Transcript Let`s Do Algebra Tiles

Let’s Explore
Algebra Tiles
Simplifying Polynomials, Distributive
Property, Substitution, Solving
Equations, Multiplying & Dividing
Polynomials and Factoring
Modeling
Polynomials
Modeling Polynomials
 Algebra
tiles can be used to
model expressions. 600.10.35;
700.10.25
aid
in the simplification of
expressions. 700.10.40; 6.EE.3; 6.EE.4
Modeling Polynomials
=1
=x
= -1
=-x
= x2
= - x2
Modeling Polynomials
1) 2x + 4
2) -3x + 1
Modeling Polynomials
3) 2x2 – 5x -4
Simplifying
Polynomials
Students need to use the same idea
of zero pairs with variables
Simplifying Polynomials
1) 2x + 4 + x + 2
simplified: 3x + 6
2) -3x + 1 + x + 3
simplified: -2x + 4
More Polynomials
try:

3) 3x + 1 – 2x - 4
This process can be used with
problems containing x2.
(2x2 + 5x – 3) + (-x2 + 2x + 5)
More Polynomials
How would you show/demonstrate:
1) (3x + 5) – (2x + 2)?
2 ) (2x2 – 2x + 3) – (3x2 + 3x – 2)?
Substitution
Using Algebra Tiles for evaluating expressions
600.10.25; 6.EE.1; 6.EE.2
Substitution

Algebra tiles can be used to model
substitution.
Represent original expression with tiles.
 Then replace each rectangle with the
appropriate tile value.
 Combine like terms.

For example:
3 + 2x
let x = 4
Substitution
3 + 2x
Therefore when x=4,
3 + 2x = 11
let x = 4
Substitution
3 + 2x
Simplify
Therefore when x=-4,
3 + 2x = -5
let x = -4
Substitution
How would you show/ demonstrate?
3 - 2x
let x = 4
3 - 2x
let x = -4
Distributive Property
Using Algebra Tiles to demonstrate
the Distributive Property
600. 60.65(numbers only); 800.60.30; 6.EE.3
Distributive Property
Use the same concept that was applied
with multiplication of integers, think of the
first factor as the counter.
 The same rules apply.
3(x+2)
 Three is the counter, so we need three
rows of (x+2).

Distributive Property
3(x + 2)
simplified 3x + 6
Distributive Property
3(x - 2)
simplified 3x - 6
Distributive Property
Try these:
1. 3(x – 4)
2.
-2(x + 2)
3.
-3(x – 2)
Solving Equations
Using Algebra Tiles to show the
steps for solving equations
Solving Equations

Algebra tiles can be used to explain and
justify the equation solving process. The
development of the equation solving model
is based on two ideas.


Equations are unchanged if equivalent
amounts are added to each side of the
equation.
Variables can be isolated by using zero
pairs.
Equations are unchanged if
equivalent amounts are added to
each side of the equation.
x+2=3
Show using symbols
x +2=3
-2
-2
x=1
Solving Equations
2x – 4 = 8
Show using symbols
2x  4  8
 4  4
2x 12

2
2
x6
Solving Equations
2x + 3 = x – 5
2x  3  x - 5
Show using symbols
- x  -x
x  3  -5
 3  3
x-8
Algebra tiles
Questions at this point?
How can you use this in your
classroom?
Advanced
Polynomials
Using Algebra Tiles in higher level
math courses
More Advanced Polynomials
 Algebra
tiles can also be used to:
 Multiply polynomials,
 Divide polynomials, or
 Factor polynomials.
Multiplying Polynomials
(x + 2)(x + 3)
x+3
Does it matter which factor
goes on top and which
factor goes on the side?
x+2
(x + 2)(x + 3)=x2+5x+6
Multiplying Polynomials
(x + 2)(x + 3)
x+2
x+3
(x + 2)(x + 3)=x2+5x+6
Multiplying Polynomials
(x – 1)(x +4)
(x – 1)(x +4)=x2+3x-4
Multiplying Polynomials
Try:
(x + 2)(x – 3)
(x – 2)(x – 3)
Dividing Polynomials

Algebra tiles can be used to divide
polynomials.
Use tiles and frame to represent
problem. Dividend should form array
inside frame. Divisor will form one of
the dimensions (one side) of the
frame.
 Be prepared to use zero pairs in the
dividend.

Dividing Polynomials
x2 + 7x +6
= x+6
x+1
Dividing Polynomials
x2 + 5x +6
x+2
Dividing Polynomials
x2 + 5x +6
x+2
Dividing Polynomials
x2 + 5x +6
x+2
Dividing Polynomials
x2 + 5x +6 = x+3
x+2
Dividing Polynomials
Try:
x2 - 5x +6 = x-3
x-2
Dividing Polynomials
Try:
x2 - 5x -6 = x-6
x+1
Factoring Polynomials
Algebra tiles can
be used to factor
polynomials. Use
tiles and the frame
to represent the
problem.
 Use the tiles to fill
in the array so as
to form a rectangle
inside the frame.

3x + 3
2x – 6
Factoring Polynomials
x2 + 6x + 8
We need
to make a
rectangle
that uses
all of the
Algebra
tiles
Factoring Polynomials
x2 + 6x + 8 = (x+2)(x+4)
Factoring Polynomials
x2 – 5x + 6 = (x-2)(x-3)
Factoring Polynomials
x2 – x – 6 (harder) = (x+2)(x-3)
Factoring Polynomials
x2 - 1 (even harder) = (x+1)(x-1)
Factoring Polynomials
Try these:





x2 + x – 6
x2 – 4
2x2 – 3x – 2
2x2 + 3x – 3
-2x2 + x + 6
Questions???????