My students have a basic understanding of prime and composite numbers. We'll soon dive in a little deeper. I plan to do the sieve of Eratosthenes and have found a few other resources as well. My question is, do you think it's valuable/appropriate to have them memorize the prime numbers? If so, how high?

From a high school perspective, probably not. It should be enough that they've memorized their times tables, and the rules for factorization (like if the sum of the digits add up to a multiple of 3, it's a multiple of 3. LIkewise with 9. If the last 2 digits are a multiple of 4, so is the whole number. Multiples of 5 end in 5 or 0. Even numbers are divisible by 2....) That would be enough to tell whether a number, even a large number, was likely to be prime. For example, you can tell, pretty much by inspection, that 34,624 is a multiple of 4 and that 374,234,421 is a multiple of 3.

It's more important that they know what a prime number is and how to figure out if a number is prime than to memorize them. It's also important that they realize not all odd numbers are prime. So many of my sixth graders still fight me on 9 and 21 and 27. Those pesky multiples of three kill them.

Try a hands-on activity first. Use tiles or other manipulative. Have them separate the total group (start with 1 as the total) into 2 identical groups, then 3, then 4, etc. Similar to what you might do when they are learning division. Discuss which totals could not be divided equally. Have them record results on a table. Highlight the totals which cannot be divided into equal groups. Discuss that every even numbered total can be divided into two equal groups. When you introduce the concept of prime numbers (the highlighted ones), note that only odd numbers, except for 2, are prime. Then, review the divisibility rules (not just 2 and 5), as Alice suggested, and make sure they know them. We used to review them daily in calendar, listing the factors of each day of the year. Eventually, they will know, not just memorize, the first ten primes, at least.

"Memorizing" the prime numbers drives me crazy! First of all, they aren't really "memorizing" them. They are remembering them for the quiz, then forgetting. It is 1000 times more useful for them to know and be able to apply divisibility rules. I never remember if 57 or 59 is prime, but knowing divisibility rules, I can quickly determine that. I do prime numbers with our calendar math (Everyday Counts). We make arrays for each number 1-60. Beyond that, we apply divisibility rules and patterns to determine if a number is prime. By the time we hit 100, they've got the process down.

Right, what everyone else said. Focus on getting them to understand what it means to be a prime and composite number. Once they have a deep understanding of it, just looking at a number should be enough to figure out if it's prime or composite. No need to memorize them. Plus, JUST memorizing the numbers won't help them to understand what it means to be a prime. If they know what being a prime number means, then that's more important than memorizing a list.

Thank you for your responses. I was not fond of memorization either, but I doubted myself because it seems like such a popular thing to do.