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So, 1-7 is the lesson where you solve equations and Jr knows the concept very well, and he enjoyed reading through  and discussing that lesson again. He really seemed to like the way the book illustrated the concept very simply with scales. He's very familiar with the number-balance so I guess he just enjoyed seeing that same concept in a book?

However, he hit a road block with the last part of the exercises. Where the equations have a fraction as the coefficient of x.

At first he said he couldn't do it. Then we talked some about what fraction notation tells us. We looked at the problems he had already done with ease.

Because of the way that he was first introduced to fraction notation as symbol for division he had no problem with and easily understood that with equations like: (1/3)x = 4, he had to multiply by 3. But when the equations took the form (2/3)x = 6, he got confused for a moment.

We used the whiteboard, factored those types of fractions into a form that he could easily "see" and did a few examples.

He was finally able to tell me that (2/3)x means that x is multiplied by 2 and divided by 3. We did a few examples on the board, then he was able to do the first 2 practice problems from the textbook. Algebra 1 CH1 - JR_3.pdf

However, when we discussed the problems afterwards he mentioned that you have to divide first! 🤯

So, we spent the next day looking at more of these equations--factoring and commuting the fractional coefficients and discussing the problems.
When we had 3 examples on the board--both of them done 2-ways--I asked him to tell me what he noticed, he explained to me that since you factor, and multiplication is commutative, it doesn't matter if you divide first or not when removing the fractional coefficient. Whew!

Finally, he used his notebook time solving the rest of the practice problems in this lesson. Algebra 1 CH1 - JR_4.pdf

Today, we move on to 1-8, but I will keep these fractional equations in the rotation for a little while.

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That was a nice little jolt to remind me that just because he's getting the individual problems "right" doesn't mean that he he is getting the underlying concept "right".

It's so important to keep the dialogue going so that misunderstandings like this are taught and dealt with promptly.

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So, I've been thinking a lot about algorithms in arithmetic or algebra.

@square_25 I could probably guess what you're going to say, but what do you think? Is there a right time to insist on the use of an algorithm? If so, when?

Jr is very fluent with manipulating numbers. He understands the arithmetic operations and the basic properties. He has methods of performing each of the operations, but there is no sure-fire pattern to how he solves various problems.  Mostly, he uses the distributive property to perform multiplication or division, but sometimes he does the "ink-saving" way (which is just a form of the standard algorithm). He uses integers to subtract as often as he regroups. He does mental math or finger-abacus to add/subtract a lot.

He has a mix of strategies that he uses. He understands the operations and he is really clever with numbers, but should I insist on an algorithm at a certain point? If so, when?

 

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@square_25 Jr, "discovered" reciprocals.

We've been working a few fractional equations each day since last week and I've been discussing with him about the equations.
2 days before Yesterday, he generalized that: whenever there is a fraction you just have to undo the multiplication and the division.
A day before yesterday I asked him about that insight again and as we worked on a few problems he said "Division and multiplication are the same level, so you could actually do them both at the same time."

When I pushed him on that idea some more he seemed confused so I backed off of it. Yesterday we reviewed what he'd come up with so far and continued to work a few problems, I coach him to keep using his words and I kept using his words from the last couple of days as we worked through the problems.

Today he told me that we can use multiplication by another fraction--"the opposite fraction"--  to clear the fraction in one move. I pushed back on that language (he knows integers and that -3 and 3 are "opposites") so we discussed what it would mean for a fraction to be oppostie and after that rabbit hole, he agreed that it's wrong to call it an opposite fraction and changed the wording to "the inverting fraction".

We did several equations with the "inverting fraction" shortcut.

 

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7 minutes ago, mathmarm said:

@square_25 Jr, "discovered" reciprocals.

We've been working a few fractional equations each day since last week and I've been discussing with him about the equations.
2 days before Yesterday, he generalized that: whenever there is a fraction you just have to undo the multiplication and the division.
A day before yesterday I asked him about that insight again and as we worked on a few problems he said "Division and multiplication are the same level, so you could actually do them both at the same time."

When I pushed him on that idea some more he seemed confused so I backed off of it. Yesterday we reviewed what he'd come up with so far and continued to work a few problems, I coach him to keep using his words and I kept using his words from the last couple of days as we worked through the problems.

Today he told me that we can use multiplication by another fraction--"the opposite fraction"--  to clear the fraction in one move. I pushed back on that language (he knows integers and that -3 and 3 are "opposites") so we discussed what it would mean for a fraction to be oppostie and after that rabbit hole, he agreed that it's wrong to call it an opposite fraction and changed the wording to "the inverting fraction".

We did several equations with the "inverting fraction" shortcut.

 

Neat!! Does he think of it as multiplying by 1/f, where f is the original fraction?

Or do you mean he figured out to undo multiplication by a/b by multiplying by b/a, but not necessarily that one is 1 over the other? Either way, that’s a lovely insight!!

As a mathematician, I’d say that both are reasonable definitions of “opposites” — they are just inverses for different operations! 😉

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1 minute ago, square_25 said:

Neat!! How did he figure out to multiply by the reciprocal? (Does he think of it as the reciprocal, that is, as 1/f, where f is the original fraction?)

As a mathematician, I’d say that both are reasonable definitions of “opposites” — they are just inverses for different operations! 😉

Can you clarify what you mean by "how did he figure out to multiply by the reciprocal"?
From what he's told me, thus far, he hasn't mentioned that the reciprocal is 1/f, where f is the original fraction. We haven't done work with complex fractions in a while, so I personally really doubt that he's noticed that or would make that leap.

It could be semantics, but I really disagree that it's okay to call 1/3 the opposite of 3. Each number can only have a single opposite.

I understand his thinking and instinct for the language-choice--it isn't unreasonable to dub it "opposite fraction"--but, it is wrong.

If 3/4 is the opposite (-3/4), it can't be the opposite of (4/3) too.
 But, I acknowledge that I much prefer precision and consistency in mathematical language.

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9 minutes ago, mathmarm said:

Can you clarify what you mean by "how did he figure out to multiply by the reciprocal"?
From what he's told me, thus far, he hasn't mentioned that the reciprocal is 1/f, where f is the original fraction. We haven't done work with complex fractions in a while, so I personally really doubt that he's noticed that or would make that leap.

It could be semantics, but I really disagree that it's okay to call 1/3 the opposite of 3. Each number can only have a single opposite.

I understand his thinking and instinct for the language-choice--it isn't unreasonable to dub it "opposite fraction"--but, it is wrong.

