Fractions Number & Operations -- Fractions

Division by 1/4

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?

Fractions Number & Operations -- Fractions

Classic Mistakes: Adding Fractions

I arbitrarily designate some mistakes to be “classics,” and this here is one of them.

No need to identify the mistake. It’s right there. But let’s get some wisdom in the comments. What is the significance of this mistake? Does knowing that this mistake is common change the way that you do (or should) teach?

[I didn’t exactly know how to tag this post, but it’s from a 9th grade classroom. For CCSS I tagged 5th grade.]

Doesn't really fit anywhere Number & Operations -- Fractions

Whole numbers and 1

This one is really interesting. We’ve got both a sample of student work, and on the left the student has crossed out \frac{5}{1}. We also have a record of the following student question:

Kid: [Raises hand.]

Teacher: “What’s up?”

Kid: “Are you looking for a 1 or, like, the number 1?”

What is the student’s question, and what does it reflect about what he knows and thinks? How would you help?

[The submission also comes with an answer, yielded by the teacher through further questioning, revealed in the comments.]