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

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

MathNotations Soaring With Eagles or Just For the Birds? Updates 5-28-10

NOTE: I added a new solution (see (e) below). Also, read the comments to see even more solutions. Thanks to Jonathan for pointing out my error in (d) of my results.

I'll get to that cryptic title in a moment (may be obvious to some)...

1.  Remember the challenge problem I posted in the tribute to Martin Gardner a few days ago? Well, we rec'd several excellent replies and I have an additional response from a very sharp high schooler as well. Here was the problem:

Can you form 95 using each of the digits 5-2-2-1-0 exactly once? No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc...  You may combine the digits to form numerals like 12 or 120.



Mr. Lomas: 5! - (2+2)! - 1 - 0   Perhaps the most elegant since it uses the individual digits in the given order.


Robot Guy: (21-2)*5+0


Nate (high schooler): 120-5^2   Oh, the simplicity of that one! Combining digits is not the first way I thought of...


Mine so far:


(a) 102 - (5+2)  Pretty simple but I wasn't thinking much of combining digits until I saw Nate's


(b) 120 -25 (Shameless plagiarism from Nate's but I couldn't resist!)


(c) (2^5)(2+1) - 0! (I posted this one already)


(d) 10^2 - 5 x (2 - 0!)   (I knew there had to be a way using 100 - 5)
NOTE: JONATHAN POINTED OUT MY ERROR HERE. SEE COMMENTS.


(e) A new one: (2 + 2)! x (5-1) - 0!  I felt I needed to atone for my error in (d)!


I suspect Mr. Lomas has even more! It was definitely the spirit of Martin Gardner at work here!

Keep these coming if you can find more. I'd like to see us get to 10 ways.

2. Remember the hens -a- layin' problem I posted a few days ago? The video on YouTube gave the answer for 6 hens in 6 days: 24 eggs.

The problem on the blog was:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Here the answer is: 6 eggs

Here's a black-box method, i.e., work shown but no explanation:

(2/3) egg per (hen⋅day) x 3 hens x 3 days = 6 eggs.

This is how most solutions are given online and in the literature. It has little to do with middle schoolers actually learning the underlying principles. See the video for details.

3. Now for something completely different as M.P would say!

I've decided for now to tweet a daily (SAT) Problem of the Day.  "SAT" is in quotes because you can use these in your class as regular warm-ups or students can try these on their own to prepare for the upcoming SAT on June 5th and beyond.

Answers to each question will generally appear the next day, just before I tweet the new question. I've posted two problems thus far and the answers are up there today. Today's question will appear shortly.

My Twitter address is naturally dmarain.
Get the RSS feed for this at Twitter/dmarain if you want to see the daily problems.
If you have a question about the problems or want more details about solutions, send me a Direct Message in Twitter or email me.

Follow me if you'd like. These questions will not appear on this blog, so you will need a Twitter account or subscribe to the RSS feed above. Let your students know about it as well if you'd like.

Let me know by commenting here or replying on Twitter (Direct Message) if you like these and want me to continue next fall. Last SAT Problem of the Day on Twitter for this school year will be 6-15-10.

Requiescant in Pacem, Martin...

"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

Martin Gardner - The Original 'Riddler'

Today and over the next few days, you will find, in the media and on many math and science blogs, many touching, almost reverential, tributes to the greatest puzzler of our generation. How I looked eagerly to the next edition of Scientific American when I was younger. We didn't have much money but my dad insisted on purchasing a subscription to this classic magazine, intended for those scientists and non-scientists who wanted to know what was happening in the forefront of modern science and mathematics. Of course, I turned immediately to the back page to tackle another set of Mr. Gardner's challenging puzzles. I was so proud of myself if I could solve even one of these! Many of his puzzles had an almost magical quality to them. Now you see it -- now you don't. My forte was the logic type of puzzle but I tried them all.

Martin Gardner died Saturday, 5-22-10, at the age of 95. (See the puzzle  created below in dedication to Mr. Gardner).
By the way, 95 = 19x5, 94 = 47x2, 93 = 31x3.
It is only fitting that he left us at an age which is the largest 2-digit number with exactly two prime factors.

For you puzzlers out there, here is my conundrum dedicated to Mr. Gardner. Feel free to submit your solution, but only one,  in the comments to this post. Our readers can choose which one they think is the most elegant. I found one way, but I'm certain there are others!

