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mathmarm

Algebra Textbook for Younger Students

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33 minutes ago, square_25 said:


I want her to use her logical understanding of the operations. I don't mind if she uses logic that only works on integers, but something using the meaning of the operations is key. 

In this case, I'd want her to justify her work using the fact that (ab)c = a(bc) and the fact that ab = ba, and I'd want her to also be able to explain those properties logically. I find that kids can almost always explain better with specific numbers than with a's and b's, although I'm certainly happy to hear explanations with a's and b's. So I'm happy to take the explanation that (ab)c means you take ab copies of c, and that a(bc) means we take a copies of bc, and since each bc contains b copies of c, that's a total of b + b + ... + b copies of c, added up a times, so ab copies of c again!

I can't see how the stepping through this with the associative property is especially useful for this problem.

(xy)^3
(xy)(xy)(xy)
(xyx)(y)(xy)
(xyxy)(xy)
(xyxy)(x)(y)
(xyxyx)(y)
(xyxyxy), All this is legitimate use of the associative property, but it's not particularly useful. If she was going down this path, I'm not surprised that your DD got confused in this "setting it up with the associative property" phase. At this point, you still haven't commuted anything...

I mean, that's 5 lines of associating before you can commute. If you commute throughout this process, it just looks even more cluttered and messy.

Maybe I'm being dense, but ideally what would you have wanted her to write down? I mean that really, what would this solution have looked like in the perfect scenario?

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So, if you were to jot-down the "ideal" response to the (xy)^3 what is it? I'd really like to see what you feel is an appropriate/ideal response to this? Do you ever use color-coding?
 

 

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20 minutes ago, mathmarm said:

I can't see how the stepping through this with the associative property is especially useful for this problem.

(xy)^3
(xy)(xy)(xy)
(xyx)(y)(xy)
(xyxy)(xy)
(xyxy)(x)(y)
(xyxyx)(y)
(xyxyxy), All this is legitimate use of the associative property, but it's not particularly useful. If she was going down this path, I'm not surprised that your DD got confused in this "setting it up with the associative property" phase. At this point, you still haven't commuted anything...

I mean, that's 5 lines of associating before you can commute. If you commute throughout this process, it just looks even more cluttered and messy.

Maybe I'm being dense, but ideally what would you have wanted her to write down? I mean that really, what would this solution have looked like in the perfect scenario?

Well, you don't have a choice, lol. You HAVE to use the associative property for this question. How else could you do it?

I agree that it's clunky. I was hoping she basically had the intuition that you can multiply numbers in any order and way you choose, and she clearly doesn't. However, if you're going to be rigorous about it, you have to use the associative property. But I wasn't all that concerned about her justifying it perfectly: the issue was that she really wasn't sure how to manipulate the numbers. I would have probably been fine with her writing down x^3y^3 without any manipulations, to be honest -- it's just that she kept write down outrageous stuff. (I think her first attempt was x^{x+y} y^{x+y}, which really just makes no sense.)

By the way, the reason I say that you need both the properties is that it's simply not true that 

(xy)^3 = x^3 y^3

in non-commutative or non-associative settings. 

Edited by square_25

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10 hours ago, mathmarm said:

How do you proceed, when the kid in question doesn't like manipulatives, and/or seems to be going through an anti-visualizations phase?

Erm, I don't really know. 🤷‍♀️ My one AL who resists using manipulatives is the weakest in these kinds of concepts, and my one who used manipulatives most extensively is strongest in all things math. You can approach math from a verbal standpoint, but I'm not sure its the most efficient way. I'm coming from an engineering (not real math) background, though. Language was always more of a frilly decoration to be added to math in my mind, but I may be limited by my own visually-based thought processes.

I haven't figured out how to help my one kid who doesn't like manipulatives or visualizing, but instead have found it most effective to let him explore concepts in his own way, which takes significantly longer and seems to result in less robust conceptual understanding. Consequently, he'll be getting to algebra at 10-11 vs. my heavy-manipulatives, super-visual kid who did basic algebraic concepts like (x+y)^2 at 5 and AoPS algebra at 7. The two boys have the same cognitive abilities, so I tend to think the difference is largely due to personality factors and learning habits/preferences.

