Those "Function" Questions on the SATs - Practice, Tips

PLS NOTE THE EDIT TO THE PROBLEM BELOW. THE ORIGINAL WORDING WAS INACCURATE.

The following is not a classic function question even though it uses function notation. This is an original problem I wrote but it is the kind of question that might appear. The level of difficulty would be medium. The math content is middle school level but the wording and notation are the challenge for most students. Beyond preparing students for a test like the SATs, my strong belief is that such questions should be included in textbooks from middle school on (even with that function notation!). This question reviews basic math concepts (primes, factors, gcf) and can also be used as a springboard for discussion of the concept of "relatively prime", Euler's phi function, π(x) and other number-theoretic topics.
Note: The "For example" hint may or may not be included in the question. It certainly makes the notational issue less formidable.

If n is a positive integer greater than 1, then the sets F(n) and P(n) consist only of positive integers and are defined as follows:

A positive integer, k, belongs to the set F(n) if k ≤ n and the greatest common factor of k and n equals 1.
A positive integer, k, belongs to the set P(n) if k ≤ n and k is prime.
For example, F(6) contains the numbers 1 and 5 and therefore has two elements. P(6) contains the numbers 2, 3 and 5 and therefore has three elements.

What is the ratio of the number of elements in F(20) to the number of elements in P(20)?

Click Read more below to see answer (suggested solution will be posted later).

Answer: 1
Explanation: Not yet...

...Read more

Pi Day, SAT Day - Last Minute Reminders and a Problem to Try...

It's fitting that many juniors will be taking their SAT on 3-14! The following are some thoughts for students to take with them on Saturday morning.

Each question on the Math section is worth about 11 points. Six to seven careless errors (which is common) can cost you up to 90 points! How many students would like to increase their score 90 points without that high-powered SAT Course? It is within you! So what can you do to minimize the damage?

• HIGHLIGHT (circle, underline) KEY WORDS AS YOU READ THE QUESTION.
  
• LOOK FOR WORDS/PHRASES LIKE EVEN, POSITIVE INTEGERS VS. INTEGERS, NOT, LEAST POSSIBLE, ...
• SLOW DOWN! WRITE OUT DETAILS - DO NOT SKIP STEPS; DON'T WORRY ABOUT FINISHING EVERY QUESTION. IT'S THE 6-7 CARELESS ERRORS THAT HURT MOST!
• IF YOU HAVE ANY DOUBT ABOUT DOING THE PROBLEM ALGEBRAICALLY, PLUG IN SIMPLE WHOLE NUMBERS LIKE 1, 2 OR 3. See Example below.
  
• REMEMBER:
  INTEGERS: ...,-3,-2,-1,0,1,2,3,...
  EVEN INTEGERS: ...,-4,-2,0,2,4,..
  PRIMES: 2,3,5,7,11,... NOTE THAT ONE IS NOT PRIME!!
  
• ZERO IS THE MOST IMPORTANT NUMBER ON THE SATS AND MATH IN GENERAL! LEARN THE TRUTH ABOUT ZERO:
  WHOLE, EVEN, INTEGER, NOT POSITIVE, NOT NEGATIVE, 0/5 = 0, 5/0 IS UNDEFINED!

Example 1: If m > 0, the average (arithmetic mean) of 2⋅3^m and 2⋅3^m+2 can be written as c⋅3^m+1. What is the value of c?

Possible solution: Unless you're a math team whiz, you should immediately substitute a simple whole number for m and not worry about c. Also, ignore the phrase "arithmetic mean" which is math terminology for the common average.

"Plug in" m = 1: We want the average of 2⋅3¹ and 2⋅3¹⁺². This translates to the average of 6 and 54 which equals 30. If you're prone to any careless arithmetic errors (order of operations, exponent issues), do this on the calculator even though your math teacher would cringe!
We want to express 30 as c⋅3¹⁺¹ or 9c. Thus 9c = 30 or c = 30/9 = 10/3 = 3.33 if you're gridding in.

Additional Comments

• The above example would normally be among the last 5 questions of a section, therefore, considered to be more difficult. Students should not give up too quickly on these. They often can be solved by straightforward methods like the one described above.
• For the mathematical purists out there who are offended by "plug-in" methods (btw, I'm one of those purists!), this post was about providing test-taking strategies or 'survival' techniques. For the classroom development of algebraic skills, I would certainly have demonstrated an algebraic method:
  2⋅3^m+2 = 2⋅3²⋅3^m = 18(3^m). The average of 2(3^m) and 18(3^m) = 10(3^m) = 10(3^-1)(3^m+1) =
  (10/3)(3^m+1) ,etc. Wonderful review of exponents and operations but not necessarily for everyone taking this test...
  

A Middle School Mental Math 'Practical' Problem

Look here for links to MathNotations π-Day Resources, Activities and Investigations!


You're on a superhighway in the middle of nowhere at 10 PM, your fuel tank shows about a quarter of a tank remaining, your trip odometer shows you've gone about 180 miles since the last fill-up and you just passed a fuel rest stop. The sign reads "Next Rest Area and Gas - 58 miles." Are you in trouble or will you make it?

