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47

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 ...

36

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?"...

18

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/...

16

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: $$\frac{... 16 I think there's a countervailing issue that the book you're describing is trying to deal with. I teach college students, so I don't know what the particular approach it's taking is age appropriate for a 6th grader, but it is trying to solve a real problem. What I would do about this is introduce them as separate topics, probably spaced far in time. Let ... 14 One reason is that mathematics was not handed down by the gods fully formed and unambiguous. It is a human construction over a very long time and mathematical notation even more so. Any time a notation doesn't "work" for all possible contexts, it's an opportunity for us to talk about this human side of mathematics, and about the pros and cons of notation ... 14 I find this diagram helpful when relating the two: It comes from a model curriculum unit on Rates and Ratios for 6th graders (you can see them all here after registering), and I have found this particular graphic very helpful with math content professional development with 6th grade math teachers. 14 I have no experience teaching fractions, but I think moving away from using the divide symbol makes things easier. It doesn't get used at university level (but exponents start being used, so there are still two notations). I would do$$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

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 ...

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 ... 10 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?... 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 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 Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that \sqrt{2}/2 should be preferred to 1/\sqrt{2} struck me as unmotivated when I was a student, and now seems to me problematic to motivate. The situation is different with respect to writing rational numbers or rational ... 9 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 slices by ... 8 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, ... 8 "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 ... 8 This depends on what meaning you give to fractions. One approach is to give fractions the meaning of shares or portions of some unit e.g. a rectangle. Multiplication is mostly equal to the word of. Examples: \frac{5}{8}=\frac{5}{8}\cdot 1 are five eights of the unit. \frac{2}{3}\cdot\frac{3}{4} means two thirds of \frac{3}{4}. Show a rectangle, ... 8 This is strictly context dependent. Consider the following sentence: Is banana a fruit? Well, is it? The answer, of course, depends on what exactly you're talking about. It can be interpreted in two ways: 1. Is the physical object known as banana a fruit? In this case, yes, it obviously is. But consider this interpretation: 2. Is the string of ... 8 First, a disclaimer: I am a mathematician, and not a math educator (at least, not beyond tutoring, and teaching algebra, statistics and some calculus as a grad student); thus, my answer is going to be colored by the experience of someone who has learned a lot more math than I have taught. The answer depends on what you mean by "higher math". If by higher ... 8 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 ... 8 I suspect the main trick will be going from division by unit fractions to division by non-unit fractions. I would begin by considering that you have 4 groups of a certain size, and you want to know how many groups you'll have if you make a new set of groups \frac{1}{7} the size of the current groups. This can be simplified further by considering each group ... 8 I would say you're doing your student a disservice if you were to seriously disallow a negative denominator. A fraction is simply a ratio of two integers (where the denominator is not allowed to be zero). I disagree with @yoniLavi that we never need such fractions. Since division by negative numbers makes sense, such a fraction with a negative denominator ... 8 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 ...

8

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 $30$ ...

8

We have two cookies. We divide them into pieces of 1/2 cookie each and end up with four pieces. Thus 2 divided by 1/2 equals 4. We have two cookies. We take 1/2 of the collection which is one cookie. Thus 2 multiplied by 1/2 equals 1. Each of those examples can be criticized. In the first example, one could claim that it shows that 2 divided by 4 equals 1/...

7

If you have some apples and chop them into seven pieces, how many pieces do you have? Well, seven times as many as you had apples, of course! What if you pair those pieces two-by-two? Well, okay, you now have half as many pairs. ("X divided by Y" means "how many Ys can you get out of what you started with".) (Did it matter that we used seven? Well, no, ...

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