If 3/4 is the opposite (-3/4), it can't be the opposite of (4/3) too.
 But, I acknowledge that I much prefer precision and consistency in mathematical language.


I guess I don’t really think “opposite” is proper mathematical terminology, anyway — I think you’d want to use “negative” and “reciprocal” for that.

However, it is true that -3 is the inverse of 3 with respect to addition, and 1/3 is the inverse of 3 with respect to multiplication, and if you’re using “opposite” to loosely mean “inverse with respect to some operation,” then both are opposites of 3 :-D. I probably wouldn’t encourage that language, but it’s a wonderful observation :-). Could be a fun segue into group theory!!

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3 hours ago, square_25 said:


I guess I don’t really think “opposite” is proper mathematical terminology, anyway — I think you’d want to use “negative” and “reciprocal” for that.

Point. However "opposite" is a widely-used, well-understood term in mathematics in the US.
In the US the term "opposite" is used to mean the "additive inverse" of a number. In my experience, I have never--not once--seen "Opposite"  used for the multiplicative inverse--we only call that the "reciprocal" or "multiplicative inverse"

More importantly, Jr. already knows the concept of "opposite" as most (US educated) students know it-- as a word that is compatible only with "Additive inverse".
I'll teach him the word "reciprocal" soon enough and pretty soon we'll get to to the terms additive-inverse and multiplicative-inverse anyway but my experience has taught me that it's always best to nip "sloppy" in the bud quickly.

3 hours ago, square_25 said:

Sounds like he’s having fun with the book!! I’m glad :-). I love the updates!! Thank you for writing them up.

I think he is. I've slowed him down. I want to preserve the enjoyment of the written book-work,I want the activity to fit within a certain time-slot and it leaves more time for other threads 🙂

6 hours ago, square_25 said:

Neat!! Does he think of it as multiplying by 1/f, where f is the original fraction?

Or do you mean he figured out to undo multiplication by a/b by multiplying by b/a, but not necessarily that one is 1 over the other? Either way, that’s a lovely insight!!

As a mathematician, I’d say that both are reasonable definitions of “opposites” — they are just inverses for different operations! 😉

He has not summarized the reciprocal in terms of complex fractions at all and it's been a good long while since he's done any real complex fractions so I doubt that he's made that connection yet. But, he does know complex fractions fairly well, so he might have that little nugget rolling around in his brain somewhere, just waiting for him to unearth it. 🤷🏿‍♀️

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15 minutes ago, mathmarm said:

  Point. However "opposite" is a widely-used, well-understood term in mathematics in the US.
In the US the term "opposite" is used to mean the "additive inverse" of a number. In my experience, I have never--not once--seen "Opposite"  used for the multiplicative inverse--we only call that the "reciprocal" or "multiplicative inverse"

 

Huh, interesting! I Googled it, and you're right, I do see people using those words. 

I don't think this is a US versus European disconnect as much as a mathematician versus not mathematician disconnect (as I said, I've done math only in English since I was 11, which was a while ago) -- I've really never seen any math contest people or mathematicians use the terminology this way. I just asked my mathematician DH, who was born in Boston and is certainly US educated, what the opposite of 3 was, and he actually said 1/3 :-P. He said he'd never heard anyone use that wording for negatives, either. 

15 minutes ago, mathmarm said:

More importantly, Jr. already knows the concept of "opposite" as most (US educated) students know it-- as a word that is compatible only with "Additive inverse".
I'll teach him the word "reciprocal" soon enough and pretty soon we'll get to to the terms additive-inverse and multiplicative-inverse anyway but my experience has taught me that it's always best to nip "sloppy" in the bud quickly.

 

I think if you already have an explicit meaning for "opposite," you should stick to it and not let him use it sloppily! But I'd probably gently move to "negative" and "reciprocal" for clarity in higher math. Or additive and multiplicative inverse! Those work, too 🙂

15 minutes ago, mathmarm said:

I think he is. I've slowed him down. I want to preserve the enjoyment of the written book-work,I want the activity to fit within a certain time-slot and it leaves more time for other threads 🙂

He has not summarized the reciprocal in terms of complex fractions at all and it's been a good long while since he's done any real complex fractions so I doubt that he's made that connection yet. But, he does know complex fractions fairly well, so he might have that little nugget rolling around in his brain somewhere, just waiting for him to unearth it. 🤷🏿‍♀️

 

By complex fractions, do you mean things like 2/3 as opposed to 1 over something? We're definitely still working on fraction arithmetic around here, alongside of algebra -- DD gets it and can do it, but she isn't automatic about it, and she occasionally has trouble taking her knowledge and applying it to situations with variables (I asked her to figure out general formulas for the sum, difference, product and quotient of a/b and c/d, and I think it took an hour, even though she can do it numerically quite quickly.) Fractions are tricky!! 

We seem to be solidly in "how to write a proof" around here. DD7 is suddenly writing tons and tons of proofs -- sometimes, she's using words where she really should just write down the sequence of algebraic manipulations :-P. We've been working on solving equations by completing the square and graphing a few lines (although for now, I only have her plot some points.) I'm not entirely sure how long we'll stay in this "early algebra" phase -- I need to go back and teach her long division and about lowest common multiples and greatest common factors and some geometry and whatnot! 

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On 4/5/2020 at 10:04 PM, mathmarm said:

So, I've been thinking a lot about algorithms in arithmetic or algebra.

@square_25 I could probably guess what you're going to say, but what do you think? Is there a right time to insist on the use of an algorithm? If so, when?

Jr is very fluent with manipulating numbers. He understands the arithmetic operations and the basic properties. He has methods of performing each of the operations, but there is no sure-fire pattern to how he solves various problems.  Mostly, he uses the distributive property to perform multiplication or division, but sometimes he does the "ink-saving" way (which is just a form of the standard algorithm). He uses integers to subtract as often as he regroups. He does mental math or finger-abacus to add/subtract a lot.

He has a mix of strategies that he uses. He understands the operations and he is really clever with numbers, but should I insist on an algorithm at a certain point? If so, when?

 

 

Hmmm, I don't have a hard and fast rule about this one :-). I tend to work more on algorithms after understanding is solid, but you're obviously there already. I would generally nudge towards algorithms if it's helpful and faster -- like, it really doesn't make sense to multiply out 323*45 or something like that in your head, you know? I guess I wouldn't exactly insist as much as introduce problems which make the use of an algorithm necessary! 

