Back ] Up ] Next ] [Timeline]

Grade 9:  The Learning Equation Math

22.10 Factoring Polynomials with Tiles

2icon.jpg (3565 bytes)

Variables and Equations

Refresher pp 50-51

 

SeekAWord2ech.class Author

Prerequisite Skills:

Key Terms           

polynomial, trinomial, constant, variable, factor

Learning Outcomes:      Review     Enrichment      Key Terms     Prerequisite Skills

The student will:

Algebra Tile Review

Algebra tiles have a variety of other names: algetiles, math tiles, and virtual tiles.  Algebra tiles can help you visualize equivalent algebraic expressions and/or equations.  Many of the real tiles have a positive and a negative side. The virtual algebra tiles activities will use multi-coloured positive tiles. The reverse (negative) side of each of these tiles is red. It helps to understand that negative tiles have one positive and one negative dimension.

Some algebra kits use black tiles for positive areas and red for the flip (negative) sides. Click the "Black/Red Tiles" button for additional information.

 

Factoring Trionomials with and Without Algebra Tiles

This activity will create the polynomial, then show the intermediate steps to find the factors., If you have algebra tiles, build the equation generated.  "Click "Algebra Tiles" below to use the virtual tiles if you do not.  Make the appropriate changes in the tiles as you work through each of the steps in the solution.

 

  • check factoring using the equivalence of standard and factored forms of trinomials.

wpe6.jpg (3621 bytes)

=

wpe8.jpg (4218 bytes)

(x + 1)(x + 2) = x2 + 3x + 2
factored form = standard form

 

Review        Learning Outcomes       Enrichment        Key Terms        Prerequisite Skills

 

Algebra Tiles Pre Test

Vocabulary and ("translation")

Example(s)/Translation

Explanation

algebra ("bind together)

2x + 37y + 7

3x + 4 = 7

The study of mathematics using letters.  The terms are the "words" of an algebraic expression or equation.

terms

-12, 5, -4x0, 15x0y0,

6x4, 5x3, 4x2y, -x2,  x

Terms include numbers (numerical coefficients) and letters (variables).  Terms are connected by addition or subtraction.

monomial ("one term")

-12, 15x0y0, 6x4, -x2,  x

one term polynomial

longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes) = 3x

binomial ("two terms")

x2 + 3, 3y3 - 2x

two term polynomial

longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes) smlred.jpg (946 bytes)smlred.jpg (946 bytes) = 3x - 2

trinomial ("three terms")

5x3 + 2xy - 4

three term polynomial

bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes) longtall.jpg (1108 bytes) smlred.jpg (946 bytes)

= 2x2 + x - 1

polynomial ("many terms")

15x0y0

x2 + 3

5x3 + 2xy - 4

-6x4 + 4x3 - 7xy - 9

Polynomial can be used to represent any number of terms.   Polynomial is often used to represent more than three terms.

polynomial degree

6x5 + 5x3 + 3

3x2y3 + 3

The polynomial degree for both examples is 5.  The degree is determined by the term with the greatest power.  If there is more than one variable in a term, add the exponents to get the degree.

like terms

2x, 3x, -6x

Like terms have the same variable(s) and the same degree for each variable.

6x4, 4x4, -3x4
3x2y3,  5x2y3,  -2x2y3

adding like terms

longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)    longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)

2x + 3x

= (2+3)x = 5x

  • Add coefficients of like terms (use the Distributive Property): 

    ax + bx = (a + b)x

  • Watch your signs!!

  • Remember:

x = x1      1x = x       -1x = -x

 

bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes)   bigsqr.jpg (1371 bytes)

3x2 + x2

= 3x2 + 1x2 = (3+1)x2 =4x2

3x2y3 + 5x2y3

= (3+5)x2y3 = 8x2y3

subtracting like terms

redtall.jpg (1206 bytes)redtall.jpg (1206 bytes) - redtall.jpg (1206 bytes)redtall.jpg (1206 bytes)redtall.jpg (1206 bytes)redtall.jpg (1206 bytes)redtall.jpg (1206 bytes)redtall.jpg (1206 bytes)
-2x - 6x
 
= redtall.jpg (1206 bytes)redtall.jpg (1206 bytes) + longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)longtall.jpg (1108 bytes)
= -2x + 6x

= (-2+6)x = 4x

  • Subtract coefficients of like terms (use the Distributive Property:

ax - bx = (a - b)x

  • Watch your signs!!

  • Remember:

x = x1      1x = x        -1x = -x

 

bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes) - bigred.jpg (1379 bytes)bigred.jpg (1379 bytes)bigred.jpg (1379 bytes)
2x2 - 3x2
 
= bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes) + bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes)bigsqr.jpg (1371 bytes)
= 2x2 + 3x2

= (2+3)x2 = 5x2

3x2y3 - 4x2y3

= (3-4)x2y3 = -1x2y3

= -x2y3

 

 

Enrichment        Learning Outcomes       Review        Key Terms        Prerequisite Skills

 

 

Algebra Tile Practice:

Algebra Tile Basics

  • Use all of the pieces to make a rectangle.

  • To flip a rectangular tile, select it with the mouse and press the space bar.

Rules

  • Big squares can't touch little squares.
  • Little squares must all be together.
  • Only equal length sides may touch.
  • You may not lay two equally sized tiles of different colours next to each other.

AlgebraTilesPractice applet author:  Arlen Strader

 

Factoring Trinomials (Positive Coefficients)

Algebra Tiles - Factoring Trinomials

  • Use all of the pieces to make a rectangle.
  • To flip a rectangular tile, select it with the mouse and press the space bar.
  • Once you have correctly arranged the tiles into a rectangle, the factors of the quadratic are the length and width of the rectangle.

Rules

  • Big squares can't touch little squares.
  • Little squares must all be together.
  • Only equal length sides may touch.
  • You may not lay two equally sized tiles of different colours next to each other.

AlgebraTiles1 applet author:  Arlen Strader

 

Factoring Trinomials (Positive and Negative Coefficients)

  • Use all of the pieces to make a rectangle.
  • To flip a rectangular tile, select it with the mouse and press the space bar.
  • Once you have correctly arranged the tiles into a rectangle, the factors of the quadratic are the length and width of the rectangle.

Rules

  • Big squares can't touch little squares.
  • Little squares must all be together.
  • Only equal length sides may touch.
  • You may not lay two equally sized tiles of different color next to each other.

AlgebraTiles2 applet author:  Arlen Strader

 

For each of following use the color codes to help you find the roots of the following:

1x2 + 4x + 4 = 0

Quadratic Equation:
Ax2 + Bx +
C = 0
A =
B =
C =
X1 =
X2 =

 

 

Back ] Up ] Next ] [Timeline]

Comments to:  Jim Reed
Started September, 1998. Copyright 1999, 2000

Hit Counter visitors since September 3, 2000