Can you form 95 using each of the digits 5-2-2-1-0 exactly once. No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc...  You may combine the digits to form numerals like 12 or 120.

He was not a mathematician, nor a professor, nor a scientist. Yet I feel strongly that he deeply influenced all of these groups as well as anyone who enjoyed the satisfaction of challenging the mind. Read about him in the Wikipedia article and in the many tributes. If you're too young to have experienced the sheer joy brought to so many of us then discover it for yourself by looking at the annals of Scientific American or reading one of Mr. Gardner's many books.

Martin Gardner was more than a maker of puzzles of course. He was also known as a debunker of quackery and pseudoscience. He was an amateur magician, a philosopher, a lover of knowledge, a true Renaissance Man - a man for the ages.

Dr. Gardner - thank you for making a difference in my life and the life of so many others. Now if only I could remember how to get the cherry out of the martini glass by moving two matches...

On behalf of all my fellow bloggers, my sincerest condolences to your family.

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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

Uh Oh -- The 2-yr old turns 3 -- or does he!

Hey, these are precious moments and, anyway, I can't do more than one video a week!  So to take a respite. here's another anecdote, and, by the way, his 6-yr old brother and 3-yr old cousin are now demanding equal time.

So, he turned three the other day and when he went to bed that night his mommy, aka my daughter, told him: "____, now that you're three, I'm just going to tell you a story, give you a hug and kiss and say goodnight" (rather than staying with him until he fell asleep).

Knowing how this young man's mind operates, what do you think his reply was?

"Mommy, I'm only three during the day, I'm still two at night!"


Good luck to my daughter and all of his teachers!


Ok, fair's fair...
When the 6-yr old brother who was then 5, got his new bed (full size like his parents) my daughter and son-in-law heard him pacing and sighing loudly outside their door after they had tucked him in. Finally, his mommy came out and asked him what was wrong. Here was his response:

"I don't want to sleep in that bed. I'm not ready to be married!" 

This young man definitely marches to a different drummer, perhaps an entirely different band! He does remind me so much of myself at that age. I'm not sure if that's good...

Finally, my son's beautiful 3-yr old daughter went to Disney World a month or so ago. My wife asked her what her favorite ride was and she replied, "the rollercoaster." Then my wife asked if she was afraid. This was her reply:


"I was not afraid. I just screamed the whole way."


May their innocence remain forever...

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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

Challenging Geometry Assumptions: Review for SAT I/II

The video below presents a more challenging 3-dimensional geometry problem which would be at the upper end of SAT I or SAT II - Subject Tests (Math I/II). The key here is to challenge students' assumptions about a quadrilateral being a square because it has 4 congruent sides, a common error. This question will also review a considerable amount of geometry: Pythagorean Theorem, Volume of cube, spatial reasoning, 45-45-90 triangles, area of a rhombus, etc.

As always, the focus is on the art of questioning, suggested instructional strategies and pedagogy, although this problem may be interesting enough to capture the attention of some students who are preparing for upcoming standardized tests. For students who need help with spatial visualization, a model could be provided or have enough empty boxes available (they don't have to be cubes!).  

I strongly urge using learning partners or pairs for the discussion. 

Benefits include:

(1) Students feel less tentative when offering ideas to one other person or in a small group.

(2) Instead of posing conceptual questions to individuals, receiving little or no response except from the most confident or capable, you can pose a question to a learning pair: "Julie and Jason, what is needed to insure that ABCD is a square?" They should be given a few moments to think and confer before responding. The stronger student will usually explain it to the other. If neither can respond, they can say, "Pass!"

(3) The biggest advantage of student dialog is that often our explanations simply don't click with several students, but they do make sense to others. Those who "get it" can usually explain it in terms that their peers understand better, a benefit to both the "explainer" and the "explainee"!



By the way, the question posed near the end of the video is worth pursuing if time permits:

"Without calculating the areas, is the area of the non-square rhombus less than or greater than the area of the square?"

The answer is less for many reasons, but we would hope they would recall the base x height formula for a rhombus. The height is maximized when the angle between the sides is 90°. Why? Interestingly, the areas are quite close: 19.6 vs. 20. I believe strongly that this is the type of higher-order question that not only reviews important concepts but promotes deeper thinking, or should I say, thinking more than one inch deep!