I will say that since DD7 has learned how to add, subtract and multiply column by column, her mental math got a bit worse. So I do think letting him use mental math when it makes sense isn't a bad idea! (Right now, I'm having more troubling getting her NOT to use algorithms occasionally.) 

Does that make sense? What's your take? 

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9 hours ago, square_25 said:

a- mathematician versus not mathematician disconnect (as I said, I've done math only in English since I was 11, which was a while ago) --
b-I've really never seen any math contest people or mathematicians use the terminology this way.

c-I just asked my mathematician DH, who was born in Boston and is certainly US educated, what the opposite of 3 was, and he actually said 1/3 :-P. He said he'd never heard anyone use that wording for negatives, either.

 

d-But I'd probably gently move to "negative" and "reciprocal" for clarity in higher math. Or additive and multiplicative inverse! Those work, too 🙂.

a- School math vs math is an excellent point. The word "opposite" is firmly in the school-math vernacular in the US.
NOTE: I specify US, not because you're European, but because I thought I remembered you saying that you went to school in Canada at 11.
So while it's uber-common for US-educated 11-14 year olds to be expected to learn, understand and know the word "opposite" for the additive-inverse, I can't say what they do in Canada.
I really doubt that "opposite" is a universally an English-math word because I have had Anglophone students from other countries who learned integers as "directed numbers" and with other terminology. But I do expect that US-educated students in that age-range would learn and know the word "opposite"

b-That's curious. I know that "opposite" is used in BA5 and AoPS Prealgebra, so I'd be really curious to poll 5th-9th grade students in math clubs to see if they really just don't know the term of if the formality of being in a math contest/club means that they are inclined/encouraged to use "proper" math terms only.

The concept of the additive inverse is often introduced by the label of "opposite".    -7 is the opposite of 7 and 8.4 is the opposite of -8.4
The concept of the multiplicative inverse is often introduced under the label of "reciprocal" 4/7 is the reciprocal of 7/4 and 5 is the reciprocal.

c-No, I think your DH simply made a mistake. In the realm of school-math--at least in the US--saying that 1/3 is the opposite of 3 is wrong.
It's possible  that he never learned the term, but I think it's  more likely that he's long since forgotten the word--since it is dropped early on in most Algebra 1 classes--but if he was educated in the US and is speaking the expected math-vernacular then he is wrong to say that 1/3 is the opposite for 3.

d-I want him to have the "expected understanding" of reciprocal and opposite before I transition to using additive-inverse and multiplicative-inverse a little more exclusively.

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9 minutes ago, mathmarm said:

a- School math vs math is an excellent point. The word "opposite" is firmly in the school-math vernacular in the US.
NOTE: I specify US, not because you're European, but because I thought I remembered you saying that you went to school in Canada at 11.
So while it's uber-common for US-educated 11-14 year olds to be expected to learn, understand and know the word "opposite" for the additive-inverse, I can't say what they do in Canada.
I really doubt that "opposite" is a universally an English-math word because I have had Anglophone students from other countries who learned integers as "directed numbers" and with other terminology. But I do expect that US-educated students in that age-range would learn and know the word "opposite"

b-That's curious. I know that "opposite" is used in BA5 and AoPS Prealgebra, so I'd be really curious to poll 5th-9th grade students in math clubs to see if they really just don't know the term of if the formality of being in a math contest/club means that they are inclined/encouraged to use "proper" math terms only.

The concept of the additive inverse is often introduced by the label of "opposite".    -7 is the opposite of 7 and 8.4 is the opposite of -8.4
The concept of the multiplicative inverse is often introduced under the label of "reciprocal" 4/7 is the reciprocal of 7/4 and 5 is the reciprocal.

c-No, I think your DH simply made a mistake. In the realm of school-math--at least in the US--saying that 1/3 is the opposite of 3 is wrong.
It's possible  that he never learned the term, but I think it's  more likely that he's long since forgotten the word--since it is dropped early on in most Algebra 1 classes--but if he was educated in the US and is speaking the expected math-vernacular then he is wrong to say that 1/3 is the opposite for 3.

d-I want him to have the "expected understanding" of reciprocal and opposite before I transition to using additive-inverse and multiplicative-inverse a little more exclusively.

 

Yeah, I have no idea. He just didn't know what it meant and that was the first thing that came into his head. I actually didn't know he'd say 1/3: I was curious what he'd say and whether he knew this terminology. We don't have BA5 and I've never taught AoPS Prealgebra online (I probably should, for experience and to get some teaching ideas!), so I can't comment on that. I really had no idea this phrasing was so prevalent. 

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11 hours ago, square_25 said:

E-By complex fractions, do you mean things like 2/3 as opposed to 1 over something? We're definitely still working on fraction arithmetic around here, alongside of algebra -- DD gets it and can do it, but she isn't automatic about it, and she occasionally has trouble taking her knowledge and applying it to situations with variables (I asked her to figure out general formulas for the sum, difference, product and quotient of a/b and c/d, and I think it took an hour, even though she can do it numerically quite quickly.) Fractions are tricky!! 

F-We seem to be solidly in "how to write a proof" around here. DD7 is suddenly writing tons and tons of proofs -- sometimes, she's using words where she really should just write down the sequence of algebraic manipulations :-P. We've been working on solving equations by completing the square and graphing a few lines (although for now, I only have her plot some points.) I'm not entirely sure how long we'll stay in this "early algebra" phase -- G- I need to go back and teach her long division and about lowest common multiples and greatest common factors and some geometry and whatnot! 

E-Complex fractions are just fractions, within fractions. Like these here.

F-I really look forward to when Jr. begins writing proofs, we're not there yet, but I think he'll take to it ok when we start.
Like you daughter, he's very confident with the numerical-only, but he talks his way through everything with variables. He can verbally explain the process to me fluidly, and talk me through the process, and gives the justifications verbally (and informally) but I don't think he can do it himself while he's solving the problem.

G-I don't think your DD will need to go over long-division for long. Do you have an approach for long-division that you want to use with her, or are you open to suggestions?

I have found that many kids find "long division" easier after using the distributive property for a while (some kids take a long time, some kids take a short time). Jr transitioned  to the "Standard" algorithm for long division quite easily after doing "long division" with the distributive property for a bit.

We

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1 minute ago, mathmarm said:

E-Complex fractions are just fractions, within fractions. Like these here.

F-I really look forward to when Jr. begins writing proofs, we're not there yet, but I think he'll take to it ok when we start.
Like you daughter, he's very confident with the numerical-only, but he talks his way through everything with variables. He can verbally explain the process to me fluidly, and talk me through the process, and gives the justifications verbally (and informally) but he can't do it himself.

