Use an area model to write an expression equivalent to 𝑥 plus 𝑦, 𝑥 minus 𝑦.
So to help us answer the question, we’re given a diagram of an area model. We can see that, horizontally, we have the values 𝑥 and 𝑦. This represents the first set of parentheses with 𝑥 plus 𝑦. Vertically, we have the values 𝑥 and negative 𝑦, which represents our second set of parentheses.
So using the area model gives us a quick way to multiply our parentheses. We can do this by treating the values of our 𝑥s and 𝑦 and negative 𝑦 like they’re the lengths on the edge of a rectangle. And to find the area of this rectangle, we would multiply our length and width values, in this case just the values representing the lengths.
So taking our first value, this would be worked out by calculating 𝑥 times 𝑥, which we can write as 𝑥 squared. The area of our next rectangle will be calculated by 𝑥 times 𝑦, which we can simply write as 𝑥𝑦. For our next rectangle, we calculate 𝑥 times negative 𝑦, which can be simplified as negative 𝑥𝑦. And for our final rectangle, we would have negative 𝑦 times 𝑦, which is negative 𝑦 squared.
So now we have worked out the area of each individual piece in our model. To find the total area, we would add all of these pieces together. This would give us 𝑥 squared plus 𝑥𝑦 minus 𝑥𝑦 minus 𝑦 squared. And it’s very important that we remember our negative signs in this. We can then look at our expression and simplify. We can see that we have a plus 𝑥𝑦 and a negative 𝑥𝑦. So our expression would simplify to 𝑥 squared minus 𝑦 squared, which is equivalent to our original expression.