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.
I do the "Crossing the River" problem with my 5th graders which came from Fostering Algebraic Thinking by Mark Driscoll (new on Amazon for $19.99 or used for $12.78). I did find the activity on-line HERE. The book is geared for 6-10 grade, but we used it in a class I took that was filled with elementary teachers. We were able to easily adapt most of the problems.
The goal of the activity is to get 2 children and 8 adults across the river with only one boat. The rules for crossing are the following: one adult, one child or 2 children in a boat. You can NOT have one child and one adult in the boat at the same time. The students have to figure out the least amount to trips it takes to get across the river. Eventually they figure out an equation to figure out how many trips for any number of children and adults.
What are some of your favorite problem solving activities to do with your students? I'm always looking for new activities and would love for you to share you favorites with us.