nwahomeschoolmom 171 Posted October 7, 2020 Report Share Posted October 7, 2020 My 2e kid is just now starting CLE 5 in 3rd grade. (Turned 8 this summer.). I bought him a simple algebra book (By scholastic or Barron’s) from a used bookstore this weekend and he LOVED it. He is great conceptually in math even if he makes mistakes in calculations at times. He does get A’s on pretty much all of his tests and quizzes in math though. So from at the beginning of CLE 5 what is a good way to get him to algebra? Should I try to condense/skip/switch curriculums or just let him unschool algebra in his free time when he wants to? (He does want to get to algebra and asked me to condense if I can.) I have heard CLE can drag out arithmetic and just make it harder. We both like the structure of CLE because it is organized, routine oriented and lets him be independent. He is really a self learner. He is doing IEW level A this year and he is thriving with me just editing his paper 5 minutes a week and watching the videos. Maybe I should be doing more but he doesn’t need it and I have a toddler. I was looking at Jacob’s Alegbra today and it got me excited because I think he would really like it. Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 7, 2020 Report Share Posted October 7, 2020 I'd let him unschool algebra in his own time, if he wants to, and if he doesn't wind up having the time, I'd maybe make some time for learning algebra alongside whatever else he's learning 🙂 . A lot of early algebra doesn't require anything from the later elementary grades, anyway, and it's good to give it time to penetrate. 1 Quote Link to post Share on other sites

Noreen Claire 2,002 Posted October 8, 2020 Report Share Posted October 8, 2020 I have DS8 working in BA3 this year during 'math time', but he unschools math aaalllll day long on his own, reading from upper-level books that we have around the house, algebra, geometry, calculus, logic, whatever. I think it works better this way for him, as he's very abstract random and loves to follow his own rabbit trails off into who-knows-where, but he will still sit down every day and work for 30 solid minutes in a logical progression of the 'easier' stuff. We reevaluate as we go. 4 Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 8, 2020 Report Share Posted October 8, 2020 I would not skip ahead to algebra. There is a big leap in concepts between 5th grade math and alg. I would supplement with something like Hands On Equations Verbal Problems (algebra word problems for elementary age kids). https://www.borenson.com/Products/Verbal-Problems-Book You could also add in fun books. I checked this one out of the library yesterday for my dd: https://www.amazon.com/Book-Perfectly-Perilous-Math-Mathematicians/dp/0761163743/ref=cm_cr_arp_d_product_top?ie=UTF8 3 Quote Link to post Share on other sites

nwahomeschoolmom 171 Posted October 8, 2020 Author Report Share Posted October 8, 2020 Thanks for the tips! 56 minutes ago, Noreen Claire said: I have DS8 working in BA3 this year during 'math time', but he unschools math aaalllll day long on his own, reading from upper-level books that we have around the house, algebra, geometry, calculus, logic, whatever. I think it works better this way for him, as he's very abstract random and loves to follow his own rabbit trails off into who-knows-where, but he will still sit down every day and work for 30 solid minutes in a logical progression of the 'easier' stuff. We reevaluate as we go. I guess its normal, then for 2e/advanced...I'll just keep going how things are. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 8, 2020 Report Share Posted October 8, 2020 For the record, my gifted 8 year old asked to learn algebra last year, so we did 🙂 . She loves it. I wouldn't assume he can't do it or that you can't get started on it because he's young. I'd start slow, that's all, and be mindful of whether he understand or not. DD8 read about algebra in Murderous Maths, by the way -- that's why she wanted to learn it! Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 8, 2020 Report Share Posted October 8, 2020 WHen older students struggle in upper level math, it is bc they missed mastering a foundational elementary math concept. If you are well-versed in math and can teach in such a way you know you won't leave foundational gaps, you don't need to be tied to a math program. If you aren't, there is absolutely no need to skip an 8 yr old over 5th and 6th grade math programs, both of which typically reinforce all those elementary concepts. You could definitely have him start playing with alcumus. It's free. https://artofproblemsolving.com/alcumus 2 Quote Link to post Share on other sites

nwahomeschoolmom 171 Posted October 8, 2020 Author Report Share Posted October 8, 2020 9 hours ago, 8filltheheart said: WHen older students struggle in upper level math, it is bc they missed mastering a foundational elementary math concept. If you are well-versed in math and can teach in such a way you know you won't leave foundational gaps, you don't need to be tied to a math program. If you aren't, there is absolutely no need to skip an 8 yr old over 5th and 6th grade math programs, both of which typically reinforce all those elementary concepts. You could definitely have start playing with alcumus. It's free. https://artofproblemsolving.com/alcumus I am quite good at math, but I already know what you mean because it happened to me. In 9th grade, I switch to a rigorous private school and went straight into Algebra II and struggled because of gaps and lack of mastery from previous grades. But my son is solid (except for a couple of random asynchronous things that may never rectify and may talk longer for him to master.) I'm not talking about skipping anything...just going over certain things faster if he has them down and wonder if another program could be better for hat. Both my son and I are really into "box checking" and routine so not being tied to a program would be hard! Lol! Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 9, 2020 Report Share Posted October 9, 2020 5 hours ago, nwahomeschoolmom said: I am quite good at math, but I already know what you mean because it happened to me. In 9th grade, I switch to a rigorous private school and went straight into Algebra II and struggled because of gaps and lack of mastery from previous grades. But my son is solid (except for a couple of random asynchronous things that may never rectify and may talk longer for him to master.) I'm not talking about skipping anything...just going over certain things faster if he has them down and wonder if another program could be better for hat. Both my son and I are really into "box checking" and routine so not being tied to a program would be hard! Lol! I wouldn't skip anything, I would just let him start on basic algebra 🙂 . You don't NEED to skip anything. But if a student is excited to learn some new concept... well, that's a low-hanging fruit right there. Embrace it! 1 Quote Link to post Share on other sites

