 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum          A  Notes for Session 6, Part A Note 2 The greatest common factor is equivalent to the greatest common divisor. The greatest counting number that evenly divides a and b is both the greatest common factor and the greatest common divisor of both a and b. Note 3 The LCM and GCF can be difficult concepts to understand because we hear the words in the opposite order of their importance: For example, for the LCM, first we hear "least," then we hear "common," and last we hear "multiple." However, the most important word of the three is "multiple." The multiples of 24 are 1 • 24, 2 • 24, 3 • 24, and so forth, and the multiples of 36 are 1 • 36, 2 • 36, 3 • 36, and so forth. The next-most important word is "common." We are looking for a number that is common to both numbers. The third most important word is the one we hear first, "least." So the number we want is a multiple, common to both numbers, and the least of all such numbers. This would be the same case for the GCF, for which we want a number that is a factor, common to both numbers, and the greatest of all such numbers. It would be worth taking the time to have a class discussion of the three words when introducing the LCM and GCF. Note 4

When you're asked to find factors, pay careful attention to the specific question that is posed to you. Were you told to find the prime factors, the prime factorization, the number of factors, or all the factors? These are four very different questions.

For example, for the number 36, the following statements are all true:

 • The prime factors are 2 and 3. • The prime factorization is 22 • 32. • The number of factors is nine. • Those nine factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36 -- 1 • 36, 2 • 18, 3 • 12, 4 • 9, and 6 • 6. (Note that we only count the 6 once. This is why squares have only odd number of factors.) Note 5 The area model shows how to fill a rectangle with squares (to find the GCF) or make a square with rectangles (to find the LCM). It can be a useful method for visual learners.   