Brendan Kelly Pub. Patterns,Functions,Algebra

DESC.- BRENDAN KELLY PUBLISHING

Features: * By Dr. Brenden Kelly

• This module addresses the expectations for grades 6 through 8 in the Patterns, Functions, & Algebra strand of the NCTM's Principles and Standards for School Mathematics 2000. There are two units with five activities per unit. The content of the activities in each unit is described below.
• Unit 1: In Activity 1, students are encouraged to explore the sequence of triangular numbers, first with manipulatives and then using tables.
• In Activity 2, students encounter Pascal s triangle and learn of the contributions from the Chinese culture in the 14th century. They fill in the missing numbers in the template for Pascal s triangle, and discover the diagonal line containing the sequence of triangular numbers. They are guided toward the discovery that the sum of the integers from 1 to n is the nth triangular number.
• In Activity 3, students learn that the Arabs and Hindus contributed to the mathematical development of algebra during the European dormancy of the Dark Ages. Students discover, by cutting squared paper, that the nth triangular number is merely half the number of squares in a rectangle of n rows and n+1 columns. They are then guided toward expressing this relationship in algebraic notation. Thus they discover an algebraic formula for the nth triangular number.
• Activity 4 visits Gauss, the great 19th-century German mathematician, who incidently discovered how to sum the numbers from 1 to n at an early age. The students are prompted to use Gauss s method to find an expression for the sum of the numbers from 1 to n. This brings them to an alternative way to derive an algebraic formula for the nth triangular number.
• Unit 2: In this unit, Jennifer and Steve, two middle-school students, and their committee are organizing a fundraising party. They contact three different locations to ascertain the price of renting a hall and providing snacks. The quotes they receive can be represented by three different linear functions of n where n denotes the number of guests.
• Activity 1 deals only with the quote from the Galaxy Inn. Students are asked to explain the meaning of a minimum charge and to write the linear expression 17n to represent the cost for n > 23. They form a table of values for this linear expression and then graph it for values of n that are multiples of 10 between 20 and 70.
• Activity 2 explores the linear function 14n + 90 associated with quote from Noble Pines C. C., and compares this quote with that of the Galaxy Inn. By investigating the value of n at which both quotes yield the same cost, students explore the solution of the equation 17n = 14n + 90. Similarly,
• Activity 3 compares the linear functions 14n + 90 and 500 + 12(n 25).

* Finally, Activity 4 consolidates all the previous work,
requiring students to express, as inequalities, the ranges of n
for which each quote is optimal, and use this to choose a site for
the party. This unit was particularly well-received by the students
who field tested these activities.