Division of Fractions Before and After Common Core
by Bridget Adam
Before Common Core
Math Standards from Massachusetts (2000):
MA6.N.9 Select and use appropriate operations to solve problems involving addition, subtraction, multiplication, division, and positive integer exponents with whole numbers, and with positive fractions, mixed numbers, decimals, and percents.
MA6.N.14 Accurately and efficiently add, subtract, multiply, and divide positive fractions and mixed numbers. Simplify fractions.
An assignment before the Common Core in my class would likely consist of the following:
- Focus on computation, precision
- Single step word problems
- Highly scaffolded Open Response Problems
In the years before the Common Core was adopted by Massachusetts, I used direct instruction to teach operating on fractions. Specifically, I taught dividing fractions only using the standard algorithm. I would present the steps to divide fractions, as a class we would come up with a way to remember the steps (KCF- Keep, Change and Flip), practice as a group, and then students would spend a day or two proving they were proficient with the computation. My students were masters of precision, rulers of proficiency. But, my students often struggled to determine if a word problem inferred division because they only had practice computing with fractions, and not understanding or applying division of fractions. The picture of the assignment is from a classwork I gave just a few years ago.
Moving towards Common Core
2011 Massachusetts Curriculum Frameworks that incorporate the Common Core:
CCSS.Math.Content.6.NS.A.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. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.
An assignment shifting toward the Common Core in my class would likely consist of the following:
- Focus on problem solving
- Pushing toward deeper understanding
- Multi-step problems
- Emphasis on developing the concept of division
- Fluent and precise computation
Prior to shifting instruction towards the Common Core I noticed that my students never fully mastered division concepts. They were unsure how to set up or read division expressions, cannot determine the dividend or divisor in a problem, and cannot differentiate between the types of division problems. Overall, they had a very narrow understanding of division.
Knowing this, my students now spend more time looking at various types of division word problems. I begin my division unit with identifying the dividend within a word problem. I do not expect them to give me the number value (because they still believe that the dividend is the number with the larger value), but simply identify the “thing” in the problem that was getting broken up. Moving on, students are expected to read and write division expressions in various forms. Gradually, I move from problems with whole numbers to problems that incorporate fractional values, but I continue to use the same language I used when working with whole numbers, “What is getting broken up? Don’t tell me a number, tell me a ‘thing’. Use the context of the problem, it will help you understand.” This familiar language helps them think through the problem, and they are surprised when they find out that the dividend can actually be a smaller value than the divisor! Next, we drew pictures of the process of division as an introduction to the modeling process. Here is a look at some of the new work I’m doing:
I was very nervous when I first introduced modeling division of fractions using visuals. Having taught strictly procedure in previous years, this was an uncomfortable change for me. But now, it completely makes sense, and gives students who struggle with precision an access point and provides all students a way of understanding what division of fraction accomplishes and how it works. We began by dividing whole numbers by unit fractions. The language I use is consistent, “how many ___(unit fraction) fit into ____(wholes)?” or “how many ____(unit fractions) are in ____(wholes)?” Next, we divided unit fractions by whole numbers. “What is the size of one piece when I divide _____(unit fraction) into _____(whole number) pieces?” Students began to recognize these question types and ones like them as they start to solve multi-step and conceptual problems. Here are two examples of students applying and interpreting different models to solve problems:
I have made many changes to my curriculum and I know I have plenty more to make in the future. A place for continued improvement: Students continue to struggle to create word problems on their own. I was extremely surprised when doing a group project how many groups were unable to write their own division problem. Even after 2 weeks of seeing and solving teacher created division problems, they struggled to self-create a situation where division of fractions is necessary. In the future, I would like to incorporate more time for students to model and create their own division of fractions problems.
I am always proud when I see my struggling students revert to models when they are confused. I am excited that the Common Core is a chance for me to improve both my teaching and help my students truly “get it,” rather than just “doing it.” Students can now not only use the standard algorithms to perform operations with fractions, but they understand how it works. I have found that making these moves to address the Common Core State Standards pushes students to be persevering problem solvers, compute accurately and efficiently as well as understand what they are doing.