lets explore algebra tiles simplifying polynomials, distributive property, substitution, solving...
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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
= -1
= x
= - 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 + 62) -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 expressions600.10.25; 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 let x = 4
Therefore when x=4,
3 + 2x = 11
Substitution
3 + 2x let x = -4
Simplify
Therefore when x=-4,
3 + 2x = -5
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 Property600. 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
6x2
12
2
2x
44
8 42x
Solving Equations
2x + 3 = x – 5
Show using symbols
8 -x
33
-53x
-xx -
5-x32x
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 + 2)(x + 3)=x2+5x+6
x+3
x+2
Does it matter which factor goes on top and which factor goes on the side?
Multiplying Polynomials
(x + 2)(x + 3)
(x + 2)(x + 3)=x2+5x+6
x+3
x+2
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 + 1= x+6
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 + 2= x+3
Dividing Polynomials
x2 - 5x +6
x - 2= x-3
Try:
Dividing Polynomials
x2 - 5x -6
x + 1= x-6
Try:
Factoring Polynomials
3x + 3
2x – 6
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.
Factoring Polynomials
x2 + 6x + 8We 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???????