Application of Algebra

Collaborative Learning: Learners to work in convenient groups (ability, mixed-ability, mixed gender, etc.) to model real life problems involving quadratic inequalities and present answers.
Talk for Learning Approaches: learners brainstorm through think-pair-share, building on what others say, debates and discussions to explain how they model real life quadratic inequalities.
Project-based Learning: Engage learners to come up with graphs and solutions of quadratic inequalities modelled from real life contexts.
Experiential Learning: Engage learners in hands-on activity (learning by doing) by manipulating symbols, using excel spread sheets and ICT tools such as GeoGebra to sketch real life quadratic inequalities and explain their solution process.
Problem-based Learning: Learners investigate and explain real life models of quadratic inequalities and talk about what happens when specific conditions are varied/imposed.
Activity: Revise solutions to quadratic inequalities.
Use Talk for Learning Approaches, Problem-based Approaches, Experiential and Collaborative Learning Approaches to solve and revise solutions to quadratic inequalities in one and two variables.
Example:
- Graph the system of quadratic inequalities. 𝑦 < −𝑥 ) + 3 and 𝑦 ≥ 𝑥 ) + 2𝑥 − 3
Solution The solution region could also be found algebraically by substituting corresponding values for 𝑥 and 𝑦 in possible solution regions into both inequalities for testing We can use the origin, (0, 0) as a test point For 𝑦 < −𝑥 ) + 3, −𝑥 ) + 3 = −0 ) + 3 = 3 0 < 3 satisfying 𝑦 < −𝑥 ) + 3 and thus, (0, 0) falls in the solution region of 𝑦 < −𝑥 ) + 3
𝑦 ≥ 𝑥 ) + 2𝑥 − 3 𝑥 ) + 2𝑥 − 3 = 0 ) + 2(0) − 3 = −3 0 ≥ −3 satisfying 𝑦 ≥ 𝑥 ) + 2𝑥 − 3 and thus, (0, 0) falls is in the solution region Since (0, 0) lies in the solution region of both inequalities, the range of values of 𝑥 and 𝑦 in which (0, 0) falls in the solution region for the system

𝑅 " is the only solution region common to both inequalities. This implies that all points in that region, including but not exclusive to (0, 0), (−1.4, 0.6) (−1, 1), (0.5, 2), (1, 1) and (1, 1.6) are in the solution set for the system
- Use the graphs a and b to write an inequality in terms of 𝑓(𝑥) so point 𝑃 is a solution.
- Consider the graph of the function 𝑓(𝑥) = 𝑎𝑥 ) + 𝑏 + 𝑐. 

1. What are the solutions of 𝑎 ) + 𝑏 + 𝑐 𝑐 𝑐? 2. What are the solutions of𝑎 ) + 𝑏 + 𝑐 𝑐 𝑐? 3. The graph of 𝑔 represents a reflection in the 𝑥-axis of the graph of 𝑓. For which values of 𝑥 is 𝑔 positive?
Activity 2: Real-life Problems Involving Quadratic Inequalities. Use Talk for Learning Approaches, Problem-based Approaches, project-based learning, Experiential and Collaborative Learning Approaches to solve problems contextual real life problems.
Example Ama is instructed to make a garden plot which has an area less than 18m 2 . The length should be 3m longer than the width.
- How would you represent the width of the garden plot?
- What would be the mathematical sentence?
- What are the possible dimensions of the garden plot?
Teaching and Learning Resources:
- Graph paper
- Ruler
- A scientific calculator
Assessment (2.1.1.AS.5). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.