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Binomial theorem (AASL 1.5)

Inquiry questions

Factual Inquiry Questions
  • What is the Binomial Theorem, and for what types of expressions is it used?
  • How can you calculate a specific term in a binomial expansion using the Binomial Theorem?
Conceptual Inquiry Questions
  • Why does the Binomial Theorem work for any power of a binomial expression, and how does Pascal's Triangle relate to it?
  • How does the concept of combinations play a role in the coefficients of the terms in a binomial expansion?
Debatable Inquiry Questions
  • Is the Binomial Theorem more important for its theoretical implications in mathematics or for its practical applications?
  • Can the principles of the Binomial Theorem be extended to non-binomial expressions, and what would be the implications of such an extension?
  • How has the understanding and application of the Binomial Theorem evolved with the development of modern algebra and computational tools?
Image
Mini-Investigation: Pascal's Triangle and the Binomial Theorem Objective:Understand the relationship between Pascal's Triangle, binomial coefficients, and the binomial expansion.

Introduction: Examine the image of Pascal's Triangle provided in the applet. Notice the sum of the two numbers directly above each number. This triangle is more than a mathematical curiosity; it reveals patterns and relationships! Activity 1: Exploring Pascal's Triangle 1. Finding the Next Row: Add the next row to Pascal's Triangle. Each new number is the sum of the two numbers above it. 2. Edge Numbers: Observe the pattern of the numbers on the edge of the triangle. Predict the edge numbers for the next row. 3. Row Sums: Add up the numbers in each row. Look for a pattern related to powers of 2. 4. Triangle Symmetry: Determine if Pascal's Triangle is symmetrical and prove your findings. What other patterns can be observed in the numbers in Pascal's triangle?

Activity 2: Understanding nCr (Combinations)1. Formula Familiarity: Write the nCr formula and discuss each part of the formula.2. Calculating Combinations: Use the nCr formula to calculate the numbers in a row of Pascal's Triangle. Compare with the applet. Explain how nCr can be used to find a particular value in Pascal's triangle

Activity 3: Connecting to the Binomial Expansion 1. Expansion Basics: Change the value of n in "Expand (a + b)^n" on the applet and observe the expansion's relation to Pascal's Triangle.2. Coefficient Connection: Identify the coefficients in the binomial expansion and their positions in Pascal's Triangle.3. Pattern Prediction: Predict the binomial expansion of (a + b)^5 and verify using the formula or the applet.

Conclusion:Reflect on how Pascal's Triangle, nCr, and the binomial expansion are related. Discuss the patterns discovered and their implications. Extension:- Explore diagonal patterns in Pascal's Triangle. - Investigate the Fibonacci sequence within Pascal's Triangle. - Determine the number of different pathways from the top to the bottom of the triangle. Sharing Your Findings: After completing the investigation, share and discuss your findings. Explore other areas where these patterns might appear. Post any interesting facts or discoveries here.

Part 2 - Applying the binomial theorem

Watch following video before attempting the mini-quiz questions.
Alternatively if you feel confident with the theory, look at these worked examples of exam-style questions, ļ»æBinomial expansion - BasicĀ https://youtu.be/6hay4gEK-y8Ā Ā Binomial expansion with negative termĀ https://youtu.be/qkkA8xkINVoĀ Ā Binomial expansion past paper questionĀ https://youtu.be/Pl5BGmxQYvsĀ Ā Binomial expansion past paper question - challenging versionĀ https://youtu.be/HugZvjtv4IoĀ 

What is the coefficient of in the expansion of ?

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What is the coefficient of in the expansion of ?

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What is the coefficient of in the expansion of ?

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What is the constant term in the expansion of ?

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What is the coefficient of in the expansion of ?

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Part 3 - Exam-style questions

Q1-11 Practice style. Section A Q12-54 Short style exam question Section B Q55-56 Long style exam question

[MAA 1.6] BINOMIAL THEOREM

[MAA 1.6] BINOMIAL THEOREM_solutions

Lesson Plan- Exploring the Binomial Theorem Through Pascal's Triangle

Binomial theorem- Intuition pump (thought experiments and analogies)