The distributive property of multiplication over additions says that for all numbers, A, B, and C,
A x (B + C) = A x B + A x C
In the expression A x (B + C), we are multiplying A times the quantity of B + C. While the distributive property is often merely accepted as a working strategy with all numbers, there are ways explain why it works. This is perhaps best done by using arrays of objects.
For example, let’s use the expression 4 x (8 + 3).
There are two possible ways to show this expression by way of arrays. Since the quantity of 8 + 3 can be computed, we can find the sum of the two numbers and rewrite our expression as 4 x (11). In order to maintain our focus that the 11 in our expression is a result of adding 8 and 3, we could color code our objects in the array.
Using the definition of multiplication, we know that 4 x (11) can be thought of as 4 groups with 11 in each group. This is show in the array below. Notice that of the 11 circles that make up each horizontal row, 8 are red and 3 are black. This is what I meant by color coding our array to maintain that we added 8 + 3 to get our 11 in the expression.
If we count up all of our dots, we can see that our expression 4 x (11) is equivalent to 44, or:
4 x (11) = 44.
The second way to represent the expression 4 x (8 + 3) in the form of an array is as follows:
Instead of computing 8 + 3 and then multiplying that by 4, we distributed the 4 amongst the quantity of 8 + 3. In other words, we multiplied each digit within the parentheses by 8. As a result, we can now claim that 4 x (8 + 3) = (4 x 8) + (4 x 3).
Keep in mind that this lesson is about the distributive property. So, if you were asked to use the distributive property to solve 4 x (8 + 3), you would use our second method of solving it to do so.
Watch the following video from Khan Academy to get an interactive lesson of what we’ve already covered in this post.
If you need any supplemental help with the distributive property, here are some games and supplemental graphics that may help.
Make sure to write down any questions you still have in your math journals so that we can cover them tomorrow in class.
