Grade 9 · Grade 9: Algebra, Quadratics and Functions · Algebraic Structure, Quadratics and Growth
Expression Structure, Equations and Systems
Rewrite and factorise expressions, create equations from constraints and solve systems including inequalities.
Learning objectives
- Interpret the structure of expressions and factorise quadratics.
- Use the difference of two squares.
- Create equations and inequalities to model constraints.
- Solve systems of equations and graph systems of inequalities.
AERO Mathematics alignment
AERO.M9.AE.1
Interpret the structure of expressions and rewrite them in equivalent forms, including factorising quadratics and using the difference of two squares.
AERO.M9.AE.2
Create equations and inequalities in one or two variables to represent relationships, and use them to solve problems and graph solution sets.
AERO.M9.AE.3
Solve systems of linear equations exactly and approximately, and graph solutions of systems of linear inequalities in two variables.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A rectangular garden has area x² + 7x + 12 square metres.
Think about it
What could its side lengths be, and how does factorising reveal them?
Hint: Find two numbers that multiply to 12 and add to 7.
EXPLAIN
Structure, equations and systems
Factorised form shows the zeros and the possible dimensions: x² + 7x + 12 = (x + 3)(x + 4), so the sides are x + 3 and x + 4.
| Pattern | Form | Example |
|---|---|---|
| Common factor | ab + ac = a(b + c) | 6x + 9 = 3(2x + 3) |
| Simple trinomial | x² + (p+q)x + pq = (x+p)(x+q) | x² + 7x + 12 |
| Difference of two squares | a² − b² = (a − b)(a + b) | x² − 25 = (x − 5)(x + 5) |
| Perfect square | a² + 2ab + b² = (a + b)² | x² + 6x + 9 |
Creating equations from constraints
Turn each condition into a statement. A taxi charging 5 plus 2 per km with a 30 budget gives 2d + 5 ≤ 30, so d ≤ 12.5 km.
Systems
Solve systems exactly by substitution or elimination. For systems of inequalities, shade each region and the overlap is the feasible region: every point in it satisfies all constraints.
Where this is used
Feasible regions are the basis of resource planning, from production schedules to diet and budget optimisation.
INVESTIGATION
Try it yourself
Write two constraints for a real situation, such as time and money limits for a school event. Graph both inequalities, shade the feasible region and identify one workable plan inside it.
Watch
How to Solve Quadratic Equations by Factoring
The Organic Chemistry Tutor · 10:00
Factorising is the structural skill behind this whole lesson.
Before you watch: What two numbers multiply to 12 and add to 7?
- Factorise x² − 9 and name the pattern.
- Create an inequality for: a taxi charges 5 plus 2 per km and you have 30.
Interactive simulation · PhET
Area Model Algebra
Use the area model to expand and then factorise quadratic expressions.
While you explore
- How does the area model show (x + 3)(x + 5)?
- What does the difference of two squares look like as a rectangle?
Key vocabulary
- Feasible region
- The area on a graph where all conditions are satisfied.
- Difference of two squares
- A pattern where one square is subtracted from another.
Practice questions
0/1 correct
Level 1 · Criterion A
Factorise x² − 49.
MYP criterion tasks
Level 2 · Criterion D
A taxi charges 5 QAR plus 2 QAR per km and you have 30 QAR. Write and solve an inequality for the distance.
Reflect & track
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