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?
Answers
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.
Pre-Skills
6.EE.3
Related Concepts
Adding Integers
Inverse Function
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
7.NS.1c
7.NS.1d
7.NS.3