G-I don't think your DD will need to go over long-division for long. Do you have an approach for long-division that you want to use with her, or are you open to suggestions?

I have found that many kids find "long division" easier after using the distributive property for a while (some kids take a long time, some kids take a short time). Jr transitioned  to the "Standard" algorithm for long division quite easily after doing "long division" with the distributive property for a bit.

We

Oh, I see what you mean about complex fractions!! We do use those once in a while, but not super often, either.

Proofs are fun! We had some sort of leap where they became much easier for her. We weren’t doing them 3 months ago!!

I’m not worried about long division. She’s currently basically doing it in expanded form (finding the number of 100s, then 10s then 1s.) For a while, she was resistant to doing that even though she used the distributive property to multiply, and that’s why we didn’t cover the algorithm for it yet. I think she’ll learn the algorithm in a day or two at her current level of preparation: that’s how long the other standard algorithms took. But I do need to go back and cover it, because she’s suuuuper slow dividing in expanded form right now!!

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12 minutes ago, square_25 said:

 

Yeah, I have no idea. He just didn't know what it meant and that was the first thing that came into his head. I actually didn't know he'd say 1/3: I was curious what he'd say and whether he knew this terminology. We don't have BA5 and I've never taught AoPS Prealgebra online (I probably should, for experience and to get some teaching ideas!), so I can't comment on that. I really had no idea this phrasing was so prevalent. 

If you're only working with students in Algebra and beyond, then you could easily miss the usage of "opposite" Many Algebra 1 texts introduce more formal terminology fairly early on.

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4 minutes ago, square_25 said:

Oh, I see what you mean about complex fractions!! We do use those once in a while, but not super often, either.

Proofs are fun! We had some sort of leap where they became much easier for her. We weren’t doing them 3 months ago!!

I’m not worried about long division. She’s currently basically doing it in expanded form (finding the number of 100s, then 10s then 1s.) For a while, she was resistant to doing that even though she used the distributive property to multiply, and that’s why we didn’t cover the algorithm for it yet. I think she’ll learn the algorithm in a day or two at her current level of preparation: that’s how long the other standard algorithms took. But I do need to go back and cover it, because she’s suuuuper slow dividing in expanded form right now!!

Jr. was crazy about complex fractions at one point. When he was learning to do pen and paper math, he went through a phase where he really liked to do long calculations for all sorts of "scary looking" expressions. When he demonstrated consistency and mastery with the skills, I backed him off of so much calculation because I want to make sure his math education stays balanced. I backed him down to 2 calculations a day. Just enough to maintain the skill, but he wasn't missing any of them so I refocused him on other math-things and it's been a while since he's asked for one.

I can't lie. I'm tempted to start a move towards proofs, but since math isn't our main focus for school, I will save his writing stamina/academic energy for the "main subjects". I know that Chapter 3 of the textbook we have introduces proofs and I already have plans to revisit the main part of Ch3 at the end of this year. So for now, I'll just teach vicariously through you.

Yes, the algorithms take a day or two when the student is well-prepared with the concepts. So I wouldn't make going back for the long-division algorithm a priority.
If she's slow, then I would just work on building up her fluency with the expanding. Would you please share a sample of how your DD does long division currently?

 

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9 minutes ago, mathmarm said:

Jr. was crazy about complex fractions at one point. When he was learning to do pen and paper math, he went through a phase where he really liked to do long calculations for all sorts of "scary looking" expressions. When he demonstrated consistency and mastery with the skills, I backed him off of so much calculation because I want to make sure his math education stays balanced. I backed him down to 2 calculations a day. Just enough to maintain the skill, but he wasn't missing any of them so I refocused him on other math-things and it's been a while since he's asked for one.

I can't lie. I'm tempted to start a move towards proofs, but since math isn't our main focus for school, I will save his writing stamina/academic energy for the "main subjects". I know that Chapter 3 of the textbook we have introduces proofs and I already have plans to revisit the main part of Ch3 at the end of this year. So for now, I'll just teach vicariously through you.

Yes, the algorithms take a day or two when the student is well-prepared with the concepts. So I wouldn't make going back for the long-division algorithm a priority.
If she's slow, then I would just work on building up her fluency with the expanding. Would you please share a sample of how your DD does long division currently?

 

I want long division for repeating decimals and the factor theorem, anyway :-). I’m not in a rush, but it naturally comes next.

She solves a multiplication equation right now. She needs to do some estimating instead just going up by 100 each time, lol, but otherwise it’s fine. She’s started estimating now (my picture is from more than a week ago), so I’d actually say long division is on the near horizon as a low hanging fruit. 
 

ETA: the picture is for 36087/69, in case you can’t read her equation on top.

8160B0F5-07FF-44B9-A64E-2F878FA853EB.jpeg

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6 minutes ago, square_25 said:

I want long division for repeating decimals and the factor theorem, anyway :-). I’m not in a rush, but it naturally comes next.

She solves a multiplication equation right now. She needs to do some estimating instead just going up by 100 each time, lol, but otherwise it’s fine. She’s started estimating now (my picture is from more than a week ago), so I’d actually say long division is on the near horizon as a low hanging fruit. 

8160B0F5-07FF-44B9-A64E-2F878FA853EB.jpeg

Very interesting approach. Though now I'm curious why you're subordinating the arithmetic, to the algebraic.

Did you make the conscious decision to go in this order? (I'd love to hear your rationale if so.)

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1 hour ago, mathmarm said:

Very interesting approach. Though now I'm curious why you're subordinating the arithmetic, to the algebraic.

Did you make the conscious decision to go in this order? (I'd love to hear your rationale if so.)

I'm not sure what you mean. How am I subordinating the arithmetic to the algebraic? 

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30 minutes ago, square_25 said:

I'm not sure what you mean. How am I subordinating the arithmetic to the algebraic? 

I may have misunderstood, but I thought you plan to use the factor theorem to develop the algorithm for long-division with your DD.

I just find this approach of using algebra to explore and build-out a childs understanding of arithmetic fascinating.

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4 minutes ago, mathmarm said:

 

I may have misunderstood, but I thought you plan to use the factor theorem to develop the algorithm for long-division with your DD.

I just find this approach of using algebra to explore and build-out a childs understanding of arithmetic fascinating.

No, no, we’ll do it numerically first!! That’d be interesting but I’m not that original!!

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8 minutes ago, mathmarm said:

Oh too bad. It would make a neat experiment, :laugh:. But Hubby would not be happy with me if I conducted that experiment on the kids.

