Spent ~32 minutes. I did all the practice. I got none wrong and found nothing confusing.
I had watched this video previously: Eternity Async Tutoring - #42 by Eternity.
Great. Take a look at Multi-Factor Decision Making Math and see if you can find any math parts that you might not understand.
Done. Spent ~1hr 10 minutes reading it (not including breaks). I read this before around eight months ago along with the Malcom Gladwell article.
In terms of the math used in the article, I donāt think there was anything I didnāt understand. There was some math referenced that I donāt understand such as set operators, unions, and vector math.
Was there anything non-mathematical in the article that you didnāt understand?
No, nothing that Iām aware of.
What are some areas of math, related to arithmetic or algebra, that you think might be good to work on? A long list would be good if you can think of more than a couple.
I donāt know what relates to philosophy and what doesnāt. Iāll just list what areas of math related to arithmetic and algebra I donāt have much experience with/I know I mess up a lot with.
I just kinda of looked at various sources to break down these subjects.
Arithmetic:
- From your philosophy outline from May 2022.
In the sub-tree numbers, I donāt have much experience working with number lines. In LMDās tutoring category, he used a definition of addition based on number lines and successors and stuff. First time hearing of that. LMD Async Tutoring - #75 by LMD
I also remember having a pretty high error rate for percents and ratios for the problems in the textbook I was using(Art of Problem Solving - Pre-Algebra). Some of it could be attributed to how hard the problems are but I remember having some conceptual difficulties.
- From the Pre-Algebra textbook
I remember really sucking at word problems.
āWhenever a quantity changes by a certain amount in a fixed unit of time, we have a rate.ā - ch 7.6
Rate problems I was pretty bad at. I think i struggled on all the basic problems and skipped the challenge problems for this section.
Huh, thereās a chapter on counting.
I never got to it because the basic geometry before it made me drop the textbook for a while. I also saw counting was a node on the arithmetic tree. I donāt know much on counting problems I guess.
Algebra;
In general iād say Iām rusty at doing algebra.
In Cycle Between Learning Critical Fallibilism and Its Prerequisites you say
(Note: Algebra classes commonly teach quadratics, but thatās not a core part of algebra and shouldnāt be taught to most high school kids at all. Quadratics arenāt relevant for CF. More core parts of algebra, that are more relevant to CF, include understanding variables, equations, substitution, grouping and functions.)
So Iām just ignoring quadratics. I think I was pretty mediocre at them. Mostly just memorized stuff.
- From the philosophy outline
The whole functions sub-tree is stuff Iām vaguely aware of and pretty unfamiliar with. Graphing and coordinate plane are probably fine, but sequences and mapping I got no clue on. Sequences I remember vaguely and I did them in a memorized manner. Same with mapping.
- From AOPS Intro to Algebra textbook
I have not used this textbook. Iām just glancing over the table of contents as reference for stuff in algebra.
Iām unfamiliar with this proportion stuff:
I suck at optimization problems.
My understanding of complex numbers is pretty non-existent. My math knowledge of them is purely on the memorization level and even then not that much.
Hmm, looking at the functions chapter and the problems. I think I can do the problems pretty ok. Could I explain to you functions? Nope.
My knowledge of logarithms is all memorized moving stuff around.
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This isnāt the topic weāre going to work on, but since you already did a relevant problem recently, letās check if graphing and coordinate plane are fine:
Without making the graph, doing calculations, or scrolling up to find your earlier answer, can you describe in short, broad, general terms what the graph of the equations would look like and how to find the solution by looking at the graph?
After answering that, can you make the graph? And explain how an equation relates to a graph.
Those are both linear equations. They would make lines. You find the point where the two lines intersect and you take that (x,y) value as your solutions to the system of equations.
i did it by hand. The graph has an x-axis and a y-axis. In an equation you put in a value for a variable, get an output value and then you graph the values. So with y = 6 - 3/2x, you put an x-value and it outputs a y-value. For example, you put in x at 0 and you get y=6.
Oh yeah, isnāt this kind of related to functions. I remember functions having something to do with putting some input in and getting some kind of output out.
Can you give some examples of what you have trouble with for fractions, decimals and percents? Maybe do some problems that youāll get some of wrong?
Fractions
I donāt think I mentioned having trouble with fractions per se. I did some fraction stuff for ~35 minutes. I donāt know how many problems it was. Probably around 20. I got one wrong, shown below and that was due to mis-understanding what the question asked (I thought it said how much does he have left). The question below it is something I almost got wrong until I realized my answer was complicated. AOPS doesnāt shy away from complicated answers, but I doubled checked the question and realized I misunderstood what it was asking.
Any question related to just math with fractions went fine. Its when it becomes a word problem and requires me to understand what itās asking of me do I mess up. For both problems the stuff I got āwrongā was mathematically fine. I solved my misunderstanding correctly as far as I can tell.
Decimals
Same with fractions I donāt think I mentioned having a conceptual difficulty with this. The math, or I guess the working more so, I did was right for the below problem. However, I didnāt quite understanding what it was asking. I did ~13 problems of varying type and difficulty for ~22 minutes. Only the below wrong.
Now that I think about it. This is a problem I just kind of internalized for a while, but Iāve always been kind of sloppy in reading questions. I remember a good amount of questions Iād get wrong in school were from small misunderstandings or from me being lazy in writing out my work and steps. I have a bad habit of trying to do a lot of mental math even when its something I should just write out.
Related: I found this in my online textbook notes
Percents
I did mention having issues with this. Reviewing my notes and problems most of the stuff was fine. The stuff I did with percent increase and decrease in something was where I really struggled. This particular challenge problem bothered me at the time. I had certain assumptions of how it should go, they didnāt match and I think I got tilted/confused by the explanation and just moved on. I will do some problems and re-attempt the challenge and share tomorrow.
