Binomial Expansion -Shortcut Please

Learning Resource Type

Lesson Plan

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This lesson is an introduction to Binomial Expansion and the Binomial Theorem. Students begin by expanding binomials using multiplication. They will examine the expansions looking for patterns. These patterns will be used to develop the Binomial Theorem. Both Pascal's Triangle and Combinations will be used to complete the Binomial Expansion.

This lesson results from the ALEX Resource Gap Project.

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.17
Know and apply the Binomial Theorem for the expansion of $(x + y)^n$ in powers of x and y for a positive integer, n, where x and y are any numbers.

Vocabulary

Binomial Theorem
Pascal's Triangle
Combinatorial Argument
Mathematical induction

Knowledge

Students know:

Distributive Property of multiplication over addition for polynomials.
The generation pattern for Pascal's Triangle and which binomial expansion term has coefficients corresponding to each row.
Simplification procedures for expressions involving the number of combinations of n things taken r at a time.
The patterns of coefficients and exponents in a binomial expansion.

Skills

Students are able to:

Accurately perform algebraic manipulations on polynomial expressions.
Generate rows of Pascal's Triangle.
Accurately perform simplification procedures for expressions involving the number of combinations of n things taken r at a time.
Apply the patterns of coefficients and exponents to expand any binomial raised to a power.

Understanding

Students understand that:

Regularities noted in one part of mathematics may also be seen in very different areas of mathematics, (i.e., Pascal's Triangle from counting procedures and the Binomial Theorem). These regularities are useful in computing or manipulating mathematical expressions.
The regularities that are seen in exponents and coefficients in a binomial expansion will be generalized to all binomials to aid in identifying specific terms.

Primary Learning Objectives

Students will expand binomials using multiplication.

Students will examine and develop patterns for binomial expansion.

Students will use Pascal's Triangle to find the coefficients in a binomial expansion.

Students will use combinations in conjunction with the Binomial Theorem to expand a binomial.

Procedures/Activities

Before:

Give students the Binomial Expansion Worksheet. Have them work in pairs to complete problems 1-3. These problems ask the students to expand an increasing number of binomials (2, 3, then 4). The purpose of this activity is to review binomial expansion and to show the need for a quicker method. Students will attempt to answer Question 4 by examining patterns they find from working on questions 1, 2, and 3. At this point, they may or may not find the correct answer. You will be discussing the patterns with them in the during phase of the lesson. You will also give them another chance to find the correct answer to question 4.

During:

Open the PowerPoint Presentation. This presentation will progress through the patterns in the binomial expansion. These patterns, along with Pascal's Triangle, will be introduced as a method for expanding any binomial. Finally, the use of combinations will be incorporated into the binomial theorem. Have students make notes and work problems from the slides in their notebook or on the Binomial Theorem PowerPoint Presentation Notes Handout if notebooks are not used.

Slide 1: Introductory Slide. Be sure to explain to students that we need a quicker and more efficient method for finding a binomial expansion.

Slide 2: This slide contains the answers to the worksheet questions 1 - 3. Question 4 will be answered in slide 9.

Slide 3: In this slide, you will discuss patterns and the first and last term of the expansion. These are always the first and last term of the binomial raised to the power of the binomial.

Slide 4: The discussion of the sum of the exponents of each term in the expansion is included on this slide. Students should see that the sum of the exponents of each term is always the exponent of the binomial.

Slide 5: The exponents of the first binomial term are discussed on slide 5. The exponents always begin at the binomial exponent and decrease to 0 in the expansion.

Slide 6: The exponents of the last binomial term are discussed on slide 6. The exponents begin at 0 and increase to the binomial exponent.

Slide 7: Ask students to make some conjectures about the coefficients. You want students to realize the coefficients are from Pascal's Triangle.

Slide 8: This slide presents Pascal's Triangle, you can use it to introduce this to your students, if necessary.

Slide 9: Ask the students to return to question 4 on the worksheet and find the expansion. The answer is provided when you click.

Slide 10: Provides another example for students to try.

Slide 11: Presents an example that has a coefficient in the binomial.

Slide 12: Now we need to transition to the Binomial Theorem using combinations. Point out to students that at times we don't want to have to write all the rows of Pascal's Triangle to find our expansions. Make sure they know that the numbers in the triangle are simply a set of combinations. The formula for a combination is included.

Slide 13: Students are presented with an example using combinations.

Slide 14: This slide contains a problem for students to work. Answers are included.

After:

Ask students to complete the Exit Ticket before leaving class.

Assessment Strategies

Informal:

Binomial Expansion Worksheet

Students will be presented with problems to work on throughout the lesson. Use these problems found in the PowerPoint presentation as an assessment tool.

Formal:

Binomial Expansion Exit Ticket - This will allow you to see what students know and how they feel about the material presented.

Homework problems can also be assigned and used as an assessment.

Acceleration

Allow accelerated students to explore interesting facts about Pascal's Triangle found at Math Is Fun.

These students can share what they learn with the class through discussion or by creating a digital presentation.

Intervention

Students who need extra help can be directed to this tutorial. This tutorial provides extra practice and explanations when needed.

Total Duration

31 to 60 Minutes

Background/Preparation

Teachers:

Review the Binomial Theorem PowerPoint Presentation. Be sure that you are familiar with the ideas presented on each slide.

Make copies of the Binomial Expansion Worksheet (1 copy per student).

Make copies of the Binomial Expansion Exit Ticket (1 copy per student).

Student:

Be familiar with binomial multiplication.

Materials and Resources

Teachers:

Computer connected to Projector

Binomial Theorem PowerPoint Presentation

Copies of Binomial Expansion Worksheet (1 per student)

Copies of Binomial Expansion Exit Ticket (1 per student)

Copies of Binomial Theorem PowerPoint Presentation Notes Handout (1 per student if not taking notes in notebook)

Homework Problems (Optional Assignment 1 copy per student)

Students: