Adding Fractions with Unlike Denominators

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

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

    Possible Response

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

    Possible Response

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

    Possible Response

  4. 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?

    Possible Response

  5. 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?

    Possible Response

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

    Possible Response

  7. When adding, it is important to have a common unit. What roles do units play in multiplication?

    Possible Response

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 non-numeric 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 counter-example 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 twenty-one 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 cats-and-grapes 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

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

    Possible Response

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

    Possible Response

  3. 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?

    Possible Response

  4. Using a number line, how can you think of fifths and halves as units?

    Possible Response

Related Mathematics Tasks

  1. 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?

    Possible Response

    B. If 4 cups are in 1 quart, how many cups are in 2 quarts? What is 2 quarts plus 1 cup?

    Possible Response

    C. Why could you add 2 quarts and 1 cup in part B but not in part A?

    Possible Response

  2. Consider the following example (Note: allons, bobbers, and coffs are made-up words):

    A. What is 3 allons plus 5 bobbers? Can you add the two numbers together, why or why not?

    Possible Response

    B. If there are 5 coffs in an allon and 10 coffs in a bobber, what is 3 allons plus 5 bobbers?

    Possible Response

    C. Why could you add 3 allons plus 5 bobbers in part B but not in part A?

    Possible Response

  3. Consider the following example:

    A. Is 2 fifths plus 1 half equal to 3 sevenths? Why or why not?

    Possible Response

    B. If there are 2 tenths in 1 fifth and 5 tenths in 1 half, what is 2 fifths and 1 half?

    Possible Response

    C. What did you do in part B that made it easier to add 2 fifths and 1 half?

    Possible Response

    D. Write a word problem that would require you to add 2 fifths and 1 half and explain how you would solve the problem.

    Possible Response

    E. Rewrite all the conversions and work you did in part B using fraction notation.

    Possible Response

 
 
  • MP 1: Make sense of problems and persevere in solving them.

    MP 1: Make sense of problems and persevere in solving them.

    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

  • MP 2: Reason abstractly and quantitatively.

    MP 2: Reason abstractly and quantitatively.

    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation  process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

  • MP 3: Construct viable arguments and critique the reasoning of others.

    MP 3: Construct viable arguments and critique the reasoning of others.

    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their  conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though  they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

  • MP 6: Attend to precision.

    MP 6: Attend to precision.

    Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Comments

Submitted by donna.chevaire on
This illustration adresses a very common misconception that students have regarding adding fractions with unlike denominators. This helps students understand the meaning of (making sense of) adding fractions with unlike denominators by making sense of the problem (MP 1) and critiquing the reasoning of others (MP #3) when Dana says that 2/5 + 1/2 is "Just 3/7. isn't it?" Anita in line 3 says that 3/7 is less than 1/2. Sam saying, "So how do you do it"? shows perseverence. The students connect their knowledge of place value (adding 1000 and 100) and combining feet and inches ( 2 feet and 1 inch). The students use of the cats and grapes example is confusing because you cannot add cats and grapes. There is no common denominator. It doesn't make sense. In Teacher Reflection Q # 2, the teacher might scaffold the problem for ELL students since using fifths and halves is difficult because they don't sound alike. Fifths and tenths might help ELL students to understand a little easier. ( They sound similar because of the ths ending). They can also relate this to money (nickels and dimes). Then, the students can go back and change the three-fifths to tenths and the one-half to tenths and solve the original problem. It is important to note that you don't need the lowest common denominator to add fractions with unlike denominators. The new CC does not stress simplification. The illustration does not finish the problem. (Line 13 Dana: So, we need to make the fifths and the halves the same somehow so we can add them together."

Submitted by rkelly on
A. I believe that this problem addresses MP3 and the idea that students must construct viable arguments and critique the reasoning of others. This whole dialog is representative of this. I agree with Donna (previous post) that this is a major misconception of 5tth grade students but I believe this might be due to their possibly only knowing an algorithm (line 5 Dana) and not really understanding the concept of how to handle adding different sized pieces of wholes. (Sam in line 9 hints at this) B. I actually like the us e of the students saying "cats and grapes" because it point out the difference between them. This helps to illustrate the fact that we are in fact trying to combine pieces that are different. Maybe say" cats and dogs" and then we can rename them both as "animals - common denominator" and the same point can be illustrated. But this part needs to remain to shed light on the "difference" between what you are trying to add and how you can make that happen. C. I would look at including (in the directions or prompt?) that the students draw a model, in addition to explaining, how to add these fraction. Having visuals help clear up some misconceptions and create a nice picture around which a deeper discussion can happen. It will also get at addressing the needs of the ELL, IEP, or struggling students.

Submitted by lmarcet on
A. The two math practices that seem to really apply here are MP1 and MP3. Students need to have an understanding of the value of each fraction in order to reason through the problem and determine if their answer makes sense. Dana's response in Line 2 is a clear indication that she has not made sense of what 2/5 or 1/2 represent and does not have an understanding of what the whole is or what adding these fractions together means. Anita's response in line 3 shows that she was able to make sense of the problem and critique Dana's reasoning. B. I like Dana's reasoning in line 10. Her comparison of feet and inches and hundredths and thousandths helps to illustrate the concept of having a common denominator when adding fractions. She is really making sense of the problem and using similar examples to illustrate her point. C. I agree with Richard's post of adding in having students represent the problem with a visual model. This piece can help students understand what is really happening in the problem and can help them to visualize what the sum of the two fractions should be. This can also help them to see the need for a common denominator.

Submitted by brucemallory on
When I solved this problem I used pie charts to diagram out 2/5ths and 1/2. Which (in my mind) is MP5, using "tools strategically." In one of the other comments on this dialogue, someone pointed out that transferring the problem to a money context is quite helpful here. The question of how much $ is 2/5ths of a dollar, quickly gets you to a common denominator of "dimes."

Submitted by brucemallory on
Throughout this dialogue the "kids" are trying to bring this problem back to something that they already know . . . adding like variables, adding inches and feet, adding thousands and hundreds. If I were categorizing the MP's I would have an MP that was the "what's this remind me of" MP. As I read through the MP descriptions, I see that my "what's this remind me of MP" is tucked into MP1 ("They consider analogous problems, . . . ").

Submitted by brucemallory on
7. When adding, it is important to have a common unit. What roles do units play in multiplication? I LOVE this question. In thinking about this, it occurred to me that SO OFTEN the math that we do has NO units involved. Multiplication is the perfect example. The problem 2x3=6 changes units from inches to square inches, but we NEVER notice this. For me thinking about the problem 2/5 + 1/2 became a whole lot easier conceptually if it was 2/5th's of a dollar + a half dollar. Clearly with addition, following the units helps to understand the underlying mathematics. Following the units with multiplication, less so. Or rather, it takes one to a much deeper understanding of multiplication. A level of understanding that we as teachers tend to stop talking about once we move beyond 2 groups of 3 and get the kids to memorize their multiplication tables. Thanks for the question - it's a really helpful question to ponder!