Transcript x + 2 - SDmaths

USING ALGE-TILES
IN MATHEMATICS
Adapted from: SOURCE: maths.slss.ie/resources/Algebra Tiles
Full Show.ppt
Presented by:
Baisden
Kenneth
John
Roopchan
Alge-Tiles
For all Alge-Tile work it is essential to
remember that
RED means minus
and
Any other colour means plus.
E.g.
+1
-1
= 0
What are AlgeTiles?
Coloured Tiles that
can be (are) used as resources for
developing students’ understanding
of Algebra
Defining the Variables
1
x2
x
-1
-x2
-x
N.B.: The width of the x-tile is assumed to be 1 which for our
purposes do not visually connect equitably to the variable x
Example
Represent 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 )
Answer x2+5x+1
A CONCRETE IDEA FOR CHANGING SIGNS.
Practice
Simplify the following:
Simplify the following:
1) 6-7
10)2-8-1
2) 3-2-4-1
11)-5-1-4+1
3) 5x2+2x
12)x2+2
4) 2x2+4x+2x2-x
13)x2+5x+x2-2x
5) 3x2-2x+4+x2-x-2
14)2x2-x+1 - (2x2-2x-5)
6) x2-3x-2-x2-2x+4
15)x2- 2x2-2x+4 - (x2+2x+3)
7) 2x2-2x-1-3x2-2x-2
16)3x2-4x+2 - (x2+2)
8) x2+2x+1- 3x2-x
17)x2+x-2 - 2(x2+2x-3)
9) x2-x+3-2x2+2x+x2-2x-5
18)-4x-3 - (2x2-2x-4)
Multiplying & Factorising
General 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
Practice
Multiply 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
Section 5. Factorising Quadratic
Trinomials
- a geometrical approach
Review Multiplication
Again
Factors and Area
Show (x+1)(x+3) by arranging the tiles in a rectangle.
x
+
3
Now Arrange them into a
Rectangle
x
+
Remember the little guys go in the
1
bottom right corner
Rearrange the tiles to show the expansion:
x2
+
4x
+
3
How it works
Factorise x 2 + 6x + 8
x2
+
6x
+
8
To factorise this expression form a rectangle with the pieces.
x
+
4
x
+
2
The factors are
( x + 4 )( x + 2 )
Factorise x2+6x+8
Show (x+3)(x-1) by arranging the tiles in a rectangle.
x
+3
x
1
NOTE: REDS ARE NEGATIVE
NOW COMPLETE THE RECTANGLE WITH NEGATIVE SQUARES
Rearrange the tiles to show the expansion:
x2
(x+3)(x-1)
+ 3x
-1x
-3
= x2 + 2x - 3
Factorise x 2 - 4x + 3
x2
- 4x
+3
x-3
x-1
The factors are
( x - 3 )( x - 1 )
Factorise x2-4x+3
Factorise x 2 - x - 12
x2
-12
-x
?
Clearly there is no way to accommodate the
in in
Zero
the form
of hand
+x and –x.
12 You
smalladd
guys
theinbottom
right
corner.
Whatdoing
do you
do?complete the rectangle.
And Keep
it to
Factorise x2-x-12
Factorise x 2 - x - 12
x-4
x+3
The factors are ? ( x + 3 )( x - 4 )
Factorise x2-x-12
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
Practice
Solve 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