If you have to ask the question, then the answer is “yes.”
Great work on the student's part. I would give 90%, withholding the 10% for missing explanation. Communication is important.
"if we asked students which of their classes feels most concerned with rightness and wrongness, most concerned with precision, and least concerned with their personal subjectivity," then I hope to god most of them answer math. why would we want to undo this? Why do you associate requiring students to do it the right way with the idea that getting to the right answer is important? this seems quite a logical fallacy.
I agree in this case that the student is showing how to use the Pythagorean Theorem to find distance. The distance formula is just a mechanization of the Pythagorean Theorem. Other situations you are assessing the standard. A 5th grade response to an 8th grade question doesn’t show mastery of 8th grade standards and may point to a lack of algebras thinking.
Re: tasks explicitly getting at understanding of the distance formula. Keep it simple! Ask them!
"Describe why the distance formula works. Feel free to use sketches or representations in your explanation".
"Gina claims that she doesn't need to bother with the distance formula because she knows pythagoras's theorem. Explain what she means".
Task to test more specifically their ability to use the distance formula.
"There is a circle defined by the set of points (x,y) such that the *distance from (x,y) to (5, -2)* is equal to 4." Write that as a number sentence using the distance formula.
My default tends to be to follow the written directions I provide. Sometimes my directions could use improvement, so I don't want to fault a student for my poor directions. Based on what I see in the image I would give the student full credit. Seems the student applied the formula, but did not show the reader how they got there which is a valid point, but is the point appropriate in this case.
Dan's question about how to provide students an opportunity to show their understanding of the distance formula is well asked. My first thought would be to have students create right triangles then use a coordinate grid to measure the sides. I wonder what students would do? I can imagine a discussion along the lines of "Is there anything that might be done to make our triangles work 'nicely' with the grid?"
This student did in fact show plenty of work. They calculated the difference in the y values and even labeled that as the height of the triangle. Same for the x values. Why would they need to write down that calculation when it’s easy to do in your head. Calculating the hypotenuse using Pythagorean theorem mentally is not that difficult either. In fact, I often finish that lesson by getting my students to do the distance between two points mentally. They enjoy the challenge! Then again I loathe using the distance formula! We derive it but I’d rather they didn’t use it. This kid gets it! And I’m guessing they never once reached for their calculator. I am more concerned with the student who pops points into the distance formula. Do they really understand?
As others pointed out on Twitter, “the distance formula” IS Pythagoras. Why would you create a distinction?
I absolutely agree that students should generally not be penalized for using alternative methods (and I really try to *almost never* insist that a problem be done my some specific method, not to mention the fact that the distance formula and the Pythagorean theorem are not actually different methods!)
That being said, math is not just about solving problems but *communicating solutions* to problems, and if the expectation has been set in the class is that reasoning should be clearly shown, this answer might be a little lacking.
For example, would one of this student’s classmates be able to look at that answer and understand what was happening? I’mnot so sure. (This is the standard I usually tell students to consider when they ask “how much” work or justification to show.)