3 hours ago, square_25 said:

She knows the why, she just stopped keeping it in the back of her head when she used the pattern. I think it’s important to learn the difference between “I learned the pattern” and “I can backtrack the pattern to properties I can easily understand.” The former approach is considerably more fragile.

See, and I don't think my DS 8 uses the pattern at all for things that can be visualized. The way he talks when he explains his work, it sounds like commentary on visuals, and he uses his hands to shape ideas in the air.

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42 minutes ago, Cake and Pi said:

Erm, I don't really know. 🤷‍♀️ My one AL who resists using manipulatives is the weakest in these kinds of concepts, and my one who used manipulatives most extensively is strongest in all things math. You can approach math from a verbal standpoint, but I'm not sure its the most efficient way. I'm coming from an engineering (not real math) background, though. Language was always more of a frilly decoration to be added to math in my mind, but I may be limited by my own visually-based thought processes.

I haven't figured out how to help my one kid who doesn't like manipulatives or visualizing, but instead have found it most effective to let him explore concepts in his own way, which takes significantly longer and seems to result in less robust conceptual understanding. Consequently, he'll be getting to algebra at 10-11 vs. my heavy-manipulatives, super-visual kid who did basic algebraic concepts like (x+y)^2 at 5 and AoPS algebra at 7. The two boys have the same cognitive abilities, so I tend to think the difference is largely due to personality factors and learning habits/preferences.

See, and I don't think my DS 8 uses the pattern at all for things that can be visualized. The way he talks when he explains his work, it sounds like commentary on visuals, and he uses his hands to shape ideas in the air.

 

Well, you can visualize it, but you still tend to visualize it with "sample numbers," right? You can't actually visualize an x or a y. I know sometimes you visualize sort of... a fuzzy version of a number? But still, there has to be SOMETHING to visualize, even if in some sense you're visualizing something general. 

My kiddo doesn't like manipulatives and is mixed on visuals. She definitely thinks of SOME things visually -- if you get her to explain why ab = ba, she does it visually. But she didn't want to use visuals for fractions at all. I don't know why. She's strong with fractions, though: it took her a bit of time, but she figured out formulas for

a/b + c/d, a/b - c/d, a/b*c/d, a/b/(c/d)

from first principles. And she can do most fraction calculations very quickly. 

We've done a bunch of mental math, which may be why she does some things verbally: we started fractions verbally, and it may be that it lingered. And we're overall a very verbal as well as a very mathy family. So that may be part of it. 

Edited by square_25

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Of course, you know your DD far better than I do,, but it seems like she got confused around this red-part:

1 hour ago, square_25 said:

As for plugging in numbers, I mostly wanted her to slow down when she got (xyy)(xxy)(xy) and realize there's no way this could work. I think that the step of "Am I sure that's right? If I'm not, let's try some numbers!" is really crucial in working out new operations and it also helps you avoid overgeneralizing.

I am not sure what lead to that, or what came after, but to me (with minimal context) it looks like she bungled the associative property. Perhaps she concluded at this point that what she'd been trying was wrong. That's why I asked you what kind of leading questions you asked during this exercise?

At this point, I would' have clarified and affirmed her intent, but pointed out that she'd made a mistake because she picked up an extra factor of x and an extra factor of y.
It might have been helpful to get her to verbalize her intent by asking her which definitions and properties she's going to use?
So if she IDed "associating property and commuting property" then restart the problem.
"Babe, you said you're going to do associating and commuting. Which one do you want to do first?
(if they begin associating in pain-staking step by step fashion, I might interject to ask "what will you get when you finish all the associating?"
if they know I would encourage them to right down what it looks like when all the pain-staking step-by-step associating is done.
If they don't know then I'd say "Oooh, we'll keep going 🙂 Lets find out what it will look like after all the associating is done." and put up with them doing it in pain-staking step by step fashion.

When she was done associating, I might say
"Good job with that. Now, you said you were going to use associating and commuting, you've associated. So what do you have left to do?"
(repeat the same process outlined above for if she begins commuting in pain-staking step-by-step fashion)
 

Once she has completed associating and commuting, and has some form of xxxyyy, I'd ask her to check that she didn't lose or pick up any factors. (because associating and commuting won't change the # of x-factors or y-factors that she has). Then I'd prompt her to think about what she could do with those repeated factors and I'd expect her to get x^3y^3.

