Learn on PengiBig Ideas Math, Course 1Chapter 3: Algebraic Expressions and Properties

Lesson 4: The Distributive Property

In this Grade 6 lesson from Big Ideas Math, Course 1, students learn how to apply the Distributive Property to multiply sums and differences, using it as a mental math strategy and to simplify algebraic expressions such as 4(n + 5) = 4n + 20. The lesson covers both numerical examples with whole numbers and fractions and algebraic applications involving expressions like a(b + c) = ab + ac and a(b - c) = ab - ac. Students also connect the property to real-life contexts by writing and simplifying variable expressions.

Section 1

Multiplying with an Area Model

Property

To multiply using an area model, decompose one factor by place value to determine the dimensions of the partitioned rectangle.
The total product is the sum of the partial products (the areas of the smaller sections).
This visually represents the distributive property:

a×(b+c+d)=(a×b)+(a×c)+(a×d)a \times (b + c + d) = (a \times b) + (a \times c) + (a \times d)

Examples

Section 2

Distributive Property with Variables

Property

When multiplying a number by a sum or difference in parentheses, you can distribute the multiplication to each term inside the parentheses.

For algebraic expressions:

a(b+c)=ab+aca(b + c) = ab + ac
a(b−c)=ab−aca(b - c) = ab - ac

Section 3

The Distributive Property and Factoring

Property

The distributive property of multiplication over addition states that for any numbers AA, BB, and CC:

A(B+C)=AB+ACA(B+C) = AB + AC
This property is also used to factor expressions.
For example, you can use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor, like expressing 36+836+8 as 4(9+2)4(9+2).

Examples

  • To expand 6(y+4)6(y+4), distribute the 66 to both terms inside: 6⋅y+6⋅46 \cdot y + 6 \cdot 4, which simplifies to 6y+246y+24.
  • To factor 21x+1421x+14, find the greatest common factor (GCF), which is 77. Rewrite the expression as 7(3x)+7(2)7(3x) + 7(2), which factors to 7(3x+2)7(3x+2).
  • To mentally calculate 8×238 \times 23, think of it as 8(20+3)8(20+3). Distribute to get 8⋅20+8⋅3=160+24=1848 \cdot 20 + 8 \cdot 3 = 160 + 24 = 184.

Explanation

The distributive property lets you 'share' a multiplication across terms inside parentheses. Multiplying a number by a group is the same as multiplying the number by each part of the group individually and then adding the results.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Multiplying with an Area Model

Property

To multiply using an area model, decompose one factor by place value to determine the dimensions of the partitioned rectangle.
The total product is the sum of the partial products (the areas of the smaller sections).
This visually represents the distributive property:

a×(b+c+d)=(a×b)+(a×c)+(a×d)a \times (b + c + d) = (a \times b) + (a \times c) + (a \times d)

Examples

Section 2

Distributive Property with Variables

Property

When multiplying a number by a sum or difference in parentheses, you can distribute the multiplication to each term inside the parentheses.

For algebraic expressions:

a(b+c)=ab+aca(b + c) = ab + ac
a(b−c)=ab−aca(b - c) = ab - ac

Section 3

The Distributive Property and Factoring

Property

The distributive property of multiplication over addition states that for any numbers AA, BB, and CC:

A(B+C)=AB+ACA(B+C) = AB + AC
This property is also used to factor expressions.
For example, you can use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor, like expressing 36+836+8 as 4(9+2)4(9+2).

Examples

  • To expand 6(y+4)6(y+4), distribute the 66 to both terms inside: 6⋅y+6⋅46 \cdot y + 6 \cdot 4, which simplifies to 6y+246y+24.
  • To factor 21x+1421x+14, find the greatest common factor (GCF), which is 77. Rewrite the expression as 7(3x)+7(2)7(3x) + 7(2), which factors to 7(3x+2)7(3x+2).
  • To mentally calculate 8×238 \times 23, think of it as 8(20+3)8(20+3). Distribute to get 8⋅20+8⋅3=160+24=1848 \cdot 20 + 8 \cdot 3 = 160 + 24 = 184.

Explanation

The distributive property lets you 'share' a multiplication across terms inside parentheses. Multiplying a number by a group is the same as multiplying the number by each part of the group individually and then adding the results.