Cake and Pi 725 Posted October 9, 2020 Report Share Posted October 9, 2020 You can absolutely continue through the curriculum, condensing and accelerating as appropriate, while introducing algebra concepts. I big puffy heart Algebra Lab Gear. One of my kids went through the ALG Basic Algebra book when he was still otherwise working on 3rd grade math. Hands on Equations is frequently recommended, but it never moves beyond single-variable linear equations, so I'm not really a fan. Seems a bit boring and repetitious to me, but go with whatever works for your kid. You could also let your kiddo try out DragonBox Algebra 12+ if you haven't already. Once they beat the game there's a whole other side B or something with really algebra-y algebra puzzles. And, if you think Jacobs would be a big hit, go for it. You can run two math curricula in parallel, letting him practice those foundational arithmetic skills in between playing with variables and such. Sounds like fun to me! 4 Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 9, 2020 Report Share Posted October 9, 2020 6 hours ago, Cake and Pi said: You can absolutely continue through the curriculum, condensing and accelerating as appropriate, while introducing algebra concepts. I big puffy heart Algebra Lab Gear. One of my kids went through the ALG Basic Algebra book when he was still otherwise working on 3rd grade math. Hands on Equations is frequently recommended, but it never moves beyond single-variable linear equations, so I'm not really a fan. Seems a bit boring and repetitious to me, but go with whatever works for your kid. You could also let your kiddo try out DragonBox Algebra 12+ if you haven't already. Once they beat the game there's a whole other side B or something with really algebra-y algebra puzzles. And, if you think Jacobs would be a big hit, go for it. You can run two math curricula in parallel, letting him practice those foundational arithmetic skills in between playing with variables and such. Sounds like fun to me! I would agree about running Jacob's and CLE simultaneously if he had a stronger background, but this child has only finished 4th grade math. 4th grade math to alg is a huge leap. It's one thing to be excited about ideas and another to have a strong enough background to be able to successful in its pursuit. Introducing basic alg concepts that match ability level will do a lot more to encourage than trying to use a high school text with a huge gap in required background. Quote Link to post Share on other sites

Cake and Pi 725 Posted October 9, 2020 Report Share Posted October 9, 2020 2 hours ago, 8filltheheart said: I would agree about running Jacob's and CLE simultaneously if he had a stronger background, but this child has only finished 4th grade math. 4th grade math to alg is a huge leap. It's one thing to be excited about ideas and another to have a strong enough background to be able to successful in its pursuit. Introducing basic alg concepts that match ability level will do a lot more to encourage than trying to use a high school text with a huge gap in required background. So you're saying that ALG, DragonBox, HOE, and/or some other similar supplement would be a better way to play with algebra alongside the standard curriculum? That would probably be my first pick as well, but I'd likely use whatever the child is most enthusiastic about if they have strong opinions. Some kids have really surprising definitions of "fun" math, lol. What would you think about including Jacobs if they move at the child's pace and only work on what he's grasping easily and while the enthusiasm holds? I mean, as long as they follow his lead, the worst case scenario seems like it would be to wind up shelving Jacobs until more basic math is mastered. I haven't really looked through the Jacobs algebra textbook, though. I spent a year teaching algebra to remedial students, some of whom were working at about a 4th grade level in math, at a public high school. That makes me think it's completely possible to move ahead with basic algebra concepts but at a drastically reduced pace while running an arithmetic thread on the side. In high school this is extremely unideal, but in 3rd grade it's no biggie if a kid takes three or four years to master the equivalent of algebra I while they continue shoring up the basics. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 9, 2020 Report Share Posted October 9, 2020 4 hours ago, 8filltheheart said: I would agree about running Jacob's and CLE simultaneously if he had a stronger background, but this child has only finished 4th grade math. 4th grade math to alg is a huge leap. It's one thing to be excited about ideas and another to have a strong enough background to be able to successful in its pursuit. Introducing basic alg concepts that match ability level will do a lot more to encourage than trying to use a high school text with a huge gap in required background. I actually think intro algebra can be understood well with a good understanding of integer operations. I can’t speak to Jacobs’ Algebra (I bought it, but haven’t used it yet), but lots of early algebra doesn’t require a sophisticated understanding of fractions or division, which is what I think of as later elementary math 🙂 Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 9, 2020 Report Share Posted October 9, 2020 (edited) 4 hours ago, Cake and Pi said: So you're saying that ALG, DragonBox, HOE, and/or some other similar supplement would be a better way to play with algebra alongside the standard curriculum? That would probably be my first pick as well, but I'd likely use whatever the child is most enthusiastic about if they have strong opinions. Some kids have really surprising definitions of "fun" math, lol. What would you think about including Jacobs if they move at the child's pace and only work on what he's grasping easily and while the enthusiasm holds? I mean, as long as they follow his lead, the worst case scenario seems like it would be to wind up shelving Jacobs until more basic math is mastered. I haven't really looked through the Jacobs algebra textbook, though. I spent a year teaching algebra to remedial students, some of whom were working at about a 4th grade level in math, at a public high school. That makes me think it's completely possible to move ahead with basic algebra concepts but at a drastically reduced pace while running an arithmetic thread on the side. In high school this is extremely unideal, but in 3rd grade it's no biggie if a kid takes three or four years to master the equivalent of algebra I while they continue shoring up the basics. 3 hours ago, Not_a_Number said: I actually think intro algebra can be understood well with a good understanding of integer operations. I can’t speak to Jacobs’ Algebra (I bought it, but haven’t used it yet), but lots of early algebra doesn’t require a sophisticated understanding of fractions or division, which is what I think of as later elementary math 🙂 Alg programs aimed toward elementary children fit the description posted by above. I have not seen a high school alg program that doesn't have an expectation of mastery of those topics and doesnt immediately incorporate them. Even MUS's alg (which is the easiest alg I have ever seen) starts with the expectation of fractions, negative numbers, order of operations being already mastered. This is a pg from day 1. This is meant to be review. I think introducing basic alg concepts with a child starting 5th grade math is great. I dont think that a high school math text is the best option. Edited October 9, 2020 by 8filltheheart 2 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 9, 2020 Report Share Posted October 9, 2020 10 minutes ago, 8filltheheart said: Alg programs aimed toward elementary children fit the description posted by above. I have not seen a high school alg program that doesn't have an expectation of mastery of those topics and doesnt immediately incorporate them. Even MUS's alg (which is the easiest alg I have ever seen) starts with the expectation of fractions, negative numbers, order of operations being already mastered. This is a pg from day 1. This is meant to be review. I think introducing basic alg concepts with a child starting 5th grade math is great. I dont think that a high school math text is the best option. Yeah, I agree that if some of those aren’t mastered, you’d have to edit heavily. So you may be right. Do you have Jacobs’ algebra? I should look through mine... Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 9, 2020 Report Share Posted October 9, 2020 (edited) 1 hour ago, Not_a_Number said: Yeah, I agree that if some of those aren’t mastered, you’d have to edit heavily. So you may be right. Do you have Jacobs’ algebra? I should look through mine... No, I don't have Jacobs. My understanding is that Jacobs actually starts gently with review of pre-alg. Foerster's first chpt is also simple review but again expects fractions, order of operations to mastery, etc. Foerster just increases in difficulty rapidly and goes into more depth equally rapidly compared to MUS. (This is the chpt 1 test (this is from the TE hence the answers.)) Edited October 9, 2020 by 8filltheheart Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 9, 2020 Report Share Posted October 9, 2020 13 minutes ago, 8filltheheart said: No, I don't have Jacobs. My understanding is that Jacobs actually starts gently with review of pre-alg. Foerster's first chpt is also simple review but again expects fractions, order of operations to mastery, etc. Foerster just increases in difficulty rapidly and goes into more depth equally rapidly compared to MUS. (This is the chpt 1 test (this is from the TE hence the answers.)) Interesting. It looks like you could actually do a LOT of it without fractions, but you need exponents in a pretty serious way. Or are those introduced within the text? Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 9, 2020 Report Share Posted October 9, 2020 5 minutes ago, Not_a_Number said: Interesting. It looks like you could actually do a LOT of it without fractions, but you need exponents in a pretty serious way. Or are those introduced within the text? They are covered in 1 section (basically a single day's lesson). No way I would use Foerster's with a student who wasn't 100% solid on all elementary math. My kids all go through 6th grade math and MUS's alg before they do Foerster's. Unless you are dealing with exceptionally gifted kids (and I don't consider my kids exceptionally gifted even though my 5th grader is doing MUS's alg this yr), there is not enough review and repetition of concepts in the alg book for mastery of basic elementary concepts. He introduces concepts in a way that are simple enough to master quickly, but everything is premised on full mastery of basic math so concepts are built on that understanding, not teaching it. Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 9, 2020 Report Share Posted October 9, 2020 Just now, 8filltheheart said: They are covered in 1 section (basically a single day's lesson). No way I would use Foerster's with a student who wasn't 100% solid on all elementary math. My kids all go through 6th grade math and MUS's alg before they do Foerster's. Unless you are dealing with exceptionally gifted kids (and I don't consider my kids exceptionally gifted even though my 5th grader is doing MUS's alg this yr), there is not enough review and repetition of concepts in the alg book for mastery of basic elementary concepts. He introduces concepts in a way that are simple enough to master quickly, but everything is premised on full mastery of basic math so concepts are built on that understanding, not teaching it. Yeah, I don't love using higher level books even with gifted students, because they tend to be able to do the concepts but often are asynchronous enough that they can't handle the higher levels of organization and writing needed. That being said, I'd want something other than linear equations for a bright kid who wanted to learn algebra. So if I wanted a reference, I'd probably just pull the more approachable problems out of a book. Quote Link to post Share on other sites

