What are your favorite problems to use with students?
At the beginning of the year, I think it's important to foster a culture where solving problems (not just completing exercises) is a critical element. Great problems help to create a culture where kids are allowed to think, struggle, and talk about their mathematical discoveries.
Here is my all-time favorite problem:
Assign each letter a value equal to its position in the alphabet, that is, A = 1, B = 2, C = 3, and so forth. The product value of a word is equal to the product of the value of the letters. For instance, the product value of CAT is 60, because C = 3, A = 1, T = 20, and 3 * 1 * 20 = 60.
The lessons on the Illuminations website contain a lot of great problems, too. Here are my favorites at three levels…
Create A House Number, http://illuminations.nctm.org/LessonDetail.aspx?id=L225
This Sweet Problem, http://illuminations.nctm.org/LessonDetail.aspx?id=L766
I love your problem for the beginning of the year. The students would be totally engaged! And, I agree you have selected some great resources from Illumiinations.
I love story problems that make students think and can be solved using a variety of strategies. Students work together to solve the problems thus building relationships with one another right away. Then having students share these solution strategies with the class gets them talking also. Make sure you have those manipulatives out there too. :-)
I might recommend some great story problems we have used with our teacher leaders to get them thinking. They in turn have taken them back to their students where the students have been totally engaged. Here are some of my favorites to get them working together and thinking critically:
My favorites from Illuminations include:
Elementary - Fraction Game - http://illuminations.nctm.org/ActivityDetail.aspx?id=18
Middle School - Barbie Bungee Lesson - http://illuminations.nctm.org/LessonDetail.aspx?ID=L646 - students have to figure out the line of best fit for dropping Barbie with a bungee and not letting her head hit the floor - talk about engaging!! And think about all that they learn - includes a great Line of Best Fit interactive and a spreadsheet to create the graphs.
High School - Compound Interest Simulator - http://illuminations.nctm.org/ActivityDetail.aspx?ID=172 - my all time favorite because of the real world aspect of money, credit, savings and the stock market. Kids always want to spend money, buy cars, etc. This gives them a real world eye opener which would be a great beginning of the year activity to talk about since most worked during the summer.