I’ve been teaching a unit on ‘quadrilaterals’, which I would describe as investigating the relationships between the different types of quadrilaterals – parallelograms, rectangles, squares, trapezoids, kites, etc. One of the AZ state standards is to ‘use the hierarchy of quadrilaterals in deductive reasoning’, which means being able to answer questions like ‘A rectangle is (always, sometimes, never) a square’.

That’s what the state thinks. I, on the other hand, knew at the beginning of the unit that I wanted my kids to be able to answer the following problem: “A = (3, -1), B = (1, -3), C = (7, 3) and D = (4, 3). What kind of quadrilateral is it? Justify your answer – be sure to give evidence about why you believe you shape is what it is, as well as why your shape isn’t something else”. As I was teaching, I also realized part of this unit is teaching how to understand a hierarchical relationship used to classify things – ie the Kingdom/Phyllum/…/Species classifications in biology, or the organization of the countries of the world (The 5 continents, then the many countries, then within each country there are territories/provinces, and then individual cities/towns), or object-oriented computer programming (which they won’t care about and I can’t use as examples in class, but man is this a useful skill to have if any of my kids become Java programmers). The point is: there is a method of organizing and categorizing objects such that they split into defining categories, all of which have a common property or defining characteristic – and, anything underneath these categories have all the properties of the objects above them and continue to become specialized. It is useful, in my opinion, to make this method of organization explicit and teach my students how to understand this method of classification.

The Coordinate Stuff: I just love the problem of being able to give students 4 points and ask them what the shape is. I actually just taught this stuff today and it went great – we had finished talking about the definitions and properties of the quadrilaterals we care about and I told them “You are now an architect designing a building. All the rooms in the floorplan must be rectangular – otherwise the rooms just feel weird. The drafting program you’re using only lets you create shapes if you give it the vertices of the polygon. You’ve type in (…..). Is this a rectangular room?”. I ask for opinions, but no one really has any – only the kids who like attention speak up but they can’t justify their answer. Eventually someone asks if we can graph it (sometimes this ‘someone’ is me), so I show it on Geogebra. I’ve cleverly picked a parallelogram that is tilted and looks like a rectangle, but really isn’t. Now more people enter the debate – everyone has an opinion, but it starts off based on ‘how it looks’. Finally someone starts to say it’s because it ‘looks’ like the sides are congruent or that the sides are parallel or the angles are 90 degrees – but again, they’re not sure. Next comes the guided connection between parallel/perpendicular and slope, congruence and distance, and bisect with midpoint. At this point, we’re ready to collect some data:

Update: I can’t figure out how to embed box.net stuff into WordPress, and Scribd completely messes up the formatting of my worksheets. So, sorry that you have to add a few extra clicks in order to see what I’m walking about. Anyway – here’s the ‘Here are 4 points, what shape is it?’ worksheet I made

So now they’re finding distances and slopes and midpoints… then turning that data into geometric properties… then turning those geometric properties into arguments. I really dig it. By the way, there is an ‘efficient’ method that doesn’t require so many calculations – it’s outlined in this Flowchart (right-branches mean ‘yes’, left-branches mean ‘no’), which I may or may not give to my students at some point later. My honors kids are already thinking about ‘the most efficient method’, while my regular kids are still focusing on making 100% sure that they know what (-1 – 3) equals.

The Hierarchy/Relationship Stuff: I told my students that this unit was really about relationships – things these shapes have in common, ways in which these shapes are different, how we go from a general definitions to specific definitions, etc. I then told them that they’re already intricately involved in something that involves relationships: Facebook. Then I assigned them a project: to make 4 fake facebook pages for 4 of the quadrilaterals we’ve been learning about (minor reflective note: next year, I think I’ll only assign 3). Below are the resources for this project

Facebook Template (for making the physical pages)