About Tangram Tangrams come from China. They are thousands of years old. The Tangram is made by cutting a square into seven pieces. The puzzle lies in using all seven pieces of the Tangram to make birds, houses, boats, people and geometric shapes. In each case you have to use all the seven pieces - no more, no less. Tangrams have fascinated mathematicians and lay people for years. You might be wondering why only the solutions are given. Well, you could just blacken the white lines to create problems! Watch out as Tangrams are known to be addictive. With these Seven Little Wonders the whole family can have hours of fun!

Pola untuk tangram

| Dari tangram tersebut buat menjadi bentuk- bentuk yang memungkinkan.

The Algebra Grid Sumber dari Dr.Cresencia Laguerta Ateneo de Naga University Philippines

The Algebra grid visualizes 1.Concept of – Algebraic expression – Similar terms 2.Product of – A monomial and binomial – Two binomials with similar terms 3.Factoring/Factorizing a – Product with a common monomial factor (CMF) – Quadratic trinomials

Some preliminary concepts Unit of length Unit of area

Unit of length xyabxyab

Length of segments x y x + y 2x

Length of segments a b 2b + 3a a + 2b

Unit of area : Let and x y x2x2 xy y2y2

Area of squares x2x2 y2y2 (2x) 2 (x+2y) 2

Area of rectangels 2x 2 3xy (2x 2 +3xy)

Challenge. What geometric figure can visualize the following algebraic expressions? 2x + 3y 8y 2 10y 2 6xy X+5y 9xy 6y+3x

Challenge.. 4x 2 + 2xy 5xy + 6y 2 X 2 +5xy +6y 2 10y xy + 3x 2 3x 2 + 5xy + 2y 2

Area of Rectangels X 2 + 2xy+y 2 x 2 + 4xy+3y 2 2x 2 + 4xy+4y 2

Challenge.. 4x 2 + 2xy 12 xy+ 6x 2 4x 2 +12xy+9y 2 10y xy+3x 2

The Algebra Grid

Uses of the algebra grid 1.Finding product of a monomial and a binomial – 2x(3x+2y) – 4x(5x-3y) Two binomials with similar term – (2x+3y)(4x-2y) – (x+2y)(3x+y) – (x-4)(2x-3y)

…. 2. Factorizing/factoring – Polynomial with a common monomial factor 2xy-3x 2 6xy + 12 y 2 5y 2 – 15xy – Quadratic trinomial X 2 +5xy+6y 2 5x 2 +11xy+2y 2 X 2 – 4xy – 5y 2 2x 2 + 3xy – 2y 2

… 3. Proving algebraic identities – (x+y) 2 =x 2 + 2xy + y 2 – (x+y) 2 + (x-y) 2 =2x 2 + 2y 2

Example 1 Finding product : 2x (3x + 2y) Consider : The factor 2x as the width and 3x+2y as the length of a rectangle The product 2x(3x+2y) as the area of the recangle

Finding product of two polynomials is finding the area of a rectangle

2x(3x+2y) =

(4x+y)(2x-3y) =

(x-4y)(2x-3y) =

Practice : Product of a monomial and a polynomial X (x+y)X(x-y)-x(x-y) 2x(x+y)2x(x-y)-2x(x-y) 3x(2x+5y)3x(2x-4y)-3x(2x+y) 2x(3x+2y)2x(3x-2y)-2x(-3x+2y) 5y(2x+3y)5y(2x-3y)-6y(2x-5y)

Practice : Product of two binomial with similar term (x+y)(x+2y)(X-y)(x-2y)(x-y)(x+2y) (2x+2y)(2x+3y)(x-2y)(x-3y)(x+2y)(x-3y) (2x+3y)(3x+2y)(2x-y)(3x-2y)(2x+y)(3x-2y) (2x+3y)(3x+4y)(3x-2y)(x-4y)(3x+2y)(x-4y) (3y+x)(7y+3x)(2x-3y)(3x-4y)(2x-3y)(3x+4y)

Practice X (x+y)X(x-y)-x(x-y) 2x(x+y)2x(x-y)-2x(x-y) 3x(2x+5y)3x(2x-4y)-3x(2x+y) 2x(3x+2y)2x(3x-2y)-2x(-3x+2y) 5y(2x+3y)5y(2x-3y)-6y(2x-5y)