I've talked about this lesson before, but wanted to bring it up again because my students love it so much. My 6th graders are actually going to start this lesson on Monday and I can't wait to see what they come up with.
We usually spend the last week of school doing nothing but problem solving activities during our math time. One of my favorite activities to do with my 6th graders is called Paper Pool which can be found on the Illuminations website. The students would spend a whole week on this project if I let them.
I start by going through THIS worksheet, which explains the activity.
We go through both examples so that they understand exactly what the rules are. Usually about half of the students have trouble understanding how to draw the lines (they have to be corner to corner), so I usually have to spend time working with them. If I'm lucky, the partner they have chosen understands and can show them how to do it. Once everyone understands how the "ball" moves on the board, they try different size rectangles.
When the students see a pattern, they write the "rule" on the board (they have to have found at least 3 rectangles that fit their rule to write it on the board).
The next day, we test the "rules". Every group takes a different rectangle that fits the rule to test it. If it passes the test for all rectangles tested, it goes on the proven rules list. We had approximately 25 rules when we started and it went down to only 7 after testing each one. I then wrote down whether each number was odd or even and we narrowed our rules down even further.
If you are looking for more lessons that your students will love check out the Loved That Lesson! linky by clicking on the picture below.