The relationship between multiplication and addition can also be seen in various mathematical properties.
The commutative property is one that applies to both multiplication and addition problems. The implications of this property are that in a multiplication equation, one can multiply the numbers in any order to get the same product, and in an addition equation, one can add the numbers in any order to get the same sum. An example of the commutative property being used in addition is the equation, "10 + 2 = 12." If the numbers 10 and 2 were to be switched (2 + 10), the sum would still be 12. The equation, "2 x 5," can be utilized to demonstrate the commutative property in multiplication. Two times 5 equals 10, and when the numbers switch places, 5 times 2 still yields a product of 10. The commutative property is connected to the thinking strategy of thinking about multiplication in terms of adding groups of numbers. When students see 5 x 2 they may first think that means 5 groups of 2, which is 2 + 2 + 2 + 2 + 2. The commutative property lets them know that it can also mean 2 groups of 5, which is a much simpler 5 + 5.
The associative property is similar to the commutative property, but applies to equations that have more than two numbers and have at least two of the numbers grouped. Like the principle of the commutative property, the associative property dictates that the numbers can be multiplied or added in any order, regardless of the groupings. An example of the associative property in addition is the equation, "3 + (2 + 8)." The grouping of 2 + 8 implies that these two numbers must be added first, but the associative property allows the numbers to be added in any order without the sum being changed. Whether you add 2 + 8 + 3 or 3 + 8 + 2 or 8 + 3 + 2, the sum is always 13. The equation, "(4 x 2) x 3," can be utilized to exemplify the associative property as it applies to multiplication. The grouping of 4 x 2 implies that these two numbers must be multiplied first, but the associative property allows the numbers to be added in any order without the product being altered. Whether you multiply 4 x 2 x 3 or 2 x 4 x 3 or 3 x 2 x 4, the product is always 24. Similar to the commutative property, the associative property helps students use the thinking strategy of adding together groups of numbers, because it allows them to solve the problem in the order that is easiest. When students multiply (2 x 6) x 5, which is 12 + 12 + 12 + 12 + 12, it may be easier for them to multiply (5 x 2) x 6, which is 10 + 10 + 10 + 10 + 10 + 10. It is easier to count by 10's than 12's.
The distributive property involves breaking down multiplication problems, and it uses addition as a crucial tool. While many students are able to memorize the products of multiplying numbers 1 through 10, numbers greater than 10 that aren't multiples of 10 may begin to pose difficulties. The distributive property makes this sort of multiplication easier, and can be exemplified by the equation, "4 x 56." The distributive property holds that (4 x 50) + (4 x 6) = 4 x 56. To make 56 an easier number to multiply by, 56 can be separated into two numbers that add up to 56, and then each can be multiplied by 4 and added together. Four times 50 is an easy product for students to multiply because it is 50 + 50 + 50 + 50, or 200. Four times 6