tag:blogger.com,1999:blog-8231784566931768362.comments2014-05-29T08:47:31.780-04:00MathNotationsDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger2684125tag: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.comtag:blogger.com,1999:blog-8231784566931768362.post-16491236795745049582013-11-17T16:55:59.743-05:002013-11-17T16:55:59.743-05:00If mean and median of a data set are equal, this i...If mean and median of a data set are equal, this implies that the data is uniformly distributed on both sides of the mean value. <br /><br />See www.ipracticemath.com for more information on means and medians.Tayyabhttp://www.blogger.com/profile/12927165566194080039noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-64414148528235491642013-11-13T18:20:45.670-05:002013-11-13T18:20:45.670-05:00Beautiful bar model explanation! I've missed t...Beautiful bar model explanation! I've missed this type of interaction. Thank you!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-14842795057959715512013-11-13T17:22:33.144-05:002013-11-13T17:22:33.144-05:00I love ratio problems, because they are so confusi...I love ratio problems, because they are so confusing at first and then a little bit of work can make it all clear. In bar models, this is a two-unit puzzle that simplifies to a single unit.<br />First unit = girls:<br />[---]<br />Boys are double that:<br />[---][---]<br />Second unit = women:<br />[-]<br />Men are double that:<br />[-][-]<br />What we are really interested in, however, is females, so we can tie these units together:<br />[---][-]<br />And males are double that:<br />[---][-][---][-]<br />Of course, the big trick is to get students to see that "half as many girls as boys" is the same as a 1:2 ratio, one unit of girls for every two units of boys. As you said, it often helps to ask students first which group is largest, and then encourage them to try simple numbers until they can see the relationship. In Singapore math, kids get lots of experience with ratio-like word problems in the elementary years working up to the study of ratios in middle school.letsplaymath.nethttp://letsplaymath.net/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-21255921797134314492013-10-31T14:26:09.968-04:002013-10-31T14:26:09.968-04:00Good hint, Jameel. I'm a huge believer in with...Good hint, Jameel. I'm a huge believer in withholding hints for awhile. We would like youngsters to draw the conclusion that the tens' digit cannot be even or a '5' but we know some groups would need more promoting. I'd rather err on saying less! Afterwards these would be great discussion points for developing conceptual understanding.Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-10914990598702512182013-10-31T14:14:59.227-04:002013-10-31T14:14:59.227-04:00good question! I think a good hint might be to as...good question! I think a good hint might be to ask which sets of 10 would we eliminate from our lists. Ex: since no double digit prime can end in 2, none of the 20s primes can be considered.Jameel Misbahuddinhttp://www.blogger.com/profile/02426900673024173478noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-72762395782903421122013-08-23T07:18:56.021-04:002013-08-23T07:18:56.021-04:00Burt,
Thank you for your detailed and thought-prov...Burt,<br />Thank you for your detailed and thought-provoking comments. I read the first few pages of Schmittau's excellent paper. <br />Some thoughts...<br />1. I recalled Vygotsky's work and how it contrasted with Piaget's. I remembered thinking back in the day that Piaget's essentially linear model of learning fails to capitalize on young children's innate ability to compare and estimate quantities. Using algebraic symbols to represent these relationships at an early age (even before 7!) makes perfect sense to me.<br />2. Schmittau's interpretation and implementation of the Russian model is exceptional and I would like to see this in action in the classroom. I'm tempted to contact her.<br />3. That being said, I need to come back to the empirical model. In the ABSENCE of the abstract foundation of the Russian model, my recommendations are intended to help teachers working with middle and high schoolers who have not been exposed to this quantitative/abstract model It is difgicult to "unlearn" what they have been taught. However I believe it is possible to incorporate some of Vygotsky's and Davydov's ideas into instruction with older students.<br />This is best exemplified by the classic "There are twice as many girls as boys" problem, the post which has more views on my blog than any other. I suggested we start by asking "Are there more girls or more boys?" From there move to G>B and how do We make the quantities equal, G=2B or 2G=B? This is IMO is similar to the Russian approach.<br />3. My generalization of using a concrete inductive approach is based on working with older students. If I could begin from the ground up I would blend the 2 approaches. Ironically when I've worked with my own children and grandchildren from an early age I probably used Vygotsky's approach most often. It's the algebraic representation however which distinguishes his ideas and which is worth exploring further. <br /><br />I hope to continue this conversation as I become better informed by Schmittau's paper.<br />Thanks again, Burt. We do not disagree!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.com