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We will use a “zoomed-in” view of the grid to study fractional multiplication and division.
Notice that the grid to the left now has dashed lines in addition to solid lines. The dashed lines mark fractional amounts. For example, if every third horizontal line is solid, every dashed horizontal line counts the fraction $$1/3$$.
The green rectangle you see to the left covers exactly one unit square, so it represents the number $1$.
Multiplying two fractions on the grid is exactly like multiplying two whole numbers: it means finding an area. The only difference is that the area may involve fractional pieces of grid squares. You can count those fractional pieces because they’re set apart by dashed lines.
For example, the rectangle currently shown on the grid has width $$5/2$$ and height $$4/3$$, so it gives a picture for the multiplication $$5/2(4/3)$$.
So, if you want to multiply two fractions:
When dividing two fractions, it’s useful to think of division problems as asking you to find the height of a rectangle, when you already know its area and the width of its base. Unlike in the rest of the course, we’ll use the division sign (÷) in this lesson.
So $${3/4} ÷ {1/2}$$ should be read as “$$3/4$$ divided by $$1/2$$.” As before, you can picture this division problem as a rectangle with area $$3/4$$, and base width $$1/2$$.
In order to divide fractions in the way you did in the last two problems, there need to be enough fractional (dashed) grid lines on the grid. Sometimes the fractions you start with won’t line up perfectly with the dashed lines, and then you’ll need to find equivalent fractions that do.
In the next question, we’ll be looking at the division problem $${2/1} ÷ {3/2}$$. Notice that the top line of the rectangle currently shown for that problem does not match up with the dashed lines on the grid.
You probably learned to solve division problems with fractions by cross-multiplying. For example, you could divide two fractions like this: $${2/3} ÷ {5/4}={2(4)} / {3(5)}=8/15$$. You can think of cross-multiplying as a way of making sure the grid is always split up into small enough pieces.