The list of fractions between 0 and 1 with denominators between 1 and 3 looks like this: $$\frac{0}{1}, \, \frac{1}{1},\, \frac{1}{2},\, \frac{1}{3},\, \frac{2}{3}$$We can put them in order like this:$$\frac{0}{1} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{1}{1}$$

Now let’s expand the list to include fractions with denominators of 4. We won’t include $\frac{2}{4}$, because $\frac{1}{2}$ is already on the list. $$\frac{0}{1} <\frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}$$

- Expand the list again to include fractions that have denominators of 5.
- Expand the list you made to include fractions have have denominators of 6.
- When you add a new fraction to the list, you put it in between two “neighbors.” Go back and look at your work. Do you see a relationship between a new fraction and its two neighbors?