6.1.2 Folding Paper

Take a piece of blank notebook paper and fold it in half, creasing along the fold. Open it up and shade the left-hand side.

A piece of paper is shown being folded in half, and then opened up with the left hand half shaded.

Since the paper has been divided into two equal parts, each piece is half of the original. This fraction is written as one divided by two and read as “one-half.”

The fraction “one-half” is shown as a one over a two, with a horizontal line between the two numbers. An arrow points to the one with the comment “top number is called the numerator and is the number of shaded pieces.” An arrow points to the two with the comment “bottom number is called the denominator and is the total number of pieces.”

Now fold the paper back along the original crease and then in half again along the long side. If you open up the paper, you should see four pieces of the same size, with two of them shaded. The paper is now divided into quarters (or fourths), and the fraction of the paper shaded is two divided by four.

A piece of paper is shown folded in half, then in quarters and then opened up with the two left-hand quarters shaded.

Since you haven’t altered the shading in any way, this demonstration shows that one half is equal to two quarters: equation left hand side one divided by two equals right hand side two divided by four. Now, fold the paper back into quarters along the crease lines and then fold into three equal pieces or thirds along the long side. If you now open the paper up, you can see that there are 12 equal pieces. These pieces are “twelfths,” of which six are shaded, so six divided by 12 of the paper is shaded. This fraction also represents the same amount as one divided by two.

A piece of paper is shown folded in quarters, then in twelfths and then opened up with the six left-hand twelfths shaded.

You can continue to fold the paper into smaller and smaller pieces. Each time you open up the paper, it will be divided into smaller fractions, but half of it will still be shaded. The fractions which represent the shaded part are all equivalent to each other.

6.1.3 Equivalent Fractions