Tag Archives: conversation

‘Twas the day before break…

My goal for my Geometry classes today was to be as mathematically productive as possible, given that it was the last day of classes before break. The plan was to review the last assignment, take the quizlet (what my department calls formative assessment), do an extension problem, then make a Koch Snowflake if there was still time.

The extension problem was the “Shortest Path Problem” which I highly recommend.

Shortest path problem

It turned out to fit perfectly with what we’re learning right now. It also sparked some rich conversations and good reasoning, and everyone could at least venture a guess, even if they didn’t really know what to do to figure out the exact shortest path.

This plan was carried out differently in each of my three Geometry classes. In the first class, I reviewed several problems and concepts, kids followed along, asked questions, the usual. They took the quizlet. I passed out a half-sheet with the scenario typed out on it. I didn’t include a diagram, thinking that the kids should make the diagram. That was a mistake because the wording isn’t exactly clear, so some kids drew the tent and camper on opposite sides of the river and all sorts of random things. So I had to draw the diagram up on the board for everyone, which slightly killed the magic, but at least we were all on the same page.

A few kids calculated the distance of a path, but not the shortest, and then wanted to be done. I needed a way to motivate them to keep working. In a rare moment of brilliance, I decided to keep score. I announced “Jesse found a path that’s 1,518 feet, can anyone beat that?”. Then I’d write the student’s name and their shortest path on the board. It became a competition to see who could find the shortest path. I let things linger too long in my first class because a few students were really getting into it and asking wonderful questions like, “how do you know that’s 450 feet” and “can you show me how you got that”. So unfortunately several kids had checked out, but at least everyone did something with the problem.

In my other two classes, I skipped the homework review and went straight to the quizlet because there was no way they were going to sit and listen to me blah blah blah about their homework problems on dilations and scale factors. In my first afternoon class, student behavior dictated that decision. In my second afternoon class, I asked them what they wanted to do, and almost everyone said, “let’s just take the quizlet”. So in those classes, there was plenty of time to do both the shortest path problem and the Koch Snowflake.

This time I just asked them to read the problem on their own, and then I read it aloud and drew the diagram as I read so everyone started out with the correct diagram. In one class several kids said they didn’t know what to do to get started, so I said “guess and check” or “if you were the camper, where would you go if you wanted the shortest distance”. This was an excellent starting point for those kids.

The snowflakes were fun too. We saw the Sierpinski Triangle this year, so I brought that up again as a reminder of what a fractal is, but then said that the fractal they were about to make was going to be more holiday-themed. I gave everyone some triangle graph paper to help them with their triangles. At first I thought maybe I’d have them construct the equilateral triangle, but using the graph paper was a good call.

photo (3)

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Filed under fun, Geometry

Always, Sometimes, Never

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.

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Filed under conversations, Geometry, group work