Distributive Property

In this video, let's use balanced moves to solve more complex equations.

In this example, the equation is written in a different format. Three parentheses x minus four equals 2x plus two.

The left side of this equation is an example of the distributive property. The number just outside of the parentheses tells how many copies to make of the group within the parentheses.

It is easiest to understand using cups and cubes. To start, I will first set up everything shown inside of the parentheses. X minus four. That means one cup and four black cubes.

That three just outside of the parentheses tells me I need to make three copies of the cups and the cubes. There's two copies, and now I have three copies. Let me regroup these cubes to make them more organized.

And now I can set up the right side of the equation. Two cups and two green cubes.

Even though this equation looked rather complex when it was first written, after setting it up with cups and cubes, I can see that the equation can be simplified to three x minus twelve or three x plus negative twelve, they are the same thing, equals two x plus two.

Equations written in this format are easier to understand.

Now let's make some balanced moves to reach our two objectives, getting all of the cups on one side and the cubes on the other.

Using balanced moves, I can remove two cups from each side of the equation. All cups are on one side? Check.

And after this balanced move, the equation now shows x plus negative twelve equals two.

Now for the second objective. I need all of the cubes on the other side. In order to do that, I will need to make zero pairs and cancel out the value of the negative twelve cubes.

Whatever I do to one side, I must do to the other side, so I will go ahead and add twelve green cubes to each side.

Each green and black cube make a zero pair. I can remove all of them from the left side of the equation because they have no value.

Cubes on the other side, check. And now the math shows one cup balanced with fourteen green cubes.

X equals fourteen. The equation is solved.

Of course, I should prove the answer by inputting fourteen for x back into the original equation. Three parentheses fourteen minus four equals two times fourteen plus two. Always start by solving what's inside of the parentheses first. Thirty equals thirty. It checks out.

And if you are asked to solve this type of problem using pencil and paper, no worries. You can still do it.

Draw large circles for each x, small squares for positive numbers, and shaded in squares for negative numbers. Then use balanced moves, crossing out cups on each side until the variable is isolated.

To get the squares isolated on the other side, make zero pairs by matching each of the squares with its opposite. Whatever you do to one side, you must do to the other. Then solve for x.

And now you're ready to use balanced moves to solve equations using the distributive property.