Above are three different answers to 4 divided by 1/4. What understanding stands behind each of those wrong answers? How do you teach division by fractions?

I know it’s not what you’re highlighting here, but I really want to understand what the first person was thinking for 1a.

I’d tell something like the following story:

The kid sees multiplication of fractions which connects to flipping the second fraction and multiplying. So he flips and multiplies, and then multiplies incorrectly, since 60 and 80 are similar results in this context. Then he recants, realizing that 20 / 4 might be the more appropriate operation. He still feels tied to the 1/something result, so he goes on with that, but the size of the number 5 in “1/5” indicates that he isn’t really feeling the numerator. With a moment’s more thought, he convinces himself that 5 really is the right answer and he just goes with that. The equals signs are being used in appropriately as just “the next stage in thought,” which is plausible to me (I see that all the time, and I even fall into that habit myself.)

So would you have given credit (as it looks like this teacher did)? Just curious.

I don’t like having that sort of argument with kids, so I tend to give them points for right answers, no matter how they were derived. I figure that my grading system will catch people who don’t get stuff but limped their way towards an answer, and I don’t have a non-authoritarian comeback to a kid who says “yeah, I was thinking about it in the right way.” (And I don’t like requiring that kids write things in a clear way, which is another way to release this pressure.)

… and that is why human teachers will never be replaced by algorithms.

Looks like the kid has operations pretty consistently confounded (which I see often, too — the person who said that the value of 2 in 625 was 4, because 2 x 2 is four…)

I introduce division by fractions by starting with division of whole numbers…. with a dozen eggs; specifically, the plastic ones that you can take apart. Walk through 12 divided by 1, 4, whatever… then once they’ve remembered what division is, to show the pattern (with the eggs): 12 /12 is 1…. one group of 12… two groups of six, three groups of four, four groups of three, six groups of two, 12 groups of one… but always connecting back to the divisionproblem language, ’cause they can totally grok the grouping but consistently forget that that’s what division means.

And then… 12 divided into groups with 1/2 an egg… and voila! 24 is the answer… dividing *by* something smaller than one means you’ll get a bigger answer.

Plastic eggs- good idea! Thanks, I’m getting some now.
Why did the first person not get credit for its , I wonder. I wish the student had shown some work.
Why did the second person not get credit for the last question? This grading is pretty inconsistent.
I feel sorry for the third person, who did the difficult one ( okay,did not simplify the fraction), showing the work, then sadly multiplied 4×4 to get 8 instead of 16 (but showed their work) and then apparently did not get credit for the last one, for some reason we don’t know.
I like a simple grading scheme with 1 point for showing correct, self-consistent work and 1 point for the correct answer. No points for what someone might have been thinking at the time. (I’m currently using a rubric which includes a point for the correct use of the equals sign, in 10th grade. It’s hard to undo this opposite knowing.)

What I don’t understand is how the first example in the first picture could have been marked correct!!! That student was conscientious enough to show his/her work, and whoever marked the paper, didn’t even take the time to look at it. Is this an example of students marking each others’ papers and not being able to distinguish between a right answer and a wrong procedure?

I can explain this one. The kid was checking his answer against an answer key for the quiz, ala Noschese.

## 10 replies on “Division by 1/4”

I know it’s not what you’re highlighting here, but I really want to understand what the first person was thinking for 1a.

I’d tell something like the following story:

The kid sees multiplication of fractions which connects to flipping the second fraction and multiplying. So he flips and multiplies, and then multiplies incorrectly, since 60 and 80 are similar results in this context. Then he recants, realizing that 20 / 4 might be the more appropriate operation. He still feels tied to the 1/something result, so he goes on with that, but the size of the number 5 in “1/5” indicates that he isn’t really feeling the numerator. With a moment’s more thought, he convinces himself that 5 really is the right answer and he just goes with that. The equals signs are being used in appropriately as just “the next stage in thought,” which is plausible to me (I see that all the time, and I even fall into that habit myself.)

So would you have given credit (as it looks like this teacher did)? Just curious.

I don’t like having that sort of argument with kids, so I tend to give them points for right answers, no matter how they were derived. I figure that my grading system will catch people who don’t get stuff but limped their way towards an answer, and I don’t have a non-authoritarian comeback to a kid who says “yeah, I was thinking about it in the right way.” (And I don’t like requiring that kids write things in a clear way, which is another way to release this pressure.)

… and that is why human teachers will never be replaced by algorithms.

Looks like the kid has operations pretty consistently confounded (which I see often, too — the person who said that the value of 2 in 625 was 4, because 2 x 2 is four…)

I introduce division by fractions by starting with division of whole numbers…. with a dozen eggs; specifically, the plastic ones that you can take apart. Walk through 12 divided by 1, 4, whatever… then once they’ve remembered what division is, to show the pattern (with the eggs): 12 /12 is 1…. one group of 12… two groups of six, three groups of four, four groups of three, six groups of two, 12 groups of one… but always connecting back to the divisionproblem language, ’cause they can totally grok the grouping but consistently forget that that’s what division means.

And then… 12 divided into groups with 1/2 an egg… and voila! 24 is the answer… dividing *by* something smaller than one means you’ll get a bigger answer.

Plastic eggs- good idea! Thanks, I’m getting some now.

Why did the first person not get credit for its , I wonder. I wish the student had shown some work.

Why did the second person not get credit for the last question? This grading is pretty inconsistent.

I feel sorry for the third person, who did the difficult one ( okay,did not simplify the fraction), showing the work, then sadly multiplied 4×4 to get 8 instead of 16 (but showed their work) and then apparently did not get credit for the last one, for some reason we don’t know.

I like a simple grading scheme with 1 point for showing correct, self-consistent work and 1 point for the correct answer. No points for what someone might have been thinking at the time. (I’m currently using a rubric which includes a point for the correct use of the equals sign, in 10th grade. It’s hard to undo this opposite knowing.)

What I don’t understand is how the first example in the first picture could have been marked correct!!! That student was conscientious enough to show his/her work, and whoever marked the paper, didn’t even take the time to look at it. Is this an example of students marking each others’ papers and not being able to distinguish between a right answer and a wrong procedure?

I can explain this one. The kid was checking his answer against an answer key for the quiz, ala Noschese.