Check out my previous post for an introduction to my school and idea for where this came from. Student name was changed for privacy, and some spelling/punctuation errors were fixed (but I’m sure not all). Also, since we are writing in real time, there are several places where I posted something after the student did that would probably make more sense if it was posted before the student. I did not alter these situations, but if you have a question on anything, please ask.

Chad Lower:Hello Student. How may I help you?

Student:I have a math question.

Student: The following set of coordinates most specifically represents which figure?(-4, 5), (-1, 7), (1, 4), (-2, 2) (5 points) Parallelogram Rectangle Rhombus Square

Student:I know that it is either a square or rhombus, but I don’t know which one

Chad Lower:Student, how do you know that part of it?

Student:I graphed it on graph paper and the figure had 2 pairs of parallel sides and all right angles

Chad Lower:how do you know they are right angles? They may look it, but may not be exactly right angles

Student:Oh I guess I don’t know if they are right angles

Chad Lower:okay, so that is one part we need to check.

Chad Lower:(and checking that will also verify if the opposite sides are parallel)

Chad Lower:the other part is the distance of the sides

Chad Lower:you can have 4 right angles, but then it could be a rectangle or a square

Student:yes

Student:so should i do the distance formula?

Chad Lower:exactly!

Chad Lower:for the length part anyway.

Chad Lower:you will need a different tool to show right angles

Student:they are all equal sides

Chad Lower:oh cool! so then we can narrow down to square or rhombus

Student:yup

Student:what should I use to find the right angles

Chad Lower:so think “perpendicular”

Chad Lower:how can I tell if two lines are perpendicular to each other?

Student:they make a right angle when they intersect

Chad Lower:okay–try this related question — how can you tell if two lines are parallel?

Student:can I use one of the proofs to prove perpendicular lines

Student:they never intersect each other

Chad Lower:so would you draw them forever and ever and see if they ever touched?

Chad Lower:I don’t think a proof will be helpful for this question.

Student:okay

Student:I guess you could

Chad Lower:that would take a long time

Student:but that wouldn’t work out

Chad Lower:and you would never finish — right

Student:hahah right

Chad Lower:so is there another way we could check if they were parallel?

Student:you can find their slopes?

Chad Lower:BINGO!!!!

Chad Lower:so now go back to the previous question. How can you tell if two lines are perpendicular?

Student:the slopes would be opposite reciprocals

Chad Lower:and that is what you need to prove or disprove to determine if it is a square or rhombus

Student:yes thank you!

Chad Lower:(and by finding slopes, you can also tell if opposite sides are parallel)

Chad Lower:but if the side lengths are the same, we get that as a bonus

Chad Lower:without needing to prove it separately

Student:so if they are all right angles, does that mean it is a square?

Chad Lower:correct

Chad Lower:since we know the side lengths are the same

Student:okay thank you!

Chad Lower:My pleasure, Student!

Chad Lower:Anything else today?

Student:nope!

Chad Lower:Sweet! Be sure to tell your teacher you stopped by the Live room. Thank you for your effort

So when I reflect on this question in hindsight, there are some things I like about it. I never did any of the number crunching. In addition, the student came up with the ideas for what to do next; I merely verified if the idea was correct.

If I could do it again, I might not be as blunt about her ideas being correct–maybe ask her to justify why she thought that might help (like the idea of the distance formula). I also didn’t do the problem myself; I only relied on her answers and assumed them to be correct. If she made an error somewhere, then I verified that her incorrect answer was correct. This one doesn’t bother me as much as the first, but I know in the past I *have* caught student mistakes, so there could have been a mistake here as well.

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