Timeline for Best way to explain the thinking steps from x² = 9 to x=±3
Current License: CC BY-SA 4.0
41 events
| when toggle format | what | by | license | comment | |
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| Feb 5 at 17:00 | comment | added | zipirovich | @SueVanHattum : Sure, I can understand that. I mean, it's your book, so you do what's right for it. :) But then the problem is still there: it's not true that Mom took the square root of the LHS. Maybe something could be rephrased in that part? | |
| Feb 4 at 19:56 | comment | added | Sue VanHattum♦ | @zipirovich I am trying to avoid absolute value. I agree that that makes it mathematically correct, but I don't see it as creating a simple solution. | |
| Feb 3 at 6:41 | comment | added | zipirovich | (cont'd) So doing the same thing to both sides would result in $x^2=9$ implies $\sqrt{x^2}=\sqrt{9}$ implies $|x|=3$, which gives us both roots. However, I don't know to explain this well to a 12-year old. So, back to the beginning of this two-part comment. I guess I should've said that the problem is not with Mom's words, but with the fact that she didn't do what she said she did. | |
| Feb 3 at 6:41 | comment | added | zipirovich | I think there's a problem with Mom saying "You know how, with algebra, you do the same thing to both sides when you’re simplifying an equation? … If you think of this as an algebra step, it would be taking the square root of both sides." But then Mom did not do the same thing to both sides. She didn't take the square root of the left-hand side — she merely erased the little superscript of "2" from there. Because applying the square root to the LHS would result in $\sqrt{x^2}$, not $x$. And in reals, $\sqrt{x^2}=|x|$. (to be cont'd) | |
| Feb 2 at 2:42 | answer | added | David Elm | timeline score: 1 | |
| Oct 20, 2024 at 0:17 | comment | added | Sue VanHattum♦ | Like I said, it's not for the average reader at those ages. It's not something that would be taught in school at that age. (Art of Problem Solving, meant for the very best students, teaches beginning algebra in 7th grade (12 and 13 year olds), and has factoring toward the end of the course.Actually, I'd love for you to read one of the books and tell me who you think my readers are! | |
| Oct 18, 2024 at 19:54 | comment | added | Robbie Goodwin | Broadly, I'm suggesting your target audience it too young, if that mathematical understanding is the point of the story. If it's not the main point, but a treasure to picked up and understood by the more sophisticated reader then by all means, go for it! | |
| Oct 18, 2024 at 19:50 | comment | added | Robbie Goodwin | I might be over-simplifying, yet from 55-years-old memory 13 - not 12 - was the very lowest limit of British schools trying to explain equations with multiple solutions. I guess the reason, the way the 'wrong' solution was discounted was considered too complex for most kids that age to grasp. That not the methods of resolution but even the idea of multiple answers, is far too much for most 'educated' adults to understand has in my limited experience been too much for many educators or administrators to take on board. | |
| Oct 18, 2024 at 19:25 | comment | added | Sue VanHattum♦ | @RobbieGoodwin, the characters are 12 and 13. As a much older adult, I love reading YA fiction myself. I want anyone who's ready for the math (or at least some of the math) to qualify as a possible reader. I don't think about average capability. Not everyone will want to read this book (and the other 3). You can see more about this book and the others at: sites.google.com/d/1vDRLBNeUjK1aqPgeHjRMA6TsmAny8hwV/p/… | |
| Oct 18, 2024 at 19:17 | comment | added | Robbie Goodwin | Can you specify how old your 'young adults' are, please? To me, that range from about 12-13 to perhaps 24-25… and surely if we halved that 12-year difference, the ends of the spectrum would still sit far apart in terms of average ability at arithmetic,let alone maths. | |
| Sep 6, 2024 at 14:57 | answer | added | SoupDragon | timeline score: 3 | |
| Sep 5, 2024 at 17:50 | answer | added | Nullius in Verba | timeline score: -1 | |
| Sep 3, 2024 at 19:41 | answer | added | user52817 | timeline score: 1 | |
| Sep 3, 2024 at 16:28 | comment | added | Sue VanHattum♦ | @ Malady, yep. I didn't care for the history of it, and considered changing it, but somehow (my characters feel real to me) her name just had to be Althea. | |
| Sep 3, 2024 at 16:04 | comment | added | Michael G | @Malady Why it's that very gutenberg version that I've read! I think but am not certain that they fixed some weird hypertext links glitches in the old epub version from long ago. | |
| Sep 3, 2024 at 15:48 | answer | added | Michael G | timeline score: 4 | |
| Sep 3, 2024 at 2:59 | comment | added | Malady | So, the protag's name is Althea, a.k.a Healing? en.wikipedia.org/wiki/Althea Not Aletheia or Aleth or Alethea? en.wikipedia.org/wiki/Alethea? A.k.a Truth? | |
| Sep 3, 2024 at 2:57 | comment | added | Malady | @MichaelG - What's your thoughts on gutenberg.org/ebooks/26908 | |