Sarah0000 786 Posted October 10, 2020 Report Share Posted October 10, 2020 22 hours ago, Not_a_Number said: Yeah, I don't love using higher level books even with gifted students, because they tend to be able to do the concepts but often are asynchronous enough that they can't handle the higher levels of organization and writing needed. That being said, I'd want something other than linear equations for a bright kid who wanted to learn algebra. So if I wanted a reference, I'd probably just pull the more approachable problems out of a book. This is the problem we're having with HOE. The equations are so basic and its becoming busywork too. Upper Elementary Challenge Math and some lessons in BA5 are still an appropriate challenge at least. However, doing the easier algebra with HOE does allow for learning how to write calculations down in an organized way and the student can gain practice with self-checking answers. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 10, 2020 Report Share Posted October 10, 2020 2 minutes ago, Sarah0000 said: This is the problem we're having with HOE. The equations are so basic and its becoming busywork too. Upper Elementary Challenge Math and some lessons in BA5 are still an appropriate challenge at least. However, doing the easier algebra with HOE does allow for learning how to write calculations down in an organized way and the student can gain practice with self-checking answers. The way we've done this is to start with a variety of equations with integer coefficients via both guess and check and manipulations. So we've expanded fairly complicated expressions like (x+3)^2 and solved quadratics as well as systems of linear equations. I've found all this quite approachable with my gifted student. Quote Link to post Share on other sites

Guest Posted October 10, 2020 Report Share Posted October 10, 2020 Another option that worked well here for a kid who was dying to do algebra and was completing placement tests and leaving them on my computer to show me she was ready-Keys to Algebra. DD did those at age 7/8 along with Life of Fred Fractions/Decimals/Percents/PreAlgebra I and II before AoPS PA. She had done SM through 5b, Along with HoE and some playing with Algeblocks, but wasn’t quite ready for AoPS as far as formatting, etc went. Quote Link to post Share on other sites

