Number and… · SHS1 Additional Mathematics

Lesson notes

Learning Objectives

Indicator: 1.1.1.LI.6 - Use the combination approach and other approaches to determine the coefficient and exponent of a given term in an expansion.

By the end of the lesson, learners can:

State the general term formula for the expansion of (a + b)ⁿ using combination notation: T(r+1) = ⁿC(r) × a⁽ⁿ⁻ʳ⁾ × bʳ, where n is a positive integer.
Use the combination approach to find the coefficient of a specified term (e.g., the term in x⁴) in a binomial expansion without writing out the full expansion.
Determine the constant term of a binomial expansion by setting the exponent of the variable to zero and solving for r.
Apply the general term formula to identify both the coefficient and the exponent of a given term in expansions such as (1 + x)¹² and (2x + 3y)⁸.
Verify results using multiple approaches (full expansion, Pascal’s triangle from the previous lesson, and the combination formula) and explain why the combination method is more efficient for large powers.

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Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Number and Algebraic Patterns (1.1)
Content standard: 1.1.1.CS.1 - Demonstrate knowledge and understanding of binary operations, sets and binomial theorem and solve related problems in real life situations. 1.1.1.LO.1 Solve problems involving properties of binary operations. 1.1.1.LO.2 Model and solve real life problems on sets. 1.1.1.LO.3 Expand binomials with positive integral indices and simplify coefficients of the terms. 1.1.1.LO.4 Perform basic operations on surds as well as solve simple indicial and logarithmic equations.
Indicator: 1.1.1.LI.6 - Use the combination approach and other approaches to determine the coefficient and exponent of a given term in an expansion.
Suggested placement: Semester 1, Week 3 (Week 3 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

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Curriculum reference: NaCCA curriculum document, p. 51

Exemplars (from the NaCCA curriculum)

Collaborative Learning: Learners will be working in convenient groups (ability, mixed-ability, mixed gender, or pairs etc.) to explore the combination strategies for determining various terms of a binomial expansion with positive integer indices in different context.
Talk for Learning Approaches: Learners will brainstorm using participatory activities such as think-pair-share/square and debate to discuss the combination strategies for determining the various terms in a binomial expansion with positive integer indices in different contexts. Experiential Learning: Learners will collaboratively engage in hands-on activity (learning by doing) to create set problems and apply personal and conventional strategies to determine the coefficients of the terms in a binomial expansion with positive integer indices in different contexts.
Activity 1: Coefficients and terms in binomial expansions Use Talk for Learning Approaches (building on what others say, managing Talk for Learning, structuring Talk for Learning), collaborative learning approaches and experiential learning approaches to explore combination strategies for determining the coefficients of the terms in a binomial expansion with positive integer indices in different contexts.
Small group discussions/Think-pair-share: Learners in small convenient groups (e.g., mixed gender, mixed-ability, etc.) establish that if 𝑛 is a positive integer, the expansion (𝑎 + 𝑏) % is given by (𝑎 + 𝑏) % = [𝑛𝐶 ( 𝑎 % 𝑏 ( + 𝑛𝐶 ! 𝑎 %*! 𝑏 ! + 𝑛𝐶 ) 𝑎 %*) 𝑏 ) + ⋯ + 𝑛𝐶 % 𝑎 %*% 𝑏 % ]
which can also be written using the summation
% 𝑛 (𝑎 + 𝑏) = t ui j 𝑎 %*3 𝑏 3 v % 𝑟 34!
Example 1: Find the coefficient of 𝑥 1 in the expansion (1 + 𝑥) !( ) !( Example 2: Find the constant term in the expansion of i𝑥 − 0 ! j
Teaching and Learning Resources:
- Textbooks
- Rectangular paper cut-out
- Addition pyramid
- Curriculum
- Cardboards
- Reading resources
- Colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools.
Assessment (1.1.1.AS.6). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.
Provide appropriate responses (True or False) to the following tasks a) For every positive integer 𝑛, (3𝑛)! = (3)! (𝑛)! b) There are ten terms in the expression (1 − 𝑥) !( c) The middle term of the expansion of ! 1 i1 + 0 j is 70. Find the coefficient of the term 𝑥 2 𝑦 . in the binomial expansion of the expression (2𝑥 + 3𝑦) !, Show that 11 2 + 9 !! is divisible by 10