44

The short answer to your question is: everyone is right. I agree with people here that in many contexts, $0.75$ or $\frac{3}{4}$ would be a more desirable answer than $\frac{45}{60}$. I also agree with several here that when the context is "out of 50 people", an unreduced fraction like $\frac{40}{50}$ makes perfect sense. Certainly, when I write ...


23

added Oct 6 The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found in road signs, cooking recipes, length measurements, and so on. (Denis Nardin commented that mixed numbers are never seen in Italy. Meters, centimeters, and ...


20

The answer to your question depends on the pedagogical goal of the exercise, and what learning outcomes you have identified. It basically comes down to the following question: Is manipulating fractions one of the skills which you are emphasizing in this class? If one of the goals of your class is to get students to more handily work with fractions, then ...


20

I am a GCSE Maths examiner. For a question like this, any correct equivalent decimal, percentage or fraction, whether simplified or not, would receive full marks. It is only specifically if it says in the question that the answer should be simplified that a simplified fraction is required to obtain the final accuracy mark. The reason for this is that the ...


12

I tell my students this story when this issue comes up: Imagine you are answering the phone at the local pizza place. Someone on the other end says "Yes, I'd like to place an order. I'd like twelve thirds pizzas." What would you think? They often come up with the following explanations: It is a prank call. Maybe I misheard them? Maybe they don't ...


11

The common core state standards definition of the fraction $\frac{N}{D}$ of a unit is to subdivide the unit into $D$ equal sized pieces. Each of these pieces is defined to be $\frac{1}{D}$ of the unit. Then $\frac{N}{D}$ is defined to be $N$ of these pieces. Under these definitions, I think there is no difference between $N$ times $\frac{1}{D}$ and $\frac{...


11

I like your term. The wikipedia article on fractions also mentions they are called complex fractions or compound fractions. Personally, I dislike the term complex fraction as it is obviously going to be interpreted as things like $\frac{3+i}{2-i}$. I think I call them "fractions of fractions" which is really in tune with your term composite or the ...


9

The student who designed this problem wasn't thinking about the different wholes. IN your students problem, there are 3 different wholes. Anna's flowers - The whole is 5 flowers and $\frac{4}{5}$ are daffodils Beatrice's flowers The whole is 3 flowers and $\frac{2}{3}$ are daffodils The flowers of Anna and Beatrice combined. The whold is 8 flowers and $\...


8

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 ...


8

I do not know of any relevant research. Here are my own not-research-informed ideas. Most people refer to fractions as parts of a whole. If someone says "I lost a fraction of a pound on my diet", you can be fairly certain that they didn't lose $\frac{23}{1}$ pounds. Since the common usage of the word and the mathematical usage differ, it is useful ...


8

I tutor maths as a full-time job, and many of the students I have tutored over the past few years are not fluent in their times tables up to $10$ or arithmetic with fractions, or both. And it's not like I select students with these low abilities on purpose. And when I say they are not fluent, I mean that they are nowhere near as good as me at them (...


8

Í am a physics teacher, not a mathematics teacher, but I would reward the mark. It was a multipart question and you did not show us the other parts. But when I formulate tests I try to have all parts independently of each other, so a mistake in a previous part does not impact the latter parts. Assuming that the parts are independent, the student does have to ...


7

Yes, he will have learnt how to do arithmetic with fractions, equivalent fractions, fractional/decimal/percentage equivalence, reciprocals, and how to solve problems involving fractions in Key Stage 3 - it is part of the National Curriculum, which can be found here. Whether he ought to be fluent in manipulating fractions really depends on whether he will be ...


7

We DON'T need to know the intricacies of UK standards to say yes students are expected to know fractions before algebra. However, your puzzlement over it (like a new concept to you that a kid who fails algebra might be weak on arithmetic!) is concerning. And similarly, you are not the first, by any means, "good at math, but weak at teaching it" ...


7

Student are introduced to fractions as part of a whole. They are then taught that improper fractions can be more than a whole - this is not ideal terminology or helpful for understanding. Improper fraction is a terrible name since it implies that there is something wrong with the fraction. Once student start to do calculations with fractions greater than one,...