 

I found in my classes that it helps if I can get the student to tell me what they are going to do (in their own words) so that I can help them track through whatever it was that they were thinking as they go. Some times they'll set up a plan that is flawed, but if they're in a space where they could learn from that error, then I might let them make it.

Some of my weaker students, I might tell them which definition and/or properties I want them to use. And we talk through what they are doing and why.

By figuring out what they intend to do for a particular exercise, it helps me to emphasize where a students planning is dead-on vs where their execution is faulty or flawed.

I find it helps them to realize that they DO have the right plan because a lot of my students will panic if they approached a problem with confidence only to have it fail. Many of them panic and change from a winning plan. So by affirming their intentions ahead of time, they don't change from a winning plan, to trying random, panicky things. such as that x{x+y} thing that you're daughter gave you.

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10 minutes ago, mathmarm said:

I am not sure what lead to that, or what came after, but to me (with minimal context) it looks like she bungled the associative property. Perhaps she concluded at this point that what she'd been trying was wrong. That's why I asked you what kind of leading questions you asked during this exercise?

As I mentioned before, her rather garbled explanation involved trying to use the distributive property. There were parentheses, and she tried "combining things from different parentheses" approach. It's not really based on anything in particular. She wasn't sure how to do it, and she defaulted to using a pattern. That is not an approach I like her using and we talked about that after -- that if she wasn't sure if something was true, she could CHECK with actual numbers. In fact, she should check with actual numbers to find a pattern whenever she was stuck. No, it's not the most efficient thing to do pretty much ever, but it keeps you from making nonsensical errors. 

I don't actually think of mathematics as applications of properties, so I don't think I'd teach it like you do. I tend to rather work on intuitive understanding of the properties instead, building things up slowly so they don't feel arbitrary. Otherwise, the soup of properties gets intimidating for a lot of kids. I also don't tend to want to scaffold math at the symbolic level much -- I find that if kids have worked with the numerical and visual patterns enough, they don't need it all that much, and if they do need it, it's best to go back and work with things numerically until the experience is gained. 

Anyway, to each their own :-). I am not particularly worried about putting away certain aspects of algebra for the time being and I'm excited to work on prime factorization and combinatorics with her. I'll post anything relevant we work on in this thread! I'm sure we'll keep using variables and occasionally distributing and solving linear/quadratic equations along the way, because I think it's pretty easy to build in spirals into new material and I think it's important. 

Edited by square_25

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44 minutes ago, square_25 said:

Well, you don't have a choice, lol. You HAVE to use the associative property for this question. How else could you do it?

I agree that it's clunky. I was hoping she basically had the intuition that you can multiply numbers in any order and way you choose, and she clearly doesn't. However, if you're going to be rigorous about it, you have to use the associative property. But I wasn't all that concerned about her justifying it perfectly: the issue was that she really wasn't sure how to manipulate the numbers. I would have probably been fine with her writing down x^3y^3 without any manipulations, to be honest -- it's just that she kept write down outrageous stuff. (I think her first attempt was x^{x+y} y^{x+y}, which really just makes no sense.)

By the way, the reason I say that you need both the properties is that it's simply not true that 

(xy)^3 = x^3 y^3

in non-commutative or non-associative settings. 

I know perfectly well why one must use both properties, for this exercise, but I think highly granular applications (step-by-step in a painstaking fashion) can open the door for error. It seems like your daughter went off the rails, then floundered writing all sorts of senseless stuff.

Feeling confident about an approach, only to have it end in error can make one a student feel doubtful and unsure.

That's why I'm so curious what your scaffolding sounds like during these types of exercises.

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2 minutes ago, mathmarm said:

I know perfectly well why one must use both properties, for this exercise, but I think highly granular applications (step-by-step in a painstaking fashion) can open the door for error. It seems like your daughter went off the rails, then floundered writing all sorts of senseless stuff.

Feeling confident about an approach, only to have it end in error can make one a student feel doubtful and unsure.

