Mathematics
Grade 8
15 min
Multiply polynomials using algebra tiles
Multiply polynomials using algebra tiles
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the different types of algebra tiles and their corresponding algebraic values.
Set up an area model grid using algebra tiles to represent polynomial multiplication.
Determine the product of two polynomial terms by correctly filling the area model with appropriate tiles, considering their signs.
Combine like terms represented by the product tiles, including identifying and eliminating zero pairs, to simplify the resulting polynomial.
Multiply a monomial by a binomial using algebra tiles.
Multiply a binomial by a binomial using algebra tiles.
Explain the connection between the algebra tile method and the distributive property.
Ever wondered how to multiply expressions like (x+2)(x+3) without just memorizing rules? 🤔 Algebra tiles make it visual...
2
Key Concepts & Vocabulary
TermDefinitionExample
Algebra TilesConcrete manipulatives used to represent variables and constants, helping visualize algebraic expressions and operations.Small yellow square = 1, green rectangle = x, blue square = x². Red tiles represent negative values.
MonomialA polynomial with only one term.$3x$, $5$, $-2x^2$
BinomialA polynomial with two terms.$x+2$, $2y-5$
PolynomialAn expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.$x^2 + 3x - 4$
Area ModelA visual method for multiplying expressions, where the factors represent the dimensions of a rectangle and the product represents its area.A rectangle with sides $x$ and $2$ has an area of $2x$, represented by two 'x&...
3
Core Formulas
Tile Value Representation
Small square = $1$ (yellow for positive, red for negative). Long rectangle = $x$ (green for positive, red for negative). Large square = $x^2$ (blue for positive, red for negative).
These are the fundamental values of the algebra tiles used to build expressions and represent products. Always use the correct color to denote positive or negative values.
Multiplication of Tiles (Area Model Principle)
The product of two tiles is the tile that completes the rectangle formed by placing them along the top and left edges of a grid.
This rule guides how to fill the interior of the area model. For instance, an 'x' tile (green rectangle) multiplied by an 'x' tile (green rectangle) results in an 'x²' tile (blue square).
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Challenging
The area of a completed algebra tile model represents the polynomial x^2 - 4x + 3. If the tiles along the left edge represent the factor (x - 1), what tiles must be along the top edge?
A.One green x-tile and three yellow 1-tiles, representing (x+3).
B.One green x-tile and three red 1-tiles, representing (x-3).
C.One red x-tile and three yellow 1-tiles, representing (-x+3).
D.One green x-tile and four red 1-tiles, representing (x-4).
Challenging
The product of two binomials is represented by a total of 10 tiles in the area model: one blue x^2-tile, some green x-tiles, and some yellow 1-tiles. If the polynomial product is x^2 + 6x + 3, is this possible? If the product is x^2 + 5x + 4, what were the factors?
A.Impossible; Factors are (x+2)(x+2)
B.Possible; Factors are (x+1)(x+3)
C.Impossible; Factors are (x+1)(x+4)
D.Possible; Factors are (x+2)(x+3)
Challenging
A student multiplies (x - 3)(x - 3) and incorrectly gets the answer x^2 - 9. They correctly found the x^2-tile and the 1-tiles, but what two major errors did they make in their process?
A.They forgot the x-terms, and they thought negative times negative is negative.
B.They used the wrong color for the x^2 tile, and they added the constants instead of multiplying.
C.They forgot the x-terms, and they used red x-tiles instead of green.
D.They set up the grid with (x+3) instead of (x-3), and they removed zero pairs incorrectly.
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