Content Area: Math

 

Index: 4.3A Grade 8 CPI 1

 

Standard: 4.3 - Patterns and Algebra

 

Strand: A - Patterns

 

Cumulative Progress Indicator: 1 - The student will recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers

·        Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

·        Finite and infinite sequences

·        Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

·        Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)

·        Generating sequences by using calculators to repeatedly apply a formula

 

Grade: 8

 

Sample Activities:

 

·        Students describe, analyze, and extend the Fibonacci sequence 1, 1, 2, 3, 5, 8, ... , where each term is the sum of the two preceding terms. They investigate applications of this sequence in nature, such as sunflower seeds, the fruit of the pineapple, and the rabbit problem. They create their own Fibonacci-like sequences, using different starting numbers.

 

·        Students read Isaac Asimov's short story Endlessness and write book reports to convey their reactions.

 

·        Students discuss how the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ... ) is related to the following problem: begin with two rabbits (one male and one female), each adult pair of rabbits produces two babies (one male and one female) each month, the babies themselves become adults (and start having their own babies) after one month, and none of the rabbits ever die. The students decide that the Fibonacci sequence shows how many pairs of rabbits there are each month. The students explore other patterns in this sequence, noting that each term is the sum of the two preceding terms.

 

·        Students look for infinite sequences in Pascal's triangle. Starting at the top 1 and moving diagonally to the left, there is a constant infinite sequence 1, 1, 1, 1, ... . Starting at the next 1 and moving diagonally to the left there is the sequence 1, 2, 3, 4, 5, ... of whole numbers. Starting at the next 1 and moving diagonally to the left, there is the sequence 1, 3, 6, 10, 15, ... of triangular numbers, which records the solutions to all handshake problems. Also the sum of the numbers in each row yield the exponential sequence 1, 2, 4, 8, 16, ... .

 

·        Additional Framework Activities