I’m doing coordinate geometry type stuff with my classes – slope, distance, and midpoint using two points. I had an idea for certain types of challenging/extension problems last year and I’m trying them out this year. The last page of an assignment has the following problem:

1) Add the points C and D such that:

• AB and CD have the same length
• AB and CD have the same midpoint
• AB and CD have different slopes

2) Add the points E and F such that:

• AB and EF have the same length
• AB and EF have the same slope
• AB and EF have different midpoints

3) Add the points G and H such that:

• AB and GH have the same midpoint
• AB and GH have the same slope
• AB and GH have different lengths

There are some curious properties of the points that satisfy each of these questions – you can think about them before I talk about them below the break.

First Off: I really like these as challenge/extension problems and as a precursor to properties of quadrilaterals. I’m excited to see what my higher achieving kids can do with these, and it’ll be some nice data for next year if I want to use these types of problems earlier in the unit.

Secondly: I’m taking a workshop that’s focuses on how to think in terms of transformations to help prepare for the Common Core – reframing common geometric problems and proofs in terms of transformations (ie: vertical angle theorem, proving lines are parallel, etc). It’s been a challenge because I’m not used to thinking ‘what’s the bigger picture in this problem’ and having it be transformations-related rather than shape related. I’m used to saying ‘do this process – what do you notice?’ and the answer being related to the properties of a particular shape rather than a particular transformation. Thinking about this problem sorta made me realize that. For example:

If I could project all of my student’s answers, let them sink in, then say ‘What do we notice about these answers?’…

This Year I Would Say:

• A, B, C, and D (#1) will always form the vertices of a rectangle because they represent two diagonals that are congruent and bisect each other.
• A, B, E, and F (#2) will always form the vertices of a parallelogram because A pair of parallel and congruent sides will always form a parallelogram.
• A, B, G, and H (#3) will always be collinear. I don’t have a more in-depth explanation of this.

Next Year I Will Probably Say:

• C and D are always the image of A and B under a rotation about the midpoint of AB.
• E and F are always the image of A and B under a translation (or, if you want to be fancy, a glide reflection).
• G and H are the image of A and B under a dilation about the midpoint of AB with the ratio of the distances is the scale factor (I think?)

Anyway – I’m writing about this because (1) I like these problems, and maybe you will too, and (2) retraining myself to see the ‘what do we notice?’ answers as relating to transformations (it’s a translation! it’s a rotation!) rather than shapes (it’s always equilateral! it’s always a parallelogram!) is a work in progress and I’m worried I still won’t be able to see the big pictures when I get to teach the new standards.

Or maybe I’m just reflecting on how the new geometry standards are changing the way I need to see problems. This year, these problems are interesting because of their relationships to properties of quadrilaterals (which we haven’t covered yet) in the coordinate plane. Next year, they’ll be interesting because of their relationship to transformations.