Mathematics
Grade 8
15 min
Distributive property
Distributive property
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1
Introduction & Learning Objectives
Learning Objectives
Define the distributive property and explain its purpose in simplifying expressions.
Apply the distributive property to multiply a monomial by a binomial or polynomial.
Correctly handle negative signs when distributing a number or variable.
Simplify algebraic expressions by first applying the distributive property and then combining like terms.
Identify and correct common errors made when using the distributive property.
Recognize when the distributive property can be used in reverse to factor out a common monomial.
Ever wonder how to efficiently calculate the total cost for multiple friends buying different items at the same store? 🛍️ The distributive property helps us 'share' multiplication across addition or subtraction!
In this lesson, you&...
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Key Concepts & Vocabulary
TermDefinitionExample
Distributive PropertyA property that states multiplying a number by a sum or difference is the same as multiplying that number by each term in the sum or difference and then adding or subtracting the products.3(x + 2) = 3x + 6
TermA single number, a variable, or a product/quotient of numbers and variables. Terms are separated by addition or subtraction signs.In the expression 5x + 7y - 2, the terms are 5x, 7y, and -2.
ExpressionA mathematical phrase that can contain numbers, variables, and operation symbols, but no equality or inequality signs.4(y - 3) or 4y - 12
CoefficientThe numerical factor of a term that contains a variable.In the term 7x, 7 is the coefficient. In the term -y, -1 is the coefficient.
Like TermsTerms that have the same variables raised to the same...
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Core Formulas
Distributive Property (Addition)
$a(b+c) = ab + ac$
To multiply a number 'a' by a sum (b+c), multiply 'a' by 'b' and 'a' by 'c', then add the results.
Distributive Property (Subtraction)
$a(b-c) = ab - ac$
To multiply a number 'a' by a difference (b-c), multiply 'a' by 'b' and 'a' by 'c', then subtract the second result from the first.
Distributive Property (Extended)
$a(b+c+d) = ab + ac + ad$
The distributive property can be extended to any number of terms inside the parentheses. Multiply 'a' by each term within the parentheses.
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Challenging
A student tried to simplify 10 - 3(x - 2). Their first step resulted in 7(x - 2). What is their error?
A.They distributed the 7 instead of the 3.
B.They subtracted 3 from 10 before distributing, which violates the order of operations.
C.They should have added 10 and 3 first.
D.They made a sign error when distributing the -3.
Challenging
If a(4x - 5) = 12x + b, what are the values of 'a' and 'b'?
A.a = 3, b = 15
B.a = 3, b = -15
C.a = 4, b = -5
D.a = 12, b = -5
Challenging
Simplify the expression: 2x(3y - 4) - 5y(x - 1)
A.xy - 8x + 5y
B.xy - 3x - 5y
C.xy - 8x - 5y
D.11xy - 8x - 5y
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