Picture this - Fun with Functions!

I've learned a lot during my first year of owning a classroom back in 2010/2011 (a quick distinction between that and teaching a class purely following others).

I am unsure if I am remembering it completely right, but my first attempt at creating a large project (as opposed to the many little ones I've done in class) was something called "Picture This - Fun with Functions."

The idea is straight forward, and it's not something new that hasn't been done before.  Students get a picture, and try to match the outlines with various functions.  When it isn't a function, students are asked to break them into several functions that would fit (i.e. a circle could be broken into the top circle and the bottom circle as two separate functions).

Here is the written instructions that was given to the kids at the time.  Feel free to take, change, use...etc.  Let me know if you made any changes.  The majority of the instructions are pretty much about how to use Geogebra for the purposes of the project.  In summary it's basically in two parts.

Part A - Students find a picture and attempt to fit the outlines with functions
Part B - Students provide an analysis of how they've fit the functions

Below is the copied and pasted instructions from the file:

Use piece-wise functions as the main form, and the geogebra functions.  Create an algebraic expression of all the functions you have used.  You must include the following in your analysis:

•          Domain and range

•          Proper representation of at least one piece-wise function

•          Include at least 3 different types of functions (feel free to use functions you’ve never seen before)

•          Explanations for why you’ve chosen certain functions to fit certain parts of your picture.  (You must provide an explanation for each function that you use.)

•          Address any relations that you have chosen to use, and how you have used piece-wise functions to avoid it “not being a function.”

•          Anything else you feel would be suitable for a proper analysis of your picture with functions.  Suggestions include things like algebraic calculations for x-intercept, y-intercept; evaluating stretches, compressions, and transformations...etc.

•          Creativity will be taken into account, but it must contain legitimate mathematics and follow mathematical logic.

There were some wonderful pieces of work that came out of it - even from students that struggled algebraically:

I am using this post as an opportunity to reflect on this project properly.

First, it's probably to clarify what this project is and isn't.

(What I think) it isn't:

• It is not making attempts to relate to the "real world."  I've struggled with attempting to create relationships to the real world for brief period of time.  I think this is prompted by the possibilities and opportunities for engagement that I've witnessed through this type of method.  This was primarily inspired by Big Al who introduced me to Dan Meyer's work a few years ago (and of course, thus sparking me on a journey to reading a ton of awesome educators).  But as I think more deeply about it, there doesn't need to be any restrictions on how we engage the students -- as long as that happens.  Dan's recent series also points to similar ideas (unless I'm reading him all wrong)
• It is not an evaluation to be considered in the same manner as a classroom assessment (I'll probably come back to this point).
• It is not an activity where students literally get their hands dirty.  Students are playing with functions - but as functions are not really tangible objects, there's nothing to really touch... unless you count keyboards and mice/touchpad.
• It is not an activity that allows for meaningful capture of the students thinking process as they are doing it.  So something like this may not apply.  But I will probably come back to this a bit later.

(What I think) it is:

• It is attempting to engage students in a relatively fun way of transforming functions for a purpose (even if the purpose has no immediate uses in "the real world."  Geogebra was/is neat in that students were able to deal with translations of functions relatively easily.
• It is providing an additional opportunity for students to explore function transformations at their own pace - and to their own goals - as well as providing an additional evaluative opportunity.
• It is an ongoing assessment that ran alongside student learning for part of the semester.  This provided students with lots of opportunities to go back to their work and tweak

Now that the above has been clarified... the real purpose of my reflection is not to showcase what it is and isn't.  It is to think about what it should/can/will be.  I think I will separate these into several different areas for improvement.

What it should/can/will be:

Implementation

• Geogebra was nice, but Desmos may have some different benefits to this project.  I probably need to think more on this.  My current thinking is that translations in Geogebra - while it provided some easy access for students that don't understand translations, I am unsure that easy access is what we want.  Desmos forces students to have to understand where the translations are coming from (and then they can set up sliders if they'd like).
• While it was a project that spanned a good chunk of the semester, it isn't long enough.  If it's within the scope of a course on functions, it is likely better to have it run alongside the entire semester.  This would amplified the previous point about what it is - as an ongoing assessment of student learning.
• The instructions need to be modified.  Expectations need to be made clearer...etc.

Assessments/evaluation

• The rubric, while useful on a generic level, could be used to match several expectations (or, in US terms, "standards") at once.  Instead of getting 1 generic achievement score, it should include the variety of expectations that students might venture into.  During the evaluation process, achievement can be identified for several different expectations at the same time.  3 years later, this is coming from a large variety of events that has shaped this suggestion of mine.
• There are opportunities for student discussions here.  It was not an activity that allowed for the capturing of students conversations - it was an activity for developing a student product.  But it can be.  During "checkpoints" students can share their progress, and that conversation can be captured.  There are some excellent opportunities for formative assessments here that I need to tap into (although, without complete control of what I am doing in the classroom due to a recent stress of consistency, this is going to be extremely difficult).  In fact, it is entirely possible to build the entire course (in Canada, our curriculum for a "functions" course is all about functions afterall...)
• It is generally not a good idea to make inferences about a students learning based on products that are created elsewhere.  I think this is partly fueled by the concern that the students may  not be creating the products themselves and therefore not actually engaging in the learning process.  "Should work from home be evaluated at all?" is a topic worth debating over, and I am sure each teacher has come to their own conclusions and compromise on this.  What I am suggesting is an incorporation of student product that contributes to some sort of defense that demonstrates student understanding.  Let me give an example: student finishes the picture; there is a day given to students that allows them to demonstrate their knowledge of function transformations based on their picture (this can be a verbal defense with them providing diagrams, or a separate written assignment taking place during class, or something else...).  At the moment, I am feeling good about that as a potential solution.

And.... that's all the reflections I have time to muster at this point...

What are your thoughts?