How do you go about calculating 2/3 × 1/4? You may have multiplied the two numbers on top (namely, numerators) to get 2 and multiplied the two numbers on the bottom (namely, denominators) to get 12, so you would get the answer of 2/12 (or 1/6 if simplified). Or you may have begun by simplifying the 2 in 2/3 and the 4 in 1/4 and then followed the previously described method. If you got the correct answer, congratulate yourself for remembering the procedure.
Now we have a different question for you: Can you write a word problem in which you would calculate 2/3 × 1/4 to find the answer? Hmm, do you feel like you need to brush up on your math skills? Perhaps your teachers never asked you to write a word problem when you were in school.
Four Problems, Four Interpretations
You can use at least four different contextual situations for your word problem, each of which embodies a different interpretation for multiplication of fractions. First, there's looking at fraction multiplication as taking a part of a part. For example, using that same problem, "There was 1/4 of a pan of brownies left from yesterday's party. If you ate 2/3 of the leftover brownies, what fractional part of the pan of brownies would you have eaten?"
Second, you might make a scale drawing in which one foot is represented by 1/4 of an inch. To determine how long a 2/3-foot-long table would be on the drawing, you'd need to multiply 2/3 × 1/4.
Third, perhaps you want to determine the probability of rain on both days when there's a 66 percent chance of rain on Saturday (approximately 2/3) and a 25 percent chance on Sunday. You'd multiply the two numbers in fraction or decimal form (2/3 × 1/4).
Finally, you might wish to find the area of a rectangular region whose dimensions are 2/3 of a yard by 1/4 of a yard. You'd do that by multiplying 2/3 × 1/4.
In our study involving 140 college freshmen, only two students were able to write a word problem that correctly represented a given fraction multiplication problem. Fifty-seven (41 percent) wrote a problem that would be answered by operations other than multiplication, such as addition. Fifty-three (38 percent) didn't even attempt to write a word problem, or their responses contained no substantial information, making them unanalyzable. Not only was the students' mathematical understanding disappointing, but there also were language issues, such as spelling, semantic, and syntactical errors, and issues using inappropriate or unrealistic contexts.
The Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) and the National Council of Teachers of Mathematics (2000) advocate using multiple methods that give students opportunities to learn mathematical ideas and demonstrate their understandings. By having students write word problems that encompass a variety of contextual situations, teachers gain insight into how students have interpreted a mathematical idea as well as their preferences for problem-solving strategies (National Mathematics Advisory Panel, 2008; Newton, 2008).
A Window into Student Thinking
We want our students to make sense of the mathematics they're learning and solve problems in sense-making ways, rather than merely applying rules and formulas. In our study, one of the common errors that students made concerning the multiplication of fractions was the result of a misconception—that multiplication always makes numbers bigger.
One student wrote, "Tom has 2/3 of an apple and needs to make a big batch for a recipe by 5/7. How many apples will he have now?" In addition to the readability issue this problem presents, adjusting a recipe by 5/7 would not yield a bigger batch, but rather a smaller one. Having students write word problems gives teachers immediate feedback about student misconceptions as well as the opportunity to develop lesson plans to both address student weaknesses and bolster student strengths.
Of Special Benefit to ELLs
When students write their own word problems, they typically make use of situations and contexts with which they're familiar. This can make the problems far more meaningful and comprehensible (Barwell, 2003; Chapman, 2006). This can be particularly helpful for English language learners (ELLs), who may find word problems difficult because they lack appropriate background knowledge, such as knowledge about U.S. currency or American football rules. By having ELLs write their own word problems using situations familiar to them, as well as language they can manage, teachers can more easily assess their mathematical abilities.
There's a prevalent myth that ELLs cannot be successful in solving word problems until they're more fluent in English (Martiniello, 2008). Because of this misunderstanding, many teachers might limit the teaching of mathematics to computation exercises instead of engaging the students in problem-solving efforts using word problems. This approach yields missed opportunities for ELLs to work toward overcoming the language demands of mathematics.
As students share their word problems with the class and invite their peers to solve those problems, they're led into discussions, both in small groups and as part of a whole-class discussion, about the meaning of their problems and how best to solve them. These discussions give students practice using mathematics language, which both the Common Core State Standards in Mathematics and the National Council of Teachers of Mathematics emphasize as an essential component in learning mathematics. A problem-posing activity can bring in many forms of communication, such as writing, speaking, reading, and listening, which benefit not just ELLs but all students.
What Students Need to Know
The Language of Mathematics
Mathematics language is semantically and syntactically specialized. Students may be familiar with the common uses of words like even, odd, and improper, but these words have a different meaning when used in mathematics. Sometimes the same mathematical word is used in more than one way within the field itself. The word square, for example, can refer to a shape and also to a number times itself (Rubenstein & Thompson, 2002). Also, mathematics language makes use of certain syntactic structures—such as greater than/less than, n times as much as, divided by as opposed to divided into, if/then, and so on (Chamot & O'Malley, 1994).
