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Case:

Exam Problem given to student at university: Give a problem/context illustrating the operation $\frac{4}{5} + \frac{2}{3}$

Answer by student: Anna and Beatrice buy flowers for grandpa for his birthday. Anna buys $5$ flowers , $4$ of which are daffodils. Beatrice buys $3$ flowers, $2$ of which are daffodils. What fraction of the flowers are daffodils?

Remarks:

  • The student seems to confuse the operation $\frac{a}{b} + \frac{c}{d}$ with the some other operation I am not sure how to denote, but my attempt is: $(a,b) \star (c,d) = \frac{a+c}{b+d}$

  • The student seems to think that "4 out of 5" could be related to $\frac{4}{5}$. However, in this case there actually is a difference between "6 out of 8" and "3 out of 4".

  • I am not sure that I am asking the right question, but I observed this case in real life and I would like to analyze it.

Question:

What property of fractions or addition of fractions could they be misunderstanding, and how would you explain to the student where they have gone wrong so that they don't repeat this in the future?

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  • $\begingroup$ I never learned any music theory, but I noticed that 3/4 beat and 6/8 beat are considered different. Is that related? $\endgroup$
    – Improve
    Commented Apr 17, 2021 at 15:32
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    $\begingroup$ (+1) Great question for those planning to teach at this level to ponder. Obviously $\frac{4}{5} + \frac{2}{3}$ is not the correct answer to the problem the student gave (e.g. clearly sum is greater than $1),$ and the student will surely see this when it's pointed out to the student, but a good teacher will want to come up with an explanation in which the error becomes intuitively clear without having to first express $\frac{4}{5} + \frac{2}{3}$ as a single fraction, or even without having to consider the value of $\frac{4}{5} + \frac{2}{3}.$ $\endgroup$
    – Dave L Renfro
    Commented Apr 17, 2021 at 19:02
  • $\begingroup$ The other operation is called mediant, but I don't know if there is a standard symbol for it. $\endgroup$
    – Massimo Ortolano
    Commented Apr 18, 2021 at 21:38

4 Answers 4

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The student who designed this problem wasn't thinking about the different wholes.

IN your students problem, there are 3 different wholes.

  1. Anna's flowers - The whole is 5 flowers and $\frac{4}{5}$ are daffodils
  2. Beatrice's flowers The whole is 3 flowers and $\frac{2}{3}$ are daffodils
  3. The flowers of Anna and Beatrice combined. The whold is 8 flowers and $\frac{6}{8}$ are daffodils.

When we take a fraction we have to be aware of the whole. You can't compare fractions of different wholes. This is why $\frac{1}{2}$ of an apple is less than $\frac{1}{2}$ of a watermelon even though: $\frac{1}{2} = \frac{1}{2}$. Furthermore, you can't add the two halves ($\frac{1}{2}+\frac{1}{2}=1$) because they are halves of different things. If we have $\frac{1}{2}$ an apple and $\frac{1}{2}$ a watermelon, we do not have 1 whole fruit.

Your student has shown a lack of awareness of the importance of the whole, by adding the fractions that are fractions of different wholes and by not specifying the whole in the question.

What fraction of the flowers are daffodils?

To correctly give a context for adding two fractions the 2 fractions must be of the same whole. .

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    $\begingroup$ 1) The whole is "5 flowers". 2) The whole is "3" flowers. 3) The whole is "8 flowers". I guess it takes some training or good understanding to realize that. $\endgroup$
    – Improve
    Commented Apr 17, 2021 at 21:51
  • $\begingroup$ @Improve I have edited my question to make the point clearer. $\endgroup$
    – Amy B
    Commented Apr 18, 2021 at 5:21
  • $\begingroup$ @Improve As for training or good understanding - I would hope by university students would grasp this point. $\endgroup$
    – Amy B
    Commented Apr 18, 2021 at 5:23
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The word you are looking for is mediant. The mediant of two fractions $\frac{a}{c}$ and $\frac{b}{d}$ is $\frac{a+b}{c+d}$.

According to Wikipedia,

It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions.

