# Video: Using an Area Model to Deduce the Distributive Property of Multiplication

The diagram shows a rectangle of sides 𝑎 and 𝑏 + 𝑐. Its area is thus 𝑎 × (𝑏 × 𝑐), also written as 𝑎(𝑏 + 𝑐). Work out the area of the two rectangles that make up the bigger rectangle to find an equivalent expression to 𝑎(𝑏 + 𝑐).

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### Video Transcript

The diagram shows a rectangle of sides 𝑎 and 𝑏 plus 𝑐. Its area is thus 𝑎 times parenthesis 𝑏 times 𝑐 parenthesis, also written as 𝑎 parenthesis 𝑏 plus 𝑐 parenthesis. Work out the area of the two rectangles that make up the bigger rectangle to find an equivalent expression to 𝑎 times parenthesis 𝑏 plus 𝑐 parenthesis.

So here we have side 𝑎 and side 𝑏 plus 𝑐 of this larger triangle. The area of a rectangle is equal to length times width. So for the entire rectangle, its area would be 𝑎 times 𝑏 plus 𝑐, which can also be written as 𝑎 parenthesis 𝑏 plus 𝑐 parenthesis.

Now we’re told to work out the area of the two rectangles that make up the bigger rectangle. So we have this one and this one. So let’s look at the striped one first. Its area will be 𝑎 times 𝑏, which could also just be written as 𝑎𝑏.

Now the polka dot rectangle, this length must also be 𝑎. So its length times width formula, we’ll actually be also be using the same 𝑎. So its area would be 𝑎 times 𝑐, or 𝑎𝑐.

So to have the area of the entire rectangle, we will just need to add these areas together. So we would have 𝑎𝑏 plus 𝑎𝑐. And this should make sense because if we would distribute the 𝑎 to the 𝑏 plus 𝑐, we would have 𝑎 times 𝑏, which is 𝑎𝑏, plus 𝑎 times 𝑐. And that’s what we have: 𝑎𝑏 plus 𝑎𝑐. So this will be our final answer.