6

You should be aware that, in standard developments, the two expressions being considered here are true by definition. The quotient $\frac a b$ is just a shorthand ("syntactic sugar") way of writing $a \cdot \frac 1 b$. For example, here it in Sullivan's Algebra and Trigonometry, Review Section R.1: You'll see that in any standard textbook on ...


6

Normally I avoid semantics, but in this case, they're quite important. Is it possible to understand decimals without understanding fractions? No. Such an understanding is literally impossible. With a strict and literal sense of the word "understanding", no. If you do not understand fractions - and tenths in particular - then the conventional ...


6

Interestingly enough, we do not have such a distinction in France. From the French Wikipedia article about fractions (emphasis mine): Dans l'enseignement français depuis la fin du xixe siècle, la fraction est définie comme le quotient de deux nombres entiers sans contrainte sur la taille du numérateur et du dénominateur (...) In French education, since the ...


6

The use of rulers with fractional inches is the first thing that springs to mind. Like this: The four keys have a width of $2 \frac{11}{16}$ inches at the tops of the key caps. If I calculated a length, and got $\frac{43}{16}$ inches, I'd have to convert it to $2 \frac{11}{16}$ to actually measure it. The "improper fraction" $\frac{43}{16}$ is ...


6

The term complex rational expression (or complex fraction) is commonly used, in U.S. algebra/college algebra texts, to refer to rational expressions where the numerator and/or denominator contain sums or differences of other rational expressions. Note that technically this would not apply to an expression $\frac{a/b}{c/d}$, which has no sums or differences ...


5

Let me start with an analogous question on phil-SE: Teacher: What is 2+2? Student: 2+2 is 2+2 What is wrong with this answer? Now it may seem to be an issue with using tautologies for communication. However as I point out law-of-identity/tautology is a red-herring to address this issue. To see that, let's change the exchange slightly: Q : What is 2+3 A :...


4

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 ...


4

I then stopped for pause and admitted, maybe they're right, but I'm not sure if they would get the mark if they do not fully simplify the fraction or give the answer as a decimal. Life is not about getting points. A person who doesn't automatically simplify 40/50 to 4/5 (or 0.8) is being silly and annoying, or showing a lack of competence. This person may ...


4

Here's another use-case that came up in my college remedial algebra class tonight (and again, this boils down to translations to mixed numbers): Finding fractions in a graph. So the specific example that presented itself tonight was a book exercise: "Graph the equation: $y = - \frac 5 3$". At that point, my students could tell me that this would be ...


4

Straightedge and compass construction is an interesting way to thing about this as an educator. This perspective would not be something you actually take into an elementary school classroom. Rather it is for the teacher to think about this from a higher viewpoint. Thinking about what we are teaching from higher viewpoints makes teaching more fun, and so it ...


4

tl;dr: the answer should be given in the reduced form. Say there are min. 40 students who are taking the exam. Would you want to find the reduced version of the fraction $394.784176044 / 493.480220054$ 40 times? (assuming every time the answer will be given differently but of same complexity) Another issue is that the rational number has infinitely many ...


4

(Adding this because I want to - the first version of this answer is near the end after a horizontal line.) I think that the acceptable answers should succeed in communicating with whoever is reading the answer. And the context (who is reading, what the answer is supposed to tell to the reader,...) should be taken into account. Here $40/50$ is ok, ...


3

As an abstract concept, I don't think they have different meanings. As with many situations, though, different notations can represent different situations in stories. As an example: $N \cdot \frac{1}{D}$ could represent having $\frac{1}{D}$ of $N$ different things, while $\frac{N}{D}$ could represent having $\frac{N}{D}$ of the same thing. We're ...


3

So much depends upon what you mean by "understand" decimals. Think about this in terms of Searle's Chinese Room. Also see this BBC Studios video with Marcus du Sautoy. In this thought experiment, we imagine writing a computer program that gives person A, locked in a room and who knows nothing about Chinese characters, enough instructions to ...


3

I did some research, which you can follow here, that explains the why of the nomenclature we use. Basically, something is proper if it is contained in something else, and improper otherwise. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6. This set excludes 12, because that represents the entire portion, e.g. no fraction occurs. Similarly, in ...


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