Content Area: Math

 

Index: 4.3A Grade 6 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 and rational numbers

·        Descriptions using tables, verbal rules, simple equations, and graphs

·        Formal iterative formulas (e.g., NEXT = NOW * 3)

·        Recursive patterns, including Pascal’s Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1,  1,  2,  3,  5,  8, . . .    (where NEXT = NOW + PREVIOUS)

 

Grade: 6

 

Sample Activities:

 

·        Students use iteration in Logo software to draw checkerboards, stars, and other designs. For example, they iterate the construction of a simple component of a pattern, such as a square, to recreate an entire checkerboard design.

 

·        Students use paper rabbits (prepared by the teacher) with which to simulate Fibonacci's 13th century investigation into the growth of rabbit populations: If you start with one pair of baby rabbits, how many pairs of rabbits will there be a year later? Fibonacci's assumption was that each pair of baby rabbits results in another pair of baby rabbits two months later - allowing a month for maturation and a month for gestation. Once mature, each pair has baby rabbits monthly. (Each pair of students should be provided with 18 cardboard pairs each of baby rabbits, not-yet-mature rabbits, and mature rabbits.) The Fascinating Fibonaccis by Trudi Garland illustrates the rabbit problem and a number of other interesting Fibonacci facts. In Mathematics Mystery Tour by Mark Wahl, an elementary school teacher provides a year's worth of Fibonacci explorations and activities.

 

·        Students use calculators to compare the growth of various sequences, including counting by 4's (4, 8, 12, 16, ... ), doubling (1, 2, 4, 8, 16, ... ), squaring (1, 4, 9, 16, 25, ... ), and Fibonacci (1, 1, 2, 3, 5, 8, 13, ... ).

 

·        Students explore their surroundings to find rectangular objects whose ratio of length to width is the "golden ratio." Since the golden ratio can be approximated by the ratio of two successive Fibonacci numbers, students should cut a rectangular peephole of dimensions 21mm x 34 mm out of a piece of cardboard, and use it to "frame" potential objects; when it "fits," the object is a golden rectangle. They describe these activities in their math journals.

 

·        Students study the patterns of patchwork quilts, and make one of their own. They might first read Eight Hands Round.

 

·        Students mark one end of a long string and make another mark midway between the two ends. They then continue marking the string by following some simple rule such as "make a new mark midway between the last midway mark and the marked end" and then repeat this instruction. Students investigate the relationship of the lengths of the segments between marks. How many marks are possible in this process if it is assumed that the marks take up no space on the string? What happens if the rule is changed to "make a new mark midway between the last two marks?"

 

·        Additional Framework Activities