# AOPS Intro to Counting & Prob. + Number Theory at the same time?

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I'm looking at the curriculum map on AoPS's website and see that they have the Intro to Counting & Probability as a core book with Number Theory as a supplement at the same time. How do you actually schedule these two together? Do you do a chapter or a section from C&P then one from Number Theory, or split your math time in half daily, so you work on both concurrently? Are there certain chapters that are supposed to match up, or sections that should be done at the same time?

I'm trying to figure out what to buy. I've already ordered the C&P book and had planned on that being the book to do until I saw the map. I need to get all of the different lessons/chapters kinda scheduled out by the end of this week though, at least drafted on how they work together. I'm going back into an office full-time starting next week so if I don't figure it out now I'm probably not going to, lol.

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I think what they intend to suggest with the map is more "you can do these in either order at this point in the sequence." You certainly COULD do both at the same time, but it's not necessary.

Both of those are outside the traditional courses covered in schools, so they're both "supplementary" in that sense. They can be done whenever after Intro to Algebra. They're also both shorter than the other books - think one-semester courses instead of full-year courses (though of course the time people spend on them varies).

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They are shorter courses so you could do them sequentially. I personally wouldn’t try them simultaneously. They are also outside the scope of the traditional mathematics pathway, and are not required for traditional high school math.

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I'd do Intro to C&P first. It's pretty useful for counting divisors in number theory.

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Thinking more, though, prime factorization is pretty useful for working with binomials 😛 .

Has anyone actually tried running these together? I think when I write my own lessons, I do actually run them together...

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We did them as semester classes more than a year apart.  We did Number Theory first but there was no academic reason - I just let kid do whichever sounded more interesting.  I might be wrong, but my remembrance is that we spent 1.5 years on Alg 1, then did a semester of Number Theory, then a year of Geometry, then a semester on C&P, then 1.5 years on Alg. 2.  We spent longer on the algebras because kid was young when we started and didn't have a lot of frustration tolerance, and then we wound up with a weird schedule that incorporated both Life of Fred and AoPS because kid liked seeing the same material in different ways.

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2 minutes ago, Clemsondana said:

We did them as semester classes more than a year apart.  We did Number Theory first but there was no academic reason - I just let kid do whichever sounded more interesting.  I might be wrong, but my remembrance is that we spent 1.5 years on Alg 1, then did a semester of Number Theory, then a year of Geometry, then a semester on C&P, then 1.5 years on Alg. 2.  We spent longer on the algebras because kid was young when we started and didn't have a lot of frustration tolerance, and then we wound up with a weird schedule that incorporated both Life of Fred and AoPS because kid liked seeing the same material in different ways.

So does the "counting divisors" chapter make any sense if a kid hasn't done counting? I feel like it'd be confusing.

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@Not_a_NumberI don't remember kid running into any issues - they thought NT was fun.  I was pretty hands-off because they never needed help (as opposed to some of the alg concepts and geo proofs, which we worked on together on a chalk board).

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Just now, Clemsondana said:

@Not_a_NumberI don't remember kid running into any issues - they thought NT was fun.  I was pretty hands-off because they never needed help (as opposed to some of the alg concepts and geo proofs, which we worked on together on a chalk board).

Maybe it's brief enough that you can just remember the formula, I guess. I've seen lots of kids do that in the classes.

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26 minutes ago, Not_a_Number said:

So does the "counting divisors" chapter make any sense if a kid hasn't done counting? I feel like it'd be confusing.

My kid who's doing NT had no problem understanding that (and I do mean understanding, not just memorizing). We are going in the same order as @Clemsondana

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1 minute ago, purpleowl said:

My kid who's doing NT had no problem understanding that (and I do mean understanding, not just memorizing). We are going in the same order as @Clemsondana

I'm sorry, but I've never found tree diagrams adequate and sufficient for the multiplication principle, and I have never seen a kid fully absorb it from seeing it once or twice. And I've taught this material a fair amount.

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Just now, Not_a_Number said:

I'm sorry, but I've never found tree diagrams adequate and sufficient for the multiplication principle, and I have never seen a kid fully absorb it from seeing it once or twice. And I've taught this material a fair amount.

My inference from your comment is that you believe I'm either lying or confused about what my own child understands. And I'm not really interested in defending myself against either charge, so, okay, you're welcome to continue believing that.

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Posted (edited)
9 minutes ago, purpleowl said:

My inference from your comment is that you believe I'm either lying or confused about what my own child understands. And I'm not really interested in defending myself against either charge, so, okay, you're welcome to continue believing that.

