Standard(s) for Mathematical Practice (MP)
MP 1  Make sense of problems and persevere in solving them. 

MP 2  Reason abstractly and quantitatively. 
MP 3  Construct viable arguments and critique the reasoning of others. 
MP 6  Attend to precision. 
Grade Level
Grades 4–5
Content Domain
Number and Operations—Fractions
Standard(s) for Mathematical Content
4.NF.B.3a  Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 

5.NF.A.1  Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 
5.NF.A.2  Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 <1/2. 
Math Topic Keywords
 fractions
 common denominator
 unlike denominator
 adding fractions
 equivalent fractions
The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task. Before reading the dialogue, work on the mathematics task. Next reflect on the mathematical practices you engaged in while working on the task. Finally read the student dialogue.
Mathematics Task
What does $\frac{2}{5}+\frac{1}{2}$ equal?
Student Dialogue
The students in this dialogue have experience comparing fractions and adding fractions that have the same denominator. In this dialogue, they are beginning to reason about the meaning of adding fractions with unlike denominators in the context of an example problem.
(1)  Sam  How do you do $\frac{2}{5}+\frac{1}{2}$? 

(2)  Dana  It’s just $\frac{3}{7}$, isn’t it? 
(3)  Anita  But $\frac{3}{7}$ is less than $\frac{1}{2}$, so it can’t be that! 
(4)  Sam  So… how do you do it? 
(5)  Dana  But we’re just adding: $2+1$ is $3$, and $5 + 2$ is $7$, so it should be $\frac{3}{7}$. 
(6)  Anita  We already know that 2 fifths plus 1 fifth is 3 fifths [writes down $\frac{2}{5}+\frac{1}{5}=\frac{3}{5}$]. It’s not 3 tenths. You can’t just add everything you see. 
(7)  Sam  So… how do you do it? 
(8)  Dana  [to Anita] Oh, right, I get it. It’s like when we were saying “2 cats plus 1 cat, 2 grapes plus 1 grape, 2 fifths plus 1 fifth.” 
(9)  Sam  Yeah, I get it, too, but how do we do 2 fifths plus 1 half?! It’s not just 3 of something, but what is it? We’re adding two different things. Like 2 cats and 1 grape; 2 feet and 1 inch. Or, maybe like 2 thousand and 1 hundred. We can add them, but they’re not three of something. 
(10)  Dana  Oh, we need the same thing. 2 feet and 1 inch is 25 inches. Or, 2 thousand plus 1 hundred is like saying 20 hundreds plus 1 hundred. [writes down 2100] That’s 21 hundred. 
(11)  Anita  Or 2.1 thousand: 2 thousand plus 0.1 of a thousand. But not 3 of anything. 
(12)  Sam  [sighs] Yeah, but I still don’t know how to do $\frac{2}{5}+\frac{1}{2}$… 
(13)  Dana  So, we need to make the fifths and the halves the same somehow so we can add them together more easily. 
The teacher reflection questions are intended to prompt thinking about 1) the mathematical practices, 2) the mathematical content and extensions, 3) student thinking, and 4) teaching practices. Begin by reflecting on each of the questions, referring to the student dialogue as needed. After you had some time to do your own reflection, read the possible responses and comment at the bottom of the page with questions or thoughts provoked by the reflection questions and/or responses. Please note some of the mathematics extension tasks are meant for teacher exploration, to prompt engagement in the mathematical practices, and may not be appropriate for student use.
Teacher Reflection Questions

What evidence do you see of the students engaging in the Standards for Mathematical Practice?

How is spoken language helpful when working with fractions with unlike denominators?

What’s wrong with the claim “you can’t add $3a^2$ and $2a$"?

Students might still try to add 2 cats and 1 grape saying it’s 3 objects. Adding 2 thousand and 1 hundred together makes it harder to see some sensible common unit. What other examples might be good to use to show the logic of needing a common unit?

At the end of the dialogue, Sam still doesn’t know how to add $\frac{2}{5}+\frac{1}{2}$. What does the student need to understand in order to add those two fractions? How could you help Sam build that understanding?

Are there any fractions whose (correct) sum can be found by adding the numerators and adding the denominators? How do you know?

When adding, it is important to have a common unit. What roles do units play in multiplication?
The mathematical overview is intended to 1) explain how students in the dialogue engaged in the mathematical practices and 2) further explore the mathematical content and its extensions. Read the mathematical overview and comment at the bottom of this page with questions or thoughts it provoked.
Mathematical Overview
Mathematical Practice  Evidence 