What are your thoughts? Would you give students the e√3 formula before a standardized test or ever?Are these videos helpful to you? If you respond both on this blog and on my YouTube Channel, MathNotationsVids, and also rate these videos, that gives me the guidance I need to improve 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

If a hen and a half can lay an egg and a half in a day and a half...

The full version in one of its many many variations:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Putting aside the silliness of the riddle, there really is some serious mathematics going in these kinds of rate/ratio/proportion problems. Rather than solve the "hen" problem for you, I'll leave it to my readers to solve it by their own favorite methods. By the way, the answer to this riddle is in the description of the video below on my YouTube channel. Sorry 'bout that!!

Instead, the video below, which appears on my YouTube channel, MathNotationsVids, presents a developmental approach to a more complicated ratio problem for middle schoolers and beyond. I'm far more interested in your thoughts about the teaching strategies than I am about the problem itself. Please understand, further, that I am not suggesting the method shown in the video is efficient nor would it make much sense for the upper level math or science student. See comments below the video for further discussion of this.


The Problem in the Video Below:


If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days? Assume all workers build at the same rate.




More Advanced and Efficient Algorithms


(1) We assume from the "constant rate" assumption in  the problem that the number of houses (H) which can be built varies jointly as the number of workers (W) and the number of days (D).
Thus, H = kWD.

Substituting, H=3, W=10 and D=60, we obtain:
3 = k(10)(60) or k = 1/200. Note that the units of k are Houses/(Workers x Days).
We can interpret k to mean that 1/200 of a house can be built by 1 worker in 1 day. Thus, k is not only a constant but actually represents a rate. Another way of expressing this rate is
(1 House)/(200 Worker-Days) or the reciprocal version:
(200 Worker⋅Days)/(1 House)

Substituting the new set of values into the relationship H = (1/200)WD, we obtain:
5 = (1/200)(W)(40) or W = 25 workers.

(2) This can be made even more efficient using the "factor-label" (dimensional analysis, etc.) format:

(200 Worker⋅Days)/(1 House)) x (5 Houses)/(40 Days) = 25 Workers!

(3)  I could also exploit the inverse variation between W and D, but that's for my readers to bring up or for another video!

I see these efficient methods as "black box" methods for some students. Developing a deeper understanding of direct and inverse variation is far more important for the younger student.



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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)

Odds and Evens Week of 4-19-10

I've decided to post a few headlines from various sources (Ed WQeek, NYTimes, other blogs, etc.) over the past few weeks. I will not post links to most of these or make comments at this time. I believe they speak for themselves...

• Stand and Deliver Teacher Dies of Cancer
• Is Teaching A Marathon or a Sprint?
• 
  
  A Letter For Anyone Concerned About K-12 Mathematics Education 
• U.S. Needs Better-Trained Math Teachers To Compete Globally
• U.S. Falls Short In Measure of Future Math Teachers -- American college students earned a C on a new test comparing their skills with their counterparts in 15 other countries.
• Teacher Training No Boon For Student Math Scores -- A major federal study concludes that intensive teacher professional development in math did not immediately improve test scores.
• The Boys Have Fallen Behind
• Merit Pay For Students Fails to Raise Score, Study Finds -- Research suggests that payments can boost achievement if they reward behavior conducive to learning, rather than test scores themselves.
• Both Value and Harm Seen in K-3 Common Standards -- While some view the proposed expectations as valuable guidance, others worry that they are inappropriate for youngsters.
• What They're Watching -- The Ten Most Popular Course Lectures Available on You Tube
• An Open Mind -- Putting free courseware online was a first step in reimagining education. What now? Wiki U’s, smart courses and, maybe, learning

A new website, LearnBoost, is launching 0n 4-19-10. It looks like the developer, Matthew Hunter, and his team put extensive thought and planning into this. It compiles some of the top news stories in education and readers can move their favorite articles up the ladder by voting on them.

So where are MathNotations math contests for 2009-10. Sorry, folks, I'm just too overwhelmed to put this together for the year. However, I have been considering having my 2-year old grandson compose some questions!

Anyone actually looking at the videos I've been posting on my You Tube channel, MathNotationsVids?  I can see a few views but, without any responses or comments, it's hard to tell if this is worth continuing. I probably will because I enjoy it!

Stay tuned!

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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