Let's place each of the fractions on the number line provided, and then justify their positions.

Fractions to place:

• $-\frac{1}{3}$
• $\frac{1}{6}$
• $-\frac{1}{12}$
• $\frac{2}{3}$
• $\frac{2}{6}$
• $\frac{2}{12}$
• $-\frac{3}{3}$
• $-\frac{3}{6}$
• $\frac{3}{12}$

Converting Fractions to Common Denominators

To make it easier to place these fractions on the number line, let's express them with the same denominator, 12.

1. $-\frac{1}{3} = -\frac{4}{12}$
2. $\frac{1}{6} = \frac{2}{12}$
3. $-\frac{1}{12}$ remains $-\frac{1}{12}$
4. $\frac{2}{3} = \frac{8}{12}$
5. $\frac{2}{6} = \frac{4}{12}$
6. $\frac{2}{12}$ remains $\frac{2}{12}$
7. $-\frac{3}{3} = -1$
8. $-\frac{3}{6} = -\frac{6}{12}$
9. $\frac{3}{12}$ remains $\frac{3}{12}$

Placement on the Number Line:

Now that we have all fractions in terms of twelfths, we can place them accordingly:

• $-1$ is located at the far left (from $-\frac{3}{3}$).
• $-\frac{6}{12}$ (equivalent to $-\frac{1}{2}$) is between $-1$ and 0.
• $-\frac{4}{12}$ (equivalent to $-\frac{1}{3}$) is between $-\frac{6}{12}$ and 0.
• $-\frac{1}{12}$ is just left of 0.
• $\frac{2}{12} = \frac{1}{6}$ is a small positive fraction just right of 0.
• $\frac{3}{12}$ is slightly further to the right.
• $\frac{4}{12} = \frac{1}{3}$ is a little further to the right.
• $\frac{8}{12} = \frac{2}{3}$ is larger, closer to 1.

Observations:

• Many fractions reduce to simpler forms, such as $\frac{2}{6}$ and $\frac{1}{3}$.
• Fractions can be compared more easily by expressing them with a common denominator.
• The number line is symmetric around 0, with negative and positive fractions placed equally on either side.

Would you like to explore specific fraction comparisons or further breakdown?

Here are 5 related questions:

1. How does finding a common denominator help in placing fractions on a number line?
2. Why is it useful to simplify fractions before plotting them?
3. How can we quickly estimate the position of fractions without full calculations?
4. What role does the symmetry of the number line play in organizing negative and positive fractions?
5. How does the concept of equivalent fractions help in visualizing their relative sizes?

Tip: Always convert fractions to the same denominator when comparing them or placing them on a number line. This makes it easier to see their relative positions.