Comments
(1) Unrealistic scenario? Rest areas rarely that far apart on the Interstate? Why didn't the driver realize that he was low on gas and just stop! Why not just get off the Interstate and look for a local station or call AAA or Onstar if needed or use a navigation system to locate another gas station? (Do all of us have access to these technologies?). Does the student have to know that one still has some fuel remaining even when the gauge points to empty? Should a discussion of these kinds of practical issues be an integral part of 'real-world' problem-solving?

(2) Is this type of question fairly common in the texts you're using?

(3) What prerequisite skills and conceptual understandings are needed for this problem? Should most 7th or 8th graders be able to do this?

(4) Why do you think I suggested a mental math approach? Should we eliminate the mental math/estimation aspect altogether or use it as a starting point for further discussion?

(5) How do you think your students would fare on this problem? How many students would recognize the 1:3 ratio between gas remaining to gas consumed? Is this part:part construct something stressed in texts and in our classrooms? Should it be? Is it worthwhile discussing several approaches to this problem?

(6) Do you have a favorite visual model for this kind of ratio problem? Your thoughts?

Updates, Math Teachers at Play, ADP Algebra Links, More Features,...

Math Teachers at Play #2
Denise has done another wonderful job of gathering some excellent posts for K-12 educators. Puzzles, articles on creative ways to develop arithmetic skills, bouncing balls, Pythagorean Day, Math Contests, vectors, integrals, a link to the 50th Carnival of Math, more Monday Math Madness and I'm just scratching the surface! And with all of this, Denise sprinkles in engaging images, graphics and intriguing quotes including one from Woody Allen! Denise, you're the best. If I can get my act together, I'll try to submit something for next time.

Recent Comments Widget in Sidebar
It's about time that this feature appeared! Wordpress and Typepad make this easy for bloggers but I had to figure out a way to do this for Blogger. Actually, it's not that hard and long overdue...

π and π-Day Links!
Over the past couple of years, I've posted several articles on π Day activities and other π factoids and investigations. The π-Day Scavenger Hunt post is definitely the most popular but you can locate all of these easily by going to the labels section in the sidebar and looking for π or π-Day. Better yet, this link will get you to all 7 of my π-Day posts!


Personal Thoughts or a Silly Math Riddle in the Sidebar?
Well, for now, I put up the latter in the sidebar. Please don't groan too much...

ADP/ACHIEVE LINKS ALGEBRA I & II
Many visitors seem interested in all of the updates I've posted about the End of Course Exams, so I've decided to keep relevant links in the sidebar. I don't think Algebra I released test items have yet been posted (as of this date) but keep checking the Achieve site or, better yet, check here! What is particularly interesting are the test specifications for Algebra I.

NATIONAL STANDARDS MOVEMENT MOVING INEXORABLY FORWARD...
I've been keeping a very low profile here as one education leader after another comes out with their latest support for this novel idea! Those of you who have been following this blog for the past two years already know my position but it is gratifying to see one's views validated even if it took 15 years! I will have more to say about how this movement is growing exponentially but what is critical here is the idea that we're talking National Standards in Math, not Federal! Subtle but critical distinction...

Another New Feature - KENKEN is Mind-Bending and Instructive!!

KENKEN® Puzzles

Invented by Japanese math teacher, Tetsuya Miyamoto, KENKEN® allows you to test your puzzle acumen and improve your math skills at the same time.
Exclusively on NYTimes.com, updated with 6 new puzzles daily.

This extraordinary new math-logic puzzle started appearing in the NYTimes about a month ago. Link here to start playing. I also placed a link in the sidebar so that you can play every day. It will take you a few minutes to catch on to the rules and then you will give up Sudoku, Kakuro, your daily crossword puzzle and Jumble! I just started playing and I'm hooked. Most importantly, it will reinforce and develop basic arithmetic skills for your students and/or children!

I also recommend that you read the Times article which introduced this new feature. The creator is a talented and unique math teacher. Here is some fascinating background from the article:

KenKen was invented in 2004 by the Japanese educator Tetsuya Miyamoto, who founded and teaches at the Miyamoto Math Classroom in Tokyo. Students attend his class on weekends to improve their math and thinking skills. Mr. Miyamoto said he believes in “the art of teaching without teaching.”

He provides the tools for students to learn at their own pace using their own trial-and-error methods. If these tools are engaging enough, he said, students are more motivated and learn better than they would through formal instruction.

About 90 minutes of class time each week is set aside for solving puzzles, usually designed by Mr. Miyamoto. The most popular one has been KenKen.

NOTE: The Math Problem of the Day for Wed 3-4-09 was deleted because I deemed that the context was inappropriate for younger readers who may visit. I'm assuming this was an aberration and this feature will resume shortly.

"Multiple" Pi Challenges for MS and HS

Consider the following table displaying the first four positive integer multiples of π rounded to 4 places:


So what's the challenge here? You will be a π-multiple investigator!

