Content Area: Math

 

Index: 4.3A Grade 7 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

·        Generating sequences by using calculators to repeatedly apply a formula

 

Grade: 7

 

Sample Activities:

 

·        Students explore and try to explain the sequence made by the numbers of diagonals in a series of polygons with increasing numbers of sides. For example, a triangle has no diagonals, a square has 2 diagonals, a pentagon has 5 diagonals, a hexagon has 9 diagonals, and so on. They examine the sequence 0, 2, 5, 9, try to extend it, and justify their conclusion.

 

·        Students read The King's Chessboard by David Birch. In this old folk tale, which has been told many times in many languages, a king is undecided about a gift to give to one of his advisors. Finally he decides that, on one day, the advisor will receive a single grain of rice on the first square of a chessboard. On the next day, that amount will be doubled and two grains will be placed on the second square. On day three, four grains will be put on the third square, and so on, doubling every day until the entire board is filled. The students use calculators to figure out the king's indebtedness on the tenth day, the twentieth day, and so on.

 

·        Students make equilateral triangles of different sizes out of small equilateral triangles and record the number of small triangles used for each larger triangle. These numbers are called triangular numbers. If the following triangular pattern is continued indefinitely, then the number of 1s in the first row, the number of 1s in the first two rows, the number of 1s in the first three rows, etc. form the sequence of triangular numbers. The triangular numbers also emerge from the handshake problem: If each two people in a room shake hands exactly once, how many handshakes take place altogether? If the answers are listed for 2, 3, 4, 5, 6, 7, ... people, the numbers are again the triangular numbers 1, 3, 6, 10, 15, 21, ... .

 

·        Students imagine cutting a sheet of paper into half, cutting the two pieces into half, cutting the four pieces into half, and continuing this over and over again, for about 25 times. Then they imagine taking all of the little pieces of paper and stacking them on top of one another. Finally, they estimate how tall that stack would be.

 

·        Students describe, analyze, and extend the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). They research occurrences 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.