alge-tiles for all alge-tile work it is essential to remember that red means minus and any other...
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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