Last summer, in this blog, I addressed important aspects of division with fractions, and considered one of the most challenging elementary school topics. Tirosh writes, “division of fractions is often considered the most mechanical and least understood topic in elementary school.” Van de Walle and Lovin write, “Invert the divisor and multiply is probably one of the most mysterious rules in elementary mathematics. We want to avoid this mystery at all costs.” The difficulty with fraction division carries over to the study of ratios and proportions in middle school, and to proportional relationships and linear functions in high school. In this post, informed by experience and research, I revisit fraction division by addressing five fundamental reasons why students have so much difficulty with it and offer suggestions for helping students overcome them (more detail and further suggestions can be found in Planting the Seeds of Algebra, 3-5: Explorations for the Upper Elementary Grades). Hopefully, the suggestions will help teachers clarify some of the mystery!
Don’t Rush to Teach Algorithms
“Knowing” fractions has long meant knowing how to perform fraction operations, and “knowing” proportions has long meant knowing how to solve proportional equations such as by cross multiplying and solving for x. Rushing to teach algorithms and procedures deprives students of precious time to explore concepts through different context interpretations, multiple representations, and the relationships among them. When the leap to procedures is premature, algorithms for manipulating symbols are memorized but students are not connecting the algorithmic procedures to their conceptual understanding, nor are they making sense of the algorithms. Abundant research has shown that students first need time to build on their informal understandings, discover the new meanings and uses of fractions, work with helpful models and contexts, and develop symbol sense before they are asked to combine fractions through any of the operations, in particular division. Ideally, the algorithms should emerge naturally through problem solving situations.
Help Students See Fractions as Numbers
Delve deeply into your students’ mental images of fractions. These mental images must be explored, visualized, verbalized, drawn, gestured and discussed. Meaningful learning takes place when we build on students’ mental images of fractions made visual. Try this exercise: Ask your students, “What mental images does the fraction one fourth bring to mind? Make a picture, drawing, or diagram of what you see with your mind’s eye.” You will be surprised. Depending on the students’ ages, answers may include:
- The shaded quarter of a circle or a square (A fraction of a continuous quantity called an area)
- One shaded star out of four stars (A fraction of a discrete quantity called a collection or set)
- One quarter or 25 cents (a measure: money)
- A quarter mile (a measure: distance)
- ¼ (the symbolic fraction form)
- 0.25 (the decimal form)
Discuss the multiple interpretations and representations of fraction and help students see the connections and relationships among them. As students move on to middle school, the list will grow to include a ratio, a percent, a rational number, a quotient, an operator, etc. Bottom line for fraction division: If we want to build on students’ prior knowledge of operations with whole numbers, they must see fractions as numbers. The fraction ¾ is a single number, just as 7 is a single number, even though two numerals make up its symbolic representation. And the most useful representation of fractions as numbers is points on the number line, on the oh-so-underused number line! The most logical way to place 4⁄7 or 11⁄7 on the number line is first to find 1⁄7 (by dividing the unit segments into 7 equal, shorter segments) and then to count 4 and 11 copies, respectively.
Solve Partitive and Quotative Division Problems Starting in Grade 3.
The Grade 3 Common Core Standard 3.OA.A2 states, “Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.” The message is clear: Explore both partitive divison (fair shares) and quotative division (measurement or repeated subtraction) problems in grades 3 through 5. Why? Because the latter interpretation is the one that will help students understand division by a fraction.
It is very common for students to enter middle school with only the former metaphor for division, namely dividing a number into equal shares or parts (partitive interpretation). When carrying over this interpretation to fraction division, say 2 ¾ ÷ ½, the fair share meaning breaks down because you cannot share 2 ¾ pizzas among a non-integer number of people. Therefore, to help students make sense of this equation we must build on their understanding of the quotative interpretation of whole number division. For example, “How many portions of 2 bars can I make from 8 bars of chocolate?” Answer: 4 2-bar portions. That is a different question from, “How many bars will each person get if I share 8 bars equally between 2 people?” Answer: 4 bars. While the two problems have different meanings, they are both modeled by the equation 8÷2=4.
Transferring this thinking to fractions, change the portion size to ½ bar and ask, “How many ½-bar portions can I make out of 8 bars?” Answer: 16 portions. The equation is: 8 ÷ ½ = 16;. [Note that “invert then multiply” makes sense: 8 ÷ ½ = 8 × 2⁄1 = 16⁄1 = 16]
Explore the Concept of the Unit or Whole
Notice that in the previous example, the unit associated with the answer 16 is not “whole bars” but rather “half bars.” When making sense of division by a fraction, students must juggle two units at the same time. Let’s return to the example: 2 ¾ ÷ ½. Offer the following context to give meaning to this expression: “Mr. Cardone, the owner of a Manhattan pizza carryout, caters to Manhattan residents. Since 50% of them live alone, his unit portions are half-pizzas. At 8:00 p.m., there are two-and-three-quarters pizzas left. Help Mr. Cardone figure out how many half-pizza portions he can make from the remaining pizzas before he closes at 9:00 p.m.?” The figure below illustrates counting the number of “half-portions” that live inside the 2 ¾ pizzas. This action is modeled by the equation 2 ¾ ÷ ½ = 5 ½. The small piece at the right, while a quarter of an entire pizza, is actually labeled ½ because it is one half of a half pizza. The unit we are counting by here is a “half pizza.” Therefore, a quarter of a full pizza represents a half of a half pizza.
This can be confusing to students. To help them become more flexible with different units, create more challenging problems. Draw a number of discs or other objects—whole or mixed number—such as two-and-a-half, and pose three kinds of questions:
- Draw ½ if the figure above represents the unit (find a fraction given the unit)
- Draw 5⁄6 if the same figure represents ⅓ (find a fraction given another fraction)
- Draw the unit if the Figure represents 1¼ (find the unit given a fraction)
Don’t Say, “Multiplying Makes Bigger and Dividing Makes Smaller”
Finally, when teaching the four operations with whole numbers, we often say to students, “Addition makes bigger and subtraction makes smaller.” And, analogously, “Multiplication makes bigger and division makes smaller.” Such golden rules are detrimental. When students encounter fractions and integers, they don’t understand why division by fractions and subtraction with negative integers can produce bigger numbers. Rather than formulating such golden rules, I suggest we use more forward-thinking statements such as: “In the universe of numbers you know so far, subtraction and division make smaller. But soon your universe of numbers will expand and you’ll come to know new numbers called negative numbers and fractions. In this expanded universe of numbers, new rules will apply. Subtraction and division may make bigger. Such statements create mystery and instill in children a sense of adventure into their ever-expanding universe of numbers. They also pique their curiosity and prompt new questions. If they ask, “What are negative numbers?” or “What are fractions?” begin the adventure with them into the expanded universe of numbers, building on their informal ideas.