A handful of glaring misconceptions stick out to me in this student's piece of work. First of all, he says the factor a factor of nine. In this problem, we are looking for factors of 18. Next, he comes up with various ways to arrive at the number 18 using multiplication. This shows me the student does not have a clear understanding of what a factor is since he does not associate it with multiplication. The student also may not be able to distinguish between addition and multiplication.
The other root to this student work could have sparked from a prior discussion with the class. Students were asked to notice patterns in multiples, and for the multiples of nine we were talking about how the digits in any multiple of nine add up to nine. The student could have been confused when we talked about adding the digits, showing me he did not understand this pattern in nines.
The student could benefit from reviewing different terms in math. He seems to be confused about what a factor, multiple and product is. If he is confusing addition and multiplication sentences now, I wonder how he will do with properties of multiplication- will he confuse those with properties of addition? The student needs to review and gain better understanding of simple multiplication facts, recognizing the patterns of multiplication we have been working on.
Yes; clearly, the student is confused as to what a "factor" means (not an uncommon problem or student misconception). But also think about the fact that he is using some "definition". You might strongly consider asking him, "What is a factor?" and have him explain. This will give you insight into what he is thinking here, as well as what types of opportunities or conversations might help scaffold his understanding towards the "textbook" definition.
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