Subtraction and the Commutative Property of Addition

[One speaker] 

[Any necessary descriptions.]

Narrator: Hi, welcome to the video on subtraction and  how it relates to the commutative property of addition. [A timeline is shown from -11 to 11. Two equations are written below: 5-4=1 & 5+(-4)] Now that’s a lot to say in one sentence, but the commutative property of addition is just the rule that says 5+4 is equal to 4+5. The numbers can change spots, or they can change position. They can commute, and the answer is still the same because if we have 5 and go to the right 4 steps, we end up with 9. Similarly, if we start with 4 and go to the right 5 steps, 1,2,3,4,5, we also end with 9. [commutative property of addition appears on the screen.] This is the commutative property of addition. But subtraction does not follow that same rule, let’s see why. If I have 5 minus 4, it’s equal to one. This is easy to see if we start here at 5 and we subtract 4. So we go to the left 4 spaces, we end up with one. But it isn’t the case for 4-5. 4-5 we start at the 4 and we go to the left 5 spaces, that leaves us at -1, which is not the same number. So subtraction does not have the same property of being able to switch the numbers. But however, if we think of subtraction as the addition of a negative number, then we can use that commutative property. Let me show you. So here we had 5-4=1.  Well if we think of this same problem as the addition of a negative number, then we have 5+-4. According to the rule of the commutative property of addition, this is equal to (-4)+5.  Now let’s see if we get the same answer. If we start here at a -4, and go to the right 5 spaces, we get the same answer of 1. So this is why it’s so powerful to understand that subtraction is really the addition of a negative number. If we can remember this rule then we can use tools, like the commutative property of addition, to manipulate the numbers in the ways that we need to use them. Let’s do another example. Let’s do 9-5.  According to our rules of subtraction, we start with 9 and we go to the left 5 numbers, for an answer of 4. So 9-5=4. Now let’s think of it as the addition of a negative number. So instead of 9-5, we’re going to say it’s 9+(-5). Again, we would do the same thing, we would start at 9 and move to the left, in the negative direction,  5 spaces which is 4. But now let’s use the commutative property of addition to see if we get the same answer. So since these are separated by addition, we can actually change the position that they are within the equation. So we can say that 9+(-5) is the same as -5+9. So we’re keeping the sign of the number 9, the same, it’s a 9, positive 9. So  -5+9 is equal to, well let’s see, we start at -5 and now we’re going to go to the right 9 numbers, putting us at 4 again.  So 9-5 is equal to -5+9.              

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