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# The Binomial Theorem

Lesson 12 of 15

## Objective: The students will connect Pascal’s Triangle to binomial expansion, make a generalization, and prove it!

*47 minutes*

Although Pascal’s Triangle is named after seventeenth century mathematician, Blaise Pascal, several other mathematicians knew about and applied their knowledge of the triangle HUNDREDS of years before the birth of Pascal in 1623. Today we will study this fascinating triangle on our way to creating a short-cut to last night’s homework assignment!

Begin by asking the students to share out what patterns emerged from yesterday’s study. Prompt them to talk about the structure of the triangle, and any conclusions that can be made. This is important when working to connect yesterday’s learning to today’s extension lesson. If the students have not yet made the connection between the coefficients in the homework assignment and Pascal’s Triangle through the course of the discussion, take time to ask them to expand simple binomials such as the following:

(x +y)^{0}

(x +y)^{1}

(x +y)^{2}

(x +y)^{3}

(x +y)^{4}

Note to the students that although Pascal’s Triangle has MANY unique features and connections, we will focus our study today on this particular connection, that is, the connection between it and the expansion of binomials.

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Now that we have proven the theorem, I ask the students to use it (in connection with the triangle) to re-do their homework assignment from last night.

After the students have time to complete a couple of the problems, I ask them if they like the theorem as a shortcut to traditional binomial expansion – the answer is always a unanimous “YES!”

Circulate Homework Assignment: Using the Binomial Theorem. With as long as this lesson was, it will most certainly be about time for the bell to ring.

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- UNIT 1: Culture Building Unit - Welcome to the New Year!
- UNIT 2: Exponential and Logarithmic Functions
- UNIT 3: Sequences and Series
- UNIT 4: Statistics: Something for Everyone
- UNIT 5: Review Lessons
- UNIT 6: Writing Prompts in Mathematics
- UNIT 7: Trig Tidbits
- UNIT 8: Functions, Problem Solving, and 21st Century Skills
- UNIT 9: Polynomials and Problem Solving
- UNIT 10: Probability
- UNIT 11: Imagine This! Imaginary and Complex Numbers
- UNIT 12: Let's Explore Radicals!

- LESSON 1: Introduction to Sequences
- LESSON 2: Arithmetic Sequences and Series
- LESSON 3: Geometric Sequences and Series
- LESSON 4: Sequences and Series Battleship!
- LESSON 5: A Double-Dose of Series Application
- LESSON 6: Mortgages and Geometric Series DAY #1
- LESSON 7: Mortgages and Geometric Series DAY #2
- LESSON 8: Mortgages and Geometric Series DAY #3
- LESSON 9: Mortgages Wrap-up and Additional Scaffolding
- LESSON 10: Investigating Infinite Geometric Series
- LESSON 11: Using Sequences and Series to Make Connections
- LESSON 12: The Binomial Theorem
- LESSON 13: Sequences and Series Partial Unit Review
- LESSON 14: Sequences and Series Partial Unit Review II
- LESSON 15: Unit Test: Sequences and Series