Teachers often do this in grading papers. For example, a test has two parts: the first has 10 items, the second has 20 items. The student got 8 items in the first part correctly (8/10), and 12 items in the second part correctly (12/20). The "total score" of the student is (8+12)/(10+20) = 20/30. Note that the "total score" is not 2/3.

I haven't tried this yet, but perhaps instead of just "explaining to the student where they have gone wrong," you might also want to show the student that this operation (mediant) is sometimes the correct thing to do (as in my example).

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    $\begingroup$ Note that the mediant operates on "fractions" (ordered pairs) and not on rational numbers. For example, the mediant of 1/2 and 2/3 is 3/5, but the mediant of 2/4 and 2/3 is 4/7, so your notation $(a,b)\star(c,d)=\frac{a+c}{b+d}$ shows a good understanding of the situation. $\endgroup$
    – JRN
    Commented Apr 17, 2021 at 16:08
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    $\begingroup$ If you have the time, you can spend a meeting talking about mediants and how they relate to Farey sequences, Stern-Brocot sequences, and Simpson's paradox. $\endgroup$
    – JRN
    Commented Apr 17, 2021 at 16:18
  • $\begingroup$ freshman sum --> ah like the freshman's dream. nice. $\endgroup$
    – BCLC
    Commented Apr 17, 2021 at 16:31
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    $\begingroup$ Interestingly the student's answer gives a good intuition for why $\frac{a}{b} \leq \frac{a+c}{b+d} \leq \frac{c}{d}$. The fraction of daffodils of the total cannot be more than the fraction of daffodils of Beatrice and cannot be less than the fraction of daffodils of Anna. $\endgroup$
    – Improve
    Commented Apr 17, 2021 at 19:31
  • $\begingroup$ Do teachers do this intentionally when grading papers? $\endgroup$
    – Rusty Core
    Commented Apr 18, 2021 at 15:50
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  • The student was conflating the sum $$\frac45+\frac23$$ and the weighted average (where the weights account for group-size differences) $$\left(\frac{\color{#00F}5}{\color{#180}{5+3}}\right)\frac45+\left(\frac{\color{#00F}3}{\color{#180}{5+3}}\right)\frac23\\=\frac{4+2}{5+3}$$ of the two given fractions/rates.
  • Clearly, the sum of two positive fractions is necessarily greater than each of them, whereas their weighted average lies somewhere between them.
  • The student's misinterpretation (“What fraction of the flowers are daffodils?”) is akin to claiming that the overall performance on an exam comprising several papers and in which every mark is worth the same value equals the $\color{#C00}{\text{sum}}$ of the individual scaled scores. $$\frac{60+80}{100+100}=\frac12\left(\frac{60}{100}+\frac{80}{100}\right)\\\color{#C00}=\frac{60}{100}+\frac{80}{100}.$$ In this example, the student would've been likelier to catch the discrepancy, since the averaging and summing are easier to contrast, since the given fractions share a common denominator.
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What property of fractions or addition of fractions could they be misunderstanding, and how would you explain to the student where they have gone wrong so that they don't repeat this in the future?

[Emphasis added.]

Short Answer

Fractions are numbers and they behave like numbers when we do operations on them. Many students never learn this.

You (and the student) may be focusing on things like common denominators and equivalent fractions and why we add numerators but not denominators and then simplifying at the and maybe converting to a mixed or whole number and...

These are all procedures and rules. Nothing in there emphasizes that a fraction is a number.

Perhaps you could ask your student to draw similar pictures based on similar context for the following two equations. See if they can determine answers by inspection. Tell them that you can.

$13 - 2 =$ _____

$13 - \frac{1}{6} = $ _____

If they're struggling to think of a context, tell them to start with 13 loaves of bread.

This has a good chance of revealing if they think of fractions as numbers.

Generally speaking, don't do any exercise ever involving two fractions until the student demonstrates understanding by doing that exercise with one fraction (new knowledge) and one whole number (prior knowledge).

Long Answers

Mine

Best answers from Hung Hsi-Wu

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