tag:blogger.com,1999:blog-8231784566931768362.comments2015-07-20T13:52:01.346-04:00MathNotationsDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger2716125tag:blogger.com,1999:blog-8231784566931768362.post-60770918487002581112015-07-20T13:52:01.346-04:002015-07-20T13:52:01.346-04:00Outstanding DEMONSTRATION, John! Desmos beautifull...Outstanding DEMONSTRATION, John! Desmos beautifully uses sliders to allow investigations with parameters. Ideally we want our students to be able to show this algebraically and to use technology to verify and explore further. I'm a huge proponent of a balanced approach. Am I alone?Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-53757519161427308122015-07-20T13:23:21.690-04:002015-07-20T13:23:21.690-04:00One of the things with dynamic math software is wh...One of the things with dynamic math software is what does show mean and will students see a need. Does this Desmos sketch show the result? https://www.desmos.com/calculator/0krudh5tl7John Goldenhttp://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-2064465957246533322015-07-19T07:16:58.917-04:002015-07-19T07:16:58.917-04:00Deeply appreciate your reasoned and detailed comme...Deeply appreciate your reasoned and detailed comment. Your suggested Investigations make far more sense for younger students. Quite a bit of multiplicative machinery needs to be developed for geometric sequences but there is a strong analogy to arithmetic sequences. Just as 3,7,11,15,19 can be expressed symmetrically in terms of its arithmetic mean: 11-8,11-4, 11,11+4,11+8 whose sum is 5•11 we can express 2,4,8,26,32 as 8/4,8/2,8,8•2,8•4 whose product is 8^5. This in itself is worth discovering even without the 5th root.<br /><br />I wasn't suggesting this Investigation could be implemented as is for younger students. But I wanted to present a less well-known application which teachers could adapt to the group. I believe some students would use their calculator to discover that the product is 8•8•8•8•8 where 8 is the median. More background enables them to go further but less machinery does not prevent some level of discovery and the satisfaction that comes with it. Our role is to set the objectives and guide the exploration. When we provide the time and opportunity for children to explore, we can't always anticipate where they will go. For every student who's disinterested and will not invest there will be one who will take the journey. We create the culture of inquiry...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-30013102063557321042015-07-19T03:14:37.528-04:002015-07-19T03:14:37.528-04:00I think it could be a nice investigation, but I wo...I think it could be a nice investigation, but I would make sure that there is ample practice beforehand for messing with whole number factorization. (I left a couple of related comments here: http://marilynburnsmathblog.com/wordpress/a-mental-math-lesson/)<br /><br />To really get the idea of this proof, I would think students should be familiar with factor pairs. In my experience: one rich, cognitively demanding task is to have students explore the sorts of whole numbers that have an odd number of factors. (Both symbolically and diagrammatically, e.g., drawing rectangular arrays for each pair of factors.)<br /><br />For example, observe that 16 has an odd number of factors: 1, 2, 4, 8, 16; a total of 5 (an odd number).<br /><br />1, 4, and 9 also have an odd number of factors...*<br /><br />Besides the foundational elements involved in factoring (how do these re-arise, e.g., when factoring quadratic expressions?) and an increased understanding of multiplicative reasoning (an important transition that many students struggle with) the factor pairs of (1, 16); (2, 8); and (4, 4) are reminiscent of the pairing up involved in tackling the "verify without a calculator" example in your post (as well as the general proof for an odd number of terms).<br /><br />You might even connect the two tasks: Suppose the whole number n has an odd number of distinct factors, and write down this list. What is the geometric mean? What is the arithmetic mean? What is the median? Which of these can you write a formula for? Which of these can be the same? When/why? (If never: Why not?)<br /><br />*Given a whole number's prime factorization, what more can we say about how many distinct factors it has? E.g., if p and q are prime, how many factors does a number of the form p^2 x q^3 have? (Why should this answer be the same for any primes p and q?) Etc.<br />mathwaterhttps://mathwater.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-34034291481146042192015-07-10T12:38:17.021-04:002015-07-10T12:38:17.021-04:00Sorry about that. Try searching my blog for "...Sorry about that. Try searching my blog for "Clocks". I'll try to find a better link.Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-8119961340876611182015-07-10T12:28:16.742-04:002015-07-10T12:28:16.742-04:00Happy to hear that we share the same views.
