I don’t think you should feel tricked when the problem description isn’t just laser focused on giving you the exact information you need with nothing else. That kind of problem is extra easy and is different than real life. The potential for confusion is something you have to deal with for real problems. It’s normal not a trick. A trick is when something is actively misleading.
So I think you’re looking at it wrong and used to school-type problems and should try to be more robust in how you approach problems (even when the problems are easy enough you don’t have to) so you get confused less easily so your skills will translate to real world problems better.
Yes. And get better conceptual understanding. And get more experience with numbers/math to develop more intuition and number sense.
I think you could have gotten this right if you had approached the problem more methodically. You skipped steps when setting up the problem. Your initial information gathering covers 60, 13, 1 and 1/8 but doesn’t have 7 or 17 which are also part of the problem. And you didn’t identify your goal before you started doing arithmetic.
~yea. Word problems/setting them up was one of the things we worked on previously. I think I got fairly good at them at the time and then I didn’t get much practice. Also, maybe because of how long ago we worked on it, I don’t really think to be methodical doing word problems anymore again.
I’ll review the steps you shared before and try them on any upcoming word problems.
A thought that came to mind that I haven’t fully checked:
I think the rule was something like you don’t have to check past a square of sorts. So like for checking if 23 is prime. You would only have to check the numbers up to the square root of the closest square greater than the number. In this case it would be 25. The root of 25 is 5. So you only need to check the numbers 2,3,4. If none of those work the number is prime. I’m too tired to work out this math right now (I just thought of this after going for a walk) but I think this is correct? But why is it like this? hmm. in my head it went something like this:
let’s say you have 23. the reason you don’t go past 5 is because I think it has to be composed of the numbers prior to 5. like let’s you checked 6 (after checking out 2-5), well you know that 2-5 won’t work, but if it isnt a number from 2-5 then it doesn’t work. because any number after will be too big. 6^2 is bigger than 5^2. and if 2-5 did work and it also did have a 6 you would have already gotten that from the prime factorization.
i think i have the right idea. i’ll flesh this out more tomorrow.
I think my idea is more or less the same from my previous post. This is more so an attempt to write it neater and have some more confidence in it.
The rule for checking whether X is prime is to check whether X is divisible by smaller numbers until the square of one of the divisors is greater than X. For example, using 31:
You would check 2, 3, 4, 5 , and 6 (though in one sense 4 and 6 could’ve been skipped because the numbers that make up 4 and 6 don’t work as divisors).
You stop at 6 because 62 is 36, greater than 31. The reason you stop here is because:
Since the number is not composed of numbers smaller than 6 it may be composed of numbers greater than 6, except
if its composed of a number greater than 6 then that number, lets say 7, has to be multiplied by a number larger than 2-6. Except if we do that we can only get a number bigger than 31 (72 = 49).
I noticed that when finding decimals that infinite decimals only came up with things that don’t divide nicely with 10 or 2 and 5. So like if the denominator was something like 3,7,9, etc. (nothing composed of 2 or 5) then you’d get an infinite decimal. I thought it had maybe something vaguely to do with our number system being in base 10 (I don’t understand other bases), so I googled it. It seems like my hunch was right. Well if this reddit post is correct that is:
Decimals are in base 10. 1/3 is a repeating decimal because 3 doesn’t divide 10. If we used base 3 instead, 1/3 is written as 0.1 and so what is and isnt ‘repeating’ is really just a symptom of how we write numbers. For this reason, mathematicians tend not to think of decimal representations of fractions as different kinds of numbers based on whether they are repeating or not - they’re all called ‘rational’.
Are you aware that you only need to check for prime divisors? If a number isn’t divisible by any primes, then it’s prime. You don’t need to check 4.
Also, you don’t need to check 6 because it’s too big. To get 31, you’d have to multiply 6 by something smaller than 6. If you multiply it by a 6 or bigger, you’d get 36 or more, so that can’t work. And you already checked the smaller numbers.
When you’re checking if X is prime and looking at number N and N^2 > X, then you can stop without checking divisibility by N.
BTW, you should memorize the primes up to at least 19 and probably higher. Among other things, this helps with factoring small numbers. And it should be easy if you’re decent at math and have enough experience with numbers. I also have the prime squares up to 13 memorized.
Yes. I was going to make that a point, but then I guess I started overthinking it in my head? Something like this:
Your checking if this number is prime. Now I know you only have to check which the prime numbers but, to me, it seems then your doing double the work. Your checking this number for being prime and checking which numbers are prime to factor this number.
Oh I think thats what I meant. You don’t have to check 6. You would stop at 6.
I know the primes up to 31. Are prime squares the square of primes? I know the squares of numbers up to 13.
Yes. Good. (BTW, to be clear, I meant up to 13 including 13 itself. I know a lot of people learn their times tables up to 10 or 12 inclusive but don’t know 13^2.)
