Free Grade 3 Introducing Completing the Square Lesson Plans
Free Grade 3 lesson plans for Introducing Completing the Square: 12 real questions with a full answer key and worked solutions. A quadratic has an x² term. Factorise by finding two numbers that multiply to the constant and add to the x-coefficient; each bracket gives a solution.
Free
Core
30 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- UnderstandExplain introducing completing the square
Explain introducing completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate introducing completing the square
Calculate introducing completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare introducing completing the square
Compare introducing completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- EvaluateJustify introducing completing the square
Justify introducing completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
Prerequisites
Close these gaps first — each has its own free lesson, worksheet and quiz.
How Introducing Completing the Square works
A quadratic has an x² term. Factorise by finding two numbers that multiply to the constant and add to the x-coefficient; each bracket gives a solution.
12 practice questions
Easy to hard, 24 marks in total. Try them first, then open the answers.
- 1.Factorise x² + 0x − 81.[1]
- 2.Factorise x² − 7x + 6.[1]
- 3.Factorise x² − 9x + 18.[1]
- 4.Factorise x² − 13x + 42.[1]
- 5.Solve x² + 1x − 2 = 0.[2]
- 6.Solve x² + 2x − 48 = 0.[2]
- 7.Solve x² − 1x − 6 = 0.[2]
- 8.Solve x² − 3x − 54 = 0.[2]
- 9.Solve x² − 2x − 63 = 0.[3]
- 10.Solve x² + 9x + 14 = 0.[3]
- 11.Solve x² − 12x + 27 = 0.[3]
- 12.Solve x² + 3x − 18 = 0.[3]
Show answers and working
1. (x − 9)(x + 9)
Find two numbers that multiply to -81 and add to 0: -9 and 9. So x² + 0x − 81 = (x − 9)(x + 9).
2. (x − 6)(x − 1)
Find two numbers that multiply to 6 and add to -7: -6 and -1. So x² − 7x + 6 = (x − 6)(x − 1).
3. (x − 3)(x − 6)
Find two numbers that multiply to 18 and add to -9: -3 and -6. So x² − 9x + 18 = (x − 3)(x − 6).
4. (x − 7)(x − 6)
Find two numbers that multiply to 42 and add to -13: -7 and -6. So x² − 13x + 42 = (x − 7)(x − 6).
5. x = -2 or x = 1
Factorise: (x + 2)(x − 1) = 0. One bracket must be zero: x = -2 or x = 1.
6. x = -8 or x = 6
Factorise: (x + 8)(x − 6) = 0. One bracket must be zero: x = -8 or x = 6.
7. x = -2 or x = 3
Factorise: (x + 2)(x − 3) = 0. One bracket must be zero: x = -2 or x = 3.
8. x = 9 or x = -6
Factorise: (x − 9)(x + 6) = 0. One bracket must be zero: x = 9 or x = -6.
9. x = 9 or x = -7
Factorise: (x − 9)(x + 7) = 0. One bracket must be zero: x = 9 or x = -7.
10. x = -2 or x = -7
Factorise: (x + 2)(x + 7) = 0. One bracket must be zero: x = -2 or x = -7.
11. x = 9 or x = 3
Factorise: (x − 9)(x − 3) = 0. One bracket must be zero: x = 9 or x = 3.
12. x = -6 or x = 3
Factorise: (x + 6)(x − 3) = 0. One bracket must be zero: x = -6 or x = 3.
Worked examples
Factorise x² + 3x − 4.
- Find two numbers that multiply to -4 and add to 3: -1 and 4.
- So x² + 3x − 4 = (x − 1)(x + 4).
- Answer: (x − 1)(x + 4)
Solve x² − 2x − 8 = 0.
- Factorise: (x + 2)(x − 4) = 0.
- One bracket must be zero: x = -2 or x = 4.
- Answer: x = -2 or x = 4
Solve x² − 7x − 8 = 0.
- Factorise: (x + 1)(x − 8) = 0.
- One bracket must be zero: x = -1 or x = 8.
- Answer: x = -1 or x = 8
Interactive practice
Free, no account required — answer on screen or print it.
Common mistakes
What learners typically get wrong, and how to fix it.
Gives x = 3 from (x + 3) = 0.
Correction: x + 3 = 0 means x = −3.
Gets the signs muddled when factorising.
Correction: Check by expanding the brackets.
Teacher tips
- · Use a product–sum grid for finding the pair.
Parent tips
- · Ask what two numbers multiply to 12 and add to 7.
Real-life applications
- · Projectile paths, maximising area.
Assessment objectives
How mastery is evidenced, and which standards it maps to.
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Frequently asked
- Is this Grade 3 Introducing Completing the Square lesson plan really free?
- Yes. Every lesson plans on FreeMathWorksheet is free to use, print and share — no account, no paywall and no watermark.
- What should a learner already know before this?
- Start with Factorising Quadratics. Each one has its own free lesson, worksheet and quiz.
- How is the lesson plan sequenced?
- Recognition first, then understanding, application, reasoning and finally mastery — the same ladder used across every free resource on the platform.
- What comes next after Introducing Completing the Square?
- Move on to Working with Completing the Square, Completing the Square in Context, Expanding Double Brackets, Factorising Quadratics, which build directly on this idea.
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