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.

Interesting read. I am at an alternative school and we run things much differently than a traditional school, but I think you have me beat for non-normal teaching duties 😀. I am excited to see the focus on the student doing the hard lifting in the online classes. I’ve had some bad experiences with math online, so I’m glad to read a good one.

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My first public school job was at an alternative school. I know in some states, the “Alternative” schools is where they send the G&T students. Not where I worked in Detroit Public Schools. In Michigan, the alternative schools was where the students who weren’t able to be successful in traditional high school were sent. Most of our students were there due to truancy or because they were on probation. Judge said “go to jail or go to school.” Many of them ended up in our halls.

Which type of alternative school are you at?

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Similar to your your example. Truancy cases, really far behind on credits or students expelled for drugs/violence. We have some that choose to come for the small classes or flexible environment and we are working on creating a community so students want to be at instead of as a last resort, but we are known as the Last Chance School. Most of our work is grounded in trauma sensitive learning. Our kids are amazing, they just have lots of life challenges I couldn’t have imagined at 16.

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