Visual models of identities

The remaining area is (a^2-b^2), and its rectangle form is ((a+b)(a-b)). In exams identify sides after rearranging the cut part.

(4x^2=(2x)^2), (9y^2=(3y)^2), and the middle term is negative. In exams connect the sign with tile color or direction.

The middle term is ((4+9)x=13x), so (p=13). In exams the sum of strips in a rectangle model gives the middle coefficient.

The total area of the two rectangles is (2\cdot3a\cdot2b=12ab). In exams find the middle area of a perfect square using the (2ab) pattern.

(9m^2=(3m)^2) and (25n^2=(5n)^2), so it forms a difference of squares. In exams first find the sides of both squares.

Two (5x) strips are removed and the (5) by (5) overlap is added back. In exams the last term in a subtraction square is the square of the strip width.

Factors with opposite signs form a rectangle with area (4x^2-49). In exams connect the sum-difference form with a rectangle.

Each side of the larger square is \(x+y\), so its area is \((x+y)^2\). Adding the tile areas gives \(x^2+xy+xy+y^2=x^2+2xy+y^2\). In \((x-y)^2\), the middle term is negative. Exam tip: two \(xy\) tiles indicate the term \(2xy\).

Three squares and six rectangles make (a^2+b^2+c^2+2ab+2bc+2ca). In exams count each pair of rectangles.

((x+3)(x+5)=x^2+8x+15) and ((x+4)^2=x^2+8x+16), so the difference is (-1). In exams compare nearby rectangles and squares by expansion.

(49=7^2) and (-14x=-2\cdot x\cdot7), so the side is (x-7). In exams use both the middle term and the last term to identify a perfect square.

Since (5y+(-4y)=y) and (5y\cdot(-4y)=-20y^2), the sides are (x+5y) and (x-4y). In exams add tiles with signs.

The corner is ((2q)^2=4q^2), and the middle area is (2\cdot5p\cdot(-2q)=-20pq). In exams the corner area can remain a positive square.

The cells of ((2x+3)(4x+5)) are (8x^2), (10x), (12x), and (15). In exams multiply rows and columns in the grid.

The large square has side a+b. Its area is a²+ab+ab+b² = a²+2ab+b², so two ab rectangles are essential. In exams, identify the two cross-shaped rectangles.

First (x^2+2xy+y^2=(x+y)^2), then (z^2) is subtracted. In exams first group the large part as a perfect square.

The square model gives (2\cdot2x\cdot z=4xz) and (2\cdot3y\cdot z=6yz). In exams each pair appears twice in a three-part square.

The corner gives (k^2=81), and (2kx=18x), so (k=9). In exams use both clues to fix the constant side.

(16x^2=(4x)^2), (25y^2=(5y)^2), and the middle term is (-2\cdot4x\cdot5y). In exams check all three parts of a perfect square.

((3x+4)(x-5)=3x^2-15x+4x-20=3x^2-11x-20). In exams add positive and negative tiles separately.