• IME
• Tucson Teachers' Circle

Fred Stevenson presented Squares within Squares

    October 6, 2009

What are the components of a good exploratory problem?
The following list comes from a talk Fred gave in 2007.

1. The problem should be attractive and easy to understand.


2. The mathematics involved should be relevant to the student.


3. The problem should allow for collection of data.


4. The data should generate a pattern.


5. The patterns should lead to conjectures.


6. The conjectures should lead to a solution.


7. The solution should allow for an explanation.


8. The problem should have natural extensions.





Squares within Squares satisfies every one of those criteria. We began with a grid of 6 rows of dots with 6 dots per row. We determined how many squares could be located such that all vertices of each square were located on one of the given dots. Fred demonstrated an efficient method of counting the squares and a notation which enabled us to organize our data ingeniously. Using the strategies that had been demonstrated, we explored all squares in a 10 by 10 array of dots. The numerical patterns and overall beauty of the results were awesome.



The ideas and strategies presented were beyond the norm of middle school mathematics, yet numerous standards were addressed. Fred has used this problem with middle school students in summer math camps for many years.





Some of the AIMS Standards included are:



M06-S5C2-03 Analyze and compare mathematical strategies for efficient problem solving; select and use one or more strategies to solve a problem.


M06-S5C2-05 Represent a problem situation using multiple representations, describe the process used to solve the problem, and verify the reasonableness of the solution.


M06-S5C2-06 Communicate the answers to a problem using appropriate representations, including symbols and informal and formal mathematical language.


M06-S5C2-07 Isolate and organize mathematical information taken from symbols, diagrams, and graphs to make inferences , draw conclusions, and justify reasoning.


M06-S5C2-08 Make and test conjectures based on information collected from explorations and experiments.