As to the actual process of measuring shapes, a vast part of higher mathematics, from geometry on up, is devoted to the task of discovering methods by which various shapes can be measured—complex methods which consist of reducing the problem to the terms of a simple, primitive method, the only one available to man in this field: linear measurement. (Integral calculus, used to measure the area of circles, is just one example.)
that I highlighted a while back. Is this accurate (I know you’re not a mathematician, but I think you know quite a bit of math)? Just curious. Looks like I was confused in the past too. I had it highlighted with a note just saying: ?.
I read this as saying that most of mathematics does this. Something like set theory, discrete mathematics, and other fields I don’t know of are smaller parts of math. I don’t know if I read it correctly. Another reading I thought of: there is a field of higher mathematics that deals with how various shapes can be measured. Without looking it up I think topology is the field that covers this.
Something I thought of while coming across a quote in my reviews from a pro charter school perspective:
People who are pro something don’t really do a good job (or don’t do a job at all) of addressing criticisms related to the thing they are pro of. They usually ignore the criticisms, pretend they are not criticisms, actually think they’re not criticisms without thinking about the criticisms, or do a poor job addressing the criticism but consider it addressed.
I thought that kind of thing sucks and that people should do a better job of addressing criticisms, but then the thought came to me that, in some contexts, that doesn’t matter. Like this isn’t an argument against addressing criticisms per se, but even if you address them in a high quality intelligent manner there are a lot of people who just won’t listen. Their response to your addressing their criticism is similar to the above: ignore it, act like they aren’t criticisms, or do a poor job refuting that criticism and then consider it refuted.
I guess this is due to, in part, tribalism. I wonder what other factors play a part here.
The primary reason to address criticism is for your own knowledge, not to influence others. If you ignore criticism, then you don’t know if your own ideas are right or not.
2.) Got this right the second time. Two issues: I kinda wanted to keep the work to one page so my work became sort of a mess and, related to that, I messed up some division on the first go leading to the wrong answer.
3.) Got this wrong. I dislike the wording of this question. Apparently we were looking for what fraction do we take out of 2/3 to get 1/4. I read it as he has 2/3 of a pizza, he then proceeds to give 1/4 of that 2/3 away. Is my reading comprehension the issue here?
Oh yeah. I don’t think it matters, but the day I share these problems is the day I did the problems. If I don’t share any problems on a day. I have’t done of them. Not like this is an assignment or anything, but, idk, maybe theres an impression that I’m doing well on problems that I’m not sharing and I only share when I mess up or something? thats not the case.
Then what happened after you found setup hard? Did you continue working on it until you had a setup you were satisfied with? Get stuck and stop? Proceed to calculate using a setup you thought was confused?
The wording is fine. You’re incorrect here. It specifically says he gives away 1/4 “of an entire pizza” not of the portion he has.
Doing work neatly and using as many pages as needed (or a computer) is an important skill or attitude. You need to improve at that to get the problems right. You also need to be able to get the arithmetic right and need to check if your answers make sense. You can’t rip paintings up into pieces. Your answer shouldn’t be a fraction. Also you wanted to give him less than one painting. But he should be getting close to a million dollars, which should clearly be more like 2 or 3 paintings. You’re treating the numbers as totally arbitrary and meaningless, and just trusting whatever your calculation says, instead of paying attention to them and thinking about them.
Sorry. Didn’t communicate that. The problem felt too hard to set-up so I got stuck and stopped.
I get that. I think my bigger confusion is the later part of the question. So if he were to give away 1/4 of an entire pizza to Marlene (so we’re giving 1/4 of an entire pizza no?), then what are we talking about with the “fraction of his partial pizza”?
Hmm. Though some issues I have with doing those:
What is neat work(I do acknowledge that the work I should their isn’t neat, but I don’t much of my work is that neat)? Idk how to put it. I’ll do some problems in a bit with my best attempt to keep work neat.
In terms of using a lot of pages: one reason I do that is because I like seeing everything involved with that question on one page (or two pages in a regular notebook). Sometimes I wanna check the steps I did and going back to other pages is annoying/hard to keep in my head(?).
So:
I agree you can’t rip a painting into pieces. This is a habit that stemmed from school that I still kinda do even though I’m not in school and Alcumus doesn’t do partial credit. The point about it not being school: I definitely had questions in school where things that didn’t make sense to be split up, were getting split up. A pizza makes sense. A painting does not. I agree, but out of habit I still assume maybe thats what the question is asking. The other part about partial credit: sometimes I would leave answers that I know to be incorrect because its better than no answer. Again it doesn’t make sense for Alcumus problems I’ve been doing, but yeah. The main reason though was definitely the first reason.
Oh yeah: I remember now, because the answer didn’t work out to an even division I just assumed it was a weird problem where the paintings aren’t actual good representations (like how I mentioned above) and I would put a decimal. That also played a part in why I just kinda ignored the issue with how many paintings he should’ve gotten. I did think why did he not get enough, but my mentality was like “Ehh crappy problem”, not that I messed up (at least initially).