| S Sep 2, 2024 at 19:03 | history | suggested | CommunityBot | CC BY-SA 4.0 | Spelling fixes |
| Sep 2, 2024 at 17:05 | comment | added | Yakk | I mean, nothing in any of the arguments presented show that -3 and +3 are the only solutions. The actual argument about uniqueness requires proving (or knowing about) properties about the real numbers that nobody is mentioning. | |
| Sep 2, 2024 at 16:05 | answer | added | Solomon Slow | timeline score: 11 | |
| Sep 2, 2024 at 12:59 | review | Suggested edits | |||
| S Sep 2, 2024 at 19:03 | |||||
| Sep 2, 2024 at 6:23 | comment | added | infinitezero | "“We need more space!”" Missed opportunity to say "We're gonna need a bigger piece of paper" | |
| Sep 2, 2024 at 4:45 | comment | added | Sue VanHattum♦ | @MichaelG, thank you! I just now took the ebook of that out of the library. | |
| Sep 2, 2024 at 2:43 | comment | added | Michael G | Apropos of nothing, I simply want to say that your dialog charmingly reminds me of Jane Marcet's "Conversations on Chemistry, Intended More Especially for the Female Sex" written in 1805. In addition to being hugely popular at the time, the legendary Michael Faraday credited Mrs Marcet's book with putting him on the pathway to becoming who we now consider one of the greatest scientists of all time. Good luck with your books! | |
| Sep 1, 2024 at 22:33 | history | edited | Sue VanHattum♦ | CC BY-SA 4.0 | added 42 characters in body |
| Sep 1, 2024 at 21:30 | answer | added | Stef | timeline score: 2 | |
| Sep 1, 2024 at 18:50 | comment | added | Sue VanHattum♦ | Yeah. This seems pretty obvious to me, so probably not worth including. We mathematicians are super careful, and that might be interesting to include somehow, but probably not with this example. | |
| Sep 1, 2024 at 16:37 | answer | added | Xander Henderson♦ | timeline score: 3 | |
| Sep 1, 2024 at 16:23 | comment | added | Dave L Renfro | For a $12$-year old, maybe by looking at some examples guess that the squares of numbers larger than $3$ will be larger than $9.$ And how can we justify this to a super-picky $12$-year old? Maybe by saying that a number larger than $3$ times $3$ will be larger than $9$ (perhaps point out the result if we add a number larger than $3$ to itself $3$ times; maybe first consider adding $6$ to a number larger than $3,$ and replacing $6$ with a sum of two larger than $3$ numbers), and so a number larger than $3$ times itself will be larger still. Now consider numbers between $0$ and $3,$ etc. | |
| Sep 1, 2024 at 16:07 | comment | added | Dave L Renfro | only that the possibilities are a subset of $\{-3,\,+3\}$ (substitution shows both are solutions, of course). (2) That's a huge reduction in possibilities, and making this reduction (when accepting arithmetic rules for negative numbers) seems to be the most significant step in solving, since we can determine whether the solution set is $\emptyset$ or $\{-3\}$ or $\{+3\}$ or $\{-3,\,+3\}$ simply by trial and error substitution. | |
| Sep 1, 2024 at 16:04 | comment | added | Dave L Renfro | What got me to think about this is that I paused when reading your question at the beginning, especially the end -- "get us from $x^2=9$ to $x = \pm 3$". I was thinking that technically, when viewed as a logical implication and translating "$\pm 3$" as "$+3$ or $-3$", that we've reduced the possibilities for $x$ from the set of all real numbers to the set $\{-3,\,+3\},$ which led me to think about a couple of things: (1) This logical implication by itself does not (necessarily) imply that BOTH $-3$ and $+3$ are solutions, (continued) | |
| Sep 1, 2024 at 15:16 | comment | added | Sue VanHattum♦ | Hmm, how do we know that no other numbers will work? I'll have to think about that. If it's an interesting question, then it might possibly fit later in the book. They discuss the situation of there being n different nth roots. But I hadn't considered why there aren't more. | |
| Sep 1, 2024 at 12:25 | answer | added | Morresh | timeline score: 5 | |
| Sep 1, 2024 at 11:34 | comment | added | Dave L Renfro | When I initially read the beginning of your question, I was thinking the issue was not going to be whether both $3$ and $-3$ work, but instead how do we know that no other numbers will work. Depending on how this discussion fits into other aspects of their interaction at this point (and with possible aftermath [pun intended] situations), it might make for a more novel [pun also intended] reader experience to go that route than down the overused path of how to multiply negative numbers. But if the intent is to have them discuss non-real numbers, then maybe stay the current course. | |
| Sep 1, 2024 at 8:06 | history | became hot network question | |||
| Sep 1, 2024 at 1:17 | answer | added | Pedro | timeline score: 11 | |
| Sep 1, 2024 at 0:42 | answer | added | fedja | timeline score: 9 | |
| Sep 1, 2024 at 0:33 | answer | added | TomKern | timeline score: 41 | |
| Sep 1, 2024 at 0:09 | history | edited | Sue VanHattum♦ | CC BY-SA 4.0 | added 155 characters in body |
| Sep 1, 2024 at 0:03 | history | asked | Sue VanHattum♦ | CC BY-SA 4.0 |