Pascal's Triangle and Binomial Expansion: A Calculator Shortcut

Algebra 2 Grades High School 1:59 Video
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Lesson Description

Learn how to efficiently use your TI-84 or TI-83 graphing calculator to generate Pascal's Triangle coefficients for binomial expansion, especially useful for higher powers.

Video Resource

TI84 TI83 Pascal's Triangle Binomial Expansion

Mario's Math Tutoring

Duration: 1:59
Watch on YouTube

Key Concepts

  • Pascal's Triangle
  • Binomial Theorem
  • Combinations
  • TI-84/TI-83 Calculator Use

Learning Objectives

  • Students will be able to generate rows of Pascal's Triangle using a TI-84 or TI-83 calculator.
  • Students will be able to use Pascal's Triangle coefficients to expand binomials raised to a power.

Educator Instructions

  • Introduction (5 mins)
    Briefly review Pascal's Triangle and the Binomial Theorem. Explain the challenge of expanding binomials with high powers and introduce the calculator shortcut.
  • Calculator Demonstration (15 mins)
    Follow the video's instructions to demonstrate how to use the TI-84/TI-83 calculator to generate Pascal's Triangle coefficients. Emphasize the 'math -> probability -> combinations' sequence and the use of the variable key. Step through an example, such as finding the coefficients for (x+2)^5, as shown in the video. Explain Table Set function, focusing on starting at 0 and going up by ones.
  • Binomial Expansion Application (15 mins)
    Using the coefficients generated by the calculator, demonstrate how to expand a binomial raised to a power. Explain how the exponents of the variables change in descending and ascending order, respectively. Reiterate the example in the video of (x+2)^5. Show the simplification steps to arrive at the final expanded form.
  • Practice Problems (10 mins)
    Students work individually or in pairs to expand binomials using the calculator shortcut. Provide problems with varying powers (e.g., (x+1)^4, (2x-3)^3, (x-y)^6).

Interactive Exercises

  • Binomial Expansion Challenge
    Divide the class into teams. Each team receives a binomial to expand (e.g., (x-3)^7, (2x+1)^6). The first team to correctly expand the binomial wins. Require use of the calculator method taught in the lesson.

Discussion Questions

  • Why is using a calculator shortcut helpful when expanding binomials with large powers?
  • How does the combination formula (n choose k) relate to Pascal's Triangle and binomial coefficients?
  • What are the limitations of using this calculator method?

Skills Developed

  • Calculator Proficiency
  • Algebraic Manipulation
  • Problem-Solving
  • Pattern Recognition

Multiple Choice Questions

Question 1:

What function on the TI-84 calculator is used to generate Pascal's Triangle coefficients?

Correct Answer: nCr

Question 2:

In the combination function nCr, what does 'n' represent?

Correct Answer: The row number in Pascal's Triangle minus 1

Question 3:

When using the table function on the calculator, what should the 'Table Start' value be for generating Pascal's Triangle coefficients?

Correct Answer: 0

Question 4:

What is the coefficient of the x^3 term in the expansion of (x+1)^5?

Correct Answer: 10

Question 5:

In binomial expansion, the exponents of the first term in the binomial will...

Correct Answer: Decrease from left to right.

Question 6:

What is the value of 5 choose 2 (⁵C₂)?

Correct Answer: 10

Question 7:

Which row of Pascal's triangle gives the coefficients for the expansion of (a + b)^4?

Correct Answer: 1 4 6 4 1

Question 8:

When expanding (x-2)^3, what sign will the last term have?

Correct Answer: Negative

Question 9:

What is the constant term in the expansion of (x+3)^2

Correct Answer: 9

Question 10:

What is the coefficient of the x term in the binomial expansion of (x + 2)^3

Correct Answer: 12

Fill in the Blank Questions

Question 1:

Pascal's Triangle is a triangular array of _________ that appear in binomial expansions.

Correct Answer: coefficients

Question 2:

The function on the calculator to calculate combination is found under MATH and then ____________.

Correct Answer: Probability

Question 3:

In the combination function nCr, 'r' represents the number of items ____________.

Correct Answer: chosen

Question 4:

In the expansion of (a+b)^n, the exponents of 'a' go in _________ order.

Correct Answer: descending

Question 5:

In the expansion of (a+b)^n, the exponents of 'b' go in _________ order.

Correct Answer: ascending

Question 6:

The coefficients of (x+y)^2 are 1, ____, and 1.

Correct Answer: 2

Question 7:

The binomial theorem provides a formula for expanding expressions of the form (a + b)^____.

Correct Answer: n

Question 8:

The calculator's table function should start at X equals ______ when generating Pascal's Triangle.

Correct Answer: 0

Question 9:

When using the calculator, to get to combinations you press the Math button and then arrow over to _________.

Correct Answer: probability

Question 10:

When expanding (x+5)^2, the constant term is ________.

Correct Answer: 25

Educational Standards

A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions. A2.A.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies.

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