Make sense of problems and persevere in solving them. 
In lines 6–11 of this dialogue, the students “consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution” (MP 1). They discuss examples of adding fifths, cats, grapes, cats and grapes, feet and inches, and thousands and hundreds as they try to make sense of what it means to add two fractions with unlike denominators. 
Reason abstractly and quantitatively. 
The cats and grapes argument (lines 8–9) uses a totally nonnumeric context to make sense of the numeric situation and to understand how the notion of a common unit might apply to fractions. Students are “considering the units involved; attending to the meaning of quantities, not just how to compute them” (MP 2). When we add two numbers, we tacitly assume that the numbers quantify the same thing. As Sam points out in (line 9), we can add 2 thousand and 1 hundred, but it is not 3 of any thing we’ve already named. The same is true of $2x + 3y$. We can add them—in fact, we just did—but they are not 5 of anything we’ve already named. (What are they five of? What are $\frac{2}{5}+\frac{1}{2}$ three of? What is $\frac{a}{b}+\frac{c}{d}$ “$a + c$” of?) In this dialogue, the students never figure out how to convert the fractions into forms (e.g., 4 tenths and 5 tenths) that give them integer counts (4 and 5) that they can add, but they do establish what is needed (Dana says, “the same thing” (lines 10, 13), and they do establish that without that, they cannot simply add numerators (let alone denominators). 
Construct viable arguments and critique the reasoning of others. 
The students are not exactly constructing and critiquing arguments in the sense implied in MP 3, in that they are not really developing much of a chain of reasoning—during the course of this dialogue they do not reach a conclusion about what they should do, only that they should not add the fractions as if the denominators were the same units. However, the students are beginning to engage in MP 3 in the conjectures they are making and their evaluation of each others’ conjectures. For example, Dana thinks that they can add fractions by adding the numerators and denominators, but Anita (lines 3, 6) evaluates the reasonableness of this claim by comparing $\frac{3}{7}$ to $\frac{1}{2}$ in one case, and by providing a counterexample in another case (i.e., “$\frac{2}{5}+\frac{1}{5}=\frac{3}{5}$. It’s not 3 tenths.”). All of the students are flirting with this practice of constructing arguments by their increasingly clear verbalization and their attention to each others’ statements. 
Attend to precision. 
The students in this dialogue work to communicate precisely with each other (MP 6) by jointly establishing what units they are talking about when answering the stated problem and how those units impact what they can do. Starting in line 8, the three students begin to highlight for each other how the units for the two fractions being added are different from each other, giving specific examples of why defining the units will matter when trying to add the fractions. 
Commentary on the Mathematics
The dialogue also illustrates the idea of students working through a problem for which none of them have a complete solution, and checking their answers rather than just jumping to a solution. Importantly, the students don’t fall back on a rule to provide an answer. The discussion of twentyone hundred and 2.1 thousand both hint at the ideas underlying a method for combining the two addends—the need for a common unit—as does their appeal to the catsandgrapes argument, but these all draw on students’ understanding and “sense making”: they are attempts to derive a logical method, rather than attempts to recall a learned (but maybe not understood) rule about “common denominators.” Students push each other to think through why the temptation to add both the numerators and the denominators—a common misconception—will not work. Though this little clip of dialogue ends before the students have solved their problem (see Sam’s disappointment in line 12), the dialogue represents an important step in learning to persevere. At some point, the content goal—figuring out how to add the two fractions—still needs to be satisfied, and waiting too long without resolution risks frustration and loss of interest, but if teaching brings closure too soon or too often, and if too many problems are resolved quickly, students don’t get a chance to stretch their ability to struggle. From line 9 on, we get many hints that they’re close to a resolution and might, in the next few minutes, hit on it by themselves. More likely, they will need help, but, having struggled with the problem and come this far, they’re now really ready for that help. The teacher who is lucky enough to have overheard their reasoning is now in a perfect position. They’ve made the necessary steps, and may just not realize (or have the confidence to believe in) the value of what they’ve done. As Dana says, the fractions need to be made “the same somehow,” and the statements by all three in lines 9 through 11 give exactly the right ideas. $\frac{2}{5}+\frac{1}{2}$ is not three of something, but if both of those fractions were made “the same somehow” by changing both into tenths or twentieths or, for that matter, fifths…. The teacher’s role may be as simple as saying: “Exactly! So how can you do that?!”
Student discussion questions and related mathematics tasks are supplementary materials intended for optional classroom use with students. If you choose to use this task and dialogue with your students these discussion questions are intended to stimulate discussion and further exploration of the mathematics. Related mathematics tasks are intended to provide students an opportunity to engage in the mathematical practices as they connect to content that is similar to or an extension of that found in the dialogue. Please note, responses are to be read by teachers only in preparation for using the questions and tasks with their class.
Student Materials
Student Discussion Questions

How does Dana get $\frac{3}{7}$ from $\frac{2}{5}+\frac{1}{2}$? What is the common mistake the student is making?

In line 3, how does Anita realize that $\frac{3}{7}$ is incorrect? How does Anita know this?

In line 8, Dana is giving several examples of addition. What do you notice about all the examples? What does this tell you about two numbers if you want to add them together to get one number?

Using a number line, how can you think of fifths and halves as units?
Related Mathematics Tasks

Consider the following example:
A. What is 2 quarts plus 1 cup? James claims he can add these together to get 3 quarts of milk. Is this true? Why or why not?
B. If 4 cups are in 1 quart, how many cups are in 2 quarts? What is 2 quarts plus 1 cup?
C. Why could you add 2 quarts and 1 cup in part B but not in part A?

Consider the following example (Note: allons, bobbers, and coffs are madeup words):
A. What is 3 allons plus 5 bobbers? Can you add the two numbers together, why or why not?
B. If there are 5 coffs in an allon and 10 coffs in a bobber, what is 3 allons plus 5 bobbers?
C. Why could you add 3 allons plus 5 bobbers in part B but not in part A?

Consider the following example:
A. Is 2 fifths plus 1 half equal to 3 sevenths? Why or why not?
B. If there are 2 tenths in 1 fifth and 5 tenths in 1 half, what is 2 fifths and 1 half?
C. What did you do in part B that made it easier to add 2 fifths and 1 half?
D. Write a word problem that would require you to add 2 fifths and 1 half and explain how you would solve the problem.
E. Rewrite all the conversions and work you did in part B using fraction notation.
Comments
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