look at it another way. When we fold a circle in half we put any two imaginary points together and crease. This give us a diameter. If we call what we can not see negative (the imaginary points) and by putting them both together will generate a line we can see (diameter), that is positive.
John Faig said...
I like a simple applet to show how subtracting negative numbers actually results in a positive. The applet is from the National Library of Virtual Manipulatives (http://nlvm.usu.edu/en/nav/frames_asid_162_g_2_t_1.html)
Once they master this idea, they should be able to grasp the idea that multiplication is this subtraction of a negative numer repeated over and over.
Anita Sharma said...
interesting discussion!
Another thought-observe the following pattern
-5 x 3 = -15 = (-5)+ (-5 )+ (-5)
-5 x 2 = -10 = -15 +5
-5 x 1= -5 = -10 + 5
-5 x 0 = 0 =-5 + 0
-5 x -1 = 5 = 0 + 5
-5 x -2 = 10 = 5 +5
using fundamental def. of multiplication as addition we can conclude product of two negatives is positive
Deborah Leslie said...
I was also taught to think about multiplication of positive and negative numbers using money analogies:
(+5) x (+3) = receiving $5 three times = $15 more in your pocket = 15
(+5) x (-3) = not receiving $5 three times = $15 less in your pocket = -15
(-5) x (3) = paying $5 three times = $15 less in your pocket = -15
(-5) x (-3) = not paying $5 three times = $15 more in your pocket = 15
It might be confusing for some people, but relating math concepts to money also helped me understand the concept faster.
Bradford said...
(-)+(-)= (+)
Since movement is a right angle function; one line added to one line, when combined at right angles is a plus sign.
(-)+(+)=(-) the second line is missing to make a plus sign, so it remains one line.
(+)+(-)=(-) this is simply the inverse of the above.
Only two things to understand; two lines make a positive symbol. One line does not make a positive symbol.
Senad Ibraimoski said:
You could use basic reasoning and explain this :
But we all know it is a convention ( -1 ) * (-1) = 1 ...
Joshua said...
It's not a convention: it's required by the distributive law and so on.
0 = 0 * 1 = (-1 + 1) * -1 = (-1) * (-1) + 1 * (-1) = (-1) * (-1) + -1,
and therefore (-1) * (-1) = 1.
I don't expect this to be the best way to explain it to students, but I do think it's important to realize that it's not an arbitrary convention but rather a requirement for the properties of addition and multiplication to carry over to negative numbers.
I like the Ask Dr. Math link from Rashmi Kathuria. There's also Teacher 2 Teacher at the Math Forum.
Number-line type arguments are good, as are payment/debt type explanations.
I tend to reason from patterns. I think you can do something like this:
4 * -3 = -12
3 * -3 = -9
2 * -3 = -6
1 * -3 = -3
0 * -3 = 0
and then they can see why -1 * -3 should be 3 to continue the pattern. (I see that someone else below said the same thing, and pointed out that using multiplication as repeated addition we can see why we're adding 3, or if you prefer taking away -3, as you go from each line to the next - that's a good connection!)
The walking and facing method is pretty nice too.
Dylan Faulin said...
This is how I teach it. I just wanted to '2nd' this suggestion. It's hard for students not to see this pattern. By the way, I do the same thing when introducing negative exponents:
5^3 = 125
5^2 = 25 = 125 / 5
5^1 = 5 = 25 / 5
5^0 = 5 / 5 = 1
5^-1 = 1 / 5
5^-2 = 1/5 / 5 = 1 / 25
This also takes care of why anything (but 0) to the power of 0 is 1.
when i was teaching in algebra class one of my students got the doubt how the product of two negatives will be positive?
Replies...
Useful Link
Senad said...
You could use basic reasoning and explain this :
For example, you are probably familiar with this one:
" The enemy(-) of my enemy(-) is my friend(+) "
" The enemy(-) of my friend(+) is my enemy (-)"
" A friend(+) of my enemy(-) is also my enemy(-)"
etc ...
But we all know it is a convention ( -1 ) * (-1) = 1 ...
Whit Ford said...
I prefer an expanded version of the (brief) number line explanation that was given:
- Multiplication represents repeated addition, so (3) x (5) represents three fives added together
- To represent this on the number line, I start at zero then move the the right by five units three times. I will land at (+15).
- If a negative sign is involved, it turns addition into subtraction... which would cause me to move in the other direction on the number line.
- To represent (3) x (-5) on the number line, I need to "add negative five" three times which is the same thing as "subtracting five three times", so I will start at zero, and move to the left by five units three times. I will land at (-15)
- To represent (-3) x (-5) on the number line, I need to "subtract negative five" three times. This reverses the previous example. I still start at zero facing left to move by negative five, but the (-3) means that instead of moving the way I am facing three times, I move backwards... to the right by five units three times, landing at (+15).
