CCSS-M G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
One of the theorems that we prove in our unit on Dilations is the Side-Splitter Theorem.
At the end of class one day, I checked to see how intuitive the Side-Splitter Theorem is.
Not so intuitive, apparently.
(I didn’t show them the correct answer.)
But we started class the next day with our diagram and our learning goal, look for and make use of structure.
What do you see that isn’t pictured?
We set the learning mode to individual. Students worked alone for a minute or two to see what they could see.
I monitored. And selected. And sequenced.
And then we talked.
Which auxiliary lines help us determine a relationship between the segments?
Our work with dilations and similarity has been mostly with triangles. Do you see any similar triangles? Why are they similar?
We generalized our results. And then I sent back the original question.
And so the journey to make sense out of what isn’t always intuitive continues …
Easing the Hurry Syndrome 
howardat58
January 21, 2015 at 12:27 pm
I like the problem. Had the picture been at the edge of a piece of paper it would have been easier for them to see that extending the crossing lines to their point of intersection would be quite useful.
This is one of the “fun” things about geometry, that the picture often does not tell all that there is. I am guessing that students feel confined to what they have in front of them, and even joining points within the picture seems rather daring.