Here is a set of middle school problems on primes that require careful reading, knowledge of prime digits, organized listing and other skills. They can also be used to prepare high school students for SATs and other standardized tests which frequently test knowledge of prime numbers. Working without the calculator is strongly recommended. The last couple of questions require a partial list of primes which could be an internet activity. None of these questions is highly challenging but one of the goals is to make children aware of the mysteries of primes, something few appreciate! I guess you could say that learning how to read critically is also a PRIME objective!
1) List the FIVE 2-digit primes whose units' digit is 1.
2) List the FIVE 2-digit primes each of whose digits is NOT prime.
3) List the TWELVE 2-digit nonprimes (composites) each of whose digits is prime.
4) Mentally, determine the largest 3-digit nonprime (composite) each of whose digits is prime.
5) A palindrome is a number like 101 or 222 which reads the same when its digits are reversed.
(a) Explain why there are ninety 3-digit palindromes without listing all of them.
(b) (Internet activity). Search for a list of primes up to at least 1000. Use this list to answer the following: If a 3-digit palindrome were chosen at random, what is the probability that it would be prime?
6) Using your list of primes, determine the largest 3-digit prime having 3 different prime digits. Anything surprise you?
7) (Additional work outside of class) Using the list of primes, devise three problems of your own about 3-digit primes to challenge your classmates. A special prize for the best questions!
Tuesday, February 27, 2007
PROBLEMS WITH PRIMES!
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6 comments:
1. One observation about prime palindromes?
2. Are there any 2-digit primes, each of whose digits is composite?
3. How many 3-digit primes are there, all of whose digits are composite?
4. Assume it takes one and a half minutes to check if a 4-digit number is prime. How long would it take 3 students to find out how many 4-digit prime numbers have all composite digits? How long would it take them to check for 5-digit primes with the same property?
Nice questions, jonathan!
My wording in question 1 was not the best. It should have read "...each having a units' digit of 1" or something like that.
Your word problem looks like fun. i wonder how long it would take me to do each part!
Actually I implemented these questions in my 'skills' class on Tue and in an Honors Algebra 2 class I happened to be in. Guess what happened! BOTH groups stuggled with the wording in questions 2 and 3. From experience I was not surprised. Here is one of the major stumbling blocks we need to overcome as teachers:
Helping students distinguish between the NUMBER itself, like 25 (which is composite) and its separate digits 2 and 5 which are prime. Once I made that clear distinction, students in both groups took off although the Honors class needed less direction (part of that is more self-confidence BTW, not just aptitude!).
The other key was beginning by reviewing the concept of prime digits. Both groups were shaky and we know how many of our best students still believe 1 is prime! By making two separate lists they did much better:
PRIME DIGITS: 2,3,5,7
NON-PRIME DIGITS: 0,1,4,6,8,9
Again, I know some of you are tackling these as challenges but I'm disappointed that I am not getting reactions from classroom teachers who might be trying these in class and could provide real-time anecdotal records...
Distinguish between number and digit? I make a big big fuss about this, when I do my first digits problem. I focus on my mistakes: htu, what's the matter with my notation? What does htu mean? How can we get it to say what we want it to say? I let them devise their own "warning label." It doesn't solve the problem, but it helps. (And its a lesson in annotating their work. "In this problem, a number written like [htu] means...."
What was supposed to be the surprising part about (6)?
One surprise to me was that I could find only two such numbers (three-digit primes with all digits being prime).
I was taught that 1 was a unique number, neither prime nor composite. Is the same terminology used at this point in the space-time continuum?
Another interesting exercise is how you can generate palindromes (reverse a number and add, and keep continuing the process). There are a few numbers that do not generate palindromes, however.
TC
TC--
The accepted convention today is that 1 is not considered prime. This appears to be arbitrary, however one explanation is that a prime number has exactly two DISTINCT factors! I've seen other justifications for this definition. In the end, all students need to know that for standardized tests, math contests and everywhere else, 1 is not considered to be prime. Actually, I was once taught what you were -- that 1 is unique. Oh, well...
Nothing that remarkable about 523. I agree, thought, that it is surprising that there are only TWO 3-digit primes whose digits are three DISTINCT primes, namely 523 and 257. They both have the property that one digit is the sum of the other 2, but that's not particularly remarkable.
The palindrome problem is an excursion of its own. I only threw that in as an afterthought. I intend to explore it further. BTW, the majority of students explained the ninety 3-digit palindromes by stating something like, "NINE GROUPS OF 10 NUMBERS." Is this more desirable, in your opinion, than direct use of the multiplication principle?
The approach seems different depending on the way the solution is given:
(1) 9 groups of 10 numbers - the student most likely tried generating some examples, arrived at a pattern and generalized.
(2) Multiplication approach: The student thought about possibilites, rather than examples, and arrived at the answer.
Method (2) generalizes better to larger problems, IMHO, so that is my preference of where the students should ultimately be, even though they might start with method (1).
An interesting followup: Which is greater, the number of 3-digit paalindromic numbers, or the number of 4-digit palindromic numbers?
TC
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