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Grade 10 · Core Mathematics · Lesson 4 of 14

Restoring a Difference-Square Area

A Grade 10 Core Mathematics CBE lesson from the sub-strand Quadratic Expressions and Equations. In it, learners represent subtraction in an area model.

Strand Numbers and AlgebraSub-strand Quadratic Expressions and EquationsLesson code 10CM-13-L440 minutes · 3 slides

KICD learning outcome (b)

Derive the quadratic identities from the concept of area.

By the end of this lesson, learners should be able to

  1. Represent subtraction in an area model.
  2. Explain the positive overlap correction.
  3. Evaluate a difference-square identity.

Key vocabulary

Overlap
A region shared by two shapes or actions.
The 13-by-13 corner is the overlap of the two removed strips.
Correction
An adjustment that repairs an overcount or undercount.
The positive 169 corrects the corner removed twice.
Remaining side
The length left after a stated part is removed.
When q=29, the remaining side q−13 is 16.

Lesson preview

Slide 1 · Remove two stripsOpening · 10 min
A large square of side length q showing two overlapping strips of width thirteen and their shared corner circled in red.
Figure 1: A large square of side length q showing two overlapping strips of width thirteen and their shared corner circled in red.

A square forest of side q requires two q-by-13 firebreak strips removed along its edges to prevent wildfires. Noticing their overlap is removed twice explains the remaining area (q−13)². Use Figure 1 to see the strips' corner overlap.

  • Start with the full q-by-q square.
  • Remove one strip of area 13q.
  • Remove the perpendicular strip of area 13q.
  • Mark the shared 13-by-13 corner.
Worked example. Begin with q². Subtract 13q twice. Add back 13×13=169 because the corner was subtracted twice. Answer: (q−13)²=q²−26q+169..

Check question: When deriving the identity for (q−13)2 using the area of a square of side q, what are the areas of the two rectangular strips that are initially removed?

  1. 13q and 13q
  2. 13 and 13
  3. q2 and 13q
  4. 13q and 169
Show answer

13q and 13q. Each strip spans the full side length q of the original square and has a width of 13. Therefore, the area of each strip is calculated as length × width, giving 13q.

Slide 2 · Explain the positive correctionModel · 18 min
A four-step sequence showing how subtracting two rectangular strips from a large square removes their shared corner twice.
Figure 2: A four-step sequence showing how subtracting two rectangular strips from a large square removes their shared corner twice.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • All 3 slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 13 lessons in this sub-strand
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All 14 lessons in Quadratic Expressions and Equations

  1. 1Building Quadratic Expressions from DimensionsForm quadratic expressions from different situations
  2. 2Four Products on a NoticeboardForm quadratic expressions from different situations
  3. 3Deriving a Square Identity by AreaDerive the quadratic identities from the concept of area
  4. 4Restoring a Difference-Square AreaDerive the quadratic identities from the concept of area
  5. 5Nearby Bases for Mental SquaringApply quadratic identities in numerical cases
  6. 6Factor Pairs and GroupingFactorise quadratic expressions from different situations
  7. 7Forming an Area EquationForm quadratic equations in different situations
  8. 8A Consecutive-Integer EquationForm quadratic equations in different situations
  9. 9Solving by FactorisationSolve quadratic equations by factorisation
  10. 10Exact Roots with Signed FactorsSolve quadratic equations by factorisation
  11. 11A Triangular Sign ModelApply quadratic equations to real life situations
  12. 12Planning a Rectangular TankApply quadratic equations to real life situations
  13. 13Exploring Quadratic Models for a Community MuralExplore the use of quadratic equations in real life situations
  14. 14Exploring a Community Greenhouse FrameExplore the use of quadratic equations in real life situations