An Extension of Simplifying Rational Expressions with Leibniz’s Harmonic Triangle
Introduction
This lesson is designed extend on simplifying rational expressions. They will use knowledge of simplifying rational expressions to use the harmonic triangle. The students will observe the pattern in the triangle.
Objectives
Upon completion of this lesson, students will:
· Have practiced simplifying rational expressions.
· Have observed the pattern of triangle numbers.
· Have observed the pattern of the harmonic triangle.
· Have practiced finding formulas for the nth term of the triangle.
Standards Addressed
The activities and discussions in this lesson address the following NCTM Principles and Standards Expectations for Grades 9-12:
Generalize patterns using explicitly defined and recursively defined functions
Place in the NC Mathematics Curriculum
The activities and discussions in the lesson fit into the high school Algebra 2 curriculum, reinforcing the following competency goals:
Competency Goal 2: The learner will perform operations with complex numbers, matrices, and polynomials.
1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.
Place in the Common Core
A-APR.1.Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Student Skills
Mathematics – Students will need to:
· Be able to simplify, multiply, divide, add and subtract rational expressions.
· Be able to write explicit formulas.
Technology – Students will need to:
· Use their calculators to verify answers.
Teacher Preparation
Background - Teachers will need to:
· Prepare to review or introduce the students to the above named skills. The lesson plan below assumes students are familiar with those skills. The lesson can be lengthened to include reviewing or introducing these skills.
Manipulatives – Teachers will need to:
· Reproduce the “Simplifying Rational Expressions Warm up” and “Extension: Leibniz’s Harmonic Triangle” handouts for each student, if desired.
Key Words:
Reciprocal, Harmonic Triangle
Lesson Outline
1. Focus and Review
It is important for students to be ready to simplify rational expressions. Have the students complete the warm up on their own. Then go over the answers as a class. Vocabulary to review includes: Reciprocal
2. Identify Objectives
Let the students know that the intent here is to find the patterns that exist in the harmonic triangle. This will require previous knowledge of simplifying rational expressions.
3. Teacher Input
a. Hand out the “Extension: Leibniz’s Harmonic Triangle” worksheet and give the students an overview of what the activity entails.
b. Have the students watch as you point out some highlights in the activity. Compare Leibniz’s triangle to Pascal’s triangle.
4. Guided Practice
a. Walk around the room and answer any questions the students may have regarding the activity.
b. Next try to help the students come to realistic answers.
5. Independent Practice
The students will complete the remainder of the activity with a partner.
6. Closure
End the lesson by pointing out the usefulness of the harmonic triangle. Have the students explain why this triangle exists.
Alternate Outlines
This activity can be shortened by cutting out some of the questions, depending on the level of students that it is being presented to. You could also give a homework assignment the day before this activity is presented for the students to research Gottfried Leibniz.
Suggested Follow-Ups or Extensions
This lesson can be lengthened by also completing the “Harmonic Mean” activity. This has no connection to the harmonic triangle, but it will build on simplifying rational expressions.
Simplifying Rational Expressions Warm Up
Simplify COMPLETELY.
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Extension: Leibniz’s Harmonic Triangle
This activity provides an opportunity to apply the techniques of simplifying rational expressions to some fraction patterns. It also provides valuable practice in proving statements.
Gottfried Leibniz (1646 – 1716) was a German philosopher and mathematician who is best known for his work on Calculus. This distinguished Dutch physicist and mathematician Christian Huygens (1629 – 1693) challenged Leibniz to calculate the infinite sum of the reciprocals of the triangle numbers:
1. Show that the nth term of this series is .
(You need to know that the nth triangle number is .)
2. a) Show that each term can be written as the difference .
b) Hence, show that the sum of the first n terms can be written:
c) Find a formula for the sum of the reciprocals of the first n triangle numbers. Write your formula as a single fraction.
d) Use your formula to find the sum of the reciprocals of the first five triangle numbers. Check your result by adding the appropriate fractions.
3. Now, think about the sum to infinity of the reciprocals of the triangle numbers. Show that, as n gets larger, the sum gets closer and closer to 2.
In the course of his work on summing infinite series, Leibniz devised a triangle which he called the harmonic triangle. Part of his triangle is
· The fractions on each edge form a sequence of unit fractions where the denominators increase by 1 each time.
· Each fraction in the triangle is the sum of the two fractions below it.
1. Verify that the sum of 1/20 and 1/30 is 1/12.
2. In their simplest form, find the fractions in the next row of the harmonic triangle.
3. What fraction appears in row 10 and diagonal 2?
4. a) Show that in diagonal 1 the kth fraction and its successor can be written as and .
b) Show that the sum of the first n fractions in diagonal 2 can be written as
c) Find a formula for the sum of the first n fractions in diagonal 2. Write your formula as a single fraction.
d) Use your formula to find the sum of the first four fractions in diagonal 2. Check your result by adding the appropriate fractions.
e) What will happen to the sum of the first n fractions in diagonal 2 as n gets larger and larger? Justify your answer.
f) Prove that the kth fraction in diagonal 2 can be written as .
g) Show that the fraction appears in diagonal 2. In which row is ?
5. a) Show that in diagonal 2 the kth fraction and its successor can be written as and .
b) Find a formula for the sum of the first n fractions in diagonal 3.
c) What will happen to the sum of the first n fractions in diagonal 3 as n gets larger and larger? Justify your answer.
d) Prove that the kth fraction in diagonal 3 can be written as . What is the 10th fraction in diagonal 3?
6. Investigate the other diagonals in the harmonic triangle.
· Can you find an expression for the nth fraction in diagonal m?
· Can you find an expression for the sum of the first n fractions in diagonal m?
· What happens to the sum of the first n fractions in diagonal m as n gets larger?
7. Prove that the sum of the reciprocals of an pair of consecutive triangle numbers Tn and Tn+1 is .