The following is the winning student solution from the original challenge. I know I was impressed!
Sunday, January 28, 2007
A Student Solution of 1-29-07 Challenge
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Look for fully developed K-14 math investigations, math challenges, and standardized test practice both for SATs and Common Core assessments. The emphasis will always be on maintaining a balance between skills proficiency and developing conceptual understanding in mathematics. There will also be dialogue on key issues in mathematics education. Use Blogger Contact Form at right for personal communication. See top sidebar to subscribe to solutions of Daily Twitter Problems.
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SAT Math Tips
READ/LEARN ALL OF THE FOLLOWING!
ZERO IS A 'WEIRDO'!
(W)hole
(E)ven
(I)nteger
(R)Rational/Real
(DO) Cannot Divide by O!
BUT Zero is NOT Positive and NOT Negative!
POSITIVE INTEGERS start from 1
PRIMES start from 2 (not 1)
INTEGERS can be NEGative (and zero!) as well as positive
MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.
Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?
Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.
Step 2: Identify the values which are given
First term or a(1) = 3
Common difference or d = 4
Number of terms or position of desired term or n = 100
Step 3: Substitute into formula and solve
a(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399
Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!
Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!
ZERO IS A 'WEIRDO'!
(W)hole
(E)ven
(I)nteger
(R)Rational/Real
(DO) Cannot Divide by O!
BUT Zero is NOT Positive and NOT Negative!
POSITIVE INTEGERS start from 1
PRIMES start from 2 (not 1)
INTEGERS can be NEGative (and zero!) as well as positive
MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.
Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?
Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.
Step 2: Identify the values which are given
First term or a(1) = 3
Common difference or d = 4
Number of terms or position of desired term or n = 100
Step 3: Substitute into formula and solve
a(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399
Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!
Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!
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2007
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January
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- Problems 2-1-07
- Problems 1-31-07
- Problems for 1-30-07
- A Student Solution of 1-29-07 Challenge
- Special Challenge Week of 1-29-07
- Problems for 1-26-07
- Problems for 1-25-07
- Problems for 1-24-07
- Problems for 1-23-07
- Another Definitive Report 'Proving' That Discover...
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- Introducing Exponents (Middle School)
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- In Observance of Dr. King...
- National Standards Dialogue with Professor Schmidt
- A Challenging Problem...
- Problems of the Day 1-12-07
- Problem of the Day 1-11-07
- Problem of the Day 1-10-07
- thanks for the links...
- Warmup #4 for Tue 1-9-07
- Warmup #3 for Mon 1-8-07 A Quickie??
- Now, You Don't Have to be Registered to Comment!
- DY/DAN - I give it an A+
- Legs of a right triangle revisited...
- 'Warm Up' #2 (not a 5-minute one, perhaps)
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- Birth of a Blog... Chapter Zero...
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About Me
- Dave Marain
- NJ, United States
- Recently retired math educator and Supervisor of Mathematics; 30 years experience as an Advanced Placement Calculus (BC) teacher; Former Author of Math Teachers of New Jersey Annual HS Math Contest; Former K-5 Chair of New Jersey Math Content Standards and Curriculum Frameworks; Former member of Math Item Review Committee for New Jersey High School Proficiency Assessment; Experienced SAT Math Instructor and author of SAT materials; speaker at many regional and national math conferences
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