## Friday, June 11, 2010

### SAT Videos: Twitter Problems of the Day 6-9 and 6-10-10

As we wind down toward the summer my SAT Problems and Videos continue to pick up steam! Below is the latest video from you YouTube channel, MathNotationsVids. I want to thank those who voted in my survey of these videos. I am gratified but I really need more specific suggestions on how to improve these. Your comments on YouTube or here are welcome!

Note: Because I am explaining two problems on one video, I am omitting details and multiple solution paths. Therefore these videos may be useful for your students who want to practice over the summer or revisit in the fall.

The percent increase problem could be asked in a variety of ways and demonstrated using multiple representations, aka The Rule of Four.  The visualization suggested in the description of the video has students physically demonstrating that doubling the edges of a rectangular solid, a cube in this case, will allow placing not only the original box inside of the bigger box, but SEVEN MORE! There's your percent increase, hands on!

I will be stopping the posted SAT Problem on Twitter on Tue 6-15-10. If I am able to sustain it, I will try to keep this up for the entire 2010-11 school year but who knows...

Finally, as posted on Twitter, I will be offering an individual or small group online course (using Skype) for the SAT or ACT Math this summer on a very limited basis. If you know of any student who might benefit from individualized instruction just email me at dmarain@gmail.com and I will provide details. This must be done ASAP however, as I will be closing this out very quickly.

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

## Sunday, June 6, 2010

### Video Solutions to Two Twitter SAT Problems of the Day

Please note correction to 2nd problem in the video. The correct answer is 4096 "real" values. The original answer, 13, applies to rational solutions only. Thanks to Nick Hobson for pointing out my careless error. Haste makes waste!!

Please vote in the poll at the right. Be candid in your opinion of these videos. It will guide me in the future to improve. Don't hesitate to share your opinions on MathNotationsVids and rate each video there as well. If you subscribe to my feed, please vote directly on the site. Only a few days left...

The title says it all so here is the video as promised:

Note: See above correction to 2nd problem! The video has not been corrected so beware!

Comments on 2nd problem:

If x is greater than or equal to 0 and less than or equal to 3, for how many values of x will 16^x be an integer?

As mentioned above, Nick pointed out my error. I should have restricted x to be of the form a/b, where a and b are integers, b ≠ 0. Normally, SAT questions avoid use of the term rational so they would spell it out. This problem however is very questionable for SATs. If real solutions were sought, this question would be more appropriate for a math contest. Here's one way of explaining why the answer is 4096 for real solutions:

16^x = k, k an integer → 2^(4x) = k
3 ≥ x ≥ 0 → 12 ≥ 4x ≥ 0 →  4096 ≥ 2^(4x) ≥ 1 since the exponential function 2^(4x) is increasing. This argument is reversible, so there are 4096 solutions for x, one of each integer value of k from 1 to 4096 inclusive. This solution could be written more concisely using log base 16 or log base 2 as Nick did, but I wanted to show a method without the log symbol.

Again, the video solution is WRONG as it shows only rational solutions! Well, at least i was thinking "rationally!"

I fully realize that the school year is over for some and about to end for others but these SAT Problems will be around for you or your students in perpetuity! Let me know if you like the questions. They are now appearing in the right sidebar of my blog so you will need to visit the page to see them.
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"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

## Saturday, June 5, 2010

### A Little Birdie Tweeting the SAT Blues, Carnivals and Other Musings

Well, today's SAT Tweet comes too late for many students taking SATs this morning (unless you're in a much later time zone) but I posted it anyway since it can certainly be used to review for final exams in Algebra 2 or whatever"∫ -ated"  name you have for it in your district!

As you can see if you visit this site (rather than get the RSS feed), I'm now posting the Twitter Problems of the Day in the right sidebar.  I'm new at this, don't have too many Twitter followers yet and I am learning that you need to get the word out there any way you can. Those who have replied to me seem to really like the level of these questions. I do feel the need to explain some methods to students who want them. If they're following me, they can simply send me a Direct Message or, if not, they can reply with @dmarain. I've also placed these questions in the  #Math and #SAT categories on Twitter so more will be able to see them, but a lot of what's there is promotion, links and personal thoughts --  so who knows. I may also post a video or two here and on my YouTube channel, MathNotationsVids, to explain a couple of these problems using a variety of approaches both for teachers and students.

If I were a faithful math blogger I would have been  announcing Denise's latest Math Teachers at Play and latest Carnival of Math 66 over at Sol's Wild About Math sites. They are in my blogroll, but I am deeply ashamed I haven't been promoting them here. So, please please please go over to Let's Play Math and Wild About Math to view the latest and greatest Carnivals!  Also, look here for Denise's ranking of her most popular posts broken down by categories, a mammoth undertaking, but well worth it. Sorry for being so negligent...

Finally, I feel the need to say something that may be provocative but is absolutely necessary for my integrity and the raison d'etre for this blog:

While I have been advocating for a standardized math curriculum for the past 25 years, I know fully well that learning outcomes depend far more on teacher effectiveness than any other factor. Yes, algebra should  cover the same topics in every district, however, there's coverage and then there's teaching. It is my observation that most professionals who've been on the job for awhile, whether in education, medicine, engineering or whatever, are open to receiving new information about the latest research and technology, but when you start making suggestions about the actual practice of their profession, you're sure to provoke strong negative reactions in many.

Bottom line, folks...
There are ways of introducing and developing ratio concepts, for example, that are more effective than other methods. I have never pretended to know what the best practices are in every case, but I sure know what has worked better for me and what has failed. We need to be open to these ideas and accept the truism that when students do not perform there is a myriad of reasons, one of which was our failure to reach these particular students. We have the obligation to vary our methods and be the researchers in the classroom. We have the obligation to learn from our students, our colleagues, our supervisors and others who have been there and done that.

Ok, I'm off the soapbox.

Also, I must say that I find it fascinating that recounting my grandchildren's latest observations on life seems to bring in far more my readership than any math post I have published! Surely this is the ultimate tribute to Art Linkletter.

Have a great "end of the year" and an even better summer!

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"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific