Misconceptions surrounding fractions are so well-studied that I feel a bit ridiculous sharing anything about them. Anyway…
I was chatting with this kid who was having a bunch of trouble with written fraction notation. She had been correctly solving problems that involved language such as “shade in four out of seven pieces” or “divide this shape into eighths,” but got stuck when she reached a problem that asked her to “shade in 4/6 of the shape.”
Alice: Oh, so that’s 5.
Me: Can you explain why?
Alice: Because it’s not six sixths.
Me: So, not quite.
Alice: Oh, it’s 2. Because that’s 6-4.
Alice: Or it’s 10?
Alice: I’m really confused here. What’s the answer?
There’s no puzzles or misunderstandings here. Alice thought that the fraction symbol was an operation between the numbers 4 and 6. And of course she did. Every other time that she’s seen two numbers and a symbol before she’s been asked to produce a third number. This is new ground for her.
I’ve been taking the advice of Brilliant Commenters Fawn, Jenny and Avery and using the language of “out of” to bridge the gap for this kid.
3 replies on “Fraction bar as an operation”
This post reminds me that Cal mathematician H.H. Wu has advocated that THE model for teaching/learning fractions is the number line and that teachers do a disservice to children by not focusing on fractions as points on that line. I won’t tip my hand quite yet as to my thoughts on his position, but let me ask this question: if we define fractions as points on the real number line, then does the fraction a/b denote “a iterations of size 1/b from zero, with positive fractions going to the right from 0 and negative fractions going to the left from 0?” Not trying to give Wu’s or anyone else’s formal definition; just thinking aloud about what his model would mean ‘intuitively’ to kids, assuming of course that it doesn’t confuse them. . . or are there other reasonable ways to interpret fractions on the number line?
That said, do you think your student would benefit from this model? I’m perfectly comfortable, by the way, with the “a out of b” interpretation for rational fractions. Not quite sure what we do with pi/sqr(2) and the like.
I am thinking out loud…
“a out of b” is a perfectly good interpretation when integers are involved. But isn’t the fraction symbol *also* an operation between a and b? The division operation. I always regarded it as such (among other interpretations) and I found it useful. So if the student has a tendency to regard it as an operation, why not lead her this way too. Sure, for exercises “shade 4/6 of the shape) the division interpretation if the least helpful, and it takes too long to get the answer. But it is still valid, and given that it is hard to get the answer, the student might be motivated to explore other interpretations.
In a way, won’t we soon be asking for two numbers here to become two more numbers? I am thinking of simplifying where 4 /6 becomes 2 /3. That’s when the confusion amplifies, doesn’t it?