Why are we learning Greatest Common Factor (GCF)?

Last week, we wrote about one of the procedure in Math in Focus workbook 4A for finding Greatest Common Factor (GCF), based on prime factorization. The other method introduced in the same chapter is to simply list down all factors, find the common ones, then identify the greatest.

Two methods of finding Greatest Common Factor in Math In Focus 4A

Besides these common methods, students at elementary level might also learn a variety of strategies to find the factors of numbers to 100, including drawing factor trees, using skip-counting, making tables, drawing diagrams etc (from The Everything Parent’s Guide to Common Core Math Grades K-5).

GCF_other_methods

As far as the standards are concerned, in 4th grade, students understand and use concepts and language in factors, multiples, prime and composite numbers but need not be fluent in finding all factor pairs (US Common Core 4.OA.4). In 6th grade, students are expected to be fluent in finding the GCF of two numbers less than or equal to 100 (US Common Core 6.NS.4)

Regardless of the method used to find GCF, asking students to find the GCF of two numbers are largely procedural-level questions not requiring the student to have much conceptual understanding (from The Everything Parent’s Guide to Common Core Math Grades 6-8)

Why learn Greatest Common Factor

However, it is important for teachers and parents to understand the applications of GCF (and Lowest Common Multiple – LCM) while teaching these procedures, in order to let students know why we’re learning the topic. Also when handling more advanced topics at higher levels, we can more effectively identify students’ lack of proficiency in the procedures for finding GCF and LCM as a possible cause of their anxiety when trying to solve problems. Below we list three areas that, among other basic mathematical skills, require procedural fluency of finding GCF and LCM to be proficient.

Fractions

GCF is most commonly used when reducing a fraction to its lowest terms, while LCM is used when adding unlike fractions. For example,

18/48 = (18÷6)/(48÷6) = 3/8 (GCF)

1/2 + 1/3 = 3/6 + 2/6 = 5/6 (LCM)

Algebra

In algebra, factorization procedure requires student to know the GCF of the coefficients, e.g.

18x+ 48x = 6x (3x + 8) (GCF)

Word Problems

At the heart of math education is problem solving. It is often beneficial to introduced challenging word problems that increase the students’ level of understanding of GCF. These examples are from The Everything Parent’s Guide to Common Core Math Grades 6-8:

  • Samantha has two pieces of cloth. One piece is 72 inches wide, and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips?
  • Bennie is catering a party and putting snack food on plates. He has 72 cheese puffs and 48 carrot sticks. He wants both kinds of food on each plate. He wants to distribute the food evenly and he doesn’t want any left over. What is the largest number of plates he can use. and how many of each type of food should he put on each plate?

Conclusion

While it is important for students to gain procedural fluency in finding greatest common factors (and least common multiples), it is important for us educators to recognize that teaching the various procedures for finding GCF and LCM is not a standalone topic, but rather a foundation skill required for more advanced applications a few years down the road. For higher grades, when dealing with students with anxiety over algebraic manipulations and fractions operations, it is also useful to identify if procedural fluency of finding GCF and LCM might be the root cause.


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