While watching a couple of students play checkers one day, I wondered, "Just how many squares are on that board?" I thought and thought and came up with a number greater than 200! Is that possible? Your task is to find out exactly how many squares are on that board. Be creative and thoughtful about your mathematics. Show all of your work, explain all of your thinking and identify all the strategies you use to solve this problem.
We had been working with exponents. My students have explored square numbers, working with sugar cubes to build squares and to visualize the growing of patterns. This task requires students to use problem-solving approaches to investigate, draw logical conclusions and generalize solutions and strategies. It requires understanding and applying reasoning processes, with special attention to spatial reasoning. It also allows me to assess whether or not students have understood the activities we have been working on in class. Most students will start with finding the number of squares with an area of one, then an area of four and so on. Some were not able to approach the task as systematically, which resulted in incomplete solutions. The students were given one week to work on this problem individually or in a group. Students had been studying the history of games. Questions we have explored include: What constitutes a successful game? A challenging game? A game that will endure over time? The unit could be further extended in mathematics to the area of probability and what constitutes a "fair game". Working with sugar cubes to build squares and visualize growing patterns is a helpful pre-assessment activity. I have also taught my students the following five steps of an investigation, which was also useful: 1. Collect your data.
2. Organize your thinking (identify strategies).
3. Represent your thinking using graphs, charts, tables, drawings, etc.
4. Explain your thinking (validate your solution).
5. Refine and/or extend your thinking. (Find another solution. Create another problem.)
- Graph paper
- Colored pencils
- Sugar cubes
- Checkerboards
- Squares
Total = 204 squares: 8 x 8 = 64 1 x 1 squares
7 x 7 = 49 2 x 2 squares
6 x 6 = 36 3 x 3 squares
5 x 5 = 25 4 x 4 squares
4 x 4 = 16 5 x 5 squares
3 x 3 = 9 6 x 6 squares
2 x 2 = 4 7 x 7 squares
1 x 1 = 1 8 x 8 squares
A Novice's response will demonstrate limited understanding of the complexity of his/her task and merely count the squares traditionally noticed. The student uses neither math language nor representation to communicate.
An Apprentice's solution may not be complete. The student may find the 64 small squares and knows that there were other size squares. However, the student's strategy may only be partially useful since it lacks a systematic approach. An Apprentice may show some mathematical reasoning in finding other sized squares. There is some use of representation and some explanation of strategy using some mathematical terminology and notation.
A Practitioner's solution demonstrates understanding of the major concepts needed to find all the squares. The student uses effective reasoning that leads to a solution. There is a clear explanation that uses effective mathematical terminology and notation. The representation is accurate and appropriate for the problem.
An Expert demonstrates deep understanding of the problem by recognizing the pattern and being able to express this pattern algebraically or exponentially. The representation accurately and appropriately communicates the student's solution.
|
