I arbitrarily designate some mistakes to be “classics,” and this here is one of them.

No need to identify the mistake. It’s right there. But let’s get some wisdom in the comments. What is the significance of this mistake? Does knowing that this mistake is common change the way that you do (or should) teach?

[I didn’t exactly know how to tag this post, but it’s from a 9th grade classroom. For CCSS I tagged 5th grade.]

  • Definitely a classic. I’ve heard teachers grumble about this one. Implausible that kids would do that. Ask those same teachers “If a student got 3 out of 5 on one quiz and 2 out of 7 on a second quiz, what is his/her grade?” and what do you think they do?

    If A-Rod went 3 for 5 in game one… never mind. Pseudo-context.

    This should change the way you teach. I think. I’m assuming that the student is used to applying memorized rules. I wonder if this student was given opportunities to develop rules him/herself through concrete experiences and pictorial representations. No evidence either way.

    Are students asked to guess? What’s an “about answer”? If there were a focus on estimation and benchmarks, then starting with 3/5 being more than 1/2 makes 5/12 unreasonable.

    • “If A-Rod went 3 for 5 in game one… never mind. Pseudo-context.”
      The hilarious bit here is that if A-Rod does go 3 for 5 in one game and 2 for 7 in the next game, his combined record is 5 for 12.

  • I was pleased to realize something the other day about this classic mistake — if you add fractions this way, then the result is dependent on which of the equivalent fractions you use to represent the rational numbers being added. In fancy lingo, the binary operation the student used is not a well-defined map on pairs of rational numbers.

    For instance, if you change 3/5 in the example to the equivalent fraction 6/10 and redo the same work, you get

    6/10 + 2/7 = 8/17,

    and 8/17 is not equivalent to the 5/12 the student got using 3/5 and 2/7.

    I don’t know if this would be of much pedagogical use, but the fact that the correct addition rule is well-defined might help us appreciate it a little more.

  • This student (and millions like him/her who make this mistake) are missing the critical concept of benchmarking. How could a number greater than 1/2, plus another positive number, equal a number less than 1/2?

  • I’m going to try it on my warm-up with 9th grade strugglers tomorrow. I’ll see what they think. Given that I’m telling them it is a mistake, let’s see if the people who make the mistake can fix it.

  • Javier

    I might suggest that getting students to turn their attention to what they have, or are counting. Here we have 3 fifths and 2 sevenths, but our unit size is different (fifths and sevenths). A picture would help, as would estimation and benchmarking as others have suggested. Knowing that our units must be the same before we can compute the sum would help (can’t change fifths to sevenths or sevenths to fifths, can I change them both to something else).

  • mr bombastic

    If a I got 3 out of 5 on quiz 1 “and” 2 out of 7 on quiz 2 then that equals 5 out of 12 all together. I was told that “and” means “+”, so I have convinced myself that the student actually did do the problem correctly. I wonder if this is the sort of thinking that is going on?

    Seems like measurement and number lines are much better for getting at the idea of adding fractions, or maybe even fractions in general.

    • As you say, 3 out of 5 “and” 2 out of 7 become, correctly, 5 out of 12. Does that mean that we’re not really adding the fractions 3/5 and 2/7 here? Clearly we’re not, by *why* not? It *feels* like we’re adding 3/5 and 2/7 to get 5/12. We’re not treating the 3, 5, 2, and 7 as proper numerators and denominators. How do we know that we can do that? What’s the rule? I feel like there ought to be a clear explanation for this, but it’s got me flummoxed at the moment.

