# Subtracting Negatives

## Concept

All integers can be subtracted from other integers. The subtraction of two integers, is simply the sum of the first one and the opposite of the second one. a - b = a + (-b). Using this rule we can compute the difference of any two integers. The model we use below is merely one way of seeing integer subtraction.

## Instructions

Drag the blue and red sliders on the left to create an integer. A blue dot = +1, a red dot = -1, so 9 blue + 6 red = +9 + (-6) = +3. Drag the green slider on the right to take away negatives. Observe how the number changes when you take away the red dots.

## Questions

What is 4 - (-3)? What is (-1) - (-3)? What is 5 + (-3) - (-3)? What does this tell you about another way to compute integer subtraction?

What is 4 - (-3)? What is (-1) - (-3)? What is 5 + (-3) - (-3)? What does this tell you about another way to compute integer subtraction? 4 - (-3) = 4 + 3 = 7. To see this, view 4 as "7 blue" plus "3 red". When we remove the "3 red", what is left is "7 blue", or +7. (-1) - (-3) = (-1) + 3 = 3 + (-1) = 2. View -1 as "2 blue" plus "3 red". When we remove the "3 red", what is left is "2 blue", or +2. 5 + (-3) - (-3) = 2 - (-3). Seeing 2 as 5 + (-3), or 5 blue plus 3 red, we get 2 - (-3) = 5. We see that adding and subtracting a number are inverse operations: adding a number and then subtracting that number always gives you the original number. Same goes with subtracting then adding.

6.EE.3

## Inspirations and Applications

[Source] https://www.geogebra.org/material/simple/id/t472usG3 [by] GreenMaths Another important use of negative numbers is that it "completes"the whole numbers: When you add two whole numbers you always get a whole number, but when you subtract two whole numbers, you don't always get a whole number. On the other hand, whenever you add or subtract integers, you always get an integer. This makes the collection of integers closed under addition and subtraction.

## Common Core

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