Lots of good stuff going on here. But I don’t think I entirely understand where 1/8 came from, though I get how that gets turned into 5.8.

[I never know whether to include all the mistakes from a class set or just a few. I feel as if it’s helpful to include more mistakes, but sometimes overwhelming. My solution today is to post one especially cool mistake largely, and the others smallerly. Let me know whether that works.]

• This reminds me of those problems involving the number of buses that are needed to travel with a group of kids. How should the students here understand the context of the question? Is it reasonable for them to think of crackers in pieces, or would we be happier with them answering that each person gets 5 crackers and there are simply four left over? Personally, I think I’d be happier breaking those crackers into equal parts and say that each person gets 5 crackers and one half of one cracker. It is frustrating to see the urge to convert fractions into decimals (Christopher Danielson is working on that right now in his decimal institute) Wish I knew how to shake free of the idea that a remainder of 4 means to add 0.4 to your answer (or in this case 1/8 = 0.8)

• A bit of a chicken and egg thing with the turning remainders into incorrect decimals and vice versa.

To go from calculator results to (mostly) incorrect remainders, the underlying assumption would be (if kids were thinking about it much) that all denominators are 10, no matter what is given in the problem to begin with. But I wonder what happens as the situations they are presented with get more complicated. For example, in the problem given here, you have a divisor of 8 and a remainder of 4. If you do the problem on a typical calculator, you’ll get 5.5 – now, we hope that kids know that .5 = 1/2 (though of course many do not). So how do they interpret this result? Each person gets: 5 1/2 crackers? 5 5/8 crackers? The more I think about possibilities for misinterpretation, the more things arise. 5 1/5 crackers (seeing John Golden’s comment)? Or do they map this to 5 crackers each with 5 crackers left over?

I thought I had something useful to add to this, but now I’m less sure. I think that to make the issues that arise truly problematic for students, it’s likely necessary to have some sort of physical or drawn model of the situation. Otherwise, misinterpretations of whole number remainders into decimals and decimal fractions into wrong whole number remainders don’t become obvious and cause no cognitive dissonance.

I think someone needs to be sitting there with 8 piles of 5 crackers each and 4 crackers left over (or a drawing of same), and then consider various numerical ways to express that result.

And then 41 crackers shared with 8 people equally should push thinking further. The calculators says “5.125” and that’s pretty crazy if you don’t know that 0.125 = 1/8th. I guess what we see here is that a lot of kids would claim that 5 r. 1 = 5. 1; 5 r. 2 = 5.2, . . . , 5 r.7 = 5.7. . . . and so working backwards, what would 5. 8 or 5.9 mean? Seems like there are a lot of ways to problematize their misunderstanding. Does that lead to “correct” understanding? I would hope it’s at least a necessary beginning along that path. But so much of this seems to me to stem from having weak understanding of fractions and failure to have a reasonable set of fraction decimal equivalents firmly in one’s head when thinking about problems of this type. Granted, if there were 31 people sharing crackers, I wouldn’t know on demand or sight which remainder is indicated by a given decimal fraction or vice versa, but I could work it out. And part of the reason I can is because I can fall back on familiar examples involving thirds, fourths, fifths, sixths, eighths, etc., coupled with understanding that a remainder of 3 can’t be interpreted out of the context of a divisor. Three extra crackers with 5 people sharing is different from three extra crackers with 8 people sharing.

• These are always interesting to me because they highlight the difference between the students who are doing the division to answer the question and those who use the question to determine that division is being asked for. If you’ve got five for each, four left over is a fine answer. 4/8 shows some comfort with the operation concept. 1/2 shows good sense making. 1/4 means I’ve got this number 4 left and I don’t know what to do with it. (Which could also be the case for good procedure followers with 4/8.)

• I would conjecture that the 1/8 remainder comes from recognizing that the four remaining crackers could be broken in half to make 8 pieces, and each person gets one of those eight pieces. So each person got 5 crackers and 1/8 of the new set of eight half-crackers.

• Julie Campos

If 8 people share 44 crackers equally, each person gets 5 whole crackers and half on one cracker (5 1/2 or 5.5 crackers). I am unsure where the 5 and 1/8 is coming from because the first picture the student wrote 5 1/2 and 5 1/8 which confuses me because they have the correct answer and then another answer that needs to be justified. The answer with 5 remainder 4 is interesting because the student went ahead and gave everyone 5 whole crackers but is leaving the 4 remainder alone and is not going to split them up like the other students. I also see 5 and 1/8 which means these two students must have similar thinking and strategies for solving this problem. I would love to hear the thinking from all the students that gave answers that were not 5 1/2.

• Tiffany Chiang

If there are 44 crackers that need to be shared equally among 8 people, then we know that you can divide 44 by 8, which gives you 5.8, the mathematically correct answer. I can see where the 5 and 1/8 is coming from because when you divide 44 by 8, you have a remainder of 4. The student must’ve broken each cracker in half, and half of each of the 4 crackers would make there be 8 pieces total. This means that each person would then get 1/8 of the half of the crackers that remained.

Besides the first one and the answer that says 5 cookies with a remainder of 4, I am not sure where the other ones are coming from, and would love to hear students’ thoughts on that.

• grace chu

the correct answer for this question is 5.5 or 5 1/2. I also not sure why the students wrote 5 1/8, my guessing is 8 is the total amount of people, so students use it as the denominator. Not sure what grade the students are in, but I feel like they are starting to learn division. Thus, they were still kind of confuse how to write in fraction. When dividing fractions, some students may have trouble writing out the answer in fraction when the answer contain remainder. I was surprised when I saw some students wrote 5 1/4 as the answer. I think that is because the remainder is 4 , and the students thought each person can get 1/4 more crackers. I think the teacher can spend more time to explain how remainder works in fractions, and teach the students how to write them correctly. Also, the teacher can encourage the students to show their working process, so the teacher may have ‘clue’ why the students answer in this way