The Dishes Puzzle

 

There were 60 guests. It is important to note that technically, there could have been 61 guests as well as this would also yield 65 dishes (59 guests would give 62 dishes and 62 guests would give 66 dishes). 

The only way I could think of to solve this question without using algebra was to draw out a table of values. I actually did this on Excel and I was thinking, well it's easy to figure out how many dishes based on guests but it's difficult to go the other way without using algebra. After 12 guests, the number of dishes always exceeds the number of guests so I was thinking you could also do trial and error by starting with (n-1) if n is the number of dishes 

It definitely is beneficial to offer these kinds of math puzzles highlighting a diversity of cultures however I can't help but think of the concept that if someone is excluded, then someone is included. It's impossible to grasp all cultures in one problem. As a teacher, if I included a problem like this then there will certainly be students from other backgrounds that may feel excluded. Obviously as I introduce more puzzles as the school year continues, I would ensure all cultures represented in my classroom are presented. It's about patience and building that trust. I do remember as students we would get giddy when some of our names appeared on test questions and the teacher made sure to rotate and use all of us. Even if we weren't on one test, we knew eventually we would be represented, and that was important. 

Personally, the reality of the question helps. I'm not too particular on imagery if the concepts are too farfetched - as we discussed in class, no, Susan will not be buying 83 watermelons. However, a good balance between imagery and reality, as well as representation always makes mathematics more fun.

Ancient Problems in Modern Ways - A Reflection

For our presentation, Caris, Brandon and I tackled the volume of a truncated pyramid. To solve it using modern mathematics, the truncated pyramid was presented as the summation of 3 different shaped, a cuboid, four corner pieces (which formed a pyramid when combined), and four triangular prisms (see picture). I was having immense difficulty condensing the formula after adding the different volumes together and I was not able to come up with the correct formula for the volume of a truncated pyramid when collecting like terms. I couldn't see what I was doing wrong and after discussing with my group mates, Brandon was initially able to solve it by using the ratio of b and a to the ratio of the height of the pyramid H to the height of the truncated pyramid h. The entire method requires solving it through a limit process by using integrals but I was really confused why I wasn't able to just add up the different volumes and combine like terms to get the correct formula. I was sitting there racking my brain over the difficulty and I couldn't help but admire the simplicity of the method of ancient Egyptians. As I discussed in class, all they did was average the areas between the base and the top and multiply it by h. Then, they realized they didn't do it properly so added a "median" area of ab and took the average of all 3 and multiplied that whole result by h. 

I recall a unit sometime either in elementary school or high school where we learned about estimation and I vividly remember sitting there and thinking this has got to be the dumbest unit. Why don't we just do the same amount of work (because educated estimates require some thinking) and actually solve for the correct value. Estimating cannot be overlooked and honestly, it might be the concept we use most in our every day lives out of everything we learned in public school mathematics. It needs to be mentioned that the Egyptians were not regular mathematicians - there is no proof for the logic behind their incredibly accurate mathematics despite the lack of modern mathematics knowledge. But their ability to lean on
"rough", educated estimates is a testament to its applicability considering its success. I went down a YouTube rabbit hole just yesterday watching a stonemason create different shaped stones using a chisel and hammer. His accuracy was remarkable. At one point I thought, "but that isn't a perfect triangular prism" and then it hit me that these stones would still be used to make driveways, stone houses and other things. And if I had seen it in real life as a completed project I would be able to admire its beauty without thinking "wow but this side of the triangle isn't perfect". There's a lot to be said about that. At the end of the day, practically we don't use high-powered lasers to create perfectly level sides when we're shaping stones - we're creating roughly accurate shapes. I don't know, it's something I'll continue to ponder on for sure. 

Market Scale Puzzle

Going through elimination (and with a hint from Saiya), the four weights have to be 1, 3, 9 and 27 grams. We can weigh up to 4 grams with the weights 1 and 3. We can weigh up to 13 grams using the weights 1, 3 and 9. We can weigh up to 40 grams using all 4. The weights would have to be used in a variety of ways such as weights on both sides, omitting some weights (ie. for 30 grams we would only use the 27 and 3 gram weights), etc. I was talking to TsáKtalay’pa a little while after receiving this problem and he said he didn't use the weight 3 (if I remember correctly) so the wording in the question "must" is interesting - I wonder if there is only one combination or if there are multiple ways to weight 40 grams. 

On a one pan scale you would need the weights 1, 2, 4, 8 and 16 to weigh up to 31 grams. This problem was surprisingly much more straight forward since there was only one pan that can be used to weigh. As soon as you maxed out the amount of weight, you knew the next weight needed - for example with the weights 1, 2 and 4 you can weigh up to 7 grams and so you know the next weight needed has to be 8. 

I feel like the two pan scale problem is really counter-intuitive in the beginning. Although I can't think of any explicit connections to the secondary curriculum I do believe it could induce a larger theme around exploratory mathematics. If students took this on in a classroom activity with a real two pan scale and different weights, I strongly believe it could challenge their initial perceptions about how to combine numbers. It's almost like a physical manipulation of left side equals right side, and all the different ways that weights (terms) can be moved around and utilized to have both sides be equal.