## Real World Applications – Algebra I

### Topic

Selling Candy

### Student Exploration

One of the most common ways that teenagers start to make money is by selling candy. If you buy a case of candy, it can cost you about $26 for a box of 30 pieces of big candy bars, which means that if you sell each candy bar for $1, you can get a $4 profit. If you and your friend Jay are both selling candy, we can represent this relationship by using and proving the distributive property.

You and Jay sold a lot of candy bars. Let’s say you sold 15 boxes in one week and she sold 8 boxes.

We can represent the total amount that the two of you earned in two different ways.

The first way to represent how much money you both made total is by adding the total number of boxes sold and then multiplying this by 4. So, we would get the sum, which is \begin{align*}15 + 8\end{align*}

Another way to represent this relationship is if you figured out how much you made first, before Jay. Since you knew that you sold 15 bars, you multiplied your earnings by 4, so \begin{align*}4 \times 15\end{align*}

These two ways actually help prove the distributive property, and why the distributive property holds true. If we were to distribute the 4 to 15 and 8, we would get the second expression above.

But, let’s say that you and Jay both decided that $4 is not a very good profit, and wanted to see what kinds of profit you can make by changing how much you want your profit to be. We can also represent this using the distributive property!

Let’s say that “\begin{align*}x\end{align*}

### Extension Investigation

Check out other ways that you can represent the distributive property and factoring in your world. When would this be useful to you?

If we went back to the example above about selling candy, could you figure out a way to sell something and find a way to make the maximum profit? If so, how? What type of relationship would this represent? (Linear? Quadratic?) If it’s quadratic, what would the point of that would yield the maximum profit be called?