Clemsondana 2,427 Posted October 10, 2020 Report Share Posted October 10, 2020 I've read this thread a couple of times over the past week and I've been perplexed about how a student could skip to any typical algebra program after completing 3rd or 4th grade math. My kids understood the concepts of variables while still very young and one of mine thought solving equations like x^2 + 7 = 56 (and more complicated equations of that sort) was a fun puzzle prior to starting K, but we still went through Singapore Math (skipping sections as indicated) because there were so many concepts that kid didn't know (like fractions, decimals, techniques for multidigit multiplication and division) and I was sure that there would be others that I hadn't thought of. Knowing about squares was something that a friend of ours had taught on a lark, not something that I had done systematically. I'm doing pre-algebra with my second. The older used AOPS and my younger is doing Jousting Armadillos. Both immediately start with fractions and exponents. The fractions are supposed to be mostly review and are covered quickly but a lot of time is spent on exponents, presumably because most algebra programs use these from the first week. I'd be hesitant to jump to algebra using a standard book without having covered fractions, exponents, factors, multiples, and other things that probably wouldn't have been covered prior to a 4th grade program. If the student just wants to work with variables, a pre-algebra program might fit the bill and move slowly enough to be able to fill in missing skills as you come across them. We're also using Life of Fred as a co-curriculum this year, and its pre-algebra is taught using physics equations. We've done the first 15 or so lessons and so far there hasn't been any exponent use but there has been a lot of variable use. Obviously the other alternative is to just write problems and teach concepts as your student finds them interesting. We've done some of that at times, too, but not as our main program (although I know it works for others). 1 1 Quote Link to post Share on other sites

4KookieKids 1,077 Posted October 11, 2020 Report Share Posted October 11, 2020 My oldest did Dragonbox algebra 12+ while still not finished with Singapore 2. I feel like it was a great introduction to algebra, and gave a lot of intuition re: fractions as well, without requiring much of those older elementary concepts/arithmetic. Elements was also a super fun introduction to geometry for him! 🙂 1 Quote Link to post Share on other sites

Sarah0000 786 Posted October 11, 2020 Report Share Posted October 11, 2020 5 hours ago, Not_a_Number said: The way we've done this is to start with a variety of equations with integer coefficients via both guess and check and manipulations. So we've expanded fairly complicated expressions like (x+3)^2 and solved quadratics as well as systems of linear equations. I've found all this quite approachable with my gifted student. Yeah, he's done some of that stuff. After reading this thread I think I've been misremembering what an algebra text would cover and I'm realizing DS is likely ready for a full algebra program. I think I'm still going to pull parts from Singapore 6 though just to check for possible gaps beforehand. Honestly, I like that I can be so hands off with math for him now because I have two younger kids learning foundational stuff. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 3 hours ago, Clemsondana said: I've read this thread a couple of times over the past week and I've been perplexed about how a student could skip to any typical algebra program after completing 3rd or 4th grade math. My kids understood the concepts of variables while still very young and one of mine thought solving equations like x^2 + 7 = 56 (and more complicated equations of that sort) was a fun puzzle prior to starting K, but we still went through Singapore Math (skipping sections as indicated) because there were so many concepts that kid didn't know (like fractions, decimals, techniques for multidigit multiplication and division) and I was sure that there would be others that I hadn't thought of. Knowing about squares was something that a friend of ours had taught on a lark, not something that I had done systematically. I'm doing pre-algebra with my second. The older used AOPS and my younger is doing Jousting Armadillos. Both immediately start with fractions and exponents. The fractions are supposed to be mostly review and are covered quickly but a lot of time is spent on exponents, presumably because most algebra programs use these from the first week. I'd be hesitant to jump to algebra using a standard book without having covered fractions, exponents, factors, multiples, and other things that probably wouldn't have been covered prior to a 4th grade program. If the student just wants to work with variables, a pre-algebra program might fit the bill and move slowly enough to be able to fill in missing skills as you come across them. We're also using Life of Fred as a co-curriculum this year, and its pre-algebra is taught using physics equations. We've done the first 15 or so lessons and so far there hasn't been any exponent use but there has been a lot of variable use. Obviously the other alternative is to just write problems and teach concepts as your student finds them interesting. We've done some of that at times, too, but not as our main program (although I know it works for others). You would obviously have to edit relatively heavily. But having worked with lots of kids who have trouble with algebra, I'll say that there are LOTS of concepts you could fruitfully work on without requiring fractions or exponents. You could do: a) Graphing functions using integer points. b) Algebraic manipulation of all sorts, along with thinking about "inverse operations." c) Multiplying out binomials. d) Work with algebraic expressions. e) Factor and solve quadratics. Do I think that a student right out of grade 4 math can do a full algebra program? No. Do I think a student who's excited about it and whose parent was willing to find approachable problems that illustrate ideas out of the textbook could get a lot of out it? Yes. As for prealgebra, you could do that, too. It would depend how well a student understood arithmetic and whether the pre-algebra program went over the topics the student was interested in. 1 Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 11, 2020 Report Share Posted October 11, 2020 17 hours ago, Sarah0000 said: This is the problem we're having with HOE. The equations are so basic and its becoming busywork too. Upper Elementary Challenge Math and some lessons in BA5 are still an appropriate challenge at least. However, doing the easier algebra with HOE does allow for learning how to write calculations down in an organized way and the student can gain practice with self-checking answers. I can see where a strong math student with a BA background would find the levels 1 and 2 in the Verbal Problems book repetitive. But, I would think that at least some of the level 3 problems incorporating fractions would be a good fit. But, BA already incorporates challenging problem-solving, so HOE less so. (My kids complete the VP book by doing 3 problems a day/2-3 times/week when working through 3rd-5th grade level math. I think for a child working through CLE that the VP book would offer a way to think through math that that they are unfamiliar with.) @nwahomeschoolmom For a student just starting 5th grade math, elementary-focused alg approaches like those already mentioned or LOF's fractions/decimals books mentioned by @dmmetler) or MM's ratio/proportions workbook https://www.mathmammoth.com/preview/Ratios_Proportions_Problem_Solving_Solving_Problems_Equivalent_Ratios.pdf would be my preferred choices before an algebra textbook. (But my perspective matches @Clemsondana's.) 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 I think the US has a certain amount of awe towards algebra, given how late kids here do it compared to lots of other countries. For what it's worth, it took us considerably longer to get basic fraction concepts down than to figure out that (x+1)^2 = x^2 + 2x + 1 and that you can take equations like 2(x+4) = 16 and solve them by doing the same thing to both sides of an equation. It also didn't take us any effort to start graphing using lattice points. My gifted student found linear equations (with or without fractions) all completely trivial and identical. I guess we are still working on the fact that you can divide 3.4x by 3.4 to get x (she tends to want to work with the numerator and denominator of a fraction separately), but I'm sure she'd be bored out of her mind doing this for more than a few weeks. So for us, having a broader range of problems was crucial for keeping her engaged 🙂 . But also, the worst thing that can happen if you hand a kid an algebra textbook is they get confused and decide they can't learn it yet. Or maybe they'll make surprising headway! You never know. 1 Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 11, 2020 Report Share Posted October 11, 2020 2 minutes ago, Not_a_Number said: I think the US has a certain amount of awe towards algebra, given how late kids here do it compared to lots of other countries. For what it's worth, it took us considerably longer to get basic fraction concepts down than to figure out that (x+1)^2 = x^2 + 2x + 1 and that you can take equations like 2(x+4) = 16 and solve them by doing the same thing to both sides of an equation. It also didn't take us any effort to start graphing using lattice points. My gifted student found linear equations (with or without fractions) all completely trivial and identical. I guess we are still working on the fact that you can divide 3.4x by 3.4 to get x (she tends to want to work with the numerator and denominator of a fraction separately), but I'm sure she'd be bored out of her mind doing this for more than a few weeks. So for us, having a broader range of problems was crucial for keeping her engaged 🙂 . But also, the worst thing that can happen if you hand a kid an algebra textbook is they get confused and decide they can't learn it yet. Or maybe they'll make surprising headway! You never know. The 2nd example is standard elementary math. Quote Link to post Share on other sites