 

I actually really do prefer to start at the least abstract level and abstract up, always. But I think we will be able to move to abstract long division relatively quickly!!

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So this brings me back around to my question about algorithms.

I'm going to ask Jr to solve the same division problem to show you an illustration of the behavior that I'm seeing in him.

He knows the long division algorithm. Most of his practice with it has been with 1-digit divisors,  but he's had some experience with 2- and 3-digit divisors as well.

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2 minutes ago, mathmarm said:

So this brings me back around to my question about algorithms.

I'm going to ask Jr to solve the same division problem to show you an illustration of the behavior that I'm seeing in him.

He knows the long division algorithm. Most of his practice with it has been with 1-digit divisors,  but he's had some experience with 2- and 3-digit divisors as well.

Please do show :-). I’ll let you know what I’d do when you show me!

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12 minutes ago, mathmarm said:

So, he spent 12 or 15 minutes doing the green page. @square_25

I asked him if he could do it another way. He spent 3 minutes doing the white page.

division_example.pdf 603.73 kB · 2 downloads

Yes, I've had the same thing with DD7 -- last year, I could FORCE her to use the place value algorithm, but when left to her own devices, she'd do something much like Jr. does. And sometimes, she'd start up from 1 and try multiplying all the way up -- not because she couldn't do it in a faster way, but because she wanted to. 

Personally, I tend to believe that they do things like this not just because they are contrary, but because they are still internalizing something about the operation. I'm sure Jr. does understanding how long division works, but sometimes seeing the whole range of possibilities helps you deepen your idea of what you're doing. 

What I personally did with DD7 is to allow her to use her own algorithms, but to show her WHY there are more efficient things to do. And sometimes, that didn't work, either, lol, like the time she wrote the whole multiplication table for 47 (or something) up to 65, because she felt like it. So personally, I'd open up a conversation with Jr. about why he's choosing the method he's choosing, and I'd gently suggest that there are other things he can do. And honestly, if he still messes around like this, I'd leave it and assume he's learning through exploration. 

This year, DD7 has been quite amenable to moving to more efficient algorithms even without a ton of prompting (although I do have to nudge occasionally.) So I assume he'll get there as well, especially if you nudge him along into use place value when he does calculations in his own way. And I'd trust that he's making important connections in his head as he's playing around :-). 

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While he first learned to divide by using the distributive property on the chunked-up-dividend, I don't know why he doesn't use the "easy way" when it's clear that chunking the dividend will be cumbersome.

I think part of it is that Jr likes "scary looking" problems. So he likes for them to be long. He will dilly-dally making the problem super complicated looking, so that he can finish it and say how he beat that problem.

I'm curious if there is a point when it would be more beneficial to insist on a particular algorithm. 

But he has strong number sense and very good visualization. When he uses the standard division algorithm he hits it on the best factor immediately.

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On 4/8/2020 at 8:00 PM, mathmarm said:

  

Point. However "opposite" is a widely-used, well-understood term in mathematics in the US.
In the US the term "opposite" is used to mean the "additive inverse" of a number. In my experience, I have never--not once--seen "Opposite"  used for the multiplicative inverse--we only call that the "reciprocal" or "multiplicative inverse"

More importantly, Jr. already knows the concept of "opposite" as most (US educated) students know it-- as a word that is compatible only with "Additive inverse".

 

I disagree.  I never used the term "opposite" (except maybe maybe  coloquially) to mean the negation of a number.  We always use additive inverse or the negation or the negative of a number.  If you had ever asked me to state the "opposite" of a number I would have no idea what you mean.  I would stay away from imprecise terms like "opposite" because I think they only serve to hide what is really going on when we mean additive inverse or negation.  

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Just now, daijobu said:

 

I disagree.  I never used the term "opposite" (except maybe maybe  coloquially) to mean the negation of a number.  We always use additive inverse or the negation or the negative of a number.  If you had ever asked me to state the "opposite" of a number I would have no idea what you mean.  I would stay away from imprecise terms like "opposite" because I think they only serve to hide what is really going on when we mean additive inverse or negation.  

 

I don't use it, either, but I did have a kid in my Intermediate Algebra class today use it, and I was reminded of the conversation with @mathmarm!! It's obviously used in some places, because I can also find it Googling. 

For me, the word opposite ties in so heavily with the idea of the inverse that I just don't think that way. I guess I think of operations as having opposite operations (that is, inverses) but not so much of numbers as having opposites themselves. Not sure if that makes sense, but I think that's why the terminology doesn't "feel" right to me. But I seem to be in the minority! 

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8 hours ago, mathmarm said:

While he first learned to divide by using the distributive property on the chunked-up-dividend, I don't know why he doesn't use the "easy way" when it's clear that chunking the dividend will be cumbersome.

I think part of it is that Jr likes "scary looking" problems. So he likes for them to be long. He will dilly-dally making the problem super complicated looking, so that he can finish it and say how he beat that problem.

I'm curious if there is a point when it would be more beneficial to insist on a particular algorithm. 

But he has strong number sense and very good visualization. When he uses the standard division algorithm he hits it on the best factor immediately.

Yeah, I have no idea why DD7 likes to do it in a roundabout way, either. It's not like she couldn't figure out what to do -- she just somehow didn't want to! (And we're still having this issue. I do have to nudge her to use the "correct" factor instead of just going up by single 10s and 100s. And it's not like she can't tell that she'll need to go up by a bunch and that her guess isn't going to be the right answer.) 

I do think they are learning when they do things like that, so I wouldn't necessarily insist. Perhaps it's comforting to learn that it works however you do it? 🙂 I'm really not sure what happens when they explore like that. But then learning isn't precisely linear...

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Posted (edited)
9 hours ago, daijobu said:

I disagree.  I never used the term "opposite" (except maybe maybe  coloquially) to mean the negation of a number.  We always use additive inverse or the negation or the negative of a number.  If you had ever asked me to state the "opposite" of a number I would have no idea what you mean.  I would stay away from imprecise terms like "opposite" because I think they only serve to hide what is really going on when we mean additive inverse or negation.  

To clarify: Do you disagree that the term "opposite" is widely used and well-understood in school-math throughout US? Or do you disagree with something else?

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So @square_25 I've had a change of plans. We're only doing 1 session of math each day with the book, because I prefer to use the second slot for some "on-level" work for him.I've scrapped my original plan to do the Algebra book in 2 short sessions.


