tag:blogger.com,1999:blog-8231784566931768362.comments2015-07-06T06:28:14.484-04:00MathNotationsDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger2699125tag:blogger.com,1999:blog-8231784566931768362.post-74083379649905408712015-07-06T06:28:14.484-04:002015-07-06T06:28:14.484-04:00Nice! When math educators share ideas, several Big...Nice! When math educators share ideas, several Big Ideas emerge! Your thought about the importance of training our students to represent mathematical information helped to motivate the name of this blog! I also believe that asking students if a "linear" equation of the form Ax+B=0 can have more than one solution leads them to uncover another Big Idea. Having them think about this both algebraically and graphically deepens that understanding. Generalizing this idea to quadratic functions is a logical next step. Can a quadratic have THREE ZEROS? Interpret that graphically!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-54926444090504047742015-07-05T22:27:55.234-04:002015-07-05T22:27:55.234-04:00One question you can ask yourself is: How do I int...One question you can ask yourself is: How do I interpret the given (e.g., how do I record it symbolically?), and what conclusions might I draw?<br /><br />For example, the given is that there are at least two values of x for which the expression on the left is 0. What does this mean?<br /><br />Well, it means there are distinct N, M for which:<br /><br />(a-3)N + (b+2) = 0, and<br />(a-3)M + (b+2) = 0.<br /><br />Introducing that bit of notation is nontrivial in and of itself!<br /><br />And now? Perhaps we can see what the two equations tell us.<br /><br />In fact, subtracting yields:<br /><br />(a-3)(N-M) = 0.<br /><br />But N and M are distinct; so we need a-3 = 0, i.e., a = 3.<br /><br />Working with our original given,<br /><br />(a-3)x + (b+2) = 0 becomes<br />b+2 = 0,<br />whence b = -2.<br /><br />Recording our thinking with non-ambiguous notation is a Big Idea, for sure.mathwaterhttps://mathwater.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-48971798572827516822015-03-12T19:02:30.971-04:002015-03-12T19:02:30.971-04:00Thanks, Bryant. Yes this post will outlive me! May...Thanks, Bryant. Yes this post will outlive me! Maybe I should have the title as my epitaph!You'll survive the math. There are so many good online math tutorials. Also feel free to send me a question via the Blogger Contact Form at the top of the right sidebar of my home page. Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-86238948164906871882015-03-12T18:20:36.075-04:002015-03-12T18:20:36.075-04:00Eh, with the internet being what it has become, I ...Eh, with the internet being what it has become, I expect you'll still be seeing responses in 30 years.<br /><br />For myself, I'm a math-challenged adult trying to get started on an engineering degree, and having a hard time making the mental jumps from natural language to mathematics. Posts like yours are incredibly helpful.<br /><br />Thanks for keeping this up!Bryant Raifordhttp://www.blogger.com/profile/00951489380257323696noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-16250597934172930272015-02-08T13:32:13.202-05:002015-02-08T13:32:13.202-05:00Just tweeted this blog post - Ihor Charischak thin...Just tweeted this blog post - Ihor Charischak thinks Green Globs is the best software for algebra EVER (I think that's what he said in an email recently. Okay, now I'll go back and ACTUALLY read what you wrote - thanks.David Wekslerhttp://www.blogger.com/profile/06317465751447109100noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-69369856023793352752014-11-07T05:21:15.879-05:002014-11-07T05:21:15.879-05:00Basically mean is easy to find but for even number...Basically mean is easy to find but for even numbers median is difficult little bit just select middle two values and add them and divide it by 2 the given answer is median.Brenda Akershttp://www.blogger.com/profile/03660839631468621396noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-65534357695347474282014-10-22T16:41:03.835-04:002014-10-22T16:41:03.835-04:00Thanks, Pat. You're a treasure trove of math h...Thanks, Pat. You're a treasure trove of math history. I seem to recall that Loyd created the "15 Puzzle" which kept me busy for hours when I was younger. And I didn't WikiP it first! Correct me if I'm wrong. Gardner always reminded me of Don Herbert (Mr. Wizard) who turned me on to science as Gardner did to math. I miss those days!