Mathematics
Grade 8
15 min
Properties of multiplication
Properties of multiplication
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1
Introduction & Learning Objectives
Learning Objectives
Identify and define the commutative, distributive, identity, and zero properties of multiplication.
Apply the commutative property to demonstrate that the order of dimensions does not affect the area of a rectangle.
Utilize the distributive property to calculate the area of rectangles with algebraic expressions for dimensions.
Employ the distributive property to find the area of composite two-dimensional figures.
Simplify algebraic expressions representing areas of geometric figures using the properties of multiplication.
Explain how the identity and zero properties of multiplication apply to scaling and non-existent dimensions in 2D figures.
Ever wondered how architects calculate the area of complex buildings or how designers scale logos? 📐 It's no...
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Key Concepts & Vocabulary
TermDefinitionExample
Commutative Property of MultiplicationThe order in which two numbers are multiplied does not change the product.For a rectangle, Area = length × width = width × length.
Distributive Property of MultiplicationMultiplying a number by a sum (or difference) is the same as multiplying the number by each term in the sum (or difference) and then adding (or subtracting) the products.Area of a rectangle with length 4 and width (x+2) is 4(x+2) = 4x + 8.
Identity Property of MultiplicationThe product of any number and 1 is that number itself.If a figure is scaled by a factor of 1, its dimensions and area remain unchanged.
Zero Property of MultiplicationThe product of any number and 0 is 0.A rectangle with a width of 0 units has an area of 0 square units.
Two-Dimensional FigureA...
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Core Formulas
Commutative Property
$a \times b = b \times a$
This property states that the order of factors does not change the product. In geometry, it means you can multiply length by width or width by length to find the area of a rectangle, and the result will be the same.
Distributive Property
$a \times (b + c) = a \times b + a \times c$ and $a \times (b - c) = a \times b - a \times c$
This property allows us to multiply a factor by each term inside parentheses. It's crucial for calculating areas of composite figures or figures whose dimensions are expressed as sums or differences.
Identity Property
$a \times 1 = a$
Multiplying any number by 1 results in the original number. In geometry, this applies when scaling a figure by a factor of 1, meaning its size does not change...
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Challenging
A rectangular pool has a width of 'w' meters and a length that is 3 meters less than twice its width. Which expression represents the area of the pool and correctly applies a property of multiplication?
A.w(3 - 2w) = 3w - 2w²
B.w + (2w - 3) = 3w - 3
C.w(2w) - 3 = 2w² - 3
D.w(2w - 3) = 2w² - 3w
Challenging
A composite figure is made of three adjacent rectangles, all with a width of 'w'. Their lengths are 'x', 'y', and '5'. Which expression represents the total area of the figure and demonstrates the Distributive Property?
A.w + x + y + 5
B.w(x + y + 5)
C.5w(x + y)
D.wxy + 5
Challenging
A rectangular window has a length of (x+5) feet and a width of (y+2) feet. The total area can be found by distributing (x+5) across (y+2), resulting in (x+5)y + (x+5)2. A second application of the distributive property gives xy + 5y + 2x + 10. This process is equivalent to breaking the window into how many smaller rectangular areas?
A.Two rectangles with areas xy and 10.
B.Two rectangles with areas x(y+2) and 5(y+2).
C.Four rectangles with areas xy, 2x, 5y, and 10.
D.Three rectangles with areas x, y, and 10.
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