Jr said that he likes doing "Couch Math" in part because it's easy and fun. However, I don't want him to zip through so fast that we miss the chance to flesh out and deepen his understanding of the concepts and solidify his mastery of the skills, But I also want to keep each day of "Couch Math" within the range of "winnable".

I spend 15-20 minutes on Couch Math with him each day.

Lesson 1-9 contains 15 word problems for students to solve. He has a little trouble unpacking them and even though we only did a small sample of problems from other sections, I want him to solve every last one of the word problems, so we're doing just 2 of them a day.
I made a "spiral" out of the book to  fill out our Couch Math time by:

  • Giving a few 1-7 problems for him to solve orally or on the whiteboard.
  • Giving a few 1-8 problems for him to solve orally or on the whiteboard.
  • Doing the 1-9 Oral Practice with him each day (I just change the problems, but I follow the format/pattern from the book)
  • Written Math work: having him do just 2 problems from 1-9 at a time.

I added in practicing the skills from 1-7 & 1-8 that he finds easier to keep the majority of the session "winnable". He's still got a weeks worth word problems from 1-9 left, so I'm going to move him ahead to chapter 2 (Ch. 2 is on integers. He already knows this material well, but will enjoy the practice.) while he works through the rest of this section a little at a time.

This week went well and we're consistently finishing the whole shebang at a good pace, so I think we can sustain it through next week. But we'll see. Because he'll be writing the answers to the word problems for 1-9, he might not wind up making many notes from the beginning of chapter 2. He knows integers pretty well, but I'll see if he finds anything that he wants to note.

Plan for Next Week is

  • Giving a few 2-1 problems for him to solve orally or on the whiteboard.
  • Giving a few 2-2/2-3 problems (adding and subtracting integers respectively) for him to solve orally or on the whiteboard.
  • Giving a few 2-4/2-5 problems (multiplying and dividing integers respectively) for him to solve orally or on the whiteboard.
  • Doing Oral Practices from 1-9 and/or Chapter 2 with him
  • Written Math work: 2 word problems from 1-9

By the time he'll be finished with his word problems we'll be about ready to start 2-6. Which is good, because there was some stuff in there that might be worth noting down, but I"ll see what he thinks.

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So, I have some mixed feelings about using a textbook but one benefit that I'm finding in using a good textbook, is it helps uncover things to revisit now that Jr is older.

Example 1: Jr. has been able to simplify convoluted-looking expressions for a long-while, he has long since internalized the order of operations.

So, because he can "do it already" and continues to exhibit mastery of the skill, it is just the sort of thing that I might neglect to revisit or check up on. But by buddy-reading and discussing the textbook, we've been discussing each of these familiar basic skills again. We have started a conversation about when a grouping symbol is/isn't needed. It possibly wouldn't have come up for a long time or at all. I like having the support of the scope and sequence (with examples and explanations) to help guide our conversations.

One thing that I don't like about the book are some of the "jokes" through out are unkind. Of course, for a 14-16 year old (the age range of the intended audience) the jokes are going to be more appropriate, but for a 6.5 year old, some of them are in poor taste and I'm going to be censoring 1 or 2 of them for sure.

For example, word problems often have pun-centric names in them. Word problems star characters with names such as
Sid Upp,  Tess T. Fye and her sister Clara, Doug Upp, etc. Those are silly, amusing puns that most students will enjoy. Students in the target audience will chuckle, roll their eyes and move on. But there is also an exercise in 2-6, where you read how Iddy Ottic solved a problem and are supposed to explain why what he did is wrong.

Again this wouldn't be discussion worthy for a teenaged student, but I will use whiteout and a pen to change that name because my kid is 6.5 and I don't want Jr. to think that calling someone who makes a mistake Iddy Otic (idiotic) is funny. When you're 6.5 you need to be told and have reinforced for you that name calling is NOT okay.

Since Jr. picked up on the puns the other day so I know he'll be on the look-out for "jokes" in the names. I'll have to come up with a list of fun names to substitute into these problems.

I'm still not really in love with Chapter 6. I haven't come up with the perfect solution there.

 

 

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On 4/9/2020 at 4:08 PM, mathmarm said:

So, he spent 12 or 15 minutes doing the green page. @square_25

I asked him if he could do it another way. He spent 3 minutes doing the white page.

division_example.pdf 603.73 kB · 9 downloads

Are you 100% sure that he is fully fluent in the multiplication and subtraction algorithms?

Since you say he's got a grip on the estimation part, then it really shouldn't take 3 minutes to do this problem with the long-division algorithm.

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Jr. is 100% fluent in the concept and really fluent with the algorithm as well.

He did take a longer-than-I'd-like amount of time with this problem, but he wasn't at his most focused because the babies were distracting him.

I will keep a closer eye on him when he's working, to make sure that he's not being distracted.

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On 4/14/2020 at 7:18 PM, Gil said:

Are you 100% sure that he is fully fluent in the multiplication and subtraction algorithms?

Since you say he's got a grip on the estimation part, then it really shouldn't take 3 minutes to do this problem with the long-division algorithm.

My kiddo also plays around like this. It's not super dependent on how fluent she is with the algorithm and is more about how willing she is to use the algorithm in the first place. I tend to not push algorithms until she's more or less approximating the algorithm on her own time for that reason.

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We've hit a bit of a snag with algebra over here -- she's been expanding products like (x+3)(x-1), (x+3)^2 and even (x+y)^2 very fluently, but I asked her to figure out (xy)^3 today and it was a symbol-shuffling disaster. (Symbol shuffling is what I call a kid trying to "figure out" what something is without ever actually thinking of the variables as standing in for numbers or trying any examples. It's weirdly common.) I then asked her to explain how to expand out (x+y)^2 using logic as opposed to pattern-matching and she struggled a bit as well. 

I think she was mostly just having an off day and she's certainly been doing very well overall, but I think we may go back and do some basic number theory and combinatorics to get her more used to properties of multiplication and division in a numerical context. We'll keep up solving linear equations and quadratics inside and outside the context of word problems to make sure she stays familiar with variables, but she's clearly not quite ready to generalize some of her knowledge and symbol-shuffling is the thing I worry about most of all, because it's an orientation that is HARD to unteach, even locally. 

@mathmarm, want me to post any of our work once we move on to the next thing? I'm probably going to round out our current foray into algebra by having her write out proofs for all of the basic operations we've been using (and find counterexamples for ones that aren't true), and then we'll dive into prime factorization :-). (We've done it before, but not in serious depth. And it can be combined nicely with combinatorics, if I decide to do that.) 

