(Sept 28) Balancing the Scales

 To solve this problem we must start by going through the cases. First 1 gram of herbs. To measure this we obviously are going to need a 1 gram weight. Then 2 grams. To measure this We can either use 2 one gram weights or if we put the 1 gram weight with the 2 grams of herb we can get use a new weight that is 3 grams. So we have 2 distinct weights. Now 3 grams we can use the weight we have. For four we can use the 3 and 1. For 5 we do not want another 1 gram weight, so if we put both weights with the 5 grams of herb, we can get a new 9 gram distinct weight. Now skipping some steps the next number we get stuck on is 14. But we only have one weight left so we need to be able to reach 40 grams with the remaining weight, so it must be 27g. Can we get 14 with this weight? If we put all 13g of weight with the 14g of herbs we can balance it with our 27g of weight. And the rest of the weights fall into place as weights can be placed on both sides. We can see that the pattern is 3^0, 3^1, 3^2, 3^3, but why? It must have something to do with how each weight has 3 paths. Leave it off, or place it on either scale which adds or subtracts weight. So a 5 weight one pan system therefore must have a similar pattern. One pan scale reduces the choice of each weight by one so each weight can only be put on or left off of the scale. So this should be in powers of base 2. 2^0, 2^1, 2^2, 2^3, 2^4. Or 1, 2, 4, 8, 16. This system can only weigh up to 31 grams as that's what it adds up to, so if we want to weigh more we would need to add a 32g weight. Checking a random number 25g of herbs we can see that this is just 16+8+1. Checking all steps this does work out. This ties directly into how ancient Egyptians used doubling to solve multiplication problems. Where they would just double numbers and add the results which is what binary decomposition is in the present day. We can extend this problem by creating a new problem with 4 choices and now students can see the pattern clear as day! Where the number of choices changes the base of your solution. Doing this type of "long" mathematics will actually help students take detours to find solutions. In schools, we are constantly taught one way or the most efficient way to solve a problem. For example, memorizing multiplication tables, or calculating derivatives with the "rule". We forget what the true meaning of multiplication is and just take it at face value. Before we move on to shortcuts we MUST understand the foundations so that when the "trick" doesn't work we can find a multitude of different ways to find the solution. That's what mathematics is about, finding different ways to the same solution.

Comments

Post a Comment

Popular posts from this blog

September 14th Reading

 Math history should be incorporated into teaching as it builds conceptual knowledge in a way that can't be replicated by shortcuts. If a student somehow memorizes every single addition problem, they still won't be able to apply this knowledge to understand the concept of multiplication. Understanding how people in the past came to understand mathematical concepts recreates the feeling of discovery and really embeds the knowledge into your head compared to just taking shortcuts. Math history can be applied in classrooms by recreating the steps and thought process used in the past to discover the math facts in the first place.  The article mentions using worksheets to facilitate this type of learning and integrating math history into the lesson, but I always believed worksheets are not very effective for retaining information, as I always found worksheets to be boring and ineffective. I think that, to facilitate a feeling of discovery, an activity would definitely offer a bette...
Read more

Sept 23 Reading

 I hated mathematical symbols. I've always thought that math should be written out in words, so the Babylonian rhetorical algebra actually makes more sense to me, disregarding the odd language of course. When I got into higher level maths that started using function notation like f(x) in grade 10, I could not wrap my head around it. It made no sense to me, and later math courses did not get any better. Once I approached calculus it was like nothing was written in words anymore, nothing made sense anymore. And discrete mathematics and mathematical proofs often used more symbols to represent things that can easily be stated in just plain old English. So, similar to how proofs can be written out without symbols, every math problem should be able to be listed out step by step without any symbols as symbols just represent words, it's just shortening a word to a universally recognizable symbol. Mathematics is all about taking small things and applying it to everything, expanding it t...
Read more

Sept 21 reading

Reading time during the day was never something I've given more than a second to think about. I would just pull out my phone and read the time straight off the phone screen. I didn't consider where it had come from as time feels inevitable and I assumed that this way of counting time by 60 seconds 60 minutes and 24 hours was inevitable as well. Reading the two articles presented offered me a new perspective on time and showed me how complicated it must have been to come up with this system that I take for granted everyday. I did not know that we had leap seconds that occasionally occurred on top of the leap years that we already have. The two articles touch upon different fields for the origins of this counting convention, as one seems to focus on theories more rooted in logic while the other focuses on theories rooted in function. Seeing how the people in the past came up with these number systems purely based on counting on their fingers or their finger joints and built all o...
Read more