SHS1 Additional Mathematics · Semester 1 Week 12: Applications of Algebra

Lesson notes

Learning Objectives

Indicator: 1.1.2.LI.8. Recognise and use appropriate algebraic notation properties of linear and non-linear functions and solve them simultaneously.

By the end of the lesson, learners can:

classify equations as linear or non-linear by examining the highest power of the variable(s) and justify their classification.
solve a system of one linear and one non-linear equation simultaneously using the substitution method, showing each step clearly.
solve a system of two linear equations simultaneously using either substitution or elimination, and verify solutions.
interpret the point(s) of intersection of two graphs as the simultaneous solution of the corresponding equations.
set up and solve a pair of simultaneous equations (one linear, one non-linear) from a real-life situation involving quantities and prices.

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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.8 - Recognise and use appropriate algebraic notation properties of linear and non-linear functions and solve them simultaneously.
Suggested placement: Semester 1, Week 12 (Week 12 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.107: 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. 107

Exemplars (from the NaCCA curriculum)

Activity 1: Investigate situations that lead to linear and non-linear equations.
Collaborative Learning: Present several functions/equations (linear and non-linear equations) in algebraic and graphical forms and have learners investigate and classify them as linear or non-linear and justify their groupings.
Examples: Differences between linear and non-linear equations.
Linear Equations Non-Linear Equations forms a straight line or represents the equation for the It does not form a straight line but straight line forms a curve. It has only one degree. It has the degree as 2 or more than 2 Lines formed in the X-Y plane can be extended in any If we increase the value of the degree, direction but in a straight form. the curvature of the graph increases. The general representation is; The general representation is; y = mx +c 𝑎𝑥 ) + 𝑏𝑦 ) = 𝑐
Where x and y are the variables, m is the slope of the Where 𝑥 and 𝑦 are the variables and line, and c is a constant value. 𝑎, 𝑏 and 𝑐 are the constant values
Examples: Examples: a)10x = 1 a) x 2 +y 2 = 1 b) 9y + x + 2 = 0 b) x 2 +12xy+y 2 = 0 c) 4y = 3x c)x 2 +x+2=25 d) 99x + 12 = 23 y
Activity 2: Solving linear and non-linear equations.
Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning Approaches and Problem-based Learning:
Collaborative Learning: Learners work in convenient groups to solve systems of linear and non-linear equations.
Example 1: Solve the linear equation 3𝑥 + 9 = 2𝑥 + 18.
Solution: Given, 3𝑥 + 9 = 2𝑥 + 18 ⇒ 3𝑥 - 2𝑥 = 18 - 9 ⇒ 𝑥 = 9
Example 2: Solve the nonlinear equation 𝑥 + 2𝑦 = 1 and 𝑥 = 𝑦.
Solution: Given, 𝑥 + 2𝑦 = 1; 𝑥 = 𝑦
By putting the value of x in the first equation we get, ⇒ 𝑦 + 2𝑦 = 1 ⇒ 3𝑦 = 1 ⇒ 𝑦 = ∴ 𝑥 = 𝑦 = ! ! " "
Example 3: Solve the nonlinear equation 𝑥 ) + 𝑥 = 2 and 10𝑥 = 1
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.8). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.