One common mistake among students when doing any type of operation on numbers is the negative sign. When does it just simply disappear, or when does it stay there. No one really ever explained negative signs to me when explaining multiplication. Many teachers have simply told students that a negative times a negative is a positive. That is it, no explanation or reasoning, probably because they did not know themselves. Also, the number zero, everyone knows that zero times any number is zero. But why? Well, in case you are unaware, mathematics are all about patterns. In math we are able to make conclusions about a lot of things in mathematics based on a pattern that we notice. Here is a pattern that will help many people see why a negative times a negative is positive, negative times a positive is negative, and zero times any number is zero.

4*0= 3*0= 2*0= 1*0= 0*0=

4*1=4 3*1=3 2*1=2 1*1=1 0*1=

4*2=8 3*2=6 2*2=4 1*2=2 0*2=

4*3=12 3*3=9 2*3=6 1*3=3 0*3=

4*4=16 3*4=12 2*4=8 1*4=4 0*4=

Notice, I left all of the answers for the multiples of zero blank. That is, because I want you to look at the pattern among the answers and tell me what you think the next answer should be off the patterns shown, not off of memorization. What patterns do you see? Now, lets look at what the rest of the chart would look like based on this pattern.

3*-3=-9 2*-3=-6 1*-3=-3 0*-3=0 -1*-3=3 -2*-3=6 -3*-3=9

3*-2=-6 2*-2=-4 1*-2=-2 0*-2=0 -1*-2=2 -2*-2=4 -3*-2=6

3*-1=-3 2*-1=-2 1*-1=-1 0*-1=0 -1*-1=1 -2*-1=2 -3*-1=3

3*0=0 2*0= 1*0= 0*0= 0 -1*0=0 -2*0 =0 -3*0=0

3*1=3 2*1=2 1*1=1 0*1=0 -1*1=-1 -2*1=-2 -3*1=-3

3*2=6 2*2=4 1*2=2 0*2= 0 -1*2=-2 -2*2=-4 -3*2=-6

3*3=9 2*3=6 1*3=3 0*3=0 -1*3= -3 -2*3=-6 -3*3=-9

What I want you to imagine is trying to figure out from all positive values what the next number is going to be. This is how the historical mathematicians viewed multiplying. They were able to notice that pattern among the rows and columns allowing them to place the negative or leave it out. They noticed, that the column starting from the bottom was decreasing by three, so once it got to zero, it would be zero, and when it got to 3*-1 it would have to be -3 because 0-3=-3 and so on. This pattern also emerges when looking at the rows going from left to right. Notice, the numbers are decreasing by 3 making 0*3=0 since 3-3=0. Now, observe the whole rows and columns of zero, this is due to the fact that 0-0=0. Therefore, the previous number subtracted the multiple is zero. I hope this chart helps make sense of the pattern among numbers as much as it has helped me!

Through patterns and consistencies, mathematicians have been able to make great discoveries, and although negatives have just been understood as a fact, we should try to make some sense of it. It is patterns and consistencies like these and many others that have allowed us to use math in so many different ways to evolve the world to what it is today. Although this is not how mathematicians found these properties, it helps us explain them.

4*0= 3*0= 2*0= 1*0= 0*0=

4*1=4 3*1=3 2*1=2 1*1=1 0*1=

4*2=8 3*2=6 2*2=4 1*2=2 0*2=

4*3=12 3*3=9 2*3=6 1*3=3 0*3=

4*4=16 3*4=12 2*4=8 1*4=4 0*4=

Notice, I left all of the answers for the multiples of zero blank. That is, because I want you to look at the pattern among the answers and tell me what you think the next answer should be off the patterns shown, not off of memorization. What patterns do you see? Now, lets look at what the rest of the chart would look like based on this pattern.

3*-3=-9 2*-3=-6 1*-3=-3 0*-3=0 -1*-3=3 -2*-3=6 -3*-3=9

3*-2=-6 2*-2=-4 1*-2=-2 0*-2=0 -1*-2=2 -2*-2=4 -3*-2=6

3*-1=-3 2*-1=-2 1*-1=-1 0*-1=0 -1*-1=1 -2*-1=2 -3*-1=3

3*0=0 2*0= 1*0= 0*0= 0 -1*0=0 -2*0 =0 -3*0=0

3*1=3 2*1=2 1*1=1 0*1=0 -1*1=-1 -2*1=-2 -3*1=-3

3*2=6 2*2=4 1*2=2 0*2= 0 -1*2=-2 -2*2=-4 -3*2=-6

3*3=9 2*3=6 1*3=3 0*3=0 -1*3= -3 -2*3=-6 -3*3=-9

What I want you to imagine is trying to figure out from all positive values what the next number is going to be. This is how the historical mathematicians viewed multiplying. They were able to notice that pattern among the rows and columns allowing them to place the negative or leave it out. They noticed, that the column starting from the bottom was decreasing by three, so once it got to zero, it would be zero, and when it got to 3*-1 it would have to be -3 because 0-3=-3 and so on. This pattern also emerges when looking at the rows going from left to right. Notice, the numbers are decreasing by 3 making 0*3=0 since 3-3=0. Now, observe the whole rows and columns of zero, this is due to the fact that 0-0=0. Therefore, the previous number subtracted the multiple is zero. I hope this chart helps make sense of the pattern among numbers as much as it has helped me!

Through patterns and consistencies, mathematicians have been able to make great discoveries, and although negatives have just been understood as a fact, we should try to make some sense of it. It is patterns and consistencies like these and many others that have allowed us to use math in so many different ways to evolve the world to what it is today. Although this is not how mathematicians found these properties, it helps us explain them.