Basic Arithmetic Rules of Addition & Multiplication
Basic Rules for Addition
The process of using the array method turns out quite nicely due to some of the basic arithmetic rules for addition and multiplication. There are 4 basic rules of each operation (addition or multiplication) in the system of real numbers, explicitly: the commutative, associative, identity and inverse rules for addition or of multiplication in the set of real numbers. In addition, the distributive rule connects the two operations. In this unit we will look closely at how the distributive (or its more general version, the extended distributive rule), commutative and associative rules justify the use of area models/box method in both numerical and polynomial form.
Before we move into the rules of multiplication, it is necessary to briefly discuss the rules of addition in order to solidify the ideas of multiplication in this unit.
The Commutative Rule of Addition: the sum of two addends will be the same no matter the order of the addends.
a + b = b + a
37 + 25 = 25 +37
62 = 62
The Associative Rule of Addition: the sum of three non-negative integers, does not depend on the combination of numbers.
(a + b) + c = a + (b + c )
(42 + 37) + 25 = 42 + (37 + 25)
79 + 25 = 42 + 62
104 = 104
The Any Which Way Rule
By repeated application of the associative rule and the commutative rule, we can justify a much more general and flexible rule that is usable in a wide variety of practical calculations. We will refer to this rule as The Any Which Way Rule. This rule states that, when you are adding a list of numbers, no matter how you combine the numbers or the order that you add the numbers you will yield the same sum.
The Any Which Way Rule:
22 + 14 + 37 + 53 = (22 + 37) + (14 + 53) = (22 +53) + (14 + 37) = 126
The Any Which Way Rule also works for multiplication, as it can be justified by the associative and commutative rules for multiplication, which will be displayed below.
The Distributive Rule
The distributive rule is a rule that students are familiar with as it tells us to multiply a number by the sum of two numbers. It is the only rule of the operations that involves both addition and multiplication.
The Distributive Rule: For non-negative integers, a, b, and c
a × (b + c) = (a × b) + (a ×c)
The distributive rule can be represented by using an array model or an area model. We can demonstrate this when a nonnegative integer is represented by a horizontal row and through multiplication, we reproduce the row a specific number of times.
Example #7:
The Distributive Rule:
5 × (3 + 4)
let a, b, c be defined as a = 5, b = 3, c = 4
(5 × 3)+(5 × 4) = (15) + (20) = 35
In the above example I demonstrate how the distributive rule is applied using single digit numbers. It is then demonstrated using an area model below.
Figure 4: The Distributive Rule as an Array Model & Schematic Model
The above array model shows that this model is very similar to the area model. However, the area model offers a more schematic way of representing the multiplication of products. Using the array model to demonstrate the distributive rule is a great way for students to see how multiplying one number by two partial products will yield the same result no matter how they numbers are multiplied.
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