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So, we wound up doing some lite proofs for the exponential properties this week.
As you know, it wasn't my plan or preference to begin proofs just yet. But he was able to work out the property for raising a power to a power, and for dividing exponents. It went pretty well. He was so excited to show his dad what he'd "discovered" afterwards.

It makes me think that he might enjoy the lessons in Ch.3 that are more proof-oriented. He's cruising along nicely in Ch2, even with me slowing him down and spreading things out.

On 4/16/2020 at 6:06 PM, square_25 said:

@mathmarm, want me to post any of our work once we move on to the next thing?

Sure! Post as much as you have time or the energy for.

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On 4/16/2020 at 4:06 PM, square_25 said:

I then asked her to explain how to expand out (x+y)^2 using logic as opposed to pattern-matching and she struggled a bit as well. 

Conceptualizing these kinds of problems is where manipulatives can be so flippin' handy. I know I've said this before, but I absolutely love our Algebra Lab Gear blocks. They have their limitations, but they're fun and useful for what they are. One of the creators has his original stuff, including printable paper versions of the Lab Gear and out-of-print textbooks, free on his website. https://www.mathed.page/

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6 hours ago, Cake and Pi said:

Conceptualizing these kinds of problems is where manipulatives can be so flippin' handy. I know I've said this before, but I absolutely love our Algebra Lab Gear blocks. They have their limitations, but they're fun and useful for what they are. One of the creators has his original stuff, including printable paper versions of the Lab Gear and out-of-print textbooks, free on his website. https://www.mathed.page/

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

 

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On 4/16/2020 at 6:06 PM, square_25 said:

I then asked her to explain how to expand out (x+y)^2 using logic as opposed to pattern-matching and she struggled a bit as well.

Really it does sound like more of an off day, than a snag.

Since your dd doesn't like manipulatives and has an on-again/off-again relationship with visuals, I'm curious what your scaffolding conversation sounded like?
I know that we both agree on the importance of getting the child conversant in talking about the concepts as a part of the teaching.

For the sake of pedagogical science, I asked Jr. to explain a similiar question and he  also stammered at first.
For us, the scaffolding conversation sounded like this: ( I recorded our exploration on my phone, then transcribed it for you)
 

Quote


Me: Explain for me, how expanding (x-y)^2 works, big-boy.
Jr: They can...uhm, it would be like this *does something with his hands then wrote down his response* the answer is going to be like this x^2 - xy - xy + y^2
Me: So, your response is right, that's correct--good job!--now, can you explain why that's the answer?
Jr: I did the problem.

Me: You did. You did a good job too. But now I need you to give me the explanation.
Jr: uhm...it's like this....*he pointed to each part of (x-y)^2, then trace his finger to the corresponding part of the answer* so then you get the answer.
Me: Ok, so tell me how to get x^2 in the answer?
Jr: Because it's x multiplied by x.
Me: Good. But how do you know that it's x multiplied by x?

Jr: *Points to the ^2 from the original problem.*
Me: I see, so there's an exponent. What does that exponent tell you?
Jr: Square the grouped-together part.
Me: Good, so you really know this!
Jr:   : D
Me: Can you be the teacher, while I'm the student? Since you're the teacher, try to explain it to me so that I understand why you have to square the grouped-together part.
Jr: (making a grown up voice) This says, (x-y)^2. Square means to use the base as a factor twice. {this is directly from how he learned exponents}
Me: Oooh, ok. How do you write that, teacher?
Jr: *wrote down* (x-y)^2 = (x-y)(x-y)
Me: Ok, I want to copy it down too. (I wrote it on a note-pad).
Me: Now what do I do teacher?
Jr: The parenthesis has a meaning too. Think about that meaning and it's...
Me: groups?
Jr: No...m-m-m-m..mul....ti...what is it?
Me: Multiplication.
Jr: Yeah mommy! I knew you were smart.
Me:.... : D. Thanks, can you finish explaining it teacher?
Jr: the parenthesis has a meaning here, it tells us to...
Me: Multiply!
Jr: so you multiply each part of this (x-y) by each part of that (x-y)
Me: Okay...but why?
Jr: ...huh?
Me: Why do I need to multiply each part.
Jr: It's the...uhm...because...the next step is to do that.
Me: Ok, so where does that step come from, teacher?

Jr: from the math.
Me: Teacher, I'm confused.
Jr: No you're not mommy!! Stop being like that!

Me: I just want you to explain it, so that it's clear.
Jr: You're not confused!

After a quick talk about why a teacher should never yell at a student, we got back on track. He was much more in-the-zone now.
Jr: I'm going to show you why (x-y)^2 is x^2 -xy - xy + y^2, okay?
Me: OK!
Jr: The exponent says to use the base as a factor...2 times.
Jr wrote: (x-y)(x-y)
Jr: The parenthesis has another job too, it tells us to do the multiplying operation.

Me: ok, I understand that part. How do I do that multiplication?
Jr thought about for a moment, traced his finger over the steps a couple of times, then said
Jr: Ok, so you know how the distributive property will work with addition or subtraction? So we can do multiply like this: (X-y)(x-y) gives Xx - Xy, and then the rest of the distributive property looks this (x-Y)(x-y) gives Yx+Yy
Me: OH! That's right, I see it now teacher.
Jr: you get 4 little parts Xx -Xy -Yx +Yy because of the way that the multiplication distributed over both parts, 2 times.
Me: Does my answer have to look like that?
Jr looked at what I had written on my notepad and thought about it for a moment
Jr: No. Commuting some of the parts can give you a lot of answers that are the same but look different. Xx can look like x^2, Xy and Yx can both be switched because they're being multiplied. and Yy can look like y^2, because y is a factor twice, which means we can use an exponent!

while he was doing this, he looked at the answer he'd given and changed it from: x^2 - xy - xy + y^2, to x^2 -2xy + y^2

So, as you see, he needed significant prompting in order to go back to the basic definitions and properties. But once he did, he was able to walk his way through the explanation.

Jr was able to solve the problem immediately, but getting to the explanation part took a while. Once we got there, he was there.


 

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On 4/16/2020 at 6:05 PM, square_25 said:

My kiddo also plays around like this. It's not super dependent on how fluent she is with the algorithm and is more about how willing she is to use the algorithm in the first place. I tend to not push algorithms until she's more or less approximating the algorithm on her own time for that reason.

Plays around like what?
Keep in mind that my comment was specifically about the white sheet of paper, where his steps were *perfect* but @mathmarm reported that it took 3 minutes  for him to find the quotient.