I am ...Happy to hear that we share the same views.<br /><br />I am unable to access the link you sent. Permission denied.rohitn_wizardhttp://www.blogger.com/profile/08445749003538707305noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-86042548129455810062015-07-10T12:15:32.811-04:002015-07-10T12:15:32.811-04:00Thank you for sharing. We seem to share many commo...Thank you for sharing. We seem to share many common perspectives. In fact I posted a clock problem a while back and attempted to guide students toward a general formula.<br /><br />https://www.blogger.com/blogger.g?blogID=8231784566931768362&pli=1#editor/target=post;postID=2943142771786438431;onPublishedMenu=allposts;onClosedMenu=allposts;postNum=0;src=postnameDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-85260634407121953502015-07-10T09:04:25.830-04:002015-07-10T09:04:25.830-04:00Thanks. Here's my website - www.rohitn.com - w...Thanks. Here's my website - www.rohitn.com - where I have posted explanations of a few formulae/pattern I discovered, along with articles of fractional dimension, etc. Feel free to direct any student who seems inquisitive enough.<br /><br />One problem in particular (How many times a clock's hour and minute hands meet in a 12 hr period) seems like a common problem. But looking for a pattern to simplify the problem led to a general formulae and an interesting discovery in symmetry. I would encourage students to follow similar thought processes when they encounter any puzzle/problem - finding a general pattern rather than being satisfied by solving the given puzzle/problem.<br /><br />Here's the link to that article.<br />http://www.rohitn.com/math/math_clockHandsMeet.aspxrohitn_wizardhttp://www.blogger.com/profile/08445749003538707305noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-78688915509622083962015-07-08T18:49:07.754-04:002015-07-08T18:49:07.754-04:00Thanks for sharing your thoughts and good luck wit...Thanks for sharing your thoughts and good luck with your work here...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-20349900686257427112015-07-08T18:48:54.911-04:002015-07-08T18:48:54.911-04:00Thanks for sharing your thoughts and good luck wit...Thanks for sharing your thoughts and good luck with your work here...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-66595759591827010712015-07-08T18:29:03.706-04:002015-07-08T18:29:03.706-04:00That sounds interesting. No...I am an Indian worki...That sounds interesting. No...I am an Indian working in the US.rohitn_wizardhttp://www.blogger.com/profile/08445749003538707305noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-70326724185846869572015-07-08T17:13:23.982-04:002015-07-08T17:13:23.982-04:00Agreed and nice analysis. My experience is that yo...Agreed and nice analysis. My experience is that younger children can be engaged by math "puzzles" particularly involving large numbers. I often them questions like "If you stack a million dollar bills wou askld it reach the ceiling, top of the Empire State bldg, the moon or???" I don't use the word "tricks" as much anymore. I want them to see math as a way of thinking which makes a seemingly hard problem into a manageable one. Are you in the UK?Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-35418106080769794312015-07-08T16:52:23.831-04:002015-07-08T16:52:23.831-04:00The students have to understand or know the follow...The students have to understand or know the following aspects<br />1. How last digit points to trivial divisibility such as last digit being even or 5 or 0.<br />2. How multiplication or raising to a power affects last digit of a number.<br />3. What it means for a number to be a prime number or a composite number.<br />4. Understand how last digit can be calculated for powers without actually finding the exact answer.<br /><br />More importantly, students need to understand not to be intimidated by large numbers or problems that use large numbers, as there are tricks available to find patterns without tedious calculations.<br /><br />I am not aware of school syllabus in the US, it seems like a problem for 9th grade.<br />Some good students in 7th grade may be able to grasp it.<br />rohitn_wizardhttp://www.blogger.com/profile/08445749003538707305noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-18767266394027853272015-07-08T16:17:19.536-04:002015-07-08T16:17:19.536-04:00Thank you rohitn_wizard. Clear concise explanation...Thank you rohitn_wizard. Clear concise explanation. So at what grade level should this be given? 6th? 7th? Of course it isn't just about exponent skill or divisibility, is it? Without having seen models of this kind of thinking, it would be challenging at any level...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-32280748199759782462015-07-08T15:06:13.905-04:002015-07-08T15:06:13.905-04:00We can find the last digit of the result using fol...We can find the last digit of the result using following method - <br /><br />(a) Last digit of 173^4 can be calculated by finding the last digit of 3^4 (=81, hence last digit = 1)<br /><br />(b) Last digit of 179^4 can be calculated by finding the last digit of 9^4 (=81^2, hence last digit = 1)<br /><br />(c) Adding the last digit of 183, i.e. 3 to the above leads to a number whose last digit is 5.<br /><br />Any number containing 2 or more digits and ending in 5 cannot be a prime.rohitn_wizardhttp://www.blogger.com/profile/08445749003538707305noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-61990669975258822202015-07-07T21:18:56.693-04:002015-07-07T21:18:56.693-04:00Thanks "mathwater". Nice analysis. Some ...Thanks "mathwater". Nice analysis. Some students might discover a gimmicky approach that can be verified algebraically. <br />Let's try 1009x1004.<br />Working with the 9 and 4 we first add, then multiply as follows: 1009+4=1013; 9x4=36<br />Put these together to obtain 1013036<br />Seems very contrived but it seems to work.<br />Let's try 1010x1010:1010+10=1020,10x10=100<br /><br />Thus 1010x1010=1020100. Not as elegant as your methods but students might find it interesting and ask why it works! When we ask them to devise their own patterns strange things can happen! Our role is to enable this discovery and provide tools for them to verify their conjectures...<br />Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-46962099764132732302015-07-07T20:53:49.708-04:002015-07-07T20:53:49.708-04:00A question: What is 1010x1010?
Since 1010 = 101x1...A question: What is 1010x1010?<br /><br />Since 1010 = 101x10, we have:<br /><br />101x10x101x10 = 101^2 x 10^2 = 10201 x 100 = 1020100.<br /><br />This requires students to figure out how to square 101.<br /><br />One way is: 101x101 = (100+1)101 = 10100 + 101 = 10201.<br /><br />Back to the main story:<br /><br />You ask about the following: 1012 x 1008.<br /><br />But this is: (1010 + 2)(1010 - 2) = 1010^2 - 2^2<br /><br />= 1020100 - 4 = 1020096, as remarked.<br /><br />(Take-home lesson: The product of two numbers with an even difference can always be re-written this way. When does it help? A nice example is 43x37...)mathwaterhttps://mathwater.wordpress.com/noreply@blogger.comtag: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.com