Grade 10 · Core Mathematics · Lesson 3 of 14

Deriving a Square Identity by Area

A Grade 10 Core Mathematics CBE lesson from the sub-strand Quadratic Expressions and Equations. In it, learners partition a square into meaningful regions.

Strand Numbers and AlgebraSub-strand Quadratic Expressions and EquationsLesson code 10CM-13-L340 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

Partition a square into meaningful regions.
Explain why two equal strips create the middle term.
Use the identity to evaluate a square.

Key vocabulary

Identity: An equation that remains true for every allowed value of its variable.; The four areas establish an identity for (n+11)².
Congruent: Having the same shape and size, so corresponding areas are equal.; The two n-by-11 strips are congruent rectangles.
Substitution: Replacing a variable with a stated value before calculating.; Substitution gives 7²+22×7+121=324, exactly matching 18².

Lesson preview

Slide 1 · Partition the enlarged squareOpening · 10 min

Figure 1: A square with side length n plus 11 is partitioned into four coloured regions to show their individual areas.

A square with side n+11 can be split into an n-by-n square, two n-by-11 strips, and an 11-by-11 corner. Areas add to the whole. The partition shows why two equal strips produce the doubled middle term.

Mark the n and 11 lengths on both sides.
Name the central square n².
Name both congruent strips 11n.
Name the corner 121.

Worked example. The central region has area n². Each strip has area 11n, so together they give 22n. The corner has area 11×11=121. Answer: (n+11)²=n²+22n+121..

Check question: When partitioning a square of side length $n + 11$ into four regions, what are the individual areas of the two rectangular strips?

$11 n$ and $11 n$
$n^{2}$ and $121$
$n$ and $11$
$22 n$ and $22 n$

Show answer

$11 n$ and $11 n$ . Each rectangular strip has a length of $n$ and a width of $11$ . The area of each strip is found by multiplying its dimensions, yielding $11 \times n = 11 n$ . Therefore, the two strips each have an area of $11 n$ .

Slide 2 · Explain the doubled middle termModel · 18 min

Figure 2: An area diagram illustrates how two rectangular strips, each with an area of 11n, combine.

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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