Example 1

In a unit square, divide the base into two equal parts and the left side into three and draw a rectangle of altitude 1/3 and base 1/2 in the bottom left corner. (We learned in Module 1 how to divide any segment into a number of equal parts by using a geometric construction, but here you can just do the division by eye.)

Then divide the 1/3(1/2) rectangle into 3 equal pieces side by side and arrange the pieces into a column on the left.

Note that six copies of that column placed side by side will fill the unit square so the area of the 1/3(1/2) rectangle is one sixth of the unit square.

Exercise 1

In a unit square, divide the base into three equal parts and the left side into two and draw a rectangle of altitude 1/2 and base 1/3 in the bottom left corner.

Then divide the 1/2(1/3) rectangle into 2 equal pieces side by side and arrange the pieces into a column on the left.

Note that six copies of that column placed side by side will fill the unit square so the area of the 1/2(1/3) rectangle is one sixth of the unit square.

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Example 2

Draw a rectangle of base 7/2 and altitude 5/3 and divide it up into 35 pieces by dividing the base into 7 parts of width 1/2 and the left side into 5 parts of length 1/3

Try to rearrange all of the blocks above height one into the strip of height 1 beside the original rectangle. (This will actually require cutting up the last two blocks into thirds as we did in Example #1)

Note that this is not the only way of rearranging the first rectangle into the second. For example, by cutting all of the little rectangles up in the same way we could rearrange the (5/3)(7/2) rectangle into 35 little vertical strips of width 1/6 and height 1

Exercise 2 – same idea as Example #2 for the case of (2/3)(5/2)

Draw a rectangle of base 5/2 and altitude 2/3 and divide it up into 10 pieces by dividing the base into 5 parts of width 1/2 and the left side into 2 parts of length 1/3

Make a copy of the rectangle and cut it into 10 small rectangles of width 1/2 and height 1/3. Then take each small rectangle and by cutting it into thirds make a column of height 1 and width 1/6. Place these columns side by side in the strip of height 1 on the original picture to fill a rectangle of height 1 and width 10/6 (which is the same as 5/3).

Note that this illustrates the rule for multiplying fractions by multiplying numerators and denominators as well as the idea of simplifying by cancellation of common factors.