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Factoring Trinomials using Algebra Tiles
Student Activity
Materials:
• Algebra Tiles (student set)
• Worksheet: Factoring Trinomials using Algebra Tiles
Algebra Tiles:
Each algebra tile kits should contain three different sizes of tiles:
x2 tile
x2
1 by x tiles
1 by 1 tiles
x
1
The red tiles represent negative quantities and all other colors represent positive
quantities. The length of an “x” tile is unknown since “x” is a variable. Therefore,
one cannot use 1x1 tiles to match the length of an “x” tile.
Terminology:
Term: A number, a variable, or a product or a quotient of numbers and / or
variables. Terms are separated by plus signs. For example, in 2x2 + 3x + 1, the
terms are 2x2, 3x, and 1.
Polynomial: An expression with more than one term. Examples: 2x + 3,
x2 + 2x + 1
Binomial: A polynomial with two terms. Example: 2x + 1.
Trinomial: A polynomial with three terms. Example: 2x2 + 3x + 1.
Product: Two or more numbers or terms that are multiplied. Example: 3(x + 2).
Factor: To factor means to rewrite an expression as a product.
Example: 3x + 6 = 3(x + 2) The parts of a product are called factors.
Example: 3 and (x + 2) are factors of 3x + 6.
Prime: If an expression cannot be factored, we say that it is prime.
Lesson:
Demonstration
1. a. In order to factor a trinomial such as x2 + 3x + 2, you first begin by
gathering your tiles:
1 - x2 tile
x2
3 - 1 by x tiles
x
2 - 1 by 1 tiles
1
b. You next arrange the tiles to form a perfect rectangle. For example,
x2 + 3x + 2 forms the following rectangle.
x2
x
x
x
1
1
d. The length of the above rectangle is (x + 2) and the width is (x + 1). The
area of a rectangle is length x width. Therefore, x2 + 3x + 2 = (x + 2) (x + 1).
Guided Practice
2. Now in groups of three, try factoring x2 + 5x + 6. What tiles do you need to
gather?
Now form a rectangle using these tiles.
What is the length of the rectangle?
What is the width of the rectangle?
What is the factored form for the area of the rectangle?
How do the numbers 5 and 6 relate to the factored form?
To check an answer, we multiply the two binomials. To multiply two
binomials, we distribute one term at a time. For example:
(x + 3) (x + 2) = x(x + 2) + 3(x + 2) = x2 + 2x + 3x + 6
We now add like terms 2x + 3x to get 5x. Therefore, our final answer is
x2 + 5x + 6, which matches what we started with in the beginning.
Practice
3. Now try factoring the following trinomials on your own using algebra tiles
and fill in the factored answer. Check your answers. Remember that some
trinomials do not factor. If it does not factor, we say that it is prime.
Trinomial
x2 + 2x + 3
x2 + 4x + 4
x2 + 8x +12
x2 + 12x + 36
x2 + 11x + 24
x2 + 17x + 42
2x2 + 11x + 12
3x2 + 14x +15
6x2 + 19x +15
Factored Answer
Reflection: Pattern Recognition
List at least three patterns that you noticed when doing the check?
Pattern 1:
Pattern 2:
Pattern 3:
Work in groups of three to determine at least three patterns you noticed when
forming rectangles with the algebra tiles.
Pattern 1:
Pattern 2:
Pattern 3:
Suppose x2 + bx + c = (x + m)(x + n). How are m and n related to b? How are m
and n related to c?
Lecture/Demonstration: Using red or negative Algebra Tiles
Remember that red tiles represent negative quantities. Also remember that
1 + (-1) = 0. Adding zero to something is referred to as the identity property of
addition since it doesn’t change the value of the original quantity. Therefore, we
can always add equal quantities of black and red tile of the same size since they
add to zero. Sometimes when factoring trinomials using algebra tiles, we need to
add equal quantities of black and red tiles. Example: When factoring x2 – 3x – 4
we need the following tiles:
1 - x2 tile
x2
3 - 1 by x tiles(red)
x
4 - 1 by 1 tiles (red)
1
However, we cannot construct a rectangle using these tiles. But if we add one more
red 1 by x tile and one more black 1 by x tile then we can construct a rectangle as
shown below.
Red
1 by x
Red
1 by x
Red
1 by x
Red
1 by x
Red x
Red x
Red x
Red x
x2
Black 1 by x
Therefore, x2 – 3x – 4 = (x – 4)(x + 1).
Signed Numbers:
Remember the following rules when dealing with signed numbers.
Addition:
1. If the signs are the same, add the numbers and keep the sign.
2. If the signs are different, subtract the numbers and keep the sign of the larger
number.
Subtraction:
1. Change to addition and use the rules for addition.
Multiplication and Division:
1. If the signs are the same, the answer is positive.
2. If the signs are different, the answer is negative.
Practice:
Try factoring the following in groups of three using algebra tiles. Check your
answers by multiplying. Remember to add equal quantities of black and red tiles if
needed.
Trinomial
x2 - 5x - 6
x2 - 7x + 6
x2 + 5x - 6
x2 - 7x - 8
x2 - x - 20
x2 – 4x - 21
x2 – 11x + 18
x2 – 2x - 15
x2 + 2x - 15
x2 - 7x +12
3x2 + x - 4
2x2 + x - 6
2x2 - 9x - 5
3x2 + x - 2
6x2 - x - 2
Factored Answer
When you are done, please fill out your evaluation form.