TANGRAM

About TangramTangrams 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

Pola untuk tangram

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

The Algebra Grid

Sumber dari Dr.Cresencia LaguertaAteneo de Naga University Philippines

The Algebra grid visualizes1. 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

xy

a b

Length of segments

xy

x + y 2x

Length of segments

ab

2b + 3a

a + 2b

Unit of area :

• Let and x y

x2 xy

y2

Area of squares

x2 y2

(2x)2

(x+2y)2

Area of rectangels

2x2

3xy (2x2+3xy)

Challenge.

What geometric figure can visualize the following algebraic expressions?• 2x + 3y• 8y2

• 10y2

• 6xy• X+5y• 9xy• 6y+3x

Challenge..

• 4x2 + 2xy• 5xy + 6y2

• X2 +5xy +6y2

• 10y2 + 17xy + 3x2

• 3x2 + 5xy + 2y2

Area of Rectangels

X2 + 2xy+y2 x2 + 4xy+3y2

2x2 + 4xy+4y2

Challenge..

• 4x2 + 2xy• 12 xy+ 6x2

• 4x2 +12xy+9y2

• 10y2 + 17xy+3x2

The Algebra Grid

The Algebra Grid

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-3x2

• 6xy + 12 y2

• 5y2 – 15xy– Quadratic trinomial• X2 +5xy+6y2 • 5x2 +11xy+2y2

• X2 – 4xy – 5y2 • 2x2 + 3xy – 2y2

…

3. Proving algebraic identities– (x+y)2 =x2 + 2xy + y2 – (x+y)2 + (x-y)2 =2x2 + 2y2

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)