Traditionally speaking, most schools in the United States teach division of fractions procedurally. Students learn the method, but never learn the meaning, which denies them the deep conceptual understanding. Building procedural fluency from conceptual understanding is one of the 8 highly effective teaching practices in NCTM’s Principles to Action; and building procedural fluency from conceptual understanding is also the underlying structure used in Eureka Math to teach division of fractions in 6th grade
Let’s examine some student work from Grade 6 Module 2 Lesson 7. At this point in Module 2, students in Mrs. K’s class understand the difference between partitive and measurement division. They can use models to illustrate the solution to a division problem, and have been taught two strategies (use common denominators and multiply by the reciprocal) to compute a quotient to a division of fractions problem. Mrs. K wants to determine how well her students are putting all these pieces together. Mary, a student in Mrs. K’s class, demonstrates an understanding of these division strategies in her work, pictured below.
Craig drank 2 ¼ liters of Gatorade after an intense basketball practice. If he drank 2/3 of a liter per minute, how many minutes did he spend drinking Gatorade?
The first thing that Mary recognizes is that this is a measurement division situation. Why? In measurement division, the divisor names the size of the group (or unit), and the quotient represents the number of groups (or units). A measurement division problem can often be solved by thinking, “How many ____are in ____?” Specifically, how many 2/3s are in 9/4?
Next, Mary knows that she needs to find a common denominator. The two fractions in this problem have different denominators. Mary knows that to divide fractions with different denominators, you can find equivalent fractions with like denominators in order to solve the problem. She used the strategy of finding a common multiple to get the common denominator.
She then uses this information to rename the original fractions to fractions that have common denominators of 12.
Notice how Mary used pictorial representations to show her understanding of division of fractions. She drew 2 ¼ units with each unit partitioned into 12ths, or 27/12. She then determined how many groups of 8/12 there are in 27/12. Notice the 1, 2, 3 groups with 3 parts of size 1/8 left.
Mary then correctly interpreted the answer to the problem as:
By examining this student work, Mrs. K can see that Mary is demonstrating both conceptual understanding of division of fractions and has also demonstrated procedural fluency by using the common denominator algorithm. Mrs. K concludes that this student is able to recognize measurement division, can use a tape diagram to illustrate the solution to the division problem, and used common denominators to compute a quotient to a division of fractions problem. Mary is also demonstrating a good understanding of Standard 6.NS.1: “Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
Providing students with opportunities to build procedural fluency in division is fundamental to their success in division, and it also creates a strong foundation for them to succeed in Algebra down the line. As we can see from Mary’s work in Mrs. K’s class, she is not only solving to find an answer, she is also learning why that answer makes sense. By learning to divide this way Mary will be able to solve more complex problems because she will always know where to begin and how to tackle a problem.
This blog entry discussed how Eureka Math gave the students in Mrs. K’s class the tools and deep conceptual understanding of division of fractions along with computational procedures that demonstrates fluency in division. It should be noted that Eureka Math gives students the tools and deep conceptual understanding of all topics, not just division, to ensure they are able to solve problems demonstrating they know the “why” behind their answer
This blog post is by Eureka Math teacher-writer Connie Laughlin