Thanks for this, Graham!

What’s interesting about this to me is the mental connection between division and subtraction. I doubt that this kid has anything like an explicit model of division that involves “taking away,” but it makes sense to me that the ideas of subtraction/division would be associated much in the way that addition/multiplication are.

All the more reason to make sure that there’s a robust understanding of multiplication that goes beyond “repeated addition,” no?

## 12 replies on “Using Subtraction to Divide Fractions”

Why do you doubt he has “anything like an explicit model of division that involves ‘taking away'”? It APPEARS from this example that that is precisely what he has, though of course with n = 1, it’s quite possible that it means nothing much at all to him. Have you asked? I started using “repeated subtraction” to explain why division by 0 is undefined back in the early 1990s, right before starting grad school in math ed. A close h.s. friend who had always hated/feared math loved it. When I mentioned it at U of M in front of a prof. of elementary math education, he gave me a “funny look.” ;^) Of course, I was 42 at the time. Maybe that’s what I deserved?

Yeah, I see what you mean. I retract the comment on that post.

I agree with Michael. Division as a repeated subtraction appears to be exactly the mental model the student has to solve this problem. The better question is why did you doubt that in the beginning and what should happen next to help this student consider this problem multiplicatively?

And, is the student prepared to resolve the 3/10 remainder and rename the unit as 4 and represent 3 out of the 4 tenths needed to make a full 4/10 measure of flour? What kind of picture, manipulative, or illustration will assist in this process?

P.S. it’s great to see problem solving like this!

I don’t think that I have a great reason for doubting the awesomeness of this approach in the beginning. I think it was just slopiness on my part — I assumed that the answer was a mistake without checking carefully, and since it was an unfamiliar strategy I made some poor assumptions. #mathteachermistakes

That’s what makes this sample even more awesome. To be humbled by a student struggling to solve a problem is the greatest gift to our teaching practice! I try to take it as a reminder to continually put myself in their place and consider their problem-solving methods before imposing my own. It is a long and constant journey! Thank you for sharing!

Frankly, I’d love it if mine made this “mistake.” Instead, I get students who automatically find a common denominator for multiplication and division–then fail to use one when they add and subtract! They tend to have the algorithms exactly backwards.

I often make common denominators for division.

I remember seeing this kind of repeated subtraction for “what day of the week will it be in 1000 days?” — on a calculator, furiously pushing enter. In a quite mathematically advanced 11th grader studying precalculus with great success. That had the advantage that the remainder was easier to interpret and that you didn’t need to keep track of the quotient.

The student needs to label their units as they work. An answer of 3 batches plus 3/10 cup of flour is correct.

I thing this student has a wrong concept on division of fraction. Since it is division of fraction, you do not need to multiply the both denominators to the same.

You should make 1 1/2 to 3/2 for calculating easily.

(3/2) / (2/5)

You should flip over the 2/5 first when you are doing division of fraction, and multiplication of fraction is easy to do the calculation. like the following:

= (3/2) * (5/2)

Then you can multiply numerator by numerator and denominator by denominator, respectively. You will get:

= 15/4

= 3 3/4

So the answer should be 3 3/4 batches of cookies or 3 batches only.

Hope this answer may help.

Thanks!

does not make sense and it`s wrong.?

wrong!!!!

Just gave this a more careful look, since new comments have made it pop up in my email again.

Everything Jacob does is correct except for handling the remainder, and I’m wondering if others realize why his answer, which is correctly calculated via repeated subtraction, doesn’t provide the same answer as with standard division of fractions. Similar errors occur with decimal fractions when students have to interpret remainders.

Jacob does repeated subtraction correctly and ascertains that there are 3 full batches possible with 1 1/2 cups of flour and 2/5 cups of flour per batch. But he’s left with a remainder (3/10 of a cup) and simply appends that to the 3 complete batches. And therein lies his error. Remainders in this context are what’s left over as a fraction of the DIVISOR. Having converted halves and fifths to tenths (which is absolutely necessary if he’s going to solve this by repeated subtraction), Jacob forgets what the remainder means: 3/10 of a cup as a fraction of a whole needed for a batch. In this context, that whole is . . . 4/10. So he has 3/10 out of a needed 4/10. Hence there is enough for 3 and (3/10) / (4/10) = 3 3/4, and that’s what we calculate more efficiently with the standard division of fractions algorithm.

We always want the remainder to be either a whole number, as we get with something like 10 divided by 7 = 1 R 3, a decimal, or the remainder divided by the original divisor. Had Jacob realized/remembered that, he’d have been fine. He was right to get a common denominator: using repeated addition or subtraction forces that choice. Where he went wrong, unsurprisingly, was in how he interpreted the remainder.