Clemsondana 2,427 Posted October 11, 2020 Report Share Posted October 11, 2020 18 minutes ago, Heigh Ho said: ime, The serious answer lies in the definition of gifted. A gifted student extends what is presented. The child who sees the number line goes both ways, and wonders why the first grade teacher is ignoring the region to the left of the zero. And wonders why the other directions aren't included. ime, By the time 4th grade is done, the g. student has extended the concepts and used the class time to work thru the possibilities, which takes care of the rest of arithmetic, with the exception of ratios (beginning trig is done in ec as they measure tree heights, sun angles etc while gaining hiking/survival skills). Often they need to acquire the mathematical vocab and few problem solving techniques that are typically taught in the robust arithmetic courses. The set of gifted students and the set of accelerated students overlap, but aren't coincident. Sure, I understand that. My older has an intuitive understanding of math that I don't have and was comfortably working with negatives, squares, and all sorts of things prior to starting K. At that point we were doing all math orally for fun and weren't really intending to do anything beyond counting and basic arithmetic, of the 'How many cookies?' sort. My only point was that if one uses a standard algebra program with a student who hadn't covered some elementary content, they will likely either need to be running a separate program concurrently, be prepared to stop and teach (and possibly practice) some material that the book considers to be a prerequsite, or be prepared for their student to do some things in a very inefficient way because they understand the concept but don't know how to do a faster algorithm. For my particular student, moving through regular math at a much faster pace (we did Singapore 2 and 3 in K, working 15-30 min/day, because we skipped so much) was the way to go because we didn't have to stop and backtrack or take breaks from pre-A or A to learn concepts. Another student might do better working through pre-A or A, learning things as they need them. My only comment was that they would likely need to stop and learn some things, like exponents and fractions, very early unless they wrote their own problems or were careful in how they chose or edited their program. My older kid usually prefers to be left alone to read the text and would likely be frustrated with having to wait for me to find a resource, puzzle through it, or teach a new arithmetic concept every few lessons, but other families and kids may work differently. Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 2 minutes ago, 8filltheheart said: The 2nd example is standard elementary math. I think it depends on the textbook 🙂 . Not all elementary textbooks do algebraic manipulations. We did much more guess and check as personal preference before we were ready to jump into more serious algebraic work. I think, realistically, there's less of a dividing line between "arithmetic" and "algebra" than people think. You could probably grab a lot of the low-hanging fruit with a bright kid working on 5th grade math, although you could NOT do a whole algebra program You could probably get to something like the quadratic formula without much trouble, though (maybe without explicit square roots.) But completing the square with integer coefficients is very approachable. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 6 minutes ago, Clemsondana said: My only comment was that they would likely need to stop and learn some things, like exponents and fractions, very early unless they wrote their own problems or were careful in how they chose or edited their program. Yeah, I think you'd have to skip those parts. I would not personally advocate backfilling those in. Rather, I'd run the problems with integer coefficients concurrently with the 5th grade program. My point was merely that you could get quite far into algebraic ideas by just using integer coefficients. Quote Link to post Share on other sites

