Application of Algebra · SHS2 Additional…

Lesson notes

Learning Objectives

Indicator: 2.1.1.LI.3 - Use the expansion for (1-x)^n or (1+x)^n to approximate exponential numbers.

By the end of the lesson, learners can:

Expand expressions of the form (1 + x)^n and (1 - x)^n for a positive integer n using the binomial theorem.
State the condition |x| < 1 required for a binomial expansion to be valid when n is negative or fractional.
Rewrite a decimal number close to 1 in the form (1 + x) or (1 - x) to enable approximation.
Use a binomial expansion to approximate powers of numbers such as (0.98)^8 correct to a stated number of decimal places.
Justify the choice of how many terms of the expansion to use in order to achieve a required level of accuracy.

Sign in with your phone number to read the full note and download the GES plan - free.

Sign in with phone number

Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Application of Algebra (1.1)
Content standard: 2.1.1.CS.1 - Demonstrate the ability Establish De Morgan's laws of set theory and use it to solve related problems. to use sets properties and theories to solve real Pedagogy and Learning Experience life problems and apply Collaborative Learning: Learners work in convenient groups (ability, mixed-ability, mixed gender, or binomial theorem to approximate numbers to a given index. 2.1.1.LO.1 Investigate De Morgan's law on sets algebraically and graphically, formulate and solve real life problems up to three sets. 2.1.1.LO.2 Model sequence recursively and explicitly, and establish the relationship between the two forms, as well as solve real life problems involving linear and exponential sequences and series. 2.1.1.LO.3 Apply indices and logarithms to solve real life problems, including logarithms with different bases, and sketch and interpret logarithmic functions. 2.1.1.LO.4 Formulate and derive appropriate strategies to solve quadratic inequalities. 2.1.1.LO.5 Graph systems of given inequality and identify the region that provides the feasible solution and apply it to real life situations. 2.1.1.LO.6 Determine the set of values for which a rational function is defined and resolve rational functions into partial fractions. 2.1.1.LO.7 Multiply matrices, determine the inverse of a 2 x 2 matrix, find the determinant up to a 3 x 3 matrix and represent matrices in linear transformations.
Indicator: 2.1.1.LI.3 - Use the expansion for (1-x)^n or (1+x)^n to approximate exponential numbers.
Suggested placement: Semester 1, Week 2 (Week 2 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:

exemplars - p.280: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
exemplars - p.281: a stacked fraction, matrix, vector or other two-dimensional construct is flattened by the text layer; all visible parts require comparison with the rendered page
Curriculum reference: NaCCA curriculum document, p. 280

Exemplars (from the NaCCA curriculum)

Pedagogy and Learning Experience Collaborative Learning: Learners work in convenient groups (ability, mixed-ability, mixed gender, or pairs etc.) to determine the terms in a given binomial expansion.
Talk for Learning Approaches: Learners brainstorm using think-pair-share/square and debates to talk about and justify the nature and or signs of the terms for a given binomial expansion.
Experiential Learning: Learners collaboratively (pair and group work) work together, create and solve binomial expansions and apply them in real life contexts.
Activity: Expand binomial expressions with a given exponent.
Using Talk for Learning Approaches (Building on what others say, managing Talk for Learning, Structuring Talk for Learning) and Collaborative learning approaches, learners adopt the combination
method and other methods to determine the terms, coefficient and exponent of a given term in an expansion.
Examples
- Expand (1 − 2𝑥) 1 and use the expansion to approximate (0.98) 1 correct to 3 decimal places Solution: (8)(7)(−2𝑥) ) (1 − 2𝑥) 1 = 1 − 8(2𝑥) + 2 (8)(7)(6)(−2𝑥) " + + ⋯ + (−2𝑥) 1 5 (0.98) 1 = (1 − 0.02) 1
- Expand (1 + 𝑥) !) in ascending powers of x up to the fifth term.
Teaching and Learning Resources:
- ICT tools and resources
- Worksheets
- Scientific calculator
- Technological tools, apps, etc.
- SHS Additional Mathematics Curriculum
Assessment (2.1.1.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.