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## Favorite Problems

After attending a session on our revised curriculum by Dr. Peter Liljedahl, several teachers have told me that they have been inspired to spend the first week of Math 10 Combined on  establishing collaborative and problem solving practices in their classrooms.  Since we will be teaching math in a different manner than teachers and students are used to, it is important to create a safe environment in which students can explore and construct meaning through rich problems.

The idea that we will spend the first week on processes and exploration, rather than on checking off the first few curricular outcomes is frightening to some teachers.  Several have asked me what types of problems they should give students during that first week.  Below are three of my favorites.  They have loose curricular fit, but all allow for rich discussion, debate, exploration and invite multiple methods of solution.  The idea is to throw these out to students before any teaching has happened, and allow them to use any method they can to solve them.  Feel free to give me feedback, and share your own favorite problems.

Problem 1 Fred’s aunt is twice as old as Fred was when she was as old as Fred is now.  How old is Fred and how old is his aunt?

Problem 2 Three businessmen check into a hotel (in the olden days, when hotels were cheaper).  The desk clerk charged them \$30, so they each paid \$10.  Later on, the desk clerk realized he had made a mistake, and the room charge should only have been \$25.  The desk clerk sent a bellboy up to the room with \$5 change.  Unsure how to divide the \$5 evenly among the three men, the bellboy gave each man \$1 back, and kept \$2 for himself.  Now each man has paid \$9, for a total of \$27.  The bellboy has \$2 which brings the total money up to \$29.  But the men originally paid \$30.  Where is the missing dollar?

Problem 3 – The Jail Cell Problem In a prison, there are 100 cells numbered 1 to 100.  On day 1, a guard turns the key in all 100, unlocking them.  On day 2, the guard turns the key in all the even-numbered cells, thereby locking them.  On day 3, the guard turns the key in all the cells numbered with multiples of 3, thereby altering their state.  For example, if the cell was unlocked, he locks it.  If it was locked, he unlocks it.  On day 4, he turns the key in all the cells that are numbered with multiples of 4, thereby changing their state.  He continues this process for 100 days, at which time all the prisoners in unlocked cells are set free.  Which prisoners are set free?