That's why I'm so curious what your scaffolding sounds like during these types of exercises.

I don't think she felt confident -- she wasn't sure what she was doing. Basically, what we discovered is that she wasn't comfortable multiplying lots of numbers together and we're going to work on that :-). I think of working on that as scaffolding -- we'll come back to this when she's ready! 

From my perspective, I'm OK backtracking if we went past her intuitive level of understanding. That's why I don't scaffold this that much :-). I'm sure we'll go back and it'll be fine next time! 

Edited by square_25

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I think part of the issue is that I've been REALLY picky about notational stuff, lol. So I don't think of an expression like ABC as necessarily well-defined unless we have associativity. And we've rarely thought about how to simplify products of more than one number except by doing it in order. So I think she's just not all that aware that she can move things around or why. And that seems to me to be a MUCH bigger priority than having her rigorously use associativity and commutativity in every question -- what I really want her to understand is HOW associativity and commutativity help you rearrange a multiplication so that she knows why it works, so then when she uses it, she has those ideas at the back of her head. 

I'm not sure why there was a "confused" reaction on my post? I'm merely explaining where I'm coming from, which sounds like it's from a slightly different place than you are. 

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23 minutes ago, square_25 said:

I'm not sure why there was a "confused" reaction on my post? I'm merely explaining where I'm coming from, which sounds like it's from a slightly different place than you are. 

Because I was confused from reading it. It's not a sign of disapproval. I simply had trouble following that post the first few times that I read it.  (I always read each post in its entirety if I'm going to respond to it.)

Initially I didn't see the parallel between 3(4*5) and (xy)^3, but in re-reading it, I realized that you might have used it because (xy)^3 yields 3 basic factors.
But where (xy)^3 I immediately see the 3 identical factors the numerical example that you used had 3 distinct factors and I didn't immediately see the connection.

Also you said that when she got (xyy)(xxy)(xy) "she should've realized that there was no way that that could work" but I perceived a path that would lead to that but make perfect sense, so again I was confused about why you'd say that there was no way that her result could work. It didn't seem to come from nowhere in my mind. If she'd followed through from

(xyy)(xxy)(xy) she would've eventually gotten x^4y^4, which would at least provide a clue as to where she went wrong and she might've been able to trouble-shoot her own process and find her mistake. I don't know.

There isn't wider-context to her solution. It could've been the result of nonsensical symbol-shuffling, but it could've been the result of a reasoned approach to the problem.

Of course, (xyy)(xxy)(xy) is completely wrong, and shouldn't even appear in the process of working out (xy)^3, but (depending on the context) it's not an utterly baseless mistake to make.
It doesn't strike me as a random, out-of-nowhere mistake. Which is why I was curious as to what lead to that line, what followed it, etc.

Especially for a small child, I would look at that think that their hand and mind had fallen out of sync during that problem.

I have seen many a highly competent student think faster than their hand, or have their hand move faster than their thinking. It happens to average and below average students as well. That they will think (but not write) a variable or operation or step.
Many negatives vanish for no reason, but when you trace back through, many times you can tell if the kid simply didn't write it/it got erased and wasn't replaced or if the kid dropped it because they're clueless.

I imagine that your daughters arrival to (xyy)(xxy)(xy)  wasn't utterly baseless. To me, it looks like she simply wrote too many x's and y's in the midst of her manipulations (which might've been very tedious or might not have been, IDK).
 

Anyway, I found that particular post very difficult to parse out. It's not that I disapprove of your approach, that post was just not very clear to me. But I acknowledge that there is a lot of context missing from your daughters approach.

But I think I get it now (probably).

 

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16 minutes ago, mathmarm said:

Also you said that when she got (xyy)(xxy)(xy) "she should've realized that there was no way that that could work" but I perceived a path that would lead to that but make perfect sense, so again I was confused about why you'd say that there was no way that her result could work. It didn't seem to come from nowhere in my mind. If she'd followed through from

Mostly because it doesn't work with any numbers you might try, and I do want her to think about numbers at the back of her head, so to speak :-). It works for almost no examples. If she isn't sure whether something she writes down is correct or not, I do want her to think about whether it's true for actual numbers -- with this one, it should have been quite clear that it's just too big. Otherwise, it's very easy to get caught in the trap of "I don't remember if my teacher told me this, so I don't know if it's true." I definitely saw this with lots of college kids. They'd add 1/a + 1/b = 2/(a+b), some of them, and they stared at me like I was crazy when I asked them to check if it actually worked. There was a serious disconnect between the numerical and the variable level for them. That's what I'm trying to avoid. 