Clemsondana 2,427 Posted October 11, 2020 Report Share Posted October 11, 2020 I think a kid could get far in algebraic concepts, but I thought the question was more of a 'What program should I use?'. Singapore 6 does line graphs with variables and in either 5 or 6 they're solving problems with variables (in which studentst do the same thing to both sides to solve for X) and doing word problems where they write simple algebra problems,. To graph a lot of other things or solve other sorts of problems students would at least need to understand squares/square roots (which aren't all that complicated, but they don't already know them after 3rd grade math). To factor, it's helpful to be fluent with multiplication facts (which a student may know from an early age, or may learn as they encounter them in their math programs, which could be 2nd or 3rd grade). I think what a lot of people are saying is that, having used various pre-A and A programs, they all have the expectation of certain arithmetic knowledge. A parent can certainly teach algebra concepts without a book (as I ended up doing some of, unintentionally, just answering questions) but there isn't likely a good textbook-style standard program that can be used as-is with a student who doesn't know fractions, decimals, etc. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 11 minutes ago, Clemsondana said: To graph a lot of other things or solve other sorts of problems students would at least need to understand squares/square roots (which aren't all that complicated, but they don't already know them after 3rd grade math). To factor, it's helpful to be fluent with multiplication facts (which a student may know from an early age, or may learn as they encounter them in their math programs, which could be 2nd or 3rd grade). Well, I wouldn't normally suggest this, but this poster suggests that her child is motivated by algebra 🙂 . And later grade arithmetic can be really arduous and boring (big numbers, lots of calculations.) So I can imagine it being FUN for a kid to factor x^2 - 7x + 12 and then to solve x^2 - 7x + 12 = 0 as opposed to figuring out 43567 divided by 23 using long division, to pull random numbers out of a hat. I would also assume that a kid who finished grade 4 math may very well know their multiplication facts. I actually don't think you need square roots for lots of early algebra. And squares and small powers are quite easy. Anyway, as someone whose older girl is motivated to learn arithmetic by harder concepts and find arithmetic by itself really dull, I'm always a fan of following what a child actually wants to do. But yes, as I've said a few times, that would probably take some editing 🙂 . If a child hasn't seen fractions, they obviously can't solve 1/3*x + 2/5 = 3/4 or whatever. Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 1 minute ago, Heigh Ho said: fractions start in school with pattern blocks...grade K after several years of part/whole with building blocks from age 18 months on. Unfortunately rulers have been dropped from the curriciulum here, so that does limit what some children can soak from the environment in the classroom to move to abstract notation, fractions only remain in the music curriculum. The intuitive understanding builds, and by grade 3 when the formal notation is introduced in the Common Core math classroom, many students are already famililar from using cup measures, rulers, tape measures and wrenches in the home plus quarter notes, etc at piano lessons. I can only speak to our experience, but we did maybe 2 years of intuitive, "spoken" fractions and then still needed a year to REALLY solidify the fraction operations. It all made sense to her, but making it algorithmic and quick was another story. Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 11, 2020 Report Share Posted October 11, 2020 (edited) 1 hour ago, Not_a_Number said: I think it depends on the textbook 🙂 . Regardless, I would consider the 2nd example on par with 4th grade math, not high school algebra like the OP is asking about in terms of a high school alg textbook. Edited October 11, 2020 by 8filltheheart 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 30 minutes ago, 8filltheheart said: Regardless, I would consider the 2nd example on par with 4th grade math, not high school algebra like the OP is asking about in terms of a high school alg textbook. Ah OK. What pops into your head when you think "high school algebra"? I didn't learn algebra in North America (I knew it by age 11, when we moved to Canada), so I have no idea how the sequences are split up, since it was old news by the time I saw it. 1 Quote Link to post Share on other sites

Clemsondana 2,427 Posted October 11, 2020 Report Share Posted October 11, 2020 1 hour ago, Not_a_Number said: Ah OK. What pops into your head when you think "high school algebra"? I didn't learn algebra in North America (I knew it by age 11, when we moved to Canada), so I have no idea how the sequences are split up, since it was old news by the time I saw it. I know that by the end of Singapore 6 (which we do in a semester, so I can't remember what was in 5 and what was in 6) the kids had covered order of operations, writing/using/solving expressions and equations so that y could vary with x (first just solving simple 'x + 5 = 12' things, then moving on to 'Bob sells 3 times as many pies as cookies. Pies cost twice as much as cookies' where they first write expressions and then solve, either being given how much $ was made, or how many pies were sold, or something of that sort). They don't teach how to solve with 2 variables - they have you set one in terms of the other, but not quite saying y = 3x, more like pies = 2 cookie, but the thought process is leading in that direction. They also cover y = mx + b style lines, but I don't remember if they actually see that equation with m. I know that they use it in word problems. Pre-algebra extends that and seems to spend a good bit of time practicing what can and can't be done with numbers, variables, and exponents. So, they get problems like (5x^5/10x^2)/(3x^7/9x^3) to help solidify the difference between 5x and x^5 in terms of simplifying...and then they move on to multiple variables, or adding/subtracting in the fractions so that the kids remember/learn that if you have (7 - x^6)/x^5 than you can't just divide one term to get 7 - x. In my remembering, factoring of quadratics, completing the square, the quadratic formula, and graphing of circles, parabolas, etc, show up in algebra, along with rate problems and more of those 'Bob is twice as old as Sally was 5 years ago' or '2 cars 57 miles apart start driving towards each other' problems. That's why I had asked whether the student wanted to do algebra or just work with variables - variables can show up in programs like Singapore or pre-algebra where the student wouldn't need (or would be taught) some things likely to be missing after completing 3rd grade math, while a high school text assumes that students already know that and writes the problems accordingly. Or, the student may like using variables to solve real-life problems, which might fit with another type of book that isn't a typical high school algebra sequence. We haven't used it, but that might be in keeping with what I've heard about Jacobs' Mathematics, a human endeavor book that I'm considering for my younger, or the Life of Fred preA books, each of which uses math applied to different subjects (I think physics, biology, and economics). 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 25 minutes ago, Clemsondana said: I know that by the end of Singapore 6 (which we do in a semester, so I can't remember what was in 5 and what was in 6) the kids had covered order of operations, writing/using/solving expressions and equations so that y could vary with x (first just solving simple 'x + 5 = 12' things, then moving on to 'Bob sells 3 times as many pies as cookies. Pies cost twice as much as cookies' where they first write expressions and then solve, either being given how much $ was made, or how many pies were sold, or something of that sort). They don't teach how to solve with 2 variables - they have you set one in terms of the other, but not quite saying y = 3x, more like pies = 2 cookie, but the thought process is leading in that direction. They also cover y = mx + b style lines, but I don't remember if they actually see that equation with m. I know that they use it in word problems. Pre-algebra extends that and seems to spend a good bit of time practicing what can and can't be done with numbers, variables, and exponents. So, they get problems like (5x^5/10x^2)/(3x^7/9x^3) to help solidify the difference between 5x and x^5 in terms of simplifying...and then they move on to multiple variables, or adding/subtracting in the fractions so that the kids remember/learn that if you have (7 - x^6)/x^5 than you can't just divide one term to get 7 - x. In my remembering, factoring of quadratics, completing the square, the quadratic formula, and graphing of circles, parabolas, etc, show up in algebra, along with rate problems and more of those 'Bob is twice as old as Sally was 5 years ago' or '2 cars 57 miles apart start driving towards each other' problems. That's why I had asked whether the student wanted to do algebra or just work with variables - variables can show up in programs like Singapore or pre-algebra where the student wouldn't need (or would be taught) some things likely to be missing after completing 3rd grade math, while a high school text assumes that students already know that and writes the problems accordingly. Or, the student may like using variables to solve real-life problems, which might fit with another type of book that isn't a typical high school algebra sequence. We haven't used it, but that might be in keeping with what I've heard about Jacobs' Mathematics, a human endeavor book that I'm considering for my younger, or the Life of Fred preA books, each of which uses math applied to different subjects (I think physics, biology, and economics). Thanks! We do everything really out of order, so it's good to know what order things go in 🙂 . We have Mathematics: a Human Endeavor, actually! (I thought I had the other book, but I was just confused. It looks very cute. That might be a good way to start, although it does require fractions, I think. We've basically never separated variables from word problems, so DD8 naturally jumped to setting up questions with variables. We've done a bit of completing the square and factoring quadratics. I actually really dislike teaching graphing starting from doing tons of lines, because I think it overteaches something easy and doesn't spend enough time on something hard. So we graphed all sorts of stuff our last go-around 🙂 . I think we were just using lattice points, but I'm not sure -- we did know at least some fractions at that point. Quote Link to post Share on other sites

