Alge-Tiles

For all Alge-Tile work it is essential to remember that

RED means minus

And

Any other colour means plus.

1

Variables

x2

-x2

x

-x-1

ExampleRepresent the following trinomials using alge-tiles:

1. 2x2+3x+5

2. x2-2x-3

Alge-Tile Uses

Algebra tiles can be used for (among other things):

Section 1. Identifying ‘like’ and ‘unlike’ terms

Section 2. Adding and Subtracting Integers

Section 3. Simplifying Expressions

Section 4. Multiplying in algebra

Section 5. Factorising trinomials

Section 6. Doing linear equations

Section 1. Like Terms

Example 1. 4x+5

Can any of these be added ? Explain your answer

Example 2. 4x+5x

Can any of these be added ? Explain your answer

Section 2. Adding and Subtracting Integers

Example 1. 4-7

Example 2. –3-6

Section 3. Adding and Subtracting Trinomials

Example 1. 2x2+3x+5 + x2-5x-1

Section 3. Adding and Subtracting Trinomials

Example 1. 2x2+3x+5 + x2-5x-1

Answer 3x2-2x+4

Section 3. Adding and Subtracting Trinomials

Example 2. 2x2+3x+2 - ( x2-2x+1 )

-

A CONCRETE IDEA FOR CHANGING SIGNS.

Section 3. Adding and Subtracting Trinomials

Example 2. 2x2+3x+2 - ( x2-2x+1 )

A CONCRETE IDEA FOR CHANGING SIGNS.

Section 3. Adding and Subtracting Trinomials

Example 2. 2x2+3x+2 - ( x2-2x+1 )

A CONCRETE IDEA FOR CHANGING SIGNS.

Answer x2+5x+1

Practice

Simplify the following:

1) 6-7

2) 3-2-4-1

3) 5x2+2x

4) 2x2+4x+2x2-x

5) 3x2-2x+4+x2-x-2

6) x2-3x-2-x2-2x+4

7) 2x2-2x-1-3x2-2x-2

8) x2+2x+1- 3x2-x

9) x2-x+3-2x2+2x+x2-2x-5

Simplify the following:

10)2-8-1

11)-5-1-4+1

12)x2+2

13)x2+5x+x2-2x

14)2x2-x+1 - (2x2-2x-5)

15)x2- 2x2-2x+4 - (x2+2x+3)

16)3x2-4x+2 - (x2+2)

17)x2+x-2 - 2(x2+2x-3)

18)-4x-3 - (2x2-2x-4)

Multiplying & FactorisingGeneral Aim

•Whether multiplying or factorising, the general aim is to generate a rectangle and have no pieces left over.

•Also the small squares always go in the bottom right hand corner

Section 4. Multiplying in algebra

Example 2. Multiply (x-1)(x-3)

Answer: x2-4x+3

PracticeMultiply the following:

1) x(x+3)

2) 2(x-5)

3) 3x(x-1)

4) (x+4)(x+3)

5) (x-1)(x+2)

6) (x-4)(x-2)

7) (3x-1)(x-3)

8) (x-1)(x-1)

9) (2x+1)2

10) (x-2)2

Factors and Area

- a geometrical approach

Review Multiplication Again

Section 5. Factorising Quadratic Trinomials

Show (x+1)(x+3) by arranging the tiles in a rectangle.

x

x 3

1

+

+

Rearrange the tiles to show the expansion:

x 2 + 4x + 3How it works

Now Arrange them into a Now Arrange them into a RectangleRectangle

Remember the little guys go in the bottom right corner

x 2 + 6x + 8

To factorise this expression form a rectangle with the pieces.

x + 4

x

+

2

( x + 4 )( x + 2 )

The factors are

Factorise x 2 + 6x + 8

Factorise x2+6x+8

x + 3

x-

1

NOTE: REDS ARE NEGATIVE

NOW COMPLETE THE RECTANGLE WITH NEGATIVE SQUARES

Show (x+3)(x-1) by arranging the tiles in a rectangle.

Rearrange the tiles to show the expansion:

x2 + 3x -1x - 3 = x2 + 2x - 3(x+3)(x-1)

Factorise x 2 - 4x + 3

x2 - 4x + 3x - 3

x - 1

The factors are ( x - 3 )( x - 1 )

Factorise x2-4x+3

Factorise x 2 - x - 12

x2 - x -12

Factorise x2-x-12

Clearly there is no way to accommodate the 12 small guys in the bottom right hand corner. What do you do?

?You add in Zero in the form of +x and –x.

And Keep doing it to complete the rectangle.

Factorise x 2 - x - 12

Factorise x2-x-12

The factors are ? ( x + 3 )( x - 4 )

x - 4

x + 3

Section 6. Doing linear equations

Solve 2x + 2 = -8

=

Section 6. Doing linear equations

Solve 2x + 2 = -8

=

Section 6. Doing linear equations

Solve 2x + 2 = -8

=

=

=

Section 6. Doing linear equations

Solve 2x + 2 = -8

=

=

Solution x = -5

Section 6. Doing linear equations

=

Solve 4x – 3 = 9 + x

You can take away the same thing from both sides

Section 6. Doing linear equations

=

Solve 4x – 3 = 9 + x

You can add the same quantity to both sides

Section 6. Doing linear equations

=

Solve 4x – 3 = 9 + x

Section 6. Doing linear equations

=

Solve 4x – 3 = 9 + x

=

=

Section 6. Doing linear equations

=

Solve 4x – 3 = 9 + x

=

=

Solution x = 4

PracticeSolve the following:

1) x+4 = 7

2) x-2 = 4

3) 3x-1 =11

4) 4x-2 = x-8

5) 5x+1 = 13-x

6) 2(x+3) = x-1

7) 2x-4 = 5x+8