Oh yeah fractions and decimals were practiced from Alcumus a problem practicing tool from AOPS. The percents question is directly from the textbook though.
Can you convert between fractions, decimals and percents easily? Convert each of these to the other two types: 4/5, 12/9, 1/3, 0.2, 0.7, 2.4, 5%, 225%, 90%.
Can you divide 3/4 by 17/23?
5 out of 25 is what percent?
What out of 30 is 40 percent?
3 out of what is 150 percent?
Looks good. Iād suggest writing leading zeroes for decimals in the future, e.g. 0.8 not .8 (it helps readability).
Also, to confirm, you did that with no calculator right?
To me, those look primarily like algebra word problems. The issue might be percents but might not.
If you want to increase a number by 28%, what do you do? What if you want to decrease by 5%? What about increasing by 350%?
For the 8.3.11 a and b problems, can you show me how youād set them up? Iām guessing your issue is more with setting up the problem than doing the calculations.
For ratios, if I put 1 cup of sugar in a bowl for every 3 cups of flour, what is the ratio of sugar to flour? What fraction of the stuff in the bowl is sugar? If thatās easy, answer anyway, and show me a ratio problem that was hard for you.
Mhm. All of it was done with no calculator.
You find 28% of that number. then add 28% to the original number. 28% of 100 is 28. Then youād add 28 to 100 and you get 128. Same process with decreasing youād just subtract. 5% of 50 is 2.5. So a decrease by 5% would be 50 - 2.5= 47.5. 350% of a number would be 3.5 times that number added to itself. 10 * 3.5 = 35. A 350% increase would be 45. Iām pretty sure thatās right, but something about increases or decreases about 100% feel weird in my head. A 10% increase in 100 to 110 feels fine. A 200% increase to 300 feels weird. I think my brain is skipping the increase part when its increasing over 100% so it subconsciously looks wrong. A 200% increase to 200 feels right even thought it isnāt.
Yeah I guess another way to put it is these particular word problems Iām getting confused by. Or I did. Earlier this morning I did 10 problems for about 20 minutes and got none wrong. I also reviewed some chapter problems and was able to do them. One issue here I forgot to mention was with the associative property and percent increases and decreases. I understand it fine now, but previously I remember getting confused with certain word problems where I thought the order in which you did your percent increases and decreases mattered. When I saw that it didnāt matter, I understood the math but it wasnāt clicking cleanly for me (kinda like increases over 100%) until today.
I attempted them earlier today. I already reviewed the solution (though I did do that last time). This time I understood what was going on.
Previously, from what I remember, I was confused as to where the p/100 came from (now I understand its just p%) and how they turned 100+p into (100^2 +100p)/100. I did part b (they follow similar steps) and got the same answer which makes since due to the associative property (I think?). My issue here is this part here feels unintuitive. Other formulas Iāve done in this textbook I remember feeling clarity and understanding. This one feels like a memorized formula in my head.
Hereās my own working after the fact (its only up to a point). I like it more then how they showed it. The end formula still feels unintuitive to me. Who know maybe Iāll get it tomorrow?
The ratio is 1 to 3 or 1:3. If you were to have 2 cups of sugar you would put in 6 cups of flour. Thereās 4 cup of stuff in total in the bowl. 1 cup of sugar and 3 cups of flour. So 1/4 of the stuff in the bowl is sugar.
Iāll practice some problems tomorrow, but its specifically rates where my brain turns to mush.
āSpeed is the ratio of distance to time.ā
I remember skipping this one:
And this one
It may be more than just the math. I just remember the previous sections went fine and then I really had a hard time understanding what was going on in these rate problems.
Thatās a good answer but thereās also a different way to do it thatās also worth knowing. If you want to increase 30 by 10%, you can do it without adding. Any ideas?
Have you been reading LMD Async Tutoring and followed the math from that topic?
One thing Iād do differently is convert all percents to fractions immediately. Itās fine to include percents when initially setting up the problem (translating the words to math), but when you start calculating/simplifying Iād get rid of them.
For those rate problems, you can start by writing down things you know (the pieces of information in the problem) and then brainstorming equations. Equations involve finding two different things that are equal. If theyāre unequal but related in some clear way, you can make them equal, e.g. if something is triple the other thing, then add a division by 3 to get them equal. If you find a way to calculate something using other information, then thatās an equation too: calculating something means being equal to it.
Do 110% of 30? It skips the adding step. If you did it with adding it would be 10% which is 3. Then youād add and get 33. You can just do 110% of 30 and immediately get your answer. 30 * 1.1 = 33.
Would this work with decrease too? Letās see: 15% decrease from 200. 15% is 30. So 170. 85% of 200 is 170.
Let me also check increases over 100%. 400% increase in 20 is 100. 5 * 20 is 100. Ok.
Is this the way you were looking for?
Iāve been following it. If youāre looking at the comment about the associative property, that knowledge didnāt come from their thread. I was already familiar with it. My confusion with the associative property more so had to with it being a word problem. I forgot where I saw this but it was someone making a comment on the associative property and how as soon it came to personal finance stuff people care about the order in which their interest is calculated even though it all, in due time, multiples out the same. In a similar way I kind of intuitively thought that for the word questions involving decreases and increases that the order mattered. I think part of the issue was not, until recently, quite understanding that percents deal with multiplication to an extent.
Yeah. The main way to think of percent increases or decreases is multiplying by something to change the amount.
Can you write it as a generic formula? Like if n is the number, and p is the percent to increase, can you set up an expression to represent it? And then you could substitute in actual numbers and get the right answer for a specific case.