Student-written math problems help students connect the mathematics they're learning to other mathematical ideas and with ideas outside the mathematics classroom. These problems also help students understand how the theoretical language of mathematics and the everyday language of word problems are related.
The Unit Whole
When using fractions, it's important to clarify what the unit whole is. In the following example, the whole is unclear: "Bill has 2/3 of oranges left in his bag. Bill would like 5/7 more. How many oranges would Bill end up with?" It's unclear what whole is associated with the 5/7: the number of oranges that Bill currently has or the number of oranges that were in the bag originally. In either case, we can't answer the question because we don't know what the original number of oranges was. Asking "what fraction of a bag" instead of "how many oranges" would be more appropriate.
A Reasonable Context
In the following example, the measurements the student used are unrealistic, although the problem is correct mathematically: "Jamie made a pan of brownies. The pan's length was 2/3cm and its width was 5/7cm. How big is the pan?" In addition, the word big is ambiguous because it could mean either the area or perimeter of the pan.
Also, a number of students merely translated the multiplication sign into words. For example, "Ben has 2/3 of his candy bar left, and Sally has 5/7 of her candy bar left. If you multiply their candy bars together, how much would they have?" But what does multiplying two candy bars really mean?
Getting It Right
Teachers need to find ways of helping students overcome the difficulties they encounter in writing word problems. Let's say you ask students to write a word problem in which they need to do the calculation from the opening of this article—2/3 × 1/4—to find the answer. A student may respond with the following: Julie ate 1/4 of a pizza. Janet ate 2/3 more. How much pizza did Janet eat?
The first problem we encounter is with the words "2/3 more." Is this (a) 2/3 of a whole pizza, or is this (b) 2/3 of what Julie ate? If the meaning is (a), then the answer to the problem—How much pizza did Janet eat?—is 1/4 + 2/3, or 11/12 of a pizza. If the meaning is (b), then the answer is 1/4 + 2/3(1/4), or 5/12 of a pizza. Neither of these interpretations uses 2/3 × 1/4 to calculate the answer.
To help students gain conceptual understanding of the word problem in question, teachers can provide a visual representation. To illustrate the problem in a way that requires a calculation of 2/3 × 1/4, you'll need to start by restating the problem correctly: Julie ate 1/4 of a pizza. Janet ate 2/3 as much pizza as Julie did. How much pizza did Janet eat?
Draw a picture of a circle (pizza) cut into 4 equal pieces; one shaded piece represents the 1/4 pizza that Julie ate. Then divide that 1/4 into 3 equal pieces (each piece now represents 1/3 of 1/4, which is 1/12). Shading two of those pieces gives the answer of 2/12 (or simplified, 1/6), which is the correct answer to 2/3 × 1/4. Illustrating the problem with an image like this one can increase students' ability to write meaningful word problems.
Tips for Teachers
Here are some things that teachers can do to help students write good word problems:
- Start by writing several word problems that involve different mathematical concepts yourself (such as the four examples we described earlier) before you use this activity in class. This will give you insight into what you might expect in student responses.
- To avoid getting problems like "Jimmy needs to find out 2/3 × 1/4. What is it?" explain to students that they can't use the words multiply or times or the multiplication symbol in their problem.
- Discuss how to choose topics in small groups or as a whole class. Be sure to introduce different contextual situations (for example, in addition to dividing a pie or a pizza, mention running a leg of a relay race, planting a garden, or making a scale drawing). This will encourage the use of different modes of representations (such as a number line model or an area model) and different interpretations (such as part-of-part or scaling). If students have trouble coming up with a topic, provide situations to which they can respond.
- If appropriate, start with problems in which a fraction is to be multiplied by a whole number and then progress to using two fractions. This will help students familiarize themselves with writing strategies and ease the transition to problems involving fraction operations.
- When students present their word problems (which may be incomplete or incorrect), encourage them to clearly explain what they mean and solve the problem as part of their explanation. Alternatively, a classmate might solve the problem and give the student feedback. This gives students the opportunity to use their math reasoning strategies and apply mathematical concepts.
- Students can write word problems that involve other operations of fractions, such as division. Or students can write problems to enhance their conceptual understanding of important mathematical ideas that are often taught as rule-based, such as a procedure for finding a common denominator.
Writing Problems, Gaining Understanding
Under the new Common Core standards, mathematics instruction emphasizes conceptual understanding, procedural fluency, multiple approaches to and models of mathematical problems, and problems requiring analysis and explanation. Having students write, solve, and talk about their own word problems is an enjoyable way to integrate communication skills, including writing, into instruction while deepening students' mathematical knowledge.