16 minutes ago, mathmarm said:

I imagine that your daughters arrival to (xyy)(xxy)(xy)  wasn't utterly baseless. To me, it looks like she simply wrote too many x's and y's in the midst of her manipulations (which might've been very tedious or might not have been, IDK).
 

This was in the midst of much flailing ;-). And when she tried to explain it, it was using something like the distributive property, which is what made me think that she was trying to use the pattern she had internalized from addition. (When we have parentheses, then we group numbers from the first parenthesis with numbers with the second parenthesis -- that would be a pattern-based, symbol-shuffling way to think of using the distributive property here, and I think she was trying to do that. I'm not entirely sure, to be honest, because she was having trouble explaining any of her thinking at all.) 

I think she was mostly feeling panicky and fumbling. She really had no idea how to approach this question at all. Which I was surprised by, I admit! But having thought about it, I THINK I know where the hole is, and it just comes from lack of experience. I don't think it'll take more than a week or two to remediate, but I figure I may as well fold it into another unit -- taking a bit of time to appreciate numerical operations again seems like a good idea :-). 

I'll let you know how our next unit goes! I think I'll keep working on at least some algebra along the way, since she's certainly internalized a lot of stuff well (she solves linear and factorable quadratic equations very proficiently at this point.) But we'll stay away from products of too many numbers at once for now, until she internalizes that a) you can do it in any order you want and b) the reason for that is the associative and commutative properties. 

I wonder if it'd be fun to introduce a new operation that is NOT commutative? Just to remind her that it's possible... sometimes kids get kind of stuck thinking anything that's a product must commute. She's not ready for matrices by any stretch of the imagination, though... I don't know what I could use! Maybe some simple group theory?? 

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1 hour ago, square_25 said:

 

Well, you can visualize it, but you still tend to visualize it with "sample numbers," right? You can't actually visualize an x or a y. I know sometimes you visualize sort of... a fuzzy version of a number? But still, there has to be SOMETHING to visualize, even if in some sense you're visualizing something general. 

My kiddo doesn't like manipulatives and is mixed on visuals. She definitely thinks of SOME things visually -- if you get her to explain why ab = ba, she does it visually. But she didn't want to use visuals for fractions at all. I don't know why. She's strong with fractions, though: it took her a bit of time, but she figured out formulas for

a/b + c/d, a/b - c/d, a/b*c/d, a/b/(c/d)

from first principles. And she can do most fraction calculations very quickly. 

We've done a bunch of mental math, which may be why she does some things verbally: we started fractions verbally, and it may be that it lingered. And we're overall a very verbal as well as a very mathy family. So that may be part of it. 

When I visualize it, it looks a great deal like the Algebra Lab Gear blocks, possibly because that's how I originally learned way back in the day. So no, no sample numbers in my head, unless you consider visualizing something like c-rods to be visualizing sample numbers. I asked DS 8 to explain his thinking on (x+y)^2 and he quickly traced out boxes in the air. He said he could "see" them in the air as a table of boxes with letters inside them. I asked him if he imagined the letters as numbers and he said they weren't any particular number but letters that acted like numbers and were sort of like all numbers at once.

He did a lot of self-discovery of properties, too, but he did it mostly with manipulatives and drawings. For example, he figured out the basics of exponents by playing with 1" plastic tiles and wood cubes. It didn't even occur to me that I should provide vocabulary like distributive property or associative property back then. He got all of the terminology through AoPS. (More proof that I'd suck at teaching without curricula to follow!)

I love how you and mathmarm focus so much on naming the properties and proving work with appropriate vocabulary. I feel like it must add an extra layer of understanding for kids with that balanced ability profile. My DS 8 is exceptionally lopsided... I guess "specialized" would be the more positive way to spin it, lol.

