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a High School Algebra Teacher
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I love love love Edulastic. You can create questions with so many options (multiple choice, multiple select, drag and drop, matching, graphing, open ended, drop downs, highlight, etc). As they take the assessments, you watch their dashboards in real time and see who is on what question, how long they are taking, if they’re getting it right or wrong, etc. Then it is auto-scored for you, but you can change the points and give students feedback on each item. There are premade questions (called “items”) but I like making my own.
For written work, I have them upload a photograph of their work. So far I have been telling them to upload it to Google classroom, but there IS a question format in Edulastic in which they upload a file… I just haven’t used it yet. I bought a pen tablet so I can essentially write on their photo page, digitally, the way I would with a normal pen (been using DocHub for that).
I am in NJ, and our state test is digital, so I’m trying to convince myself to allow them to write all their algebra (equation solving) out in an open-ended text box, they way they would on the state exam... Meh. I think the benefits of hand written work is so valuable, especially now in this virtual world... so I’m still sticking to photos of work for now. Eventually I will probably shift to text boxes (Edulastic has a built in equation editor, so they have all the symbols needed for nonlinear equation solving). 
Also everything I described is totally free (!!) but you get some additional testing bells and whistles for $100 per year.
I assign video assessments. My students record themselves solving a maximum of 5 problems using Screencastify. You see all of their work because they write on the screen and explain as they solve.
In an ideal world I am thinking just like you. In reality, that is just not going to work. I decided to make life easy. Multiple choice google forms. Nothings better than self graded exams.
If your school won’t purchase Mastery Manager or a program that’s similar
Schools are using Zoom on one device to monitor combined with Peardeck or Classkick. Pear deck is great as it will download each students work straight to a google drive folder as well
GoFormative, Quizizz are other options. Desmos Teacher is another option to consider as well.
I have used Jamboard. It is a pain to grade but they can show their work or choose to take a photo and attach it
I don’t teach high school, but I do teach math (4th grade) virtually at the moment. I have been having success with having my students write and show their work on paper and upload pictures of it.
High school students would be much better at it, but the vast majority of my students do a fantastic job.
Google Classroom
Check out Kami app
virtual assessments are a fantasy. You cannot get good assessments. accept that, and you will be better off.
Delta Math for practice and assessment. Not the perfect solution, but best we can do for now.
Same here!
I have them write the work on paper and upload a photo to Google classroom. You can also use Google slides and have them insert a photo of their work.
Flipgrid is a great way to get to hear student voice as they explain a problem.
1) Solve the problem complete on paper, in Google docs etc.
2) Write the script for HOW you did the problem.
3) Upload a photo of your work to Flipgrid and explain HOW you did the problem.
4) Take a selfie and click "submit".
I use photos of hand-written work and rarely have student create a flipgrid video where the take a photo of their written work and explain it verbally.
I love GoFormative!!! It is super easy to use and I can see student work as they write, they can show work or use short answers, multiple choice, and other types of questions. When graphing I can edit the show your work space so I can include a number line or coordinate plane and the students can graph points or draw lines with the line tool.
Free? Paid?
We use a LMS called canvas. They can take a test on canvas and also submit their scratch work to be graded.