Solve on Paper

In this example, we are going to review how to transfer your algebraic problem solving skills from the equations mat with cups and cubes to a more traditional approach using paper and pencil.

I'm going to start by recording this equation on a piece of paper. There we go.

Although I'm going to be solving this by recording each step on the paper, I'm going to think back to what I would do using cups and cubes and let those steps inform me. So if we were working on the equation, Matt, what would we do first?

There's more than one possibility, but I'm going to start by removing three cubes from each side. Okay. So how can we represent that on paper?

Well, when we remove cubes from the mat, that means we subtracted them. We can represent that on paper by writing minus three on the left side and minus three on the right side.

Now on the paper, let's write what remains after we did that step. Four x equals x plus fifteen.

What should be our next move? We have an x or a cup on both sides still, so we need to remove the x from the right side and an x from the left side.

To show that move on paper, we can write minus x on the right side of the equation and minus x on the left side of the equation. Now let's record what remains. Three x equals fifteen.

And at this point, we want to figure out the value of each individual cup or x. So I'm going to split the cups into three groups and then split the cubes into three groups.

How can we represent what we just did? We did not remove anything from the mat, so we will not use the subtraction operation. We split or divided our cups into three parts and divided our cubes into three parts.

I can show that by writing divide by three on each side. That leaves us with x equals five.

And if you record each of the steps on your paper like I did on mine, you will be left with this sort of upside down triangle.

It's always helpful to prove or check our answer by plugging in our number for x to make sure that both sides are equal. Four times five plus three equals five plus eighteen. Twenty three equals twenty three. It checks out.

Now you're ready to apply your algebraic problem solving skills to pencil and paper.