If Emily's recipe calls for $\frac{1}{4}$ of a cup of sugar for a full batch, then for half a batch, she would use half of that amount.

To find out how much sugar Emily needs, we simply divide $\frac{1}{4}$ by 2:

$\text{Sugar for half a batch} = \frac{1}{4} \div 2 = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$

So, Emily would use $\frac{1}{8}$ of a cup of sugar to make half a batch of cookies.

Would you like more details, or do you have any other questions? Here are a few related questions you might find interesting:

1. How much sugar would Emily need for a double batch of cookies?
2. What would be the total amount of sugar needed for 1.5 batches?
3. If Emily has 2 cups of sugar, how many full batches of cookies can she make?
4. How would the recipe change if Emily wants to use only 1/10 of a cup of sugar for a low-sugar version?
5. How much sugar would Emily need for a quarter of a batch?

Tip: When scaling a recipe, always multiply or divide each ingredient by the same factor to maintain the correct proportions.