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Multiplication/Division Class

The visual logic for the multiplication/division class is simpler than that for the addition/ subtraction class. Again, we will show how it is done with an example. In the steps at the beginning of this chapter, we go from

$$\frac{2x+7}{3}=5$$

in the first step, to

$$2x+7=5\times3$$

in the second step. First, visualize the equation:

Then, chunk the 3 like this:

Start to move the chunk to the other side.

As you can see, the first steps are the same: visualize the equation, chunk a symbol or expression, and start to move the chunk to the opposite side of the equation. The next steps are slightly different. When we move chunks in this way - meaning, in a way that is equivalent to multiplying both sides of the equation by that symbol - we are going to move it from the denominator or the bottom of one side, to the numerator or top of the other side. (By the way, it works the other way too. If we are moving a chunk that starts in the numerator of one side, we would move it to the denominator of the other side.)

What happens when the other side is not a fraction? Actually, it is. In this example, the right hand side is just 5. It is also the fraction $\frac{5}{1}$. Keep in mind that every expression is also a fraction in this way.

As we move the 3, we do two special things. First, we remove the horizontal line that used to be between ``2x + 7'' and ``3'':

In this image, the horizontal line is a symbol. In particular, it is an operation, in the sense described on page . Remember that an operation needs two quantities in order to exist. When we take one of those quantities away, the operation symbol vanishes. That is what has happened here.

Next, we insert a $\times$ symbol (that's ``times'', for multiplication, and not the letter x) to the right of the five:

Finally, we move the 3 into place:

There is another way to do all this. After removing the horizontal line,

we can move the 5 to the right, and put the $\times$ (multiplication sign) to its left:

Then we move the 3 into the space to the left of the $\times$:

Both ``pathways'' are fine. They are both equivalent to multiplying each side by 3, and that is all that is important.

Division is similar to multiplication. Let's go backwards, from

$$2x+7=5\times3$$

to

$$\frac{2x+7}{3}=5$$

With multiplication, we move the symbol from the denominator on one side to the numerator on the other side. With division, it is the opposite. We are taking a term from the numerator (or just from the expression, if it is not a fraction) on one side, and moving it into the denominator on the other side.

Start by chunking the 3.

Begin to move the 3 across.

As it moves, we do two things. First, we drop the multiplication symbol (the $\times$) on the right side.

Also, since we are going to make the left side into a fraction, we need to bring in the operator. In other words, we need to draw a horizontal line under the numerator:

(Note that in this situation, ``bring in the operator'' and ``draw the horizontal line'' are two ways of saying the same thing.) This done, we finish moving the 3 into the denominator of the left side.