nwahomeschoolmom 171 Posted October 11, 2020 Author Report Share Posted October 11, 2020 3 hours ago, Not_a_Number said: Ah OK. What pops into your head when you think "high school algebra"? I didn't learn algebra in North America (I knew it by age 11, when we moved to Canada), so I have no idea how the sequences are split up, since it was old news by the time I saw it. I am not as smart as my son and I did high school algebra in 7th grade in a full class at a public school. I don't think of algebra as high school level for the majority of bright kids. If I did it 7th grade, I thought I could accelerate my son a bit if he wanted to start earlier. I am really not considering skipping anything, just moving more quickly. If we don't really do anything different, he will get to algebra by 7th or a bit earlier anyway, but he's much smarter than I was, so I thought I should get him ready sooner. I remember learning subtraction with borrowing in 2nd grade and then getting so bored with having to keep doing it with larger numbers, because I got the concept. And that goes for many other topics in elementary math. I'd like to accelerate on the concepts and do less arithmetic. Arithmetic is foundational, but it's also a bit boring! ( I'm not as good as math as some of ya'll though. I like traditional math and algebra....bar charts to solve word problems not so much!) I am actually shocked that CLE 4 already teaches factoring, order of operations, fractions, etc. It's crazy...I did not learn that in 4th grade. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 40 minutes ago, nwahomeschoolmom said: Arithmetic is foundational, but it's also a bit boring! Yeah, that's what I found 🙂 . However, there's lots of stuff that's not traditional "algebra" that's fun. Have you look at Beast Academy or any of the AoPS books? Quote Link to post Share on other sites

Clemsondana 2,427 Posted October 11, 2020 Report Share Posted October 11, 2020 32 minutes ago, nwahomeschoolmom said: I am not as smart as my son and I did high school algebra in 7th grade in a full class at a public school. I don't think of algebra as high school level for the majority of bright kids. If I did it 7th grade, I thought I could accelerate my son a bit if he wanted to start earlier. I am really not considering skipping anything, just moving more quickly. If we don't really do anything different, he will get to algebra by 7th or a bit earlier anyway, but he's much smarter than I was, so I thought I should get him ready sooner. I remember learning subtraction with borrowing in 2nd grade and then getting so bored with having to keep doing it with larger numbers, because I got the concept. And that goes for many other topics in elementary math. I'd like to accelerate on the concepts and do less arithmetic. Arithmetic is foundational, but it's also a bit boring! ( I'm not as good as math as some of ya'll though. I like traditional math and algebra....bar charts to solve word problems not so much!) I am actually shocked that CLE 4 already teaches factoring, order of operations, fractions, etc. It's crazy...I did not learn that in 4th grade. Those topics sound like what Singapore covers somewhere between levels 3-5. so that seems pretty normal. There's no problem with moving more quickly if he understands things - we skipped practice if my kids were consistently getting things correct. Some concepts required little practice, and other times we did all of the exercises. With Singapore, they added more digits each year (at first just going up to tens, then hundreds, then hundred thousands, etc). When we'd have the review parts at a new level, I'd just hand it to my kid - if they could do one day's addition with hundred thousands, then I knew that they both remembered it and could do it with big numbers, so there was no reason to do 10 days of practice. 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 (edited) 6 minutes ago, Clemsondana said: Those topics sound like what Singapore covers somewhere between levels 3-5. so that seems pretty normal. There's no problem with moving more quickly if he understands things - we skipped practice if my kids were consistently getting things correct. Some concepts required little practice, and other times we did all of the exercises. With Singapore, they added more digits each year (at first just going up to tens, then hundreds, then hundred thousands, etc). When we'd have the review parts at a new level, I'd just hand it to my kid - if they could do one day's addition with hundred thousands, then I knew that they both remembered it and could do it with big numbers, so there was no reason to do 10 days of practice. The reason I like doing some ideas early in a more systematic way is that sometimes kids need time. I know your kids didn't have this issue, but the idea of variables can be kind of tricky. So can the idea of algebraic manipulations, and so can the idea of distributing for products of binomials, and don't get me started on how many kids in my college classes didn't really have a feel for graphing... So the nice thing about actually starting a high school textbook is that you can let the ideas percolate instead of just hammering on linear equations again and again. But again, you'd obviously have to edit heavily. Edited October 11, 2020 by Not_a_Number Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 11, 2020 Report Share Posted October 11, 2020 (edited) 3 hours ago, nwahomeschoolmom said: I am actually shocked that CLE 4 already teaches factoring, order of operations, fractions, etc. It's crazy...I did not learn that in 4th grade. I would consider them pretty standard 4th grade concepts to me, too. (I'm going to add pictures from my grandson's 4th grade math book) I agree with @Clemsondana in that I would also reduce the number of problems if concepts are 100% mastered. (I ask my kids to explain what they are doing and why vs their just doing. I want to make sure they really do understand the concepts.) My kids are accelerated, but I accelerate with caution. I don't skip. I reduce. They do more. I supplement daily work with more challenging problems that are equally appropriate to their abilities (like using HOE alongside a 3rd grade curriculum). FWIW, I have had a similar scenario with several of my kids (a third grader doing 5th grade work). My 10 yod is my 2nd child using MUS algebra as a pre-alg course as a 5th grader. Edited October 11, 2020 by 8filltheheart add MUS algebra so it didnt seem like a reference to their pre-alg 1 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 11, 2020 Report Share Posted October 11, 2020 1 hour ago, 8filltheheart said: I would consider them pretty standard 4th grade concepts to me, too. (I'm going to add pictures from my grandson's 4th grade math book) I agree with @Clemsondana in that I would also reduce the number of problems if concepts are 100% mastered. (I ask my kids to explain what they are doing and why vs their just doing. I want to make sure they really do understand the concepts.) My kids are accelerated, but I accelerate with caution. I don't skip. I reduce. They do more. I supplement daily work with more challenging problems that are equally appropriate to their abilities (like using HOE alongside a 3rd grade curriculum). FWIW, I have had a similar scenario with several of my kids (a third grader doing 5th grade work). My 10 yod is my 2nd child using MUS as a pre-alg course as a 5th grader. Do you guys do those with algebraic manipulations or by guess and check? (For the record, I think both are valid methods.) Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 11, 2020 Report Share Posted October 11, 2020 Algebraic manipulation and check. 2 Quote Link to post Share on other sites

