tag:blogger.com,1999:blog-8231784566931768362.comments2014-10-18T20:54:52.871-04:00MathNotationsDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger2690125tag: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.comtag:blogger.com,1999:blog-8231784566931768362.post-2618250528898816042014-04-16T08:32:48.213-04:002014-04-16T08:32:48.213-04:00Excellent point, Sue, but there will always be iss...Excellent point, Sue, but there will always be issues of developmental readiness. Mandating *proficiency* of higher-level content and deeper understanding for ALL children K-3 does not make sense which implies that STANDARDIZED TESTING of Common Core skills should start later. Further it is insane for us to be discussing high-stakes CC assessments until the curriculum and training of teachers have been entrenched for at least 3-4 years. If assessment actually had something to do with helping children learn and develop there would be no issue but we know this is not the reason states have been rushing this through. We know it is not only the students who are being assessed.<br /><br />However, *raising the bar for all* children is still reasonable. I see what is currently happening in many K-5 math classes and I'm disappointed to say the least. Of course some children will struggle. Finding ways to help them has always been our primary role. I need someone to enumerate some *specific* K-3 CC proficiencies so that I can respond directly to them. Is there a particular one that you believe is inappropriate?Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-47268330800917483702014-04-15T10:00:32.202-04:002014-04-15T10:00:32.202-04:00Dave, do you think some of the early grade level s...Dave, do you think some of the early grade level stuff is possibly developmentally inappropriate? I don't know enough developmental psych to know, but I do know that different kids develop at different rates, and having common core say they must all do x at a certain time concerns me.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-26808179537963194822013-12-19T19:29:22.696-05:002013-12-19T19:29:22.696-05:00Very nice variation and brings in rotation in a na...Very nice variation and brings in rotation in a natural manner. As you can tell I'm more interested in teacher reaction to the reflections following the problem. It's important for teachers to have a resource of problems like yours or mine but I see these as a springboard for generating dialog and developing deeper conceptual understanding. Mathematical habits of mind so to speak. <br /><br />Do you think maybe one student in a classroom would be curious about the general rotation problem or the general intersection problem? I believe it is our obligation to engender that kind of curiosity and NOT just with the gifted students.Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-14526581808004958632013-12-19T15:32:00.784-05:002013-12-19T15:32:00.784-05:00A related problem, no trig required:
http://fivet...A related problem, no trig required:<br /><br />http://fivetriangles.blogspot.com/2013/02/45-overlapping-circles.htmlCCSSI Mathematicshttp://www.blogger.com/profile/12318317536740240935noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-59613064794407396842013-12-09T10:18:21.341-05:002013-12-09T10:18:21.341-05:00Thank you, Alana. Imagine if we actually trusted p...Thank you, Alana. Imagine if we actually trusted professionals like you to determine what form of PD and preservice training would be the most effective. But that would require RESPECT for our profession. ..Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-36695678598014641472013-12-09T09:49:03.761-05:002013-12-09T09:49:03.761-05:00I completely agree about the teacher preparation. ...I completely agree about the teacher preparation. This is my 4th year teaching. The way I taught my first year was similar to my preparation from college but my teaching now is more problem solving based. I believe we need to create students to be problem solvers that can make sense of mathematical problems. Alana Gilliamhttp://www.blogger.com/profile/12506331429376323344noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-69813197195178638552013-11-19T19:38:58.272-05:002013-11-19T19:38:58.272-05:00Wow! I just saw your models for this on your websi...Wow! I just saw your models for this on your website and I commented there. My attempt was pathetic but I am limited these days by posting via email on my Nexus 7. No pretty graphics! But then I figured someone with more talent will do it for me!<br />Interestingly though the essential ideas are fairly close.<br />Thanks for sharing...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-86509508801565306262013-11-19T19:22:57.633-05:002013-11-19T19:22:57.633-05:00Thanks Denise! I made that crude attempt to bring ...Thanks Denise! I made that crude attempt to bring the'experts' to the fore!<br />I'm anxious to see your model for this problem.<br />And yes I found this problem among comments from an old post of mine. I believe it was posted by someone from Singapore who was demonstrating harder problems tackled by Grade 6B students there. I'll have to verify the details but I didn't construct this problem. <br /><br />My instincts are to use algebra of course with one or more variables because that's how I was taught. But I'm always open to learning new things. It's just heard to teach an old dog like me new tricks!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-61848134063101446462013-11-19T12:33:25.599-05:002013-11-19T12:33:25.599-05:00Did you base this on an actual Singapore Math prob...Did you base this on an actual Singapore Math problem? This is more convoluted than most bar model problems I've seen. It's probably easier to work it out with regular algebra, although the models do give a visual representation of what the variables are doing. But the models are only meant as a stepping-stone to regular algebra, and when problems start to get too complex, it's time to pull out the real thing.<br /><br />As you say in the last paragraph, what you have drawn isn't really a bar model because you haven't used the size and alignment of the blocks to visually demonstrate the number relationships. So you're just doing regular algebra with some rather awkward symbols.<br /><br />One rule for drawing bar models is to keep each bar all on one line. The fact that your brackets-and-dashes bar jumps to a new line makes your model look wrong at first glance and very confusing even when I see what you mean. This is easier done with a graphic:<br /><a href="http://wp.me/a2GNt-5I2" rel="nofollow">Bar models for "When Mom was 40"</a>Denise Gaskinshttp://www.blogger.com/profile/11928843626113889088noreply@blogger.com