Applications of Algebra

Collaborative Learning: Learners will be working in convenient groups (ability, mixed-ability, mixed gender etc.) to solve problems and present answers related to functions and relations.
Talk for Learning Approaches: Learners will brainstorm through think-pair-square and debate and discuss the types and features of functions and relations.
Experiential Learning: Learners will collaboratively engage in hands-on activity (learning by doing) to create functions.
Problem-based Learning: Learners will collaboratively undertake research inquiry functions in real life.
Activity 1: Recognising relations and functions in mathematics.
Talk for Learning: Learners discuss the historical antecedent of functions, tracing it from Rene Descartes in 1637 through Gottfried Wilhelm Leibniz, Leonhard Euler, Nicolas Bourbáki, John Tate
etc. Learners discuss the importance of functions as the building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies, keeping aeroplanes in the air, etc.
Learners work in groups to answer the question, "What is your understanding of a function?" Each group should agree upon and present one meaning on a poster and discuss the concept of function in terms of a function's definition, type, and general properties.
Example 1: Learners discuss within groups which of the relations listed below are examples of functions and justify their answers to the class.

 (5, 6) y =−x 2 (3, 2)
(5, 1) (ii) (iii) (v) (iv) (i)
Example 2: Groups investigate the following problems and make presentations to the class justifying which can be classified as functions. Let A = (a, b, c}, B = {4, 5, 6}, and f = {(a, 6), (b, 4), (c, 6)}. Is f a function from A to B? Why? Let A = {1, 2, 3}, B = {c, d, e}, and g = {(1, d), (2, c), (1, e)}. Is g a function from A to B? Why? Let M be the set of all museums, N the set of all countries, and 𝐿 = {(𝑚, 𝑛) ∈ 𝑀 × 𝑁 the museum m is in the country n}. Is L a function from M to N?
Learners reflect on their informal work in trying to describe what a function is and formalize the definition of function, taking notice of the use of the symbols in the definition.
Note: Three examples of how a function might be defined are:
Given two sets 𝐴 and 𝐵, the set 𝐴 × 𝐵 consists of all ordered pairs (𝑎, 𝑏) where 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵. A subset of 𝐴 × 𝐵 is called a relation. Thus:
- A function from 𝐴 to 𝐵 is a pairing of elements in A with elements in 𝐵 in such a way that each element in 𝐴 is paired with exactly one element in 𝐵.
- A function 𝑓 from 𝐴 to 𝐵 is a rule or relation between 𝐴 and 𝐵 that assigns each element 𝑎 ∈ 𝐴 to a unique element 𝑏 ∈ 𝐵.
- A function 𝑓 from 𝐴 to 𝐵 is a subset of the Cartesian product 𝐴 × 𝐵 = {(𝑎, 𝑏) |𝑎 𝐴, 𝑏 𝐵} such that 𝑏 is unique for each 𝑎 ∈ 𝐴}.
Activity 2: Use of mapping diagrams to establish various types of relations and establish connections between relations and functions.
Learners working in convenient groups observe different graphical representations and explanations for relations and draw conclusions. 

Conclusion: All functions are relations, BUT not all relations are functions. A function is a type of relation that assigns ONE output to each input.
Problem-based/Research Work: Have learners investigate the various types of functions (absolute functions, identity function, quadratic, cubic functions etc.) study the nature of their graphs and make presentations on them.
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.3). 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.