In order to reason successfully with fractions, students need to understand the relationships between numbers and their factors and multiples.
To introduce prime and composite numbers and help students connect these concepts to what they know about multiplication and division, have students work with real or virtual manipulatives to create arrays. If the only way to put a number of tiles into an array is with one long horizontal or vertical line, then the number is prime. If more than one array is possible, the number has more factors (rows and columns of the array) than just itself and 1, and the number is composite.
A fun activity to identify prime numbers is the Sieve of Eratosthenes. Named after the Greek mathematician from Alexandria who invented this method of listing primes, it is called a “sieve” because it is like taking all of the numbers (up to some maximum) and running them through a sieve to separate out all of the primes. Here is an explanation of the procedure, a virtual manipulatives activity, and another interactive activity. This can also be done with the interactive hundreds square in the SMART board gallery.
Click here for videos that explain divisibility rules and other topics.
This Khan Academy video does a good job of explaining WHY the divisibility rules for 3 and 9 work.
To practice prime factorization, as well as facilitate understanding and finding both the Greatest Common Factor and Least Common Multiple of two numbers, use the following resources:
~ A lesson plan with activities from the National Council of Teachers of Mathematics
~ An activity at the mathplayground site
Here is a unit from the National Council of Teachers of Mathematics Illuminations site, complete with games and activities, that helps students investigate factors, multiples, and the relationships between them.
Finally, to understand what the Least Common Multiple of two or more numbers represents in real life contexts, offer students problems like these:
1) A city holds an election for mayor every 4 years and an election for treasurer every 6 years. Elections for both mayor and treasurer were held this year. How many years will it be before the elections for mayor and treasurer both happen in the same year again?
2) Many people like Koko’s restaurant. Abe eats there every other day. Beth eats there every three days. Carol eats there every 4 days. Don eats there every 5 days. By chance, all four of them are eating at Koko’s today. How many days will it be until this happens again?
3) The two-way radio Michael wants runs on 6 batteries. The store sells batteries in packs of 10. What is the least number of packs of batteries Michael should buy so that he has sets of batteries for the radio with no batteries left over?
For Greatest Common Factor use problems like these:
1) Kailey cut 12 pink roses and 15 white roses from the bushes in her back yard. If she wants to make some identical flower arrangements with no flowers left over, what is the greatest number of arrangements Kailey can make?
2) Lucy is creating a quilt that is 10 feet by 6 feet. She wants to construct it out of just squares, with no cloth left over. What is the length, in feet, of the largest square she can use?
Here is a good worksheet with word problems that have students determine if they need the GCF or the LCM to solve.