Unit Plan

Unit Plan

*updated December 1st

Thinking About Math Textbooks

When I read through the article about first and second person pronouns, I was surprised to read that first person pronouns were not in either of the textbooks that were used for the purposes of this article. When I think back to my personal experiences with mathematics textbooks, most if not all examples of person pronouns involved the word 'we', which would fall under first person, NOT second pronoun. Perhaps it is my recency bias playing a role, and undergraduate math textbooks use these first person pronouns, whereas secondary textbooks still exclusively use second person pronouns. If so, then I must ask, why? The article states that second person pronouns helps to connect the speaker to the reader directly; does this only matter for adolescents, and not for young adults?

I understand that today there is a push to not use textbooks, and some teachers are slowly becoming less dependent on them. I've also talked to some teachers about this topic, and it's apparent to me that it is quite the controversial one. However, I still think that textbooks serve a purpose and shouldn't be rid of entirely. Of course, textbooks are expensive and a new version with minimal changes seems to come out every few years. But they also contain a wealth of information in an organized structure, with many additional resources embedded in them. I will say, I think teachers should opt to make their own examples and not 'teach to the textbook', as it allows for them to adapt to their unique class and their needs, compared to the general examples and rigid format that a textbook supplies.

The Scales Problem

We have four different weights to choose from. These will be labelled as a, b, c, d, with each letter be written in ascending order of the weight. Since the vendor sells herbs ranging in weight from 1 g to 40 g, the smallest weight must be 1 g (i.e. a = 1 g). In addition, the sum of the weights must add up to 40 g, such that

a + b + c + d = 40 g

Now, in order to measure 2 g, we could say that the second lightest weight is 2 g. However, another way of doing this is by letting a mass of 2 g be achieved via a weight on both pans. Therefore, a subtraction is needed, and so we have b - a = 2 g. Since a = 1 g, it follows that b = 3 g. With a and b solved for now, I can measure weights of 1 g, 2 g, 3 g, and 4 g. But how can I measure 5 g? Well, the third weight, c, will allow me to. If 5 g is the smallest weight I have yet to obtain, then I must subtract both a and b from c in order to achieve it. So c - a - b = 5, or c = 5 + a + b = 5 + 1 + 3 = 9. This just leaves d now, which can be solved for from the above equation: d = 40 - 1 - 3 - 9 = 27. Thus, the value of the four weights are 1 g, 3 g, 9 g, and 27 g. Looking at these values, these numbers didn't come from nowhere, as it is quite apparent that they follow a distinct pattern: 3^(n-1), where n = weight number. A quick check shows that these four weights account for all the different weights from 1 g to 40 g.

I don't believe there are several correct solutions, as no other way will allow you to account for each weight distinctly, but I could be wrong as I did not delve too deep into proving it.

This puzzle could be extended by looking at different maximum weights (1 g, 4 g, 13 g, 121 g, etc.) and noticing the pattern for which weights you need, and then trying to deduce the formula that tells you the specific weights you need. This could tie in quite nicely with a unit on sequences and series.