  • So, looking at the replies by my 11th grade ( sure,we’re a failing school) students, here was my favorite:
    They have to cross-add. The answer should be \dfrac {7} {10}
    I must confess that if had been given this answer, I would not have been able to figure out what was done. “cross-add?” Then I guess I’m grateful that I didn’t end up with \dfrac {10} {7}

  • Pingback: 5 reasons you should NOT talk about fractions as "out of" | Bridging The Gap()

  • Julie

    This student knows how to add correctly by just adding the number across for both the numerator and denominator. He or she understands the concept of adding, but the student is confused on how to add fractions together. The student does not know that the denominators have to be the same in order to add across. So the student was not wrong necessarily because the student needs to find the common denominator for both of the given fraction then the student could add across. I would help the student by teaching them a math trick that will help him or her. First, the student needs to multiply 5 X 7 to get the common denominator, which is 35. So the denominator would be 35. Now to find the numeration, the student needs to cross multiply, in this case it is 3 X 7 = 21, 5 X 2 = 10. Finally add those two numbers together, which is 31. The final answer is 31/35. I can also draw this out to help the students see how it works visually.

  • Suhyeong Jo

    I agree that this kind of mistakes is one of the one that many of children make when they first learn how to add two fraction with different denominators. I also remember that I made this kind of mistake when I was young. I think that the teacher should teach the child how to find common denominators which can match up different two denominators with one, in this case 35. Teacher should teach children that numerator has to be increased in same rate as denominator did. I also think that making mistake is very important for children to understand better from error. Children know that how to do the addition, in this case just add up denominators and numerators, so got 5/12. Since they only know how to do addition, finding common denominator may take more time to understand for them. Teacher should draw pictures to help students’ understanding how it works. Or, should explain why the common denominator is 35 which is 3 x 7, and the denominator from first fraction increased by x 7 to be 35, the numerator should be 3 x 7 = 21, and the denominator from the second fraction was increased by 5 to be 35, so the numerator should be 2 x 5 = 10. So the answer should be 31/35.

  • JHK

    This is such a common mistake when students are first learning to add fractions; I’ve seen this so many times from the students that I used to tutor. This student knows how to add for sure, but doesn’t know the right procedures to adding a fraction problem. It’s important to teach students that we can’t add fractions just as if you’re adding whole numbers because thirds and fifths are not the same; Students need to first know that the denominators need to be the same when adding/subtracting fractions. An easy way to make the denominators the same is by multiplying the two denominators together: 5*7=35. And because we can’t just change the denominator and keep the numerator the same, we multiply the different number of the denominator to both the denominator and the numerator. So for example, 3/5 would be multiplied by (7/7). Then the converted fraction would be 21 (3*7) / 35 (5*7). And the second number would be 2/7*(5/5), which would become 10/35. Having the two fractions in the same denominator, we can then add the numerator and keep the denominator the way it is. (10+21)/35= 31/35. Getting students to fully understand the need to make the two denominators the same is really important when teaching students how to add/subtract fractions.

  • Tiffany Chiang

    This is indeed a classic mistake that students make when learning about fractions. We know that in order to add 2 fractions, their denominators have to be the same, and to change the denominator, you must first find the least common denominator between the 2 denominators, which in this case is 5×7=35. We also know that multiplying the fractions by 1 will keep the numerator and denominator ratio to be equal, so here, we can multiply both 3 in the numerator and 5 in the denominator by 7, which will give us 21/35. Then, we multiply 2 in the numerator by 5 and 7 in the denominator by 5 to get 10/35. Now the denominators are equal so we can add the fractions. 21/35 + 10/35 = 31/35. It’s important for students to know that they can’t just add the numbers across like they do for whole numbers.

  • Irene Pham

    I recognised this mistake as I myself some point in my early education made the same mistake. I think the problem is it wasn’t stressed good enough that in order to add fractions, the denominator has to be the same. In other words, you can’t add the proportions from two different “wholes” together. 1/2 of 20 is different from 1/2 of 10. In my early education, the idea of the “whole” wasn’t emphasised enough because I didn’t get much exposure with visual images comparing different wholes of fractions. I also think it is important to show students why this could be wrong visually using pictures and diagrams. It is never too careful to present the common mistakes that students might make so they can avoid those.

