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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 ...

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 ...

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 ...

6

I appreciate you raising the question of improving the teaching of fractions which is certainly needed. I don't feel optimistic about your suggestion. I taught gifted elementary math students for over 25 years. I have a degree in math from MIT. I have no sense of how I would present your suggestions to my very gifted students. The elementary classroom ...

6

Typically, it is better to introduce mathematical ideas after the need for that new idea is obvious. For example: How should 2 children share 8 pieces of licorice? How should 4 children share 8 pieces of licorice? How should 3 children share 15 pieces of licorice? How should 3 children share 2 pieces of licorice? And why is this different from the prior ...

6

I would say that this book is trying to get students to think about what the calculations mean, rather than simply execute an algorithm, and I strongly believe that is something that should be done right from the beginning, not saved for later or for more advanced students. I cannot tell you how many students I have worked with as a private tutor who, if ...

5

I was surprised to see such an old question with so many answers, all of them algebraic. Here is a proof (almost) without words that $1/\frac{a}{b} = \frac{b}{a}$, which I believe is the crux of the question. It uses the definition of multiplication as the area of a rectangle with the given side lengths.

4

The ratio $a:b$ and the fraction $a/b$ are generally not synonymous and should not be treated as such, at least without making clear their interpretations. In many contexts, $a:b$ corresponds to the fraction $a/(a+b)$ or $b/(a+b)$. For example, if one cuts a pizza into $6$ slices, one of which has anchovies, the fraction of slices with anchovies is $1/6$, ...

4

7? I work in the math dept at a local high school, and students 14-18 struggle with fractions. I've found 2 things that help. Money Math Somehow, the same student struggling to take 1/2 of 2-1/2 can easily tell me that half of \$2.50 is \$1.25. For your son, 1/2 of a dollar is 50 cents. No need to stack the pennies, he knows (right?) that 50 cents is 2 ...

4

The earliest mathematical insight I remember from childhood is that the word "of" almost always means "times." Half of a dozen $= \frac12 \cdot 12 = 6$ Three-fourths of a mile $= \frac34 \cdot 5280$ feet $= 3960$ feet. I'll take 6 of those thousand-count boxes $= 6 \cdot 1000 = 6000$. I remember feeling like I had secret knowledge that no one ...

3

I would say the canonical answer for what constitutes 'simplified as much as possible' is whatever the exam board says it is. 'Simplify' isn't a mathematical function. It is a pedagogical instruction trying to require students to make use of a selection of mathematical equivalences that they are expected to know. However, it is too vague a term to make ...

3

The word "of" means $\times$. For example "half of $a$" means $\frac12\times a$. Note that the product of $a$ by a number isn't necessarily less than $a$ ; it'll be less than $a$ iff the number is less than $1$ ; in that case we can write the number as $\frac{m}{n}$ ($m<n$) so that $\frac{m}{n}\times a$ means to divide $a$ into $n$ equal parts and take $m$...

3

I'll give a brief defense of the somewhat conventional view. If I understand history correctly, part of what made Newton's advance in calculus possible was the (then) recent introduction of the decimal expansion of a real number. Calculation to arbitrary precision was something that everyone could both do and easily communicate with the introduction of ...

3

With regard to this question: Any ... books I can buy which can help me to explain this? Here is a specific book recommendation, which is a source that I have personally found helpful: National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5. ERIC. Here is the ...

3

OP (@NotAgain): "Any ideas or any books I can buy which can help me to explain this?" May I suggest games (rather than books)? For example, BrainPop's Refraction:           Or the Fraction Game:           Or MathGamesFractions:           or 7 Fun Fraction Games for Kids, or ...

2

This is the only way I've ever seen it taught or read it described in textbooks - except in my classes. I just teach students to multiply the numerator and denominator of the first fraction by the denominator of the second and vice versa. No prime factorizations, least common multiples, etc. are necessary. This is based on the idea that you don't need the ...

2

I'll mention a very simple idea which, it seems, hasn't come yet: get some intuition from addition and subtraction, to which multiplication and division are analogous. I'll run through it with the divide-by-2/3 example asked about. We can intuitively see that subtracting 2-3 is the same as adding 3-2, because subtraction reverses addition. One needn't go ...

2

The numbers in a fraction are different parts of speech. Denominators are nouns. Numerators are adjectives. Denominators say 'what' Numerators say 'how many' This becomes somewhat more apparent in words. I have three quarters of a pie. But you have to take it a step further: A fraction needs a prepostional phrase: It's always a fraction of something ...

2

One can only see that this is true just as one saw what the symbols $1,2,3,\cdots$ meant. What's happening in this case is that we are seeing a ratio. Suppose we want $n/100$ of $Q.$ Then you may explain this to them as Divide $Q$ into $100$ equal parts, and take $n$ of such parts of $100.$ That is, we're taking $n$ of something. But this simply means we ...

2

There is a theorem which says that it is impossible to decide the equivalence of two elementary functions syntactically. So there is not, and cannot, be a uniquely defined "simplest form" for a given expression. http://inst.cs.berkeley.edu/~cs282/sp02/readings/caviness.pdf

2

It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I really like it. https://www.wolframalpha.com/pro-for-educators/ Also, Math.com has a worksheet generator, which allows some specification of fraction use and ... 2 As for research on fractions education... there's a TON. As for research on fractions education that rigorously measures cause and effect through randomized controlled experiments and long-term follow up... there is, I think, little to nothing as of 2010. And, since the field of education rarely uses that kind of research design, I doubt there is any need ... 1 Something like this? $$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$ $$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \left({y \over x} + {x \over y}\right)$$ $$c. \left({n \over m-n} + {m \over {m + n}} \right) \times \left({m^2 \over n^2} + {n^2 \over m^2} - 2\right)$$$$d. \... 1 7 years old seems quite young, so my answer is not specifically thought for this age (it is much inspired by a recent twitter thread about these questions for 12 yo), but I hope it can still help. The example you give shows precisely how using "real world" interpretations can be misinterpreted, therefore I do not think that coming up with more real-world ... 1 Relation between division and fraction have to be understood to understand the relationship between fraction and ratio: 1. Division: finding how many times of divisor in the dividend, or the value of a part 1/5 means 5 is 0.20 times in 1; value of one part is 0.20. 2. Fraction: It is how many times the denominator in numerator or how many parts of ... 1 I have no teaching experience so I don't know the answer. Here's a guess on a method that will work. Take the problem "What is$\frac{3}{4} \times \frac{5}{6}?$" Then explain how once you know the answer to$\frac{3}{4} \times \frac{1}{6}$, you can then use that to calculate$\frac{3}{4} \times \frac{5}{6}$. Then you could ask "How do you calculate$\frac{3}{...

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