I debated some Always, Sometimes, Never statements with my Geometry kids today. In groups, they had to choose the word that they thought went in the blank, as well as draw a picture to explain their choice.
Some example statements (taken straight out of our textbook):
Two planes ________ intersect in a line.
Lines ________ have endpoints.
Lines that are not parallel ________ intersect.
Two points _________ determine a line.
That last one created some interesting discussions, particularly in my last period. Many students wanted to put Sometimes in the blank. I didn’t look at the textbook’s answers, but I assume the authors wanted Always in the blank.
Why did so many students think Sometimes? Well, I think the statement was kind of confusing to them. What does it mean to “determine” a line? Does “a” line mean one line or does it many any line? I tried to resolve the matter by putting two random dots on the board and drawing a line through them. “Look, I can draw a line connecting any two points.” Not particularly convincing.
The students then told me to draw a line going through each of the points (parallel lines, for example). “See,” they told me, “there’s two lines, not a line.” I didn’t really know how to respond to that. I told them yes, I can draw different lines through each point, but only one line will connect them.
Well, I think I convinced them that any two points could be connected with a line, but we just left the Always, Sometimes, Never question unanswered. Which is okay. Of course, some kids insisted, “But what’s the answer?” and I replied, “Well, I think it’s Always, but I don’t think it’s totally clear.”
Perhaps the answer would have been less ambiguous if the original statement was Two points can _________ be connected with a line. But that statement seems way less powerful. So now I am intrigued by the word “determine”. I definitely think it’s important. It’s hard to explain to the kids what is meant by “determine” though.
One instructional difference I would have made during the activity was to require new people to be the writer and the speaker for each statement. In a couple groups, it was very obvious that two or three students were doing all the work while the others checked out, so some sort of rotation would have been smart.
I want to start the next class by playing Sarah Rubin’s Draw It game because some of the drawings I saw today were definitely off the mark, but that’s okay. Visualizing lines and planes and space can be tricky. I love seeing their eyes widen when they begin to “see” it.