No, not at all! I'm very sorry for implying that.

It's just that I think you can understand an argument seeing it once but not have it fully absorbed in a way that you could use it in different cases eventually. That's what I see the most often -- not that the kids don't understand the explanation, but that they may not be able to fully reproduce it, especially if some time passes.

I was actually just doing stuff like this with a kid I'm tutoring 🙂 . She could basically understand why the formula for the number of divisors worked, but it wasn't generalizing very well in her head -- she couldn't apply it easily to counting square or cube factors, for example. (Actually, come to think of it, this came up with TWO of my students yesterday.)

I have high standards for what it means to integrate an idea into one's head, that's all. And I've found the multiplication principle quite tricky for kids to integrate.

Edited by Not_a_Number
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I think part of the misunderstanding here is the assumption that a student who has not taken Intro C&P knows absolutely zero counting and zero probability.  Same with NT.  IIRC, the AoPS Intro Algebra have brief introductions (1 chapter each?) on those topics, and if a student has already been doing math contests, they probably have had some exposure to both topics before being introduced formally in the actual classes.

I know this is the case with my kids.  I was teaching them basic C&P in the context of prepping for various math contests before they actually cracked the book for formal study.

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1 minute ago, daijobu said:

I think part of the misunderstanding here is the assumption that a student who has not taken Intro C&P knows absolutely zero counting and zero probability.  Same with NT.  IIRC, the AoPS Intro Algebra have brief introductions (1 chapter each?) on those topics, and if a student has already been doing math contests, they probably have had some exposure to both topics before being introduced formally in the actual classes.

I know this is the case with my kids.  I was teaching them basic C&P in the context of prepping for various math contests before they actually cracked the book for formal study.

I just know, having taught Intro C&P a few times now, that the multiplication principle is actually kind of hard to integrate into one's thinking. I kept having kids who could do problems using binomial coefficients, but had forgotten the multiplication principle. But then I never manage to communicate what I mean by "integrate into one's thinking" even with simpler concepts, so I doubt I'll succeed with this one...

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8 minutes ago, Not_a_Number said:

I just know, having taught Intro C&P a few times now, that the multiplication principle is actually kind of hard to integrate into one's thinking.

I believe you, but I find it hard to believe.  I mean, isn't it just a matter of creating different outfits of shirts and pants?  Once you start making outfits, doesn't the multiplication principle become obvious?

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Like Daijobu said, it's entirely possible that kid had seen some of the concepts before in other places. Kid is also intuitive about math and has been known to think of the formulas as nifty shortcuts for stuff kid was already thinking through.  Sometimes they have to struggle with a concept, but other times they can't figure out why the text is taking so many words to explain something 'simple' or why mom can't just look at the problem and see what the answer obviously is without having to write and do calculations.

I had to look up the multiplication principle to remember what it was, and kid would not have had any problem with that.  We occasionally did the Family Math projects in elementary school and I know that we did probability then, and there is also some in Singapore Math where kids draw the trees and count the outcomes and play with marbles in bags and that sort of thing.  Kid might have needed to be reminded that you can just multiply, but probably not, and they definitely would not have found this to be new.

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Having written the above post, I can see what you mean by memorizing formulas.  I have a student who for every single problem wants to throw $n \choose{r}$ at it, even when it's wildly inappropriate.

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One more point is that NaN's sees a wide variety of students in her AoPS classes.  I'm guessing many are plopped into those classes because they've proven themselves to be good math students in their BM schools, which is a low bar...equivalent to being good at pattern matching.

These students arrive in NaN's class ready to deploy pattern-matching once again and fail.

I'm guessing homeschooling parents take a more thoughtful approach to selecting AoPS, as evidenced by all the threads on this board titled "Is AoPS right for my student?"

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Posted (edited)
25 minutes ago, daijobu said:

I believe you, but I find it hard to believe.  I mean, isn't it just a matter of creating different outfits of shirts and pants?  Once you start making outfits, doesn't the multiplication principle become obvious?

I've found that just like many fundamental concepts, it's very easy to introduce to people in the most basic case:  when you have two things you're choosing and you can make an array. And then you have to spend a long time making sure it's so obvious to kids that you can extend it to things where you multiply more than 2 things and then you can extend it to things where you divide.

I've definitely seen kids get tripped up by not spending enough time on this and moving forward with other combinatorics concepts. Then you wind up having to rectify the issues by teaching shortcuts for "when you add and when you multiply." And I find that at the point kids are using verbal cues for that, they make many more mistakes.

