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McREL Standards Activity


Chessboard Series


Purpose:As a result of this activity, students will be able to see how sequences and series can be used to model and solve problems.
Related Standard & Benchmarks:
Mathematics
 Standard 8.Understands and applies basic and advanced properties of functions and algebra
   Level IV [Grade 9-12]
   Benchmark 12. Understands formal notation (e.g., sigma notation, factorial representation) and various applications (e.g., compound interest) of sequences and series
Mathematics
 Standard 2.Understands and applies basic and advanced properties of the concepts of numbers
   Level IV [Grade 9-12]
   Benchmark 3. Uses discrete structures (e.g., finite graphs, matrices, sequences) to represent and to solve problems
Mathematics
 Standard 3.Uses basic and advanced procedures while performing the processes of computation
   Level IV [Grade 9-12]
   Benchmark 4. Uses a variety of operations (e.g., finding a reciprocal, raising to a power, taking a root, taking a logarithm) on expressions containing real numbers
Mathematics
 Standard 3.Uses basic and advanced procedures while performing the processes of computation
   Level IV [Grade 9-12]
   Benchmark 6. Uses recurrence relations (i.e., formulas expressing each term as a function of one or more of the previous terms, such as the Fibonacci sequence or the compound interest equation) to model and to solve real-world problems (e.g., home mortgages, annuities)
Student Product:Written report showing results and problem solving steps
Material & Resources:Scientific calculator, chessboard or drawing with 64 squares
Teacher's Note:This activity can be done individually, in small groups, or as a class activity
Activity
Students are to consider the following situation: Once a king hired a worker to do repairs on his castle. The worker asked to be paid in a very unusual way. He told the king to put a coin on the first square of his chessboard, two coins on the second, 4 on the third, and so on, doubling the amount on each square until all 64 squares had coins on them. Ask the students to write, using appropriate notation, an expression for the terms in the sequence. Ask them to help the king decide how much he would pay the worker by selecting an appropriate summation procedure to calculate how much money will be on the whole board. What if the king makes a counter offer? He agrees to put a coin on the first square, and adds one on each square (i.e., 1, 2, 3). Students should write an expression for the terms in this new sequence. Using a correct summation procedure, the students should determine how much the worker would make according to the king’s offer. As a conclusion, either as individuals or in groups, the students should turn in their results and justifications of their procedures.