I worked on the chapter 7 (Ratios, Conversions, and Rates) review for ~1hr 45 minutes. Chapter 7 Review Problems.pdf (2.9 MB)
There were 23 problems. I missed one due to a division error. So 22/23 = 96%.
There’s also some challenge problems I plan to do tomorrow.
When it comes to practicing word problems I’ve noticed that the word problems outside of the chapter on word problems are more-or-less easy for me. Looking through the textbook:
I guess the difference is word problems require you to figure out the required math and set-up to a problem while in a regular math problem with words that is usually quite obvious?
To get more practice with word problems specifically I used the Alcumus tool on AOPS and set my focus to word problems (the idea just occurred to do this). I did three problems for 30 minutes and got two right. I changed the difficulty of the problems to easy (they were at normal before) to get a feel for the easy problems. I did 5 in 8 minutes and got all of them right. I didn’t set them up quite well thought. I was mainly trying to get a feel for the easy problems.
For the next couple of days I’m gonna spend around 15-30 minutes on practicing setting up the easier word problems and being more methodical with them. When I think I’m getting good at them I’ll retry the normal problems.
Is there anything you want me to work on in the mean time? I could also just heavily focus on word problems for a bit, though I’ll probably get bored.
I’m concerned that you’re practicing bad habits. Please retry 5.46 and 5.55, but focus on following step by step instructions not on solving them. Here’s a method with three sections.
In the first section, put useful information like what some letters stand for or what some numbers from the problem mean. Then put a section divider line.
In the second section, put equations based on the problem statement without doing any calculations. If you have extra equations, mark which ones you plan to use for solving the problem. Then put a section divider line.
In the third section, put calculations. But don’t do that yet. Stop after two sections.
Is the bad habit you’re concerned with is not doing math in a step-by-step manner? Or something else? Is the bad habit with only word problems? Or how I do math problems in general? I’ll wait on practicing and reviewing further problems for now so I don’t keep practicing bad habits.
Step by step methods are an issue for all types of math. Some other issues seem more about word problems not abstract math.
For 5.55, your equations section includes “1/3” but the problem and your information section don’t include “1/3”. I think you’re doing calculation to get it. It’s also unclear what X means (“other money” than/excluding what?) and X isn’t a meaningful variable name (it doesn’t stand for a word). Try this one again.
5.46 is better but the P equation should be P = a - J - L. The concept, as presented in the problem, is that P gets all the apples except the ones J and L got. Using numbers instead of concepts is a calculation step (substitution) that is less good at modeling the problem conceptually or directly. Does that make sense?
Also, I didn’t say this, but the goal is good information to include in the first section. It’s often simple like “find T”.
Just a little confused here on how its said. I assume you mean I have issues doing math step by step for all different types of math? I think that’s pretty accurate, just unsure if that’s what you meant.
Yes. I guess this is a part of my bad habits? Though I’m unclear as to what exactly you think my bad habits are. I know I do tend do stuff like substituting immediately or, as with 5.55, just solve for stuff immediately. Related to that, I used to start working on problems as I wrote them down (still kinda do, just not as much). For example:
As I would write this expression down on paper, I would start working on it. Meaning that the problem that goes on my paper isn’t 4(2-3r) - 1/2 (4+24r), but instead 8 - 12r - 2 -12r and then I solve it from there. This did create problems from me when I was first going through the textbook, so I did stop doing that with these kinds of problems. Though I didn’t take the issue seriously and I still do it here and there.
It does. The problem definitely didn’t and I did calculate it. I calculated it in the information section. My handwriting isn’t that great and I do have some scratches and stuff, but I wrote “…, then we keep the rest 1/3.”
Mmm, they don’t write things down and then work on it. They kind of work on it as they go which is what I’ve been doing.
I do remember that from the previous problems I did with the dogs and the track Eternity Async Tutoring - #123 by Eternity they didn’t approach the problem like you had me do at all.
you can do that with relatively easy problems like this textbook has and still usually solve them, but you’ll run into more trouble with harder math problems or applying the skills to philosophy contexts
you can do it sometimes for some other reasons too but it’s important to also know how to separate the steps
yeah. the goal now is no calculation in the first 2 sections.
your 5.55 looks better now. it could use more variables to break it down into parts more. that’s not necessary here but would be important with a more complex problem.
so you can try the calculation phase for these two now. during that phase, don’t change the first 2 sections.