(1) Determine the EIGHT positive integer multiples of π, up to 1000π, which, when rounded to THREE places, produce an integer result. For example, the decimal 17.9996 when rounded to 3 places results in the integer 18.000, which we will regard as an 'integer'. Unfortunately, this decimal is not an integer multiple of π so have fun searching! Do you notice any pattern in your results? Describe it! Can you explain this pattern? [This requires more than a Yes/No response!]

(2) In fact, the "pattern" you may have found in (1) continues beyond 1000π. Show that you can go up to SIXTEEN multiples of π before the pattern breaks down. Why do you think the pattern eventually ends?

(3) Which of your results in (1) would produce an integer when rounded to FOUR places?

(4) Can you think of any practical application for finding multiples of π which are very close to an integer? Be creative! Responses may depend on how advanced your math background is.

Comments:

• Do your students know how to program their graphing calculator to do the search? OR in Java or C++ or Python or Perl?? This would certainly facilitate the search! I may display the code for the TI-83 or -84. However, students may also find a creative way to use the TABLE feature on their graphing calculators to save time without programming. Have fun!
• Please post feedback if you use this in your classes. I will not post answers yet in case students find this post in their 'searching'!
• For most students, the full significance of this innocent looking search problem will not be apparent. You might want to give them a hint. Perhaps we should call this post:
  
  IS π ALMOST RATIONAL?
  

NEW FEATURES!

Update: I decided to delete the Math Problem of the Day for Wed 3-4-09 as I felt it was inappropriate for our younger viewers. Hopefully this was an aberration. Now you're probably all wondering what it said (except for those in other countries who already viewed it!!).

Check out the sidebar as I experiment with some new features...

• My Personal Thoughts on Education
  My own reflections - I'll change these every few days
  
  
• Math Problem of the Day
  Many of you have been commenting on these. Exceptional open-ended problems with solutions for the secondary student. Most require extensive work to be shown and/or justification.
  
  
• Amazing Fraction-Decimal Calculator
  You and your students will be mesmerized by the capabilities of this software. Wonderful for discoveries of patterns, development of conceptual understanding of repeating decimals, etc.
  
  
• Free Daily Kakuro Puzzles - a link
  For me, these are far more addictive than Sudoku. Further your students will be practicing both their addition and logic skills!
  
  
• Aristotle Quote of the Day - from the consummate logician/philosopher
  
  
• MathNEXUS Portal
  As I've mentioned often, this website is one of the best compilations of resources for math teachers and is updated weekly - an online journal!

Subscribers may need to visit MathNotations to see these new features in the sidebar. Enjoy!

Stuck on a Calculus Problem? Ask St. Patrick!!

With existing technologies and all the help available on the web I am incredulous that students still feel lost at sea the night before a major exam or just getting through an assignment. I've asked so many students what they do when they're frustrated by some math problem at 10 PM: "So who do you call? CalcBusters!"Seriously, they often just look at me as if I have two heads. Well, what are their options?

(a) Call their teacher/professor? Uh, not likely...

(b) Email their teacher/professor? Assuming one has this email address, what are the odds you will receive an immediate reply which will illuminate everything...

(c) Go into a Calc help chat room enabled by your teacher or one set up by some student. A good option if anyone is actually online at that moment. Your teacher or another student can establish guidelines for this so that students know a help session will always be available from 9 PM to 11 PM for example. See (e) below for a similar idea.

(d) Call or email a friend (remember you will then have only 2 lifelines left!). Of course this presumes you have a friend who understands it better than you and can communicate a solution over the phone or via email. Remember what time it is...

(e) Go to a Calculus forum/discussion group in which you can post your question and someone with the knowledge will reply in short order. This is a viable option as there are now many such help groups out there and I will review these and provide links in another post.

OR...

Go to YouTube and find a free video tutorial demonstrating a similar problem in detail.
For example, suppose you're floundering with "integrating by partial fractions." You can just Google "YouTube partial fractions calc video" or something like that and, presto, you are transported here. Ok, the video covers a more sophisticated problem involving a rationalizing substitution as well as partial fractions, but you immediately see dozens of related videos in the sidebar. There are many excellent free videos online from many talented teachers/professors but on this post I will feature one of the best.

If you're interested in seeing an exceptionally clear presentation of calculus or other math topics you cannot do better than Patrick's videos which he offers at no cost. Patrick does not know that I am writing this review so rest assured I am not getting any commission here! The link above is one of his lessons.

He tries to limit his lessons to 10 minutes to make the file size manageable. His writing on the whiteboard is crystal clear, his organizational skills are exemplary and his speaking voice is soooo calming. Further, his explanations are mathematically precise and include just enough rigor to make the purists out there happy without sacrificing clarity. Compare his videos to the amateurish attempts I have posted on this blog - uh, there is no comparison.

Patrick also has a website where you can find all of his free videos. Look here first on YouTube or go directly to his site.
Patrick's background (from his site)
About me: I have been teaching mathematics for over 8 years at the college/university level and tutoring for over 15 years. Currently I teach part time at Austin Community College, but have also taught at Vanderbilt University (a top 20 ranked university) and at the University of Louisville.

He runs a tutoring service in Austin and I'm quite sure he is doing well considering the quality of what he is offering for free. And his videos are not exclusively calculus. Enjoy!