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-84385414350165032912014-10-22T16:33:02.424-04:002014-10-22T16:33:02.424-04:00H. Dewdney wrote very popular puzzles, but was kno...H. Dewdney wrote very popular puzzles, but was known to steal credit from others. By contrast, Martin G was always giving credit to others.. And I'm not sure most folks would rate him ahead of either Sam Loyd in creative quality. Pat Ballewhttp://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-20094243606238837552014-10-22T11:43:49.684-04:002014-10-22T11:43:49.684-04:00Great Work Dave Marain! Thanks for this Blog that ...Great Work Dave Marain! Thanks for this Blog that it helps me a lot to clear my concepts mostly we have very little bit knowledge about the formulas and now hope that i sought it out very clearly.Brenda Akershttp://www.blogger.com/profile/03660839631468621396noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-91730905594932383302014-10-18T20:54:52.871-04:002014-10-18T20:54:52.871-04:00Agreed, Pat. In fact we could generalize further t...Agreed, Pat. In fact we could generalize further to any "kite". My suggestion was based more on the greater frequency of standardized test questions relating to squares. There is also a nice visualization for (d^2)/2 to which students should be exposed. <br />Nice to hear from you again...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-4720463658202282782014-10-18T19:50:48.141-04:002014-10-18T19:50:48.141-04:00Dave,
In regard to memorizing the area of a squar...Dave,<br /> In regard to memorizing the area of a square as d^2 / 2, I would more prefer generalizing to "the area of a rhombus is the 1/2 the product of the diagonals. Pat Ballewhttp://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-33778064881479932662014-09-29T06:58:23.497-04:002014-09-29T06:58:23.497-04:00I appreciate your kind words, Owen. This has alway...I appreciate your kind words, Owen. This has always been a labor of love for me. I will continue to post some problems, but I do have to compile these into book form and either self-publish or get it on Amazon or somewhere else. <br /><br />Of course, I'll still be hangin' around advocating for our children ...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-5424534414693348682014-09-28T21:17:01.532-04:002014-09-28T21:17:01.532-04:00/*
4) I've given away over a thousand original.../*<br />4) I've given away over a thousand original higher-order problems over 7 years on this blog and, more recently, on Twitter.<br />*/<br /><br />and i'll bet a lot of 'em have been *used*.<br />probably quite a few have been used a *lot*.<br /><br />it's an amazing run; thanks for making<br />this explicit. twitter is useless to *me*,<br />of course... and so is most of the rest of<br />math-ed-online that i know about<br />(relying as so much of it does on one<br />or another high-tech "application" i'm<br />not at all inclined to think about [or some<br />whole *cluster* of 'em, more likely; too<br />bad about the worked-before-they-broke-<br />-it telecommunications systems round here]).<br /><br />but then, i'm in no need of such exercises<br />(other than as entertaining reading)...<br /><br />anyhow.<br />thanks *twice* for keepin it plain-text...owen thomashttp://www.blogger.com/profile/09249915192605437832noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-32777199302251655152014-09-11T13:18:06.576-04:002014-09-11T13:18:06.576-04:00Thanks, Fawn. Nice website. The book on Number Sen...Thanks, Fawn. Nice website. The book on Number Sense you read and reviewed speaks to the change which must take place in our classrooms. Our math lessons need to build on the natural intuition the child brings to us. This is the spirit of the Mathematical Practices in CCSS. Another big issue for me is how we define an "open-ended" math assessment question. Should the open-ended questions we pose in a classroom essentially the same as those on a standardized test? I do not believe so. How do you feel? This is important enough to warrant its own blog post!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-7670516315478260382014-09-11T01:32:00.291-04:002014-09-11T01:32:00.291-04:00Thank you, Dave. I wholeheartedly agree. Assessmen...Thank you, Dave. I wholeheartedly agree. Assessment and implementation of CCSS are our challenges. But I want to believe that we are heading in the right direction with the standards and the math practices. Fawn Nguyenhttp://www.blogger.com/profile/03605571262680195155noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-48787293423453843392014-05-28T14:22:25.015-04:002014-05-28T14:22:25.015-04:00I am working on the harder proof, I just need some...I am working on the harder proof, I just need some Pascal Triangle Numbers... or to recall the formula, I should say.<br /><br />On this one, I used a purely algebraic, rather than inductive proof:<br /><br />a=5+k<br />b=6+k<br />c=7+k<br /><br />1) a^3+b^3?c^3<br /><br />2) (5+k)^3 + (6+k)^3 ? (7+k)^3<br /><br />3] (125 + 75k + 15k^2 + k^3)<br /> + <br /> (216 + 108k + 18k^2 + k^3) <br /> ? <br /> 343 + 147k + 21k^2 + k^3<br /><br />4] 341 + 183k + 33k^2 + 2k^3 <br /> ? <br /> 343 + 147k +21k^2 + k^3<br /><br /><br />5] 36k+12k^2 + k^3 ? 2<br /><br />if k>1 then 36k+ 12k^2 +k^3 > 2<br /><br />If k>1 then a>5.<br />QED Andrew Ciszewskihttp://www.blogger.com/profile/09398438134138448768noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-67117466174235338062014-05-14T15:09:06.704-04:002014-05-14T15:09:06.704-04:00These kind of fractions are countable infinite