I kind of broke the rules here? Just for 5.23, it doesn’t say it anywhere in the question but would it be safe to assume that cars have four wheels and motorcycles have two wheels?
5.23:
Yes, it’s making sense. I’ll work on some soon. Do you still want me to just do steps 1 and 2? Or go ahead and do all three?
I gave myself ten minutes to think about it. We still want to find the final score. The score is 93 points in total. I previously did it by using the information that Kumquate was up eight fizzles but down five globbos and immediately converted that information into points. The only thing I could think of is trying to find the exact number of fizzles scored and globbos scored. Is that roughly the other way to do it? If so I can spend a bit more time trying to make it work out.
Mmmm. Nope. I don’t know how to solve anything above two equations. Roughly I remember doing some stuff with matrices in high school to solve them? No useable memory though.
After the first 2 steps, if you think you did them right, you can do step 3 without changing the first 2. If that doesn’t work out, show what happened.
Yes you could model the problem using variables for how many of each thing were scored by each team.
You can do it without matrices by basically using the same method for 2 equations (using substitution) but scaling it up. Can you figure out how that might work? Research is fine.
I missed #4 because I forgot to count Cedric as an invited guest to the party.
I technically did six but one problems wording was confusing me so out of (bad) habit I just skipped it. I didn’t even attempt it. I couldn’t find a history function.
I’ll do some more research tomorrow and attempt a problem. From just quickly looking at some stuff: for a three variable system of equations with equations in terms of x, y and z.
1.) You’d solve one of the equations for x. This means that its in terms of y and z.
2.) You can substitute x in the other equations to get them also in terms of y and z.
3.) You’d then proceed to just solve those two equations like a regular two variable system of equations.
4.) Since you have y and z, you can easily find x now.
Mmm. How would I expand this further? Let’s say you have a four variable system of equations with w, x, y and z.
An equation for w would be made into terms of x y and z. Plug that in to the other equations. Now they’re in terms of x, y and z. You can do the steps above for the new three variable system.
Hmmm. I think I have an idea of how to do it by substitution. I’ll try it tomorrow and write out a cleaner step-by-step.
For 4, your setup is skipping steps. They basically tell you two different things in two sentences which you can model with two variables and two equations. (Or there are other ways to do it, too.) But you have one equation and one variable and it isn’t obvious how you got it. Whenever it’s hard to tell where an equation came from, that means you skipped steps. You did work to figure out that equation instead of just translating what the problem said into equations. You made it it closer to a solution instead of saving all calculation for later.
As to counting Cedric, the problem wording is ambiguous about whether he’s one of the 2/3 or not. Also, when it says “bringing the number at the party” it’s ambiguous if “number” means number of people or number of invited guests. Ambiguities are common in word problems.
I re-attempted problem 4 and it came out the same. Here’s my thought process:
it says “two-thirds of those invited”, “bringing the number at the party to 5/6 of those invited”. I only see one variable here. Guests that were invited.
There were two-thirds invited. Is that one variable? So one variable are the people who had already been there. The other would be the people who arrived after? Maybe? This doesn’t make sense to me.
The equation doesn’t feel un-obvious to me from where I got it from following the question. Probably because I am the one who came up with it.
~10 minutes
System of Equations
Did two three variable system of equations problems. I got the problems from the website mentioned in LMD’s thread. Exercises. The system of linear equations with 3 variables. Took me ~36 minutes to do both. Got both correct. The pages labeled scrap are pages to do calculations.
Seems like the process worked. Step-by-step what I did:
1.) Pick any equation and solve it for one of the three variables.
2.) Substitute that solved equation into the other two equations.
3.) Simplify the other two equations.
4.) You now have a two variable system of equations.
5.) Pick any of the two equations and solve it for one of the two variables.
6.) Substitute the solved equation into the other equation.
7.) This will give you a value for one of the variables.
8.) You can use that value to find the value of the other variable by substituting it in to one of the equations.
9.) You now have two out of three values. You can substitute in the two values in any of the three original equations to find the value of the final variable.
This should work for a four-variable system of equations. I’m going to attempt that next.
