Binomial theorem (AASL 1.5)
Inquiry questions
[table][br][tr][br] [td][b]Factual Inquiry Questions[/b][br] [list][br] [*]What is the Binomial Theorem, and for what types of expressions is it used?[br] [*]How can you calculate a specific term in a binomial expansion using the Binomial Theorem?[br] [/list][br] [/td][br] [td][b]Conceptual Inquiry Questions[/b][br] [list][br] [*]Why does the Binomial Theorem work for any power of a binomial expression, and how does Pascal's Triangle relate to it?[br] [*]How does the concept of combinations play a role in the coefficients of the terms in a binomial expansion?[br] [/list][br] [/td][br] [td][b]Debatable Inquiry Questions[/b][br] [list][br] [*]Is the Binomial Theorem more important for its theoretical implications in mathematics or for its practical applications?[br] [*]Can the principles of the Binomial Theorem be extended to non-binomial expressions, and what would be the implications of such an extension?[br] [*]How has the understanding and application of the Binomial Theorem evolved with the development of modern algebra and computational tools?[br] [/list][br] [/td][br][/tr][br][/table][br]
Mini-Investigation: Pascal's Triangle and the Binomial Theorem[br][br]Objective:Understand the relationship between Pascal's Triangle, binomial coefficients, and the binomial expansion.
Introduction:[br][br]Examine the image of Pascal's Triangle provided in the applet. Notice the sum of the two numbers directly above each number. [br][br]This triangle is more than a mathematical curiosity; it reveals patterns and relationships![br][br]Activity 1: Exploring Pascal's Triangle[br][br]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.[br]2. Edge Numbers: Observe the pattern of the numbers on the edge of the triangle. Predict the edge numbers for the next row.[br]3. Row Sums: Add up the numbers in each row. Look for a pattern related to powers of 2.[br]4. Triangle Symmetry: Determine if Pascal's Triangle is symmetrical and prove your findings.[br][br][br]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.[br][br]Explain how nCr can be used to find a particular value in Pascal's triangle
Activity 3: Connecting to the Binomial Expansion[br][br]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.[br]
Conclusion:Reflect on how Pascal's Triangle, nCr, and the binomial expansion are related. [br][br]Discuss the patterns discovered and their implications.[br][br]Extension:- [br][br]Explore diagonal patterns in Pascal's Triangle.[br][br]- Investigate the Fibonacci sequence within Pascal's Triangle.[br]- Determine the number of different pathways from the top to the bottom of the triangle.[br][br][br]Sharing Your Findings: After completing the investigation, share and discuss your findings. [br]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, [br]Binomial expansion - Basic [url=https://youtu.be/6hay4gEK-y8]https://youtu.be/6hay4gEK-y8[/url] Binomial expansion with negative term [url=https://youtu.be/qkkA8xkINVo]https://youtu.be/qkkA8xkINVo[/url] Binomial expansion past paper question [url=https://youtu.be/Pl5BGmxQYvs]https://youtu.be/Pl5BGmxQYvs[/url] Binomial expansion past paper question - challenging version [url=https://youtu.be/HugZvjtv4Io]https://youtu.be/HugZvjtv4Io[/url]
What is the coefficient of [math]x^2[/math] in the expansion of [math](2x-3)^4[/math] ?
What is the coefficient of [math]x^3[/math] in the expansion of [math](3x+2)^4[/math]?
What is the coefficient of [math]x[/math] in the expansion of [math](x-4)^5[/math] ?
What is the constant term in the expansion of [math](5x-1)^3[/math]?
What is the coefficient of [math]x^4[/math] in the expansion of [math](x+2)^5[/math]?
Part 3 - Exam-style questions
Q1-11 Practice style. [br]Section A Q12-54 Short style exam question[br]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)
