Also see Dividing Fractions With Pictures!
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Math is a language. Anyone who tells you differently is just, well, wrong. The word “of” in the language of math means to multiply. Once we know this, multiplying fractions becomes a bit easier to understand.
When we multiply fractions, we’re starting with a part, say 2/7, and finding a part of it, say 3/5. Our standard algorithm would tell us to multiply the “tops” (numerators) and the “bottoms” (denominators) separately to get the product 6/35. But why does this work?
Let’s start with a picture of 2/7:
This shaded region represents “2 out of 7 (bars)”
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Now we need to look at the next fraction, 3/5. Let’s break this same picture up into 5’s and take 3 of them…
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So now, what is the area that both fractions share, or the area of the overlapping region? It’s 6 boxes out of the total 35 total boxes, or 6/35.
To see this more clearly, we can look at the below picture and see that the area the fractions share, or the overlapped region, is 6 boxes, and the area of the entire region is 35 boxes. “6 out of 35 boxes are double shaded”.
In this last picture we can see that the area of the double shaded region is 6, or (2×3), and the area of the entire region is 35, or (7×5), which is why we multiply the numerators (2×3) and the denominators (7×5) when we find the product of two fractions.
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This one, too! My students aren’t quite at fraction multiplying yet, but maybe by the end of the year. In the meantime, I’ll enjoy explaining long division to them, especially when I get the questions! Awesome posts!
[...] (Also see Multiplying Fractions With Pictures!) [...]
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The assertion that “math is a language” is commonplace, but making it and claiming that anyone who disagrees is “wrong” strikes me as oddly arrogant. You’ve offered nothing to support the claim, and your one example, that “of” means “multiply” in mathematics is both an example of ENGLISH, not mathematics, and at best occasionally correct. Sometimes “of” indicates multiplication. Other times, it does not. Telling students that “of” MEANS “multiply” automatically is ensuring that they will be confused when they continue to hit examples when it doesn’t.
The problem with trying to reduce problem-solving to a “word-by-word” translation into mathematical symbols is that it’s mindless, just like mindless application of algorithms, mindless calculator use, and any other mindless activity in mathematics. When kids aren’t thinking, they aren’t doing mathematics, and it’s bad pedagogy to encourage them to operate in such a brainless manner.
In the case of fraction multiplication (or taking percents of percents), it might be more clear to say that, for example, 3/4ths of 2/3rds can be read as (3*2) out of (4*3) which is 6 out of 12 (followed by simplifying to 1 out of 2 or 1/2). Yes, it’s implied multiplication in this case, just as 30% of 90% is 30*90 out of 100*100 or 2700 out of 10000 which simplifies to 27 out of 100 or 27%. And yes, the diagram which translates such problems into an area model can be helpful for students to understand why multiplication makes sense in such cases. But let’s help them think flexibly and reasonably about problems rather than handing them mindless crutches (e.g., the execrable “FOIL” that leaves students unable to make sense of multiplication of algebraic expressions that DON’T involve exactly two factors, both of which are binomials). And please: if you want to make the argument that “mathematics is a language” (really? So we learn it naturally, like our native tongue? People walk around “speaking” mathematics? Or is it only or primarily a written ‘language’ just like, um, Sanskrit?), do some investigations into the arguments for and against that assertion. Because I disagree, I’m not alone, and the burden of proof that those of us who don’t accept that claim are “wrong” is on YOU.
You’re pretty angry about this. My students are excelling; my guess is that you’re not even a teacher. Try solving “6 less than the sum of a number and 4 is less than 18″ without understanding of language.
As for your beef with fractions, you’re not unlike my students who hate them. The paper I wrote and posted on here showing multiplication and division of fractions through pictures received an A at Harvard. “of” means multiply. “out of” means division. What’s your point exactly?
The annominity of the internet is great for bringing out the worst in people. I’m sure you’re not angry about this, probably about something else. Bring your anger elsewhere maybe? I’m just trying to help.
Also, anyone can tell a kid how to multiply fractions. It’s the why that’s hard and that you clearly don’t understand enough to explain.
I agree wth you on FOIL, which is why I don’t teach FOIL. Ask next time?
Fractions (check), FOIL (check), language (check). Any other complaints?
Why the hostility? But anyway, check out Frank Smith’s book: The Glass Wall: Why Mathematics Can Seem Difficult. He talks about the language of mathematics and how it is different from a natural language, but a language none the less.