I. Rules for Adding and Subtracting Integers
By conceptually understanding the addition and subtraction of integers as movements on the number line, the reasoning for the following rules should make sense (see below):
1) When adding integers that have the same sign, add the absolute value of each number and keep the sign.
2) When adding integers that have different signs, subtract the smallest absolute value from the largest and keep the sign of the number with the largest absolute value.
3) When subtracting integers, one method is to convert it into an addition problem then follow the rules for addition of integers. To do this (1) change the subtraction sign into an addition sign, (2) take the opposite of the number that immediately follows the newly placed addition sign.
1) When adding integers that have the same sign, add the absolute value of each number and keep the sign.
2) When adding integers that have different signs, subtract the smallest absolute value from the largest and keep the sign of the number with the largest absolute value.
3) When subtracting integers, one method is to convert it into an addition problem then follow the rules for addition of integers. To do this (1) change the subtraction sign into an addition sign, (2) take the opposite of the number that immediately follows the newly placed addition sign.
Example: positive + positive:
15 + 3
This of course is simply the addition of two whole numbers but following the rules above, you simply add the absolute values of each number (15 and 3) and keep the sign, which here is positive.
15 + 3 = 18
Example: negative + negative:
-15 + (-3)
Again, add the absolute values of each number (15 and 3) and keep the sign, which here is negative.
-15 + (-3) = -18
Example: positive + negative:
8 + (-10)
Following the rules for adding numbers with different signs, we subtract the smallest absolute value (8) from the larger (10). Then use the sign of the number with the largest absolute value, which in this case is negative.
8 + (-10) = -2
Example: negative + positive:
-5 + 7
Again, following the rules for adding numbers with different signs, subtract the smallest absolute value (5) from the larger (7). Then use the sign of the number with the largest absolute value.
-5 + 7 = 2
Example: subtracting integers:
-3 - (-9)
Convert the subtraction problem into an addition problem by replacing the subtraction sign with an addition sign. Then convert the number that follows the newly placed addition sign to it's opposite (-9 become 9).
-3 + 9
Then follow the rules for adding integers.
-3 + 9 = 6
II. Conceptual Model
Number Line
Addition and subtraction of integers can be illustrated with use of a number line. For instance, to calculate -1 + 4, we start at 0 then move one unit to the left. This represents -1. From here, move 4 units to the right. This leaves us at 3 on the number line. -1 + 4 = 3.
III. Properties of Addition
15 + 3
This of course is simply the addition of two whole numbers but following the rules above, you simply add the absolute values of each number (15 and 3) and keep the sign, which here is positive.
15 + 3 = 18
Example: negative + negative:
-15 + (-3)
Again, add the absolute values of each number (15 and 3) and keep the sign, which here is negative.
-15 + (-3) = -18
Example: positive + negative:
8 + (-10)
Following the rules for adding numbers with different signs, we subtract the smallest absolute value (8) from the larger (10). Then use the sign of the number with the largest absolute value, which in this case is negative.
8 + (-10) = -2
Example: negative + positive:
-5 + 7
Again, following the rules for adding numbers with different signs, subtract the smallest absolute value (5) from the larger (7). Then use the sign of the number with the largest absolute value.
-5 + 7 = 2
Example: subtracting integers:
-3 - (-9)
Convert the subtraction problem into an addition problem by replacing the subtraction sign with an addition sign. Then convert the number that follows the newly placed addition sign to it's opposite (-9 become 9).
-3 + 9
Then follow the rules for adding integers.
-3 + 9 = 6
II. Conceptual Model
Number Line
Addition and subtraction of integers can be illustrated with use of a number line. For instance, to calculate -1 + 4, we start at 0 then move one unit to the left. This represents -1. From here, move 4 units to the right. This leaves us at 3 on the number line. -1 + 4 = 3.
III. Properties of Addition
Commutative Property
Addition (as well as multiplication) is commutative, meaning the order of the numbers being added doesn't matter.
x + y = y + x
x + y = y + x
Example:
4 + (-6) = -2 is the same as -6 + 4 = -2
Associative Property
Associative Property
Addition (and multiplication) is associative, meaning that when adding more than two numbers, the order in which addition is performed does not matter. For example:
x + (y + z) = (x + y) + z
Example:
x + (y + z) = (x + y) + z
Example:
1 + (2 + (-3)) = 0 is the same as (1 + 2) + (-3) = 0
No comments:
Post a Comment