Anyway, I find it super interesting to see the different ways these mathematically precious kiddos unfold. I should share some of DS 8's messy, all-over-the-place, step-skipping work from when he was 6-7. It makes the handwritten stuff both of you guys shared look like works of art!

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3 hours ago, Cake and Pi said:

See, and I don't think my DS 8 uses the pattern at all for things that can be visualized. The way he talks when he explains his work, it sounds like commentary on visuals, and he uses his hands to shape ideas in the air.

Jr does too! It's so neat to watch him. We could never tell if it's because he's fluent in ASL, math manipulatives or both.  Nice to have that additional data point!

34 minutes ago, Cake and Pi said:

He did a lot of self-discovery of properties, too, but he did it mostly with manipulatives and drawings. For example, he figured out the basics of exponents by playing with 1" plastic tiles and wood cubes. It didn't even occur to me that I should provide vocabulary like distributive property or associative property back then. He got all of the terminology through AoPS. (More proof that I'd suck at teaching without curricula to follow!)

I love how you and mathmarm focus so much on naming the properties and proving work with appropriate vocabulary. I feel like it must add an extra layer of understanding for kids with that balanced ability profile. My DS 8 is exceptionally lopsided... I guess "specialized" would be the more positive way to spin it, lol.

Personally I sprinkled in terminology in amongst the loads and loads of manipulative-based work that we did. We've been through arithmetic a few times now, and with each cycle, I added a slightly stronger emphasis on getting him to notice and talk about concepts and then layered in the terminology for the concepts that he was seeing. One of the reasons that we're able to now give attention and focus to the terminology, and organization of written work is because everything else is in place. Largely due to the manipulative work we did.

We did so.much with manipulatives, for so long, before we stretched to include anything else.

 

Quote

 I should share some of DS 8's messy, all-over-the-place, step-skipping work from when he was 6-7. It makes the handwritten stuff both of you guys shared look like works of art!

If you're comfortable, please do! I think it's utterly precious to see kids stuff.

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1 hour ago, Cake and Pi said:

When I visualize it, it looks a great deal like the Algebra Lab Gear blocks, possibly because that's how I originally learned way back in the day. So no, no sample numbers in my head, unless you consider visualizing something like c-rods to be visualizing sample numbers.

Oh, I totally think of c-rods as sample numbers! Depending on the day, I either visualize a sort of blurry rectangle, or I remember it the way I first came up with a*b = b*a when I was little and my mom posed a question to me... I didn't think of an array at all, I instead thought of grouping all the "first" items in the a copies of b (getting a group of a), then all the second items, etc. I remember this being a real epiphany :-). 

1 hour ago, Cake and Pi said:

I love how you and mathmarm focus so much on naming the properties and proving work with appropriate vocabulary. I feel like it must add an extra layer of understanding for kids with that balanced ability profile. My DS 8 is exceptionally lopsided... I guess "specialized" would be the more positive way to spin it, lol.

Oh, I so do not name the properties to my kiddo! 😄I just say things like "order doesn't matter." I do occasionally mention the formal names, but I don't focus on them. I do work on her internalizing the properties and being able to explain them, but if anything, I err on the side of using insufficiently formal language. She still refers to the standard addition and multiplication algorithms as "stacking," pretty much ensuring that no one else will ever know what she's talking about :-P. 

1 hour ago, Cake and Pi said:

Anyway, I find it super interesting to see the different ways these mathematically precious kiddos unfold. I should share some of DS 8's messy, all-over-the-place, step-skipping work from when he was 6-7. It makes the handwritten stuff both of you guys shared look like works of art!

If you don't mind sharing, I'd love to see! DD7 is lucky that she has such lovely handwriting.  

By the way, I'd like to note that I do personally like manipulatives! I've been using them in my homeschool math classes and I think the kids have found them very helpful. (We mostly use poker chips for place value and dice and cards for numbers otherwise.) However, DD7 has always been really uninterested... partially I think because she's very symbolically-minded and visualizations work really well for her. Generally, every time I've tried to push her to use my methods over what works in her head, I've regretted it, so I've tried not to push things. I'm curious what DD4 (ooh, I gotta change my siggy, she's had a birthday!) is going to wind up liking.