I trust that mm's son has superb number-sense and his math facts rock solid, so I was curious about whatthe hold-up was time-wise.
Being distracted out of the math-problem itself makes sense.

For a child who is fluent in the prerequisites, and supposed to be fluent in the division itself, that problem should've taken 35-50 seconds, not 180.

 

 

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1 hour ago, Gil said:

Plays around like what?
Keep in mind that my comment was specifically about the white sheet of paper, where his steps were *perfect* but @mathmarm reported that it took 3 minutes  for him to find the quotient.

I trust that mm's son has superb number-sense and his math facts rock solid, so I was curious about whatthe hold-up was time-wise.
Being distracted out of the math-problem itself makes sense.

For a child who is fluent in the prerequisites, and supposed to be fluent in the division itself, that problem should've taken 35-50 seconds, not 180.

 

 

Ah, I see. I expect that age helps makes this faster! DD7 could do this quickly enough but she still writes slowly and dawdles.

14 hours ago, Cake and Pi said:

Conceptualizing these kinds of problems is where manipulatives can be so flippin' handy. I know I've said this before, but I absolutely love our Algebra Lab Gear blocks. They have their limitations, but they're fun and useful for what they are. One of the creators has his original stuff, including printable paper versions of the Lab Gear and out-of-print textbooks, free on his website. https://www.mathed.page/

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.

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3 hours ago, mathmarm said:

Really it does sound like more of an off day, than a snag.

Since your dd doesn't like manipulatives and has an on-again/off-again relationship with visuals, I'm curious what your scaffolding conversation sounded like?
I know that we both agree on the importance of getting the child conversant in talking about the concepts as a part of the teaching.

For the sake of pedagogical science, I asked Jr. to explain a similiar question and he  also stammered at first.
For us, the scaffolding conversation sounded like this: ( I recorded our exploration on my phone, then transcribed it for you)
 

So, as you see, he needed significant prompting in order to go back to the basic definitions and properties. But once he did, he was able to walk his way through the explanation.

Jr was able to solve the problem immediately, but getting to the explanation part took a while. Once we got there, he was there.


 

Yeah, I see this kind of thing often. She can certainly explain this with NUMBERS very quickly (at least if it’s a single use and not two uses of the distributive property), so the problem for me is that she seems to have temporarily stopped thinking of variables as generalizations of numbers. I saw this often in college kids and it’s a red flag for me.

She eventually explained it well with a bit of prompting, but that’s not really what I’m worried about. I think it’s a bit hard to explain what I mean by symbol shuffling, but I know it when I see it!

I’m not giving up on algebra or anything!! I just think we need a numerical interlude, so the properties she works with feel intuitive given the numerical background. I don’t think she has enough experience multiplying lots of numbers, hence the topics I chose!!

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46 minutes ago, square_25 said:

Yeah, I see this kind of thing often. She can certainly explain this with NUMBERS very quickly (at least if it’s a single use and not two uses of the distributive property), so the problem for me is that she seems to have temporarily stopped thinking of variables as generalizations of numbers. I saw this often in college kids and it’s a red flag for me.

She eventually explained it well with a bit of prompting, but that’s not really what I’m worried about. I think it’s a bit hard to explain what I mean by symbol shuffling, but I know it when I see it!

I’m not giving up on algebra or anything!! I just think we need a numerical interlude, so the properties she works with feel intuitive given the numerical background. I don’t think she has enough experience multiplying lots of numbers, hence the topics I chose!!

So, what would have been the ideal response for her to tell you for (xy)^3 using logic, not pattern-matching?
When you say that you asked she explain it with logic instead of pattern matching, would you explain what you meant?
What leading questions do you ask her?


It seems like plugging in values for x and y can make this problem  more obscure than just using the definition for exponentiation.
For example, if you plug in x = 4, and y = 3, then you get  (4x3)^3 or 12^3 many kids would say 1728. Or they might say it's 64 x 27, but either way they could lose sight of the property/definition that they are supposed to be using. This seems like you run the risk of getting tangled in numbers.

I would really like to know what sort of leading questions you find best to ask in this type of situation? When do you (expect/want) your DD to plug in numbers, to try, versus using definitions, etc?

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45 minutes ago, mathmarm said:

So, what would have been the ideal response for her to tell you for (xy)^3 using logic, not pattern-matching?
When you say that you asked she explain it with logic instead of pattern matching, would you explain what you meant?
What leading questions do you ask her?


It seems like plugging in values for x and y can make this problem  more obscure than just using the definition for exponentiation.
For example, if you plug in x = 4, and y = 3, then you get  (4x3)^3 or 12^3 many kids would say 1728. Or they might say it's 64 x 27, but either way they could lose sight of the property/definition that they are supposed to be using. This seems like you run the risk of getting tangled in numbers.

I would really like to know what sort of leading questions you find best to ask in this type of situation? When do you (expect/want) your DD to plug in numbers, to try, versus using definitions, etc?


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! 

Instead, she wound up trying to use the nonexistent distributive property of multiplication, sigh. So I think we got down to some really nonsensical stuff like (xyy)(xxy)(xy) and then after a while I realized she wasn't ready. 

I find that using numbers is a balancing act. As you say, if you give kids numbers, they often start computing. On the other hand, observations of properties are much easier on a number level, and people have an easier time thinking of 5*(4*3) as "we add up 4*3 five times, but each of those is four 3's, so we have 20 copies of 3 total, so it's also (5*4)*3." Once you're very fluent in that reasoning, you can do it with a, b and c. But I find that there's a fairly long stage where kids (DD7, but also my AoPS kids and college kids) can explain the reasoning with any given group of numbers and are not quite able to generalize it to variables. 

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. For instance, it's rather hard for kids to get used to matrix multiplication, which is NOT commutative, unless they are used to playing around with the ideas. 

Basically, as a TL;DR, I do want them to use properties, but I want to make sure that their understanding of properties is very grounded in numerical experimentation as opposed to patterns of the symbols, if you know what I mean. 

I do think she actually does understanding something like (x+y)^2 very well most of the time :-). I was just concerned that she'd stop "spot-checking" her understanding and was over-relying on patterns. 

I guess we'll get to long division before I thought we would, because I'll probably teach it to her as part of our next unit on number theory and combinatorics!! 

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So, I asked Jr to do (PB)^5 and he did it much more quickly than the (x-y)^2 example and without any real issue.

I didn't have an opportunity to ask him a leading question but when I asked him if it were right, he glanced over his work and said "yes"

PB.pdf

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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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