18 minutes ago, daijobu said:

One more point is that NaN's sees a wide variety of students in her AoPS classes.  I'm guessing many are plopped into those classes because they've proven themselves to be good math students in their BM schools, which is a low bar...equivalent to being good at pattern matching.

These students arrive in NaN's class ready to deploy pattern-matching once again and fail.

I'm guessing homeschooling parents take a more thoughtful approach to selecting AoPS, as evidenced by all the threads on this board titled "Is AoPS right for my student?"

This was also my experience with teaching my own kids, though. And it's now my experience with the kids I'm tutoring. You're right that not being taught to pattern-match helps, but then lots of kids do pattern-match naturally, especially the mathy ones.

I'm not saying kids can't understand it, though. Just that really thorough comfort takes a while... and if you do start on it earlier, I'm sure it's easier.

Edited by Not_a_Number
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Thanks for clarifying what you meant, @Not_a_Number. 🙂

So, Beast Academy and AOPS Prealgebra both include the multiplication principle, so NT is not the first time DD#1 has seen it. And her grasp of the concept is very good, based on the arbitrary problems I just sprang on her with no warning (also, she's now irritated with me for doing that 😉 ).

I think perhaps for many people there are multiple levels of "understand" - things that are more than simply memorizing formulas, but less than your standard of fully integrating into thinking. Like, I think someone could say their child understands fractions and mean that the child can recognize equivalent fractions and can add and subtract fractions, even though the child cannot yet multiply and divide them, or do anything with fractions that include one or more variables. So "understand" can mean "understand to a certain extent." (Not sure if that's the best example, but hopefully it communicates my meaning well enough.)

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2 hours ago, Not_a_Number said:

I've definitely seen kids get tripped up by not spending enough time on this and moving forward with other combinatorics concepts. Then you wind up having to rectify the issues by teaching shortcuts for "when you add and when you multiply." And I find that at the point kids are using verbal cues for that, they make many more mistakes.

OMG I am so dealing with this with my Intro C&P student right now.  It's pulling teeth just to get him to step back and understand big picture the problem we are trying to solve.

We solve 3 mutually exclusive cases, and then he wants to multiply them!  Why ?  I'm still trying to figure out what questions to ask him to get him to see the answer.

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17 minutes ago, purpleowl said:

Thanks for clarifying what you meant, @Not_a_Number. 🙂

So, Beast Academy and AOPS Prealgebra both include the multiplication principle, so NT is not the first time DD#1 has seen it. And her grasp of the concept is very good, based on the arbitrary problems I just sprang on her with no warning (also, she's now irritated with me for doing that 😉 ).

Hahahaha, OK, I'm glad! I think I forgot how many times your DD would have seen it beforehand. And I'm sorry for being a cause of irritation for her!

17 minutes ago, purpleowl said:

I think perhaps for many people there are multiple levels of "understand" - things that are more than simply memorizing formulas, but less than your standard of fully integrating into thinking. Like, I think someone could say their child understands fractions and mean that the child can recognize equivalent fractions and can add and subtract fractions, even though the child cannot yet multiply and divide them, or do anything with fractions that include one or more variables. So "understand" can mean "understand to a certain extent." (Not sure if that's the best example, but hopefully it communicates my meaning well enough.)

Yes, I absolutely agree there 🙂. I suppose I've found it valuable to work up to a really high level of understanding even early on, since it seems to make it easier for a kid to use the concept fluently whenever it comes up. But it'd be hard for me to explain what exactly I mean by "high level of understanding" -- I suppose just that the way of thinking is really fluently integrated.

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1 hour ago, daijobu said:

OMG I am so dealing with this with my Intro C&P student right now.  It's pulling teeth just to get him to step back and understand big picture the problem we are trying to solve.

We solve 3 mutually exclusive cases, and then he wants to multiply them!  Why ?  I'm still trying to figure out what questions to ask him to get him to see the answer.

He's trying to multiply them because he's not spending any time visualizing, probably. So all his cues are verbal and not logical or visual. I tend to try solve that by actually going back to the basics and making the lists.

That being said, I just asked DD8 a simple multiplication principle question and she divided by 2 at the end!! 😱 Argh. Of course, DD8 is known to be my "how little effort can I get away with??" student, but this did lead to a 20-minute interlude where we discussed a mixture of the multiplication principle and also her attitude towards questions I ask her 😛. (I do try not to ask too many random questions during her free time, but she's so noncompliant during school time already that I really don't feel like it would be useful to do that.)

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