MathNotations

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is 9ドル.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!
------------------------------------------------------------------------------

I decided to post a video solution of the Twitter problem I posted on 8-25-10:

4 red, 2 blue cards; 4 are chosen at random. What is the probability that 2 of the cards will be red?

Because of the 140 character restriction on Twitter, the questions are often highly abbreviated and I actually consider it a "fun" challenge to write the question both concisely and clearly. Of course, as we all know about human interpretation of word problems, "clear" is in the eye of the beholder!

There's no doubt that the question above needs some fleshing out and might appear on the SAT and other standardized tests something like this:

A set of six cards contains four red and two blue cards. If four cards are chosen at random, what is the probability that exactly two of these cards will be red?

I'm sure my astute readers can improve on this wording but we'll leave it at this.

A few questions naturally pop up:

(1) Could this really be an SAT/Standardized Test question? Well, as I state in the video below, a question quite similar to this appeared on the College Board website the other day as the Question of the Day.

(2) For whom is the video intended? Everyone who happens upon it! I certainly wrote it to be helpful to students who will be taking the PSAT/SAT in the near future. Rather than simply presenting a single quick efficient solution, I demo'd 2-3 methods and indicated some important strategies and reviewed key pieces of knowledge to be successful on these harder probability questions. By the way, someone who is comfortable with probability will surely not find this question so formidable, but we're talking here about high school students or even undergraduates who struggle mightily with these.

(3) I'm hoping that the video will also serve as a catalyst for dialog in your math department. From the inception of this blog, I've never even intimated that a suggested way of explaining a concept, skill or a problem solution is in any way prescriptive. I encourage you to continue using whatever instructional methods have worked for you and to share these with our readers! However, for novice teachers or those who wish to see other approaches, I hope it will have some benefit. Of course, the video is not in a classroom. There are no students asking or being asked questions. There are no interruptions and I have a captive audience (except for my dogs who bark incessantly!).

SOME KEY STRATEGIES/TIPS/FACTS FOR PROBABILITY QUESTIONS

(1) It is highly recommended that students begin by listing 2-3 possible outcomes and to include at least one that is NOT one of the desired outcomes! This will help you to decide on a plan: organized list vs more advanced counting/probability methods. Further, you can ask yourself the key question in all counting/probability problems: DOES ORDER COUNT!

(2) Although it appears difficult for most test-takers to be systematic when making a list under test-taking conditions, preparation is critical here. If one practices several of these in the weeks leading up to the test, the chances of success improve dramatically. Did I just suggest preparation and practice could make a difference!

Where do you find these problems? Any SAT/ACT review book or my Twitter Problems of the Day or my upcoming SAT Challenge Quiz book to name a few sources...

(3) The basic definition of probability should always be in the forefront of your mind:

P(an event) = TOTAL NUMBER OF WAYS FOR THAT EVENT TO OCCUR DIVIDED BY TOTAL NUMBER OF OUTCOMES.

As indicated in the video, one can and should think of this ratio as TWO SEPARATE COUNTING PROBLEMS! Do the denominator first, i.e., the TOTAL number of possible outcomes. In the Twitter problem it is 15 if order is disregarded. Whether you arrive at 15 by listing/counting or by combinations methods, the denominator is 15 and is a completely separate question from "How many ways are there to get 2 red and 2 blue cards?"

(4) Finally, there are other methods for solving this probability question using Laws of Probabilities and/or permutation methods. I was going to make a 2nd video but I'm not so sure about that now.

An important point about the video below: I used 4 Blue and 2 Red cards, the opposite of the original Twitter problem but that won't change the final result!




[埋込みオブジェクト:http://www.youtube.com/v/305z8R9d56k?fs=1&hl=en_US]



Look for my other videos on my YouTube channel MathNotationsVids . Look for all of my Twitter SAT Problems on twitter.com/dmarain .

As I develop my Facebook page further, I may start posting these questions there as well as my videos. Facebook allows up to 20 minutes videos, much less restrictive than YouTube's 10 minute limit.

Posted by Dave Marain at 10:49 AM 5 comments

Labels: counting problems, math videos, mathnotationsvids, probability, SAT strategies, SAT-type problems, systematic counting, twitter problem of the day

While we're waiting for the 60° integer triangle problem, here's an easier one for both middle schoolers and secondary students. The only fact from geometry that is needed is the all-important triangle inequality:
Any side (in particular, the largest side) of a triangle is less than the sum of the other two sides.
Of course this refers to the lengths of the sides and one can express this in other forms, but I'll leave it at that.
This type of question has become a favorite on the SATs and other standardized tests but, more importantly, it develops clear systematic thinking - the organized list....

How many different triangles have integer side lengths and a perimeter of 5? 10? 15? 20? 25?

COMMENTS/INSTRUCTIONAL HINTS:

• There are really five separate questions here. The instructor can give some or all of these depending on the time allotted. To help the group get started and for clarification, it may be helpful to demonstrate the first question for the group: For a perimeter of 5, there is only one possible triangle, which we can symbolize as {2,2,1}. If these are older students who are comfortable with the triangle inequality, you do not necessarily have to model this one, but that's your call. By modeling the first one, you eliminate some of the ambiguity of ordering the sides.
  
• Since a primary objective here is to make an organized list, you may want to stop after the perimeter of 10 and discuss it at the board. Depending on the ability level of the group, I usually have students work independently, then check each other's work in pairs after they do a couple of these questions. Sort of a think-pair-share approach. Also, don't be afraid to provoke their thinking with questions as they begin to develop their systematic lists (which can get boring for some): "So, do you expect more triangles for a perimeter of 10? Twice as many?"
  
• As each question is reviewed, encourage students to record their results in a table:
  Perimeter..................Number of Triangles
  ......5........................................... 1 ................
  ....10.......................................... 2 ................
  This is critical for middle schoolers in particular, since tables are a basic model for functions! At some point, you can use n or p for the perimeter and symbolize the number of triangles having perimeter n or p as T(n) or T(p).
• Naturally, some students will assume there is a pattern and guess there are 3 possible triangles with a perimeter of 15 - NOT! However, it is natural for all of us to ask: "WHAT'S THE FORMULA?" Well, there is one. It's fairly sophisticated and related to partitions of numbers, but I'll let our readers do their own research for this...
  

Posted by Dave Marain at 6:09 AM 0 comments

Labels: combinatorial math, geometry, organized list, SAT-type problems, systematic counting, triangles, warmup