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On 4/18/2020 at 9:12 PM, mathmarm said:

If you're comfortable, please do! I think it's utterly precious to see kids stuff.

On 4/18/2020 at 9:26 PM, square_25 said:

If you don't mind sharing, I'd love to see! 

 

Here ya go! For fun and because it's so cute, there's a page from BA that he did at 5y8m old... and I saved it so it was one of his nicer-looking pages. He turned those workbooks into "art," lol. I like it though because you can sort of see his thinking in his work.

There's a page of pre-algebra challenge problems I coped out for him at 6y6m as a makeshift worksheet during the transition from BA to AoPS. It looks like he lost his minus sign in one of the problems, but it was there in the original. It just didn't come through in the picture.

The next one is his handwritten work for a writing problem in Algebra A at 7y2m. This was significantly better organized than his work for a regular problem because he knew he'd need to reference this work when he typed out his full solution afterward.

The pdf is a picture of a regular short-response challenge problem he did last week in Intro to Geometry. Probably should have saved it as a photo instead of a pdf. It came out very light, sorry!

BA 3A - 5y8m.jpg

Prealgebra ch.2 - 6y6m.jpg

Algebra A wk8 - 7y2m.jpg

Geometry frustum problem - 8y4m.pdf

Edited by Cake and Pi
clarified, added details
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On 4/23/2020 at 2:29 AM, Cake and Pi said:

Here ya go!

Thank you for sharing! Your sons work is brilliant!

It's always so fun to look back over older work and see how the child has progressed!

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This month, we are beginning the proof-writing lesson in Foerster and going to be building his fluency with long-division and polynomials with home-made daily drill sheets for him to use this month as Table Work.

Jr got a tangential taste of proofs recently with the exponent properties so problems 41-70 from Lesson 3-6 in the text are especially timely. 

This month we are covering material from Ch 3 - 5, but I will be de-emphasizing the textbook to try and makesure that math remain firmly a secondary academic subject for Jr.

We started him on a writing program and that should take up more of his energy for seat work.

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7 hours ago, mathmarm said:

This month, we are beginning the proof-writing lesson in Foerster and going to be building his fluency with long-division and polynomials with home-made daily drill sheets for him to use this month as Table Work.

Jr got a tangential taste of proofs recently with the exponent properties so problems 41-70 from Lesson 3-6 in the text are especially timely. 

This month we are covering material from Ch 3 - 5, but I will be de-emphasizing the textbook to try and makesure that math remain firmly a secondary academic subject for Jr.

We started him on a writing program and that should take up more of his energy for seat work.

How's he enjoying the book? Any more work to share? 🙂

We're finishing up the proof-writing bit we're doing now and are about to dive into prime factorization, combinatorics, and multiplication of lots of numbers. We're gonna go over long division as well :-). DD7 has been in a somewhat frustrating stage where attempting to seriously scaffold her work results in her learning very little... she seems to only be able to follow her own train of thought nowadays, no matter how clear one is or how many questions one asks along the way. We never had this before and it was accompanied by a serious developmental leap, so it's not necessarily a bad thing... but it's weird. So I'm going to try aim the work right at the level where she can do it herself with stretching herself. 

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2 hours ago, square_25 said:

How's he enjoying the book? Any more work to share? 🙂

He is still loving the whole textbook ritual. He loves "cuddle math", and taking notes, etc.

But I'm copying the work for this next section for him because the problems are LONG. Since these are proofs so the devil is in the details and I want him to be able to focus on 1 task at a time.

We are having technical difficulties so no work samples to share.

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8 minutes ago, mathmarm said:

We are having technical difficulties so no work samples to share.

Oh, I'm sorry! How so? 

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1 minute ago, square_25 said:

Oh, I'm sorry! How so? 

The scanner is broken.

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Just now, mathmarm said:

The scanner is broken.

Aaah, too bad. I think our new printer MAY have a scanner... we haven't tried using it yet, lol. I've only posted pictures of the work I took using my phone! 

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I hate taking pictures of multipage work. Neatly aligned PDFs are the way to go.

 

When we get the scanner back online I have a stack of work to process.

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