Intro to distributive property

This array is made up of $3$ rows with $6$ dots in each row. The dots show $3 \times 6 = 18$.

If we add a line dividing the dots into two groups, the total number of dots does not change.

The top group has $1$ row with $6$ dots. The dots show $1 \times 6$.

The bottom group has $2$ rows with $6$ dots in each row. The dots show $2 \times 6$.

We still have a total of $18$ dots.

The math rule that allows us to break up multiplication problems is called the distributive property.

The distributive property says that in a multiplication problem, when one of the factors is rewritten as the sum of two numbers, the product does not change.

Using the distributive property allows us to solve two simpler multiplication problems.

In the example with the dots we started with $\greenD{3} \times \purpleD{6}$.

We broke the $\greenD{3}$ down into $\greenD{1 + 2}$. We can do this because $\greenD{1 + 2 = 3}$

We used the distributive property to change the problem from $\greenD{3} \times \purpleD{6}$ to $(\greenD{1 + 2}) \times \purpleD{6}$.

The $\purpleD{6}$ gets distributed to the $\greenD{1}$ and $\greenD{2}$ and the problem changes to:
$(\greenD{1} \times \purpleD{6}) + (\greenD{2} \times \purpleD{6})$

Now we need to find the two products:
$6 + 12$

And finally, the sum:
$6 + 12 = 18$

$\greenD{3} \times \purpleD{6} = 18$ and
$(\greenD{1 + 2}) \times \purpleD{6} =18$

Some numbers like $1, 2, 5$, and $10$ are easier to multiply. The distributive property allows us to change a multiplication problem so that we can use these numbers as one of the factors.

For example, we can change $4 \times 12$ into $4 \times (\tealD{10} + \greenC{2})$.

The array of dots on the left shows $(\tealD{4 \times 10})$. The array of dots on the right shows $(\greenC{4 \times 2})$.

Now we can add the expressions to find the total.
$(\tealD{4 \times 10}) + (\greenC{4 \times 2})$
$= \tealD{40} + \greenC{8}$
$=48$

Since $10$ and $2$ are both easy to multiply, using the distributive property for this problem made finding the product easier.

The dots represent $9 \times 4$.

The distributive property is very helpful when multiplying larger numbers. Look at how we can use the distributive property to simplify $15 \times 8$.

We will start by breaking $\blueD{15}$ into $\blueD{10 +5}$. Then we will distribute the $8$ to both of these numbers.