SHS1 Additional Mathematics · Semester 1 Week 14: Applications of Algebra

Lesson notes

Learning Objectives

Indicator: 1.1.2.LI.12 - Describe polynomial functions and perform the basic arithmetic operations on them.

By the end of the lesson, learners can:

Identify a polynomial function by its form and state its degree, leading coefficient, terms, and constant term using correct vocabulary.
Evaluate a polynomial function at a given input using function notation, such as finding f(2) or g(-1).
Add and subtract two polynomial functions by combining like terms and write the result in descending order of powers.
Multiply two polynomial functions using the distributive property (each term of the first multiplies each term of the second) and simplify the result.
Divide a polynomial function by a linear divisor using long division and express the result in the form q(x) + R, where R is the remainder.

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Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Applications of Algebra (1.2)
Content standard: 1.1.2.CS.1 - Demonstrate knowledge and understanding of applying algebraic processes and reasoning involving sequence, functions, and linear programming. 1.1.2.LO.1 Examine, analyse, determine and predict other terms in a pattern/sequence. 1.1.2.LO.2 Distinguish among various types of relations, find the domain and range of, and evaluate functions. 1.1.2.LO.3 Show that a function is injective (into) and/or surjective (onto). Find the inverse, describe the relationship between two variables and establish composite functions. 1.1.2.LO.4 Graph linear and quadratic functions and determine the intercepts. 1.1.2.LO.5 Find graphical and algebraic solutions to a system of three linear equations in three variables and apply them to solve real life problems. 1.1.2.LO.6 Perform algebraic manipulations on polynomial functions and graph polynomial functions. 1.1.2.LO.7 Find the domain, range, and zero of a rational function and state when it is undefined. 1.1.2.LO.8 Identify and describe the order of a matrix, the identity matrix and the zero matrix; find the determinant and perform basic arithmetic operations on 2 by 2 matrices (addition and subtraction).
Indicator: 1.1.2.LI.12 - Describe polynomial functions and perform the basic arithmetic operations on them.
Suggested placement: Semester 1, Week 14 (Week 14 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.111: 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
Curriculum reference: NaCCA curriculum document, p. 111

Exemplars (from the NaCCA curriculum)

Activity 1: Evaluate a polynomial in function notation.
Collaborative Learning: Learners work in groups/pairs to recall key facts about linear and nonlinear functions.
- Learners examine different forms of functions, classify them and explain their reasons for the classification.
- Learners recognise that polynomial functions comprise various combinations of constants, variables, and exponents and are of the form:
- 𝑃(𝑥) = 𝑎 % + 𝑏 %*! + 𝑐 %*) + 𝑑 %*" + ⋯ . . +𝑘 where 𝑎, 𝑏, 𝑐 𝑎 𝑑 are constant and 𝑛 > 2
Example: 𝑔(𝑥) = 4𝑥 , − 𝑥 " − 9𝑥 ) + 2𝑥 − 5
Note: Examples of types of polynomial functions are linear, quadratic, cubic, quartic and quintic functions. Learners explore the terminology of polynomial functions, including working with vocabularies such as co-efficient, leading co-efficient, term leading term and degree exponents, and constants and then analyse them by putting them all together;
Activity 2: Perform basic arithmetic operations on polynomial functions.
Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning Approaches and Problem-based learning:
Collaborative Learning: Learners work in groups/pairs to explore arithmetic operations (addition, subtraction, multiplication and division) on polynomial functions.
Example 1: If 𝑓(𝑥) = 4𝑥 , − 𝑥 " − 9𝑥 ) + 2𝑥 − 5 and ℎ(𝑥) = 𝑥 " + 3𝑥 ) − 2𝑥 + 2;
- Addition: 𝑓(𝑥)+ ℎ(𝑥) = 4𝑥 , − 6𝑥 ) − 3
- Subtraction:𝑓(𝑥) − ℎ(𝑥) = 4𝑥 , − 2𝑥 " − 12𝑥 ) + 4𝑥 − 7
Example 2: If 𝑓(𝑥) = 𝑥 ) + 𝑥 − 6 𝑔(𝑥) = 𝑥 + 3
- Multiplication: 𝑓(𝑥)× 𝑔(𝑥) = 𝑥 " + 4𝑥 ) − 3𝑥 − 18 I(0) 0 ! $0*/
- Division: = = 𝑥 − 2 J(0) 0$"
I(0) Generally: = 𝑞(𝑥) + 𝑅 J(0)
Activity 3: Examine the graph of polynomial functions.
Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning Approaches and Problem-based Learning:
Collaborative Learning: Learners work in convenient groups to identify, examine and construct graphs of polynomial functions up to degree 2 and describe their nature.
Collaborative Learning, Experiential Learning, and Problem-based Learning: Learners construct functions by hand and with an appropriate technology and investigate the minimum and maximum values and the intercepts. Learners work in groups/pairs to come out with two or more polynomial functions with degrees up to 5. Then, have them swap with a different group/pair. The new group/pair should evaluate the functions at a given value. Then, they should swap back and ensure they were accurate according to the creators. (Learners debate with each other to agree or disagree with the outcome of the task)
Teaching and Learning Resources:
- Cut out shapes of different colours and orientation
- Maths Technology learning apps, tools and devices
- Chalkboard illustrations
- Worksheets
- Cut-out geometrical shapes of different colours and orientation
Assessment (1.1.2.AS.12). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.