My unit and lesson plan

https://drive.google.com/open?id=1BVK2EUn_fWkCy_S2obX7YroTuzvbWzYV
https://drive.google.com/open?id=1Oe5rUKrhduEYMOPaZfhn8-8y2TKCEMcd
https://drive.google.com/open?id=1p1yBPucDPpVPUhoExsqrDidc6TUKTpb5: unit plan
https://drive.google.com/open?id=1dUYNyHQrrJsRRuq9UD6RoAZXw3Fa3znp

On Thinking About Math Textbooks

Textbook use is a tricky subject. On one hand hand, they can contain relevant academic examples that can be difficult to find, such as the equation of human height and femur length; also, at the grade school level, they contain plenty of exercise question for honing mechanical proficiency. On the other hand, as the authors have pointed out, textbooks can overvalue the formal and deductive approach to a fault (experiments and observations have their role in mathematics), give the impression of issuing commands to the reader in a top-down fashion, and promote detachment in spite of drawing from real-life examples (apersonification versus depersonification).

Speaking of which, when I went through junior high and high school, I do not think I saw much textbook use; custom handouts were already prevalent. Still, if the technology is inadequate or papers are difficult to come by, I suppose there is little choice.

As for the examples (height and femur equation algebra with Figure 2, hand dropping coins with Figure 1, word choice for problems),

as a student:

• Figure 2 would actually lead me astray, thinking that height is a scalar multiple of femur length, when it is not exactly so
• I actually visualized myself doing the experiment in Figure 1 without difficulty.
• Not that mechanical practice is unimportant, but I suppose I might feel like a soldier going through orders when it comes to how exercises are worded

as a teacher:

• I would actually highly value the height and femur equation- here is a Diophantine equation being used for anatomy in real life! I would try to present a diagram drawn by myself based on Figure 2 rather than presenting it as-is.
• Figure 1 could potentially be interpreted as "the teacher" or "the author" doing the experiment. Thus, it may do me well to prompt students with phrases like "visualize yourself..."
• Knowing that language is important for promoting engagement, as far as mandatory worksheets go, I would probably create my own with words chosen deliberately towards that end.

On the Scales Problem

You would have to get started somewhere; on that note, you must have a 1 gram weight. However, if you need a 2 gram weight, a 3 gram weight, et cetera, this gets nowhere quickly.

Let us proceed by a greedy approach, where we try to come up with weights such that the widest range of weight values are covered with each new weight, because why not? Besides, "higher" optimization techniques like dynamic programming are motivated by the failure of the greedy approach anyway.

What if you had a 2 gram weight: Then, with a 1 gram weight, you can measure:

• 2 grams: self-explanatory
• 3 grams: put both weights on one pan
• 4 grams: impossible

What if you had a 3 gram weight? Then, with a 1 gram weight, you can measure:

• 2 grams: put the 3 gram weight on one pan, and the 1 gram weight on the other pan
• 3 grams: self-explanatory
• 4 grams: put both the 3 gram weight and the 1 gram weight on one pan

What if you had a 4 gram weight?

• 2 grams: Impossible
• 3 grams: put the 4 gram weight one pan, and the 1 gram weight on the other pan
• 4 grams: self-explanatory
• 5 grams: put the 1 and 4 gram weights on the same pan

At this point, maybe we should keep the 3 gram weight. Let us select another.


What if you had a 5 gram weight? Then,

• 5 grams: self-explanatory
• 6 grams: 5 and 1 gram on the same pan
• 7 grams: 5 and 3 grams on one pan, 1 gram on the other
• 8 grams: 5 and 3 grams on the same pan
• 9 grams: 5, 3, and 1 gram on the same pan
• 10 grams onward: impossible

What if you had a 6 gram weight? Then,

• 5 grams: 1 gram on one pan, 6 gram on the other
• 6 grams: Billie Eilish (duh)
• 7 grams: 6 and 1 gram on the same pan
• 8 grams: 6 and 3 grams on one pan, 1 gram on the other
• 9 grams: 6 and 3 grams on the same pan
• 10 grams: 6, 3, and 1 grams on the same pan
• 11 grams: impossible

Skipping a few, what if you had a 9 gram weight? Then,

• 5 grams: Put the 9 gram on one pan, and the 1 and 3 grams on the other pan
• 6 grams: Put the 9 gram on one pan, and the 3 gram on the other
• 7 grams: Put the 9 and 1 gram on one pan, and the 3 gram on the other
• 8 grams: Put the 9 gram on one pan, and the 1 gram on the other
• 9 grams: Duh
• 10 grams: 9 + 1 on the same pan
• 11 grams: 9 and 3 on one pan, 1 on the other
• 12 grams: 9 and 3 on one pan
• 13 grams: 9, 3, and 1 grams on the same pan

What if you had a 10 gram weight? Then,

• 5 grams: Impossible

At this point, we should keep the 9 gram weight. I think we see a pattern here: as we seek the next weight to add, keep adding the candidate weight by 1 until the "minimum" becomes untenable in a sense. For example, we abandoned the 4 because there was no way to make 2, settling at 3; we abandoned the 10 because there was no way to make 5, settling at 9.

I am going to conjecture that the next weight to add is 27 grams:

• 14 grams: Put 27 on one pan, and 9, 3, and 1 grams on the other pan
• 15 grams: 27 on one pan, and 9 and 3 on the other
• 16 grams: 27 and 1 on one pan, 9 and 3 on the other
• 17 grams: 27 on one, 9 and 1 on the other
• 18 grams: 27 on one, 9 on the other
• 19 grams: 27 and 1 on one, 9 on the other
• 20 grams: 27 and 3 on one, 9 and 1 on the other 
• ...
• 29 grams: 27 and 3 on one, 1 on the other
• 30 grams: 27 and 3 on one
• 31 grams: 27, 3, and 1 on one
• 32 grams: 27 and 9 on one, 3 and 1 on the other
• 33 grams: 27 and 9 on one, 3 on the other
• 34 grams: 27, 9, and 1 on one, 3 on the other
• 35 grams: 27 and 9 on one, 1 on the other
• 36 grams: 27 and 9 on one
• 37 grams: 27, 9, and 1 on one
• 38 grams: 27, 9, and 3 on one, 1 on the other
• 39 grams: 27, 9, and 3 on one
• 40 grams: 27, 9, 3, and 1 on one

With 28 grams:

• 14 grams: Impossible

I do not think that any other solutions are possible; either you will have some weights that you cannot make, or you will end up needing more than 4. Thus, this is in fact optimal.

I suppose you can extend the puzzle and help them develop a more precise understanding by making the weight bigger, asking more questions, et cetera:

• Suppose that you need to be able to measure up to 121 grams now.
• If you have a 41 gram weight...
• How far can you measure up to?
• How many more weights would you need to go up to 121?
• Repeat the questions for a 43 gram weight, 75 gram weight.
• What is the minimum number of weights needed to be able to measure up to 121? How much should each weight weigh?