Weighing herbs

 One set of weights that works is 1 gram, 3 grams, 9 grams, and 27 grams.

The first thing I did was translate the problem into mathematical language: weights added to the pan without the herbs means we add those weights, and weights added to the pan with the herbs means we subtract those weights. So, the problem is to find four numbers that we can combine with addition and subtraction to get all numbers from 1 to 40.

One of my first strategies was noticing that, since we can use weights to both add and subtract from a given weight (by placing them on either side of the scale), if we divide the numbers from 1-40 into 3 sections 1-13, 14-26, and 27-40, and include 27 as one of our weights, than we only need to be able to find a set of three weights which can measure any quantity from 1 to 13. This is because we can add or subtract the numbers 1-13 from 27 to get the numbers 14-40. I used the guess and check method here, and was unsuccessful for a while, until I realized I could use the same strategy. By dividing 1-13 into 3 sections 1-4, 5-8, and 9-13 and including 9 as one of our weights, we only need to find two weights which give us the numbers 1-4, for the same reason. This was easier, and gave me the four numbers 1, 3, 9, and 27.

After I solved the problem, I realized something interesting. Each weight can be in one of three positions: on the pan with the herbs, on the pan without the herbs, or not on the scale at all. Therefore, with 4 weights, there are 3^4=81 possible configurations. Removing the configuration with no weights on either side, we have 80. But they come in pairs: given a configuration that measures a positive amount of herbs, we can get a configuration that measures a negative amount of herbs by switching the weights between the pans. Since we aren't interested in configurations that measure a negative amount of herbs (or equivalently, negative sums) there are really only 40 possible valid configurations, which is exactly the number of distinct quantities we need to measure. Therefore, we need to choose our weights in a way that no two configurations add to the same weight. This might be another way to approach the problem.

For students, a way to scaffold this problem might be to ask a similar problem without the possibility of subtraction (in this case, we would get the series 1, 2, 4, 8, ...). A similar but more difficult problem I saw a long time ago, and don't have a good solution for yet, is this:

It is your job to design a new set of bill denominations as currency. Your goal is to create a set of denominations that can express any integer dollar value from 1 to 100 in at most four bills. What is the smallest number of bill denominations you need to achieve this, and what are their dollar values?