- The essence of this explanation could be turned into a physical demonstration in the classroom: Have someone stand up facing "forward". Ask them to walk forward for three steps = (3) x (steps). Ask them to, still facing forward, walk backwards for three steps = (3) x (- steps). Now ask them to face "backwards" and walk "backwards" for three steps = (-3) x (- steps). They end up having moved in the forward direction, and the rule about double negatives logically cancelling each other out in an English sentence should come to mind: they did not not move forward.
Senad said...
Well even though I'm not a teacher, and I even agree with what you said, I need to disapprove the second method (walking).
The best way I think is to go with my example, it is a basic principle that even a 5 year old would understand. It requires no previos knowlage, and you don't have to walk a mile to understand it. Certanlly, as a student- I appreciate correct methods that will produce best possible results in small amount of time.
For example, if (-1) * (-1 ) would be ( -1 ), if that was a convention, ( " The enemy of my enemy is my enemy " ) would be in contradiction to all things known to man, even politics doesn't work that way :). And try explaining distribution law, using your mentioned " walk " method... Or you coud try to explaing in what cases, (-1)^n would result in 1, or in (-1).
Whit Ford said...
I cited the walking example as a way to try help someone understand *why* a double negative is a positive. The rules you cite certainly are often used in life, but if a young student is questioning why the rules are true in the first place... well, that can be a tough question to answer to their satisfaction. I cited the walking example in an attempt to connect very concrete initial steps to an abstraction in the later steps: backwards. But hopefully students have enough personal life experience with "backwards" to have a decent and growing intiutitive sense of what it means and can stand for.
Once a student accepts that the product of two negatives is a positive, and has been working successfully with it for a while with simple multiplication problems, I would abandon attempting to connect increasingly complex operations, like exponentiation, with the number line or physical examples - and base explanations on the rules of multiplication that the students have acquired a comfort with from personal experience as you advocate.
I explain the sign of (-2)^n to students studying the laws of exponents by telling them to count the number of negative signs that will be in the final product (n), and recall that a double negative makes a positive. Therefore, I can use the associative property of multiplication to group the repeated multiplication problem into pairs of negative signs which cancel one another out. Since 15 or 16 year old students should be very comfortable with the concepts of odd and even by now, they are usually quite comfortable with visualizing pairs of negative signs cancelling each other out, therefore if n is even, the answer is positive... and if n is odd, the answer will be negative.
I find it helpful to explain multiplication as repeated addition, and exponentiation as repeated multiplication. This implies that exponentiation is repeated-repeated addition... the contemplation of which gives most people a headache! While this explanation is true, I agree that trying to take a framework too far in a completely concrete way can cause more difficulties than it solves. Does that mean the initial explanation should not be used because it will not extend readily (or perfectly) to all arithmetic operations? I think it has value nevertheless... because it helps a student to gain a strong conceptual understanding of the nature of a fundamental operation, one which they will depend on as they learn about increasingly complex functions which depend on the operations they first learned in grade school.
Carolyn said...
One of the visual examples I use for algebra students is on the two dimensional coordinate plane. If they label quadrants in counter-clockwise order of I, II, III and IV, then quadrant I and quadrant III are your positive quadrants.
Since I always show multiplication as an area of length times width, then if you multiply on the coordinate plane you multiply an area of X axis times Y axis. Therefore, by way of example, (-3) x (-4) means go left 3 and then down 4. If you do that you have just created a rectangle in quadrant III. Since quadrant III is positive, you have created a positive area of 12 square untis.
You can then reverse this into divison. If you draw a rectangle 3 x 4 untis, in either quadrant I or III you have a positive area of 12 square units. You then have two choices as to what the sides will be: (3) x (4) or (-3) x (-4). Therefore the area (12) divided by the side (3) is the top (4), or the area (12) divided by the side (-3) is the top (-4).
I don't know if it's the best explanation, but I try to use as much visual as I can. And this is the only way that I can think of to actually visualize the rules. It makes more sense to them than just saying "pos x pos = pos", "neg x neg = pos", etc.
The same principals then work in quadrants II and IV which are negative quadrants.... (-3) x (4) means left 3 and up 4. You now have a rectangle of 12 square units which has a value of (-12) in that quadrant. Again you can draw this in either quadrant II or IV. Your area of (-12) will have a length of (-3) and a width of (4) in quadrant II and your area of (-12) will have a length of (3) and a width of (-4) in quadrant IV. So again, I can reverse the multiplication into division.
