57

I'm going to focus on one aspect of the question that I think has not been fully appreciated: How can one best convey to beginners—without algebra—the flipping of denominator fractions... what would convince a novice Many of the ingredients of this answer are already present in some of the other answers to this question, but are rearranged here in a ...


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 ...


40

Questions of this nature were addressed in some part by New Math in the late 50s and 60s. They were also used as a point of criticism by its opponents. Most notable is Morris Kline's book Why Johnny Can't Add in which he writes relevantly in Chapter 1 : Evidently the class is not doing too well and so the teacher tries a simpler question. "Is 7 a number?"...


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 ...


21

The other answers are good, and/but some of those points are reinforced by a diagnostic that has ever-more been important to me: is intention being communicated? And a different, more linguistic point mentioned in other answers: toleration of ambiguity, allowing context (where intent is adequately clear) to disambiguate. Thus, yes, it is perverse to entrap/...


21

I think there's a countervailing issue that the book you're describing is trying to deal with. I teach college students, so I don't know what the particular approach it's taking is age appropriate for a 6th grader, but it is trying to solve a real problem. What I would do about this is introduce them as separate topics, probably spaced far in time. Let ...


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 ...


18

There are many reasons why fractions are so hard for students to learn. Mostly, they're taught gibberish and assessed according to such gibberish. Example 1 You are a 12-year-old student who has learned that "a fraction is part of a whole, such as part of pizza". So when you look at $\frac{2}{3}\times\frac{7}{5}$, you now must multiply pizza ...


17

Here are a few comments, and then an attempt at two succinct answers. In particular, I will try to answer this using a measurement interpretation, and then again with an equal sharing interpretation. I prefer the former, but include the latter for completeness. Comments: Some of the key terms in unpacking this are measurement, equal sharing, and missing ...


16

Instead of presenting "cancelling" as an arbitrary rule (which is often how students have seen it — or at least how they learned it — before), explain it in a way that shows what's actually going on. So, if we have $\frac{5x + 5}{5}$, instead of just "cancelling the 5's" (which sounds kind of arbitrary and mysterious), write out several steps, like: $$\frac{...


15

I find this diagram helpful when relating the two: It comes from a model curriculum unit on Rates and Ratios for 6th graders (you can see them all here after registering), and I have found this particular graphic very helpful with math content professional development with 6th grade math teachers.


14

One reason is that mathematics was not handed down by the gods fully formed and unambiguous. It is a human construction over a very long time and mathematical notation even more so. Any time a notation doesn't "work" for all possible contexts, it's an opportunity for us to talk about this human side of mathematics, and about the pros and cons of notation ...


14

First, I would have them really understand equivalent fractions. There are a lot of ways to write the number represented by the fraction $\frac23$. We can call it $\frac23,\frac46,\frac{20}{30},\frac{-2}{-3},$ etc., etc. Similarly, there are many ways to write the number represented by the fraction $\frac45$: as $\frac8{10},\frac{20}{25},$ etc. Once that's ...


13

I have no experience teaching fractions, but I think moving away from using the divide symbol makes things easier. It doesn't get used at university level (but exponents start being used, so there are still two notations). I would do $$3\div\frac{2}{7} = \frac{3}{\frac{2}{7}} = \frac{3}{\frac{2}{7}} \times \frac{7}{7} = \frac{3\times 7}{\frac{2}{7}\...


12

Manipulation The mixed fraction form makes subtraction and negative values easier to parse. $2\frac{1}{2} - 1\frac{1}{4}$ is substantially easier to read, write, and understand at this stage than either $2+\frac{1}{2} - (1+\frac{1}{4})$ or $2+\frac{1}{2} - 1-\frac{1}{4}$. Students at this age have not encountered the distributive property (or even brackets?...


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 obvious (to me) source of difficulty is that fractions are just plain complicated, more so than almost anything else in elementary education. You have to operate with a pair of numbers, instead of a single one, and you have to keep the order straight. Adding is quite complicated in its own right. Things are further complicated by rules about least ...


11

When manipulating fractions, students quickly get comfortable with the idea that to combine two fractions they have to manipulate to get the denominators the same. Multiplying by 2/2 or 3/3, etc doesn't change the fraction value, obviously. I use the method below to unfraction (<< is there a word for this?) the denominator - $$\frac{3}{\frac{2}{7}}=\...


11

Comparing fractions only works when the whole is the same size. Here's a few examples to get help the 7 year old understand what happens when things aren't the same size: 1/2 will be greater than 2/4 when you compare 1/2 of a watermelon and 2/4 of an apple. 1/2 will be less than 2/4 when you compare 2/4 of a watermelon and 1/2 of an apple. If you take 1/2 ...


11

A US specific answer: The Common Core State Standards define $\frac{1}{b}$ by saying it is one of $b$ equal parts making up a whole $1$. $\frac{a}{b}$ is then defined as $a$ of these. Connecting $\frac{a}{b}$ to $a \div b$ requires some reasoning. For instance $\frac{5}{3}$ of a candy bar means you take your one candy bar, divide it into 3 equal sized ...


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 ...


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{...


10

You can find a discussion of these considerations in: Chapin, S., & Johnson, A. (2006). Math matters: Grade K-8 understanding the math you teach. Sausalito: Math Solution Publications. Specifically, pp. 99 - 131, Chapter 5, Fractions. Let me just quote from Teaching Fractions on p. 131: Fractional numbers are a rich part of mathematics. However, ...


10

I feel like I always post the same thing in these threads, but this again sounds like an issue of blocking vs interleaving. In this case, the textbook may have started interleaving different problems a little bit too early, but generally it should be introduced earlier than feels intuitively right. That's because the algorithms for $\frac{1}{6}\times 90$ ...


9

"Fraction Bars" may be helpful to you. It's a virtual space for exploring bar-shaped fraction models. Fraction Bars Page On that page you'll find a link to launch the software as well as a software guide which includes some appropriate activities. A version of this software has been used in classes to prepare 7th grade teachers in Georgia to (among other ...


9

I would say that $\frac12$ is a number and that "$\frac12$" represents (or names) that number. I should therefore not say things like "the denominator of $\frac12$ is 2", because the denominator depends not only on the number but on its name. I confess, however, to being rather careless about such things most of the time.


9

Not sure about paper references. One reason why people don't understand fractions is because they are seemingly illogical. You score one basket out of three 1/3. A little while later you try again and score 1/2. Clearly you have scored 2/5 shots? In many ways this is the correct answer. So why shouldn't $\frac {1}{3}+\frac {1}{2}=\frac {2}{5}$ People ...


9

You could try keeping 30 as the denominator throughout, that is, observing that $\frac{1}{3} = \frac{10}{30}$ and $\frac{1}{5} = \frac{6}{30}$, so the portion of the class that doesn't study chemistry is $$\frac{10}{30} + \frac{6}{30} = \frac{16}{30},$$ and the portion that does study chemistry is $$\frac{30}{30} - \frac{16}{30} = \frac{14}{30}.$$ Once they ...


9

I think this problem is much easier to handle if you refer throughout to the number of students who study each subject, rather than the fraction, only expressing the final answer as a fraction at the very end. That is: $\frac{1}{3}$ of the $30$ students study Biology — this is $10$ students, because $\frac{1}{3} \times 30 = 10$. $\frac{1}{5}$ of the $30$ ...


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