What should the answer be? What does it mean to multiply fractions? An exit card or maybe not….

“Does this make sense?”  “What do you think the answer should be?”    “Is this reasonable?”   If I had one of those buttons that you press to answer a question or my own version of a magic 8 ball, these would be my answers.   They are the answers I give when a student asks me if his or her answer is right.   They are (some of the) questions I ask when we discuss the solution to a problem in class    These are some of the comments I put on student work.   I think knowing when one has gone astray in mathematics is important and knowable.   I also think that my students have to be taught that making sense of a problem matters.

As our work with fraction/mixed number multiplication draws to a close, I know which students can correctly model the operation, which students can correctly solve a problem using an algorithm, and which students can solve a problem with an algorithm and then fake a model to fit the answer.   However, I wanted to know whether my students are really thinking about their answers and whether they make sense.    I want to know if they even have a sense of what makes sense when they step outside the realm of whole number multiplication.    Today, I gave them this exit ticket.

fraction-multiplication-always-sometimes-never

I allocated more time for an exit ticket than I normally would because I didn’t administer this like I would normally administer an exit ticket.   Maybe that means it isn’t actually an exit ticket.   I don’t know.    Anyway, I posed the first question and had students respond to it and then had a discussion with them before moving on the second question.   I wanted to do this for several reasons.   First, this is the first time my 6th graders have done an Always, Sometimes, Never this year so I wanted the chance to debrief what it means to justify something that is always true, something that is sometimes true, and something that is never true.   I wanted to be sure that they realized giving an example that something is true is not sufficient to justify that something is always true.   I also wanted to take time to expose thinking about what it means to multiply.   What happens when both factors are whole numbers?   Are all whole numbers created equal in this scenario?    I wanted to begin to draw out the misconception that some students have that multiplication always means that things get bigger when you multiply.

After the debrief, I posed the second question.    With the second and third questions, I wanted my students to consider whether all fractions are created equal.   Most students initially only considered cases with proper fractions, but reconsidered their thinking when someone raised the question of “what about improper fractions?”.

I don’t know that these are the perfect questions, but I am happy with the conversation that they produced.   My students thought a little bit deeper, asked a few more questions, and looked at things from a slightly different perspective because they considered them.   That is a good thing.