Fractions misconception

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apple2357

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How would you challenge this reasoning about adding fractions?

A student sits a test comprised of two papers. They get 9 out of 10 in the first paper, written as 9/10 and 6 out 10 in the second paper (6/10).
What do they get in total? i.e. 15/20
They conclude to add fractions it is ok to do the following as it works with their reasoning:

9/10 + 6/10 = 15/20

So to add fractions you add the numerators and the denominators...
 
You give me two halves of an apple and I'll pay back with onlyt one half of an apple (if [imath]\frac{1}{2} + \frac{1}{2} = \frac{2}{4}[/imath])?
 
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apple2357 said:
How would you challenge this reasoning about adding fractions?

A student sits a test comprised of two papers. They get 9 out of 10 in the first paper, written as 9/10 and 6 out 10 in the second paper (6/10).
What do they get in total? i.e. 15/20
They conclude to add fractions it is ok to do the following as it works with their reasoning:

9/10 + 6/10 = 15/20

So to add fractions you add the numerators and the denominators...
Click to expand...
The answer of 15/20 is correct for that particular problem, but that's because it is not about adding fractions. It isn't 9/10 of a paper plus 6/10 of a paper, but about 9/10 of the first half and 6/10 of the second half.
 
Dr.Peterson said:
The answer of 15/20 is correct for that particular problem, but that's because it is not about adding fractions. It isn't 9/10 of a paper plus 6/10 of a paper, but about 9/10 of the first half and 6/10 of the second half.
Click to expand...
Subtle distinction!
 
You also can't reduce when you add this way.

If you get 6 out of 8 right on the 1st exam and then you get 6 out of 9 correct on the 2nd exam you got 12 out of 17 correct.

so 6/8 + 6/9 '=' 12/17.

We know that 6/8 = 3/4 and 6/9 = 2/3.
However 3/4 + 2/3 = 12/17 is not correct.

I call the method you describe as a running count.
 
Last edited:
apple2357 said:
How would you challenge this reasoning about adding fractions?

A student sits a test comprised of two papers. They get 9 out of 10 in the first paper, written as 9/10 and 6 out 10 in the second paper (6/10).
What do they get in total? i.e. 15/20
They conclude to add fractions it is ok to do the following as it works with their reasoning:

9/10 + 6/10 = 15/20

So to add fractions you add the numerators and the denominators...
Click to expand...
What are the units of these quantities? Seems to me it's "points". So, is it 9 points or 9/10 points? I would say 9, not 0.9.

9/10 and 6/10 are ratios of correct questions to the total number of questions. And addition for ratios is not defined in general.

I can write any pair of numbers as a/b. It doesn't mean that addition of such pairs makes sense.
 
Adding scores the way the OP noted is not something we don't all do from time to time (and I don't mean incorrectly). As I mentioned, I call this a running count and is a useful method. Maybe you are practicing basketball and you want to know how many free throws you make out of 1000 tries. Assuming these 1000 shots will take you a few days to complete, it is reasonable to use this running count method. Today I made 75 out of 160 shots. Yesterday I made 47 out of 90 shots. Etc.
 
Jomo said:
Adding scores the way the OP noted is not something we don't all do from time to time (and I don't mean incorrectly). As I mentioned, I call this a running count and is a useful method. Maybe you are practicing basketball and you want to know how many free throws you make out of 1000 tries. Assuming these 1000 shots will take you a few days to complete, it is reasonable to use this running count method. Today I made 75 out of 160 shots. Yesterday I made 47 out of 90 shots. Etc.
Click to expand...

So would you say, the misconception is actually about trying to add two running counts using fractions/ratios? So it would incorrect to group running counts together using the notation of fractions?
 
If I was keeping track of how many basketball shots I made, then I would not have a big problem writing that a/b + c/d = (a+c)/(b+d) (for exact values for a, b, c and d). This is fine as long as you understand that you are NOT adding fractions and all the usual rules about adding fractions are not allowed anymore. If you are going to share your work with others, then you'll have to explain what you are doing.

The reason I would use fraction notation is because a/b does mean a out of b, so the fractional notation is the obvious choice to use. You can invent a new symbol for adding these strings of fractions if you like.
 
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