I do all my multiplcation/division visually with area - even when it is multiplication and division of fractions. I like to be able to see as much as possible. I dont know if this breaks down as the math get to higher levels, but it works well from very, very easy concepts on through a lot of algebra. If I am dealing with problems that have all real numbers, I use if for completing the square and quadratics, etc. Beyond that I don't know... (A friend of mine who used to work as an engineer, but now teaches math, told me that when working with imaginary numbers I need to use the "a by bi" coordinate plane instead of the "x by y". However, I don't remember ever learning anything about a plane that used a and bi to visualize what to do with your imaginary numbers. In fact, I don't remember ever being shown an "a + bi" plane at all. So I am at a loss at to how to visually demonstrate anything with with imaginary numbers - except that the parabola doesn't have any x intercepts on the (x,y) coordinate plane.)
I was not ever taught math that way when I learned it. But once I realized that I could visualize things that way, (and that was as an adult learner, years after I had a math degree) I took it as far as I could in my mind. Since it has been many, many years since I took any kind of Calculus classes, etc., and since I now teach/tutor only up through College Algebra level, I honestly don't remeber enough of what you learn in those higher level classes to know if this would work well for an explanation at higher levels and finding areas under curves, etc. But for what I show students, it seems to help them visualize how the + and - multiplication/division rules work. And they can also use it for completing the square and they can use it for multiplying binomials.
Hopefully it does not prove to be problematic for them in levels higher than Algebra.... For those of you who teach Calc, etc. let me know if you think that visualizing that way on the coordinate plane would cause any problems later.
Bradford said...
The only true way two negatives will ever make a positive is for us to make friends with our enemies. That said, let's look at it another way. When we fold a circle in half we put any two imaginary points together and crease. This give us a diameter. If we call what we can not see negative (the imaginary points) and by putting them both together will generate a line we can see (diameter), that is positive.
John Faig said...
I like a simple applet to show how subtracting negative numbers actually results in a positive. The applet is from the National Library of Virtual Manipulatives (http://nlvm.usu.edu/en/nav/frames_asid_162_g_2_t_1.html)
Once they master this idea, they should be able to grasp the idea that multiplication is this subtraction of a negative numer repeated over and over.
Anita Sharma said...
interesting discussion!
Another thought-observe the following pattern
-5 x 3 = -15 = (-5)+ (-5 )+ (-5)
-5 x 2 = -10 = -15 +5
-5 x 1= -5 = -10 + 5
-5 x 0 = 0 =-5 + 0
-5 x -1 = 5 = 0 + 5
-5 x -2 = 10 = 5 +5
using fundamental def. of multiplication as addition we can conclude product of two negatives is positive
Deborah Leslie said...
I was also taught to think about multiplication of positive and negative numbers using money analogies:
(+5) x (+3) = receiving $5 three times = $15 more in your pocket = 15
(+5) x (-3) = not receiving $5 three times = $15 less in your pocket = -15
(-5) x (3) = paying $5 three times = $15 less in your pocket = -15
(-5) x (-3) = not paying $5 three times = $15 more in your pocket = 15
It might be confusing for some people, but relating math concepts to money also helped me understand the concept faster.
Bradford said...
(-)+(-)= (+)
Since movement is a right angle function; one line added to one line, when combined at right angles is a plus sign.
(-)+(+)=(-) the second line is missing to make a plus sign, so it remains one line.
(+)+(-)=(-) this is simply the inverse of the above.
Only two things to understand; two lines make a positive symbol. One line does not make a positive symbol.
Senad Ibraimoski said:
You could use basic reasoning and explain this :
But we all know it is a convention ( -1 ) * (-1) = 1 ...
Joshua said...
It's not a convention: it's required by the distributive law and so on.
0 = 0 * 1 = (-1 + 1) * -1 = (-1) * (-1) + 1 * (-1) = (-1) * (-1) + -1,
and therefore (-1) * (-1) = 1.
I don't expect this to be the best way to explain it to students, but I do think it's important to realize that it's not an arbitrary convention but rather a requirement for the properties of addition and multiplication to carry over to negative numbers.
I like the Ask Dr. Math link from Rashmi Kathuria. There's also Teacher 2 Teacher at the Math Forum.
Number-line type arguments are good, as are payment/debt type explanations.
I tend to reason from patterns. I think you can do something like this:
4 * -3 = -12
3 * -3 = -9
2 * -3 = -6
1 * -3 = -3
0 * -3 = 0
and then they can see why -1 * -3 should be 3 to continue the pattern. (I see that someone else below said the same thing, and pointed out that using multiplication as repeated addition we can see why we're adding 3, or if you prefer taking away -3, as you go from each line to the next - that's a good connection!)
The walking and facing method is pretty nice too.
Dylan Faulin said...
This is how I teach it. I just wanted to '2nd' this suggestion. It's hard for students not to see this pattern. By the way, I do the same thing when introducing negative exponents:
5^3 = 125
5^2 = 25 = 125 / 5
5^1 = 5 = 25 / 5
5^0 = 5 / 5 = 1
5^-1 = 1 / 5
5^-2 = 1/5 / 5 = 1 / 25
This also takes care of why anything (but 0) to the power of 0 is 1.
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