1/4...These kind of fractions are countable infinite<br />1/4 = 16/64 = 166/664 = 1666/6664 =<br />...<br />1/9 = 19/95 = 199/995 = 1999/9995 = ...<br />So are the restLuluhttp://www.blogger.com/profile/03216796613011396402noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-54664720151046700922014-05-12T17:22:53.245-04:002014-05-12T17:22:53.245-04:00The Desmos link should be ok now!The Desmos link should be ok now!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-91349010100994165782014-05-12T16:34:15.106-04:002014-05-12T16:34:15.106-04:00Thanks!Thanks!Denise Gaskinshttp://www.blogger.com/profile/11928843626113889088noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-29651325136350826062014-05-12T15:15:00.192-04:002014-05-12T15:15:00.192-04:00Thanks, Denise. Now I know how to generate comment...Thanks, Denise. Now I know how to generate comments!<br /><br />I posted a temporary link below the graph but it's not live. You have to copy/paste it. When I get home I'll fix it.Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-71538821895092274462014-05-12T14:49:28.938-04:002014-05-12T14:49:28.938-04:00I don't see the lesson. Is there supposed to b...I don't see the lesson. Is there supposed to be a link somewhere?Denise Gaskinshttp://www.blogger.com/profile/11928843626113889088noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-30712735614312172872014-04-20T14:44:15.114-04:002014-04-20T14:44:15.114-04:00Thank you so much for this support! I'm an edu...Thank you so much for this support! I'm an educator first and a technophile 2nd. If the technology can enhance and deepen conceptual understanding then it makes sense to me. Desmos has this potential and this was my first attempt. I've developed activities like this in the past starting with graph paper , then onto TI. It is important for me to say that the PROBLEM preceded the technology! <br /><br />I saw the interactive/feedback advantages you picked up on and my instincts told me this could work. As I continue to work with Desmos and learn from others' examples, I'll get better. <br /><br />The learning curve for Desmos does not appear to be that steep here. Inagine if each math teacher were to contribute just one activity!<br /><br />There are some features I'd like to see that may come with updates, e.g., involving setting ranges for tables, but, hey, it's early!<br /><br />Thanks again...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-9770171897001135542014-04-20T13:59:38.351-04:002014-04-20T13:59:38.351-04:00This is awesome! I just finished. I really like ho...This is awesome! I just finished. I really like how Desmos makes this activity much less low-risk -- students can try a function and adjust iteratively until they find the right linear equation/domain, and since they get three chances to graph a side of the square they have a chance to figure out more efficient ways to identify the function and identify patterns and connections.<br /><br />It's also so critical to synthesize graphing, slope, intercept, and domain into questions like these that are grounded in a context where students can check to see if their answer makes sense right away.fivetwelvethirteenhttp://fivetwelvethirteen.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-6084158925535662152014-04-16T13:58:54.380-04:002014-04-16T13:58:54.380-04:00Sue,
Below is a copy of a couple of the Grade 1 st...Sue,<br />Below is a copy of a couple of the Grade 1 standards. They seem reasonable to me. I'll have to look again at the overall breadth and quantity of these to see if they're too ambitious for Gr 1. Again, my major concern is that when teachers are evaluated according to the progress students make in these areas, we all know the primary focus will be on 'teaching to the test'. Hey, we all saw this coming, didn't we Sue!<br /><br />Represent and solve problems involving addition and subtraction.<br /><br />CCSS.MATH.CONTENT.1.OA.A.1<br />Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.<br /><br />CCSS.MATH.CONTENT.1.OA.A.2<br />Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-13249743330945789842014-04-16T09:14:16.859-04:002014-04-16T09:14:16.859-04:00No. I have followed this through other bloggers, b...No. I have followed this through other bloggers, but I have not researched it myself. (I feel like these conversations have taken away from lots of other great conversations that we all could have been having. What a time sink for all these great teachers to be studying the CC standards.) <br /><br />I know the process standards are good. I worry that there are too many topics among the other standards.<br /><br />The next time someone posts about this as a problem, perhaps we could get good teachers, pro and con, together to discuss. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com