  • Gabriella Devi Widjaja

    I feel that it’s great if we see students do this kind of mistake as it can help to show teachers of what the students have already know and the things they haven’t fully had the full understanding of. From this problem, we can see that the student understands addition pretty well. However, not in terms of fractions. 3/5 or three fifths means that there are 5 parts to make a whole and 3 parts is what we have. Similarly, 2/7 or two sevenths means that there are 7 parts to make a whole and 2 parts is what we have. Having this understanding will help students understand of why can’t we just add the denominators (5 +7) right away. It’s because they are different. We aren’t comparing an apple to apple. We should have them have the same number of parts to make a whole. (If we have a square, make sure both of them are cut into the same pieces). This is definitely not an easy stuff to teach as it may require tons of ‘pauses’, making sure the students understand each step and why they are doing that. Starting teaching fractions by using chocolate bar, pizza, and blocks will be very helpful as well as using representation or drawing.

  • Xin (Dora) Tu

    I was in agreement that mistakes showing above is classic in that the student add the 2 numerators and 2 denominators across to get the answer, reflecting the lack of understanding of adding fractions with different denominators.
    This mistake is significant for teacher to foresee because, as teachers, we can anticipate that confusion and challenge before they occur. What I will change the way to change is, while teaching the topics on addition of fraction, I will spend more time on clarify how to add fractions with different denominator and I will use fraction bars to model why we need to split into pieces in the same size to add/subtract fractions. After students understand that the denominators need to be same to add/subtract, I was also thinking it’s important for student to practice how to find common denominator, or let common denominator. For example, how to find common denominator for 5 and 7? it should be the multiples both for 5 and 7, which 35 can fit. As we change the denominator into 35, I will ask students how do we get 5 into 35, prompting them to know the process one by one. So, we times 7 to get 35 at the bottom, we also need to times 7 at the top to keep the value.

    Finally, it still need to keep track of how students add fractions with same denominators. Do they keep the denominator same? or just add across denominators? I feel, beyond using visualized representation to facilitate student’s thinking, model the reasoning and keep track of their work.

  • Natalie Sit

    This student does not really understand the concepts of adding of fraction. I cannot imagine a 9th grade student has this mistake on the mathematics problem.
    First, he/she should know what he/she get wrong with it, and he/she should understand more concepts again. If you are adding with fraction, you have to make the same denominator before adding or subtracting in fraction. You have to multiply the same value to the numerator that make sure they are in the same value with the same base. It is easy to get confused on the multiplication and division of fraction because they are not similar to adding and subtracting. I think this student misunderstand or do not understand the concepts of fraction.

    Also, it still needs to take a look about the way of doing same numerators and different denominators. Many children think it is similar but not really actually. It is important to get clear on it for the further studies.

  • Julie Campos

    If this is a ninth grader who made the mistake, this could probably be traced back to the lack of enough knowledge on fractions. If the student never learned the “rules” of fractions, for example, that in order to add these two fractions you have to find a common denominator then this mistake is understandable. But to be in the ninth grade and probably at some type of algebra level this should be a major concern to the teacher. But, if this was an example from a younger student, like fourth grade this understandable because at this grade level according to the common core progressions they are decomposing and composing fractions with the same denominator. I believe this type of mistake in the elementary years would be considered normal.

  • Jacqueline Elder

    I think that this is a classic math mistake, because often students when doing operations with fractions will forget what fractions really “mean.” They can forget that fractions represent parts of a whole; the denominator is the whole, and the numerator is how many parts of that whole there are. When they don’t understand this concept, they will want to add everything, but once they do understand it, they know that you cannot just add different sized parts to create a new whole like that. I think that it is important for the teacher to consistently remind students about what fractions really are and how they are made up, so that they can understand conceptually why this mistake does not work. Showing them also how to find common denominators would be really helpful, and showing them that this way you are not changing the actual amount that the fractions represent but just a different way of representing might help them understand better.