Grade 5 Math: Factors, Squares, and Multiples, Oh My!

Fifth graders finished their factor sorting investigation last week.  So many exciting discoveries came out of it.  Students were sorting numbers by how many factors they have.  For example, 1 has only 1 factor; 5 has two factors (1, 5).  Eight has four factors (1, 2, 4, 8).  We wanted to see if there were any relationships that emerged based on the number of factors a number has.  We found out some things that we already knew, such as every number with 2 factors is prime and that 1 is the only number with only 1 factor, but our sorting solidified this idea.  

We also realized that every number with an odd number of factors is a square number.  This led to a discussion of why this is true.   Factors usually come in pairs.  However, with square numbers, one factor gets multiplied by itself, producing one factor, not two.

Some other ideas related to doubling numbers.  We found that when odd numbers are doubled, you always get double the number of factors.  When even numbers are double, you almost always get 2 additional factors.  This doesn’t hold true for squares and doubles of squares because of the odd factor idea above.  They only gain one factor.  

The ideas that came up are too numerous to list, and even got us into a discussion of prime numbers and prime factorization.  See below for some or the student work.  Enjoy!