Clemsondana 2,427 Posted October 12, 2020 Report Share Posted October 12, 2020 Algebraic manipulation. I think that my kids see the problem that way. My older actually got frustrated learning to write problems - when he'd get those 'Sally is older than Bob was 5 years ago' type of problems he'd close his eyes and spit out an answer and we had to work on writing them once problems got too complicated for that to work (and that point was far beyond what it was for me...kiddo would look and say 'Mom, why are you writing...the answer is 14'). Actually, he may mentally do some substitutions to realize that the answer has to fit in a certain range..I know that he knows how to do manipulations, but I can't guarantee what he actually does. I'm convinced that sometimes he figures out the answer and then writes down what he must to show what he did rather than writing to help solve the problem. But, at least in the beginning, the thought process involved the steps of the 'do the same thing to both sides' process. Younger loved Balance Benders problems, which are all done with shapes. I know that I've only taught algebraic manipulation, and even with the puzzles I'd explain by saying 'Well, since both sides have squares...'. So far this kid definitely doesn't do any 'close your eyes and visualize the answer' voodoo with math - it's just do the manipulations, remembering order of operation. They do know how to sub their answer back in to make sure it works, although we don't check every problem. 1 Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 12, 2020 Report Share Posted October 12, 2020 6 hours ago, nwahomeschoolmom said: I am not as smart as my son and I did high school algebra in 7th grade in a full class at a public school. I don't think of algebra as high school level for the majority of bright kids. If I did it 7th grade, I thought I could accelerate my son a bit if he wanted to start earlier. I am really not considering skipping anything, just moving more quickly. If we don't really do anything different, he will get to algebra by 7th or a bit earlier anyway, but he's much smarter than I was, so I thought I should get him ready sooner I wanted to come back and comment on the bolded. Just keep in mind mathematically trajectory. What would you see his long-term math sequence being? Do you have academic options available to you? For perspective, my ds took his first alg class for high school credit in 6th (Foersters). He graduated from high school with credit for multivariable, diffEQ 1 and 2, and linear alg. If we hadn't had a 4 yr university that allowed him to dual enroll after AP cal, I'm not sure what he would have taken. (Options like OHS are way out of our budget.) Just make sure that you aren't racing forward with nowhere to go. Even with alg in 7th grade, they end up with at least 1 yr of math beyond cal BC. (Depending on where you live, DE at a CC is poor fit for an advanced student. Even at a 4 yr U with an engineering school and a med school, my ds was at the top of his math classes as a high school student. CC would have been a disaster.) 2 Quote Link to post Share on other sites

Not_a_Number 11,126 Posted October 12, 2020 Report Share Posted October 12, 2020 12 minutes ago, 8filltheheart said: I wanted to come back and comment on the bolded. Just keep in mind mathematically trajectory. What would you see his long-term math sequence being? Do you have academic options available to you? For perspective, my ds took his first alg class for high school credit in 6th (Foersters). He graduated from high school with credit for multivariable, diffEQ 1 and 2, and linear alg. If we hadn't had a 4 yr university that allowed him to dual enroll after AP cal, I'm not sure what he would have taken. (Options like OHS are way out of our budget.) Just make sure that you aren't racing forward with nowhere to go. Even with alg in 7th grade, they end up with at least 1 yr of math beyond cal BC. (Depending on where you live, DE at a CC is poor fit for an advanced student. Even at a 4 yr U with an engineering school and a med school, my ds was at the top of his math classes as a high school student. CC would have been a disaster.) Well, from my perspective, there's a lot of stuff you can do using calculus and linear algebra. So actually getting to calculus does open doors to interesting fields of study, if you have an interested and motivated student. There's actually a LOT of math after calculus, if one is interested. Plus, math simply isn't linear -- there's math you can learn alongside calculus that uses no calculus. And it's all interesting and cool, if you happen to be into math. Quote Link to post Share on other sites

8filltheheart 35,262 Posted October 12, 2020 Report Share Posted October 12, 2020 Just now, Not_a_Number said: Well, from my perspective, there's a lot of stuff you can do using calculus and linear algebra. So actually getting to calculus does open doors to interesting fields of study, if you have an interested and motivated student. There's actually a LOT of math after calculus, if one is interested. Plus, math simply isn't linear -- there's math you can learn alongside calculus that uses no calculus. And it's all interesting and cool, if you happen to be into math. Sure. But if you have a mom that doesn't know the math and there is no place for them to take it, it leaves them with self-studying via MOOCs. Not much fun for a mathy teenager that wants to geek out on math. Quote Link to post Share on other sites

## Recommended Posts

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.