A positive fraction is known as a proper fraction (or simple fraction) when its numerator is less than its denominator. The fraction is a number which lies between 0 and 1.

For example, 2/3 and 3/4 are proper fractions.

A positive fraction is known as an improper fraction for the opposite reason, (i.e.) when the numerator is greater than the denominator. The fraction is a number greater than or equal to 1.

For example, 5/3 and 9/4 are improper fractions.

Imagine we want to illustrate the number 3/4 on a straight line. As it is a proper fraction, it will be found somewhere between 0 and 1. We have to divide the unit segment (0-1) into four parts and count three starting from 0.

4. Illustrate the number 3/4 in your exercise book. Follow the instructions below using set squares and your compass to divide the unit segment into four equal parts:

• Draw a horizontal line. Label the end on the left with the number 0 and the right end with the number 1. This will be our unit segment.

• Draw a longer line from 0 which is not horizontal.

• Use a compass to mark four equal divisions on this line starting from 0.

• Use a ruler to draw a line connecting the last mark made by the compass on the line and point 1.

• Use your set squares to draw lines parallel to this segment from the other three marks made by the compass.

The points where these parallel lines cut the horizontal segment divide the segment into four equal parts.

Use the arrows to change the values of the numerator and denominator.

Look carefully at the drawing in the window to see what the drawing in your exercise book should look like.

Repeat the exercise to illustrate the following numbers: 4/5, 5/6 and 2/7.

Use the window to check that your drawing is correct.

Remember that an improper fraction can always be broken down into a whole number plus a proper fraction.

For example,

13/5 = 2 + 3/5,

where 2 is the whole number we get when we divide 13 by 5, and 3 is the remainder.

Therefore, the number 13/5 is located somewhere between the numbers 2 and 3. In order to illustrate the number 13/5 we need to illustrate the number 3/5 in the segment [2,3]. In other words, we have to divide the segment [2,3] into 5 parts and count 3 parts starting from number 2.

5. Follow the instructions given above to illustrate the number 13/5 in your exercise book. Then, check that your drawing is correct by comparing it with the one in the following window.

Use the arrows to change the values of the numerator and denominator.

Look carefully at the drawing in the window to see what the drawing in your exercise book should look like.

Illustrate the following numbers in the same way: 10/3, 13/2, 17/5 and 9/4.

Use this window to check your results.