English:Using the Distributive Property
Using the Distributive Property
Introduction
The distributive property is one of the most useful rules in arithmetic and algebra. It lets you multiply a number or expression by every term inside parentheses. You can use it to rewrite expressions, make mental calculations easier, simplify algebra, factor expressions, and solve equations.
For Grades 7–8, the key idea is not just to memorize a rule. You should understand why the rule works, recognize when it applies, use signs correctly, and explain why two expressions are equivalent.
The rectangle above gives a visual model of the rule. One large rectangle can be split into two smaller rectangles. The total area stays the same whether you calculate the area of the whole rectangle at once or add the areas of the two parts.
Learning Goals
By the end of this aiMOOC, you should be able to explain the distributive property in words and symbols, expand expressions with positive and negative numbers, combine distribution with like terms, factor out a common factor, use the property when solving linear equations, and check whether two equivalent expressions have the same value.
You should also be able to connect symbolic algebra to an area model and explain your reasoning clearly.
What the Distributive Property Says
The basic rule is:
a(b + c) = ab + ac
The factor outside the parentheses, a, multiplies every term inside the parentheses. You can think of the outside factor as being shared with each term.
For example:
4(x + 3) = 4x + 12
The 4 multiplies both x and 3. A common mistake is to multiply only the first term. The expression 4x + 3 is not equivalent to 4(x + 3).
The distributive property also works when subtraction appears inside the parentheses:
a(b − c) = ab − ac
For example:
5(2x − 7) = 10x − 35
Why It Works: An Area Model
Suppose a rectangle has height a and total width b + c. Its area is a(b + c). If you split the rectangle into a part of width b and a part of width c, the two smaller areas are ab and ac. Adding them gives ab + ac. Both methods describe the same total area.
This visual reasoning is important because it shows that the distributive property is not an arbitrary trick. It follows from the fact that splitting a region into non-overlapping parts does not change its total area.
Using Algebra Tiles
Algebra tiles can make variable expressions more concrete. A long tile can represent x, while smaller square tiles can represent units. Positive and negative tiles can help you model signs.
To model 3(x + 2), imagine three identical groups, each containing one x-tile and two unit tiles. Altogether you have three x-tiles and six unit tiles, so:
3(x + 2) = 3x + 6
Distributing with Positive and Negative Numbers
Signs matter. You are multiplying the outside factor by each entire term inside the parentheses.
For a positive factor:
6(2x − 5) = 12x − 30
For a negative factor:
−3(x + 4) = −3x − 12
For a negative factor and subtraction:
−2(5x − 7) = −10x + 14
The final plus sign appears because negative times negative is positive.
A Reliable Sign Strategy
First identify the outside factor. Then identify each term inside the parentheses together with its sign. Multiply the outside factor by each signed term. Finally, simplify each product.
For −4(2x − 3), treat the inside terms as 2x and −3:
−4 · 2x = −8x
−4 · −3 = +12
So:
−4(2x − 3) = −8x + 12
A useful check is to substitute a simple value such as x = 1 into both the original expression and your expanded expression. If the values differ, you have made an error.
Simplifying Expressions After Distribution
Distribution often comes before combining like terms. Consider:
3(2x + 5) + 4x
First distribute:
6x + 15 + 4x
Then combine like terms:
10x + 15
The x-terms can combine because they are like terms. The constant 15 cannot combine with an x-term.
Another example:
5(x − 2) − 2(x + 7)
Distribute both outside factors:
5x − 10 − 2x − 14
Combine like terms:
3x − 24
Common Errors and How to Diagnose Them
Error 1: Missing a term. In 7(x + 4), the 7 must multiply both x and 4.
Error 2: Losing a negative sign. In −5(x − 2), the product of −5 and −2 is +10.
Error 3: Combining unlike terms. In 6x + 12, the terms cannot be combined because one contains x and the other does not.
Error 4: Changing an expression without preserving its value. Rewriting is valid only when the new expression is equivalent to the original expression.
To check a rewrite, choose one or two values for the variable and evaluate both expressions. Matching values do not prove equivalence by themselves, but different values prove that the expressions are not equivalent.
Factoring: Using the Property in Reverse
The distributive property works in both directions. Expanding turns a product into a sum or difference:
6(x + 4) = 6x + 24
Factoring reverses that process:
6x + 24 = 6(x + 4)
To factor, find a common factor shared by all terms. For 12x + 18, both terms are divisible by 6:
12x + 18 = 6(2x + 3)
You can check by distributing 6 back through the parentheses.
Greatest Common Factor
A useful factoring strategy is to remove the greatest common factor when possible.
For 15x − 20, the greatest common numerical factor is 5:
15x − 20 = 5(3x − 4)
Factoring is useful because it can reveal structure that is harder to see in expanded form.
Using Distribution to Solve Equations
Some equations contain parentheses. Distribution can remove the parentheses so that you can combine like terms and isolate the variable.
Example:
3(x + 4) = 24
Distribute:
3x + 12 = 24
Subtract 12 from both sides:
3x = 12
Divide both sides by 3:
x = 4
A balance scale is a useful model for equation solving: whatever operation you perform on one side of an equation must be matched on the other side to preserve equality.
A Multi-Step Example
Solve:
2(3x − 5) + 4 = 20
Distribute:
6x − 10 + 4 = 20
Combine constants:
6x − 6 = 20
Add 6 to both sides:
6x = 26
Divide by 6:
x = 13/3
A solution does not have to be a whole number. You can verify it by substituting 13/3 into the original equation.
Mental Math and Number Sense
The distributive property is also useful without variables. It can break a difficult multiplication into easier parts.
For example:
17 × 6 = (10 + 7) × 6
Distribute 6:
10 × 6 + 7 × 6 = 60 + 42 = 102
You can also break a number around a convenient benchmark:
49 × 8 = (50 − 1) × 8 = 400 − 8 = 392
This shows that the same algebraic property supports efficient mental calculation.
Real-World Modeling
Suppose 8 students each buy a notebook for $3 and a pen for $2. The total cost can be written as:
8(3 + 2)
Using the distributive property:
8 · 3 + 8 · 2 = 24 + 16 = 40
Both expressions represent the same total cost. In real problems, the parentheses often represent a repeated package or group, while the outside factor tells how many identical groups there are.
A Step-by-Step Decision Guide
When you see parentheses in an expression, ask these questions.
Is there multiplication outside the parentheses? If yes, distribution may be useful.
What are the complete signed terms inside? Identify each term before multiplying.
Can I simplify after distributing? Combine like terms only after the products are correct.
Could factoring be more useful than expanding? Sometimes the reverse direction gives a simpler form.
How can I check my work? Redistribute after factoring, or substitute a value into equivalent expressions.
Worked Examples
Example A: 7(2x + 1) = 14x + 7
Example B: −4(3y − 5) = −12y + 20
Example C: 2(5x + 3) + x = 11x + 6
Example D: 18x + 12 = 6(3x + 2)
Example E: 5(2x − 1) = 3x + 23 gives 10x − 5 = 3x + 23, then 7x = 28, so x = 4
Pause after each example and explain why every term, sign, and operation is valid. If you can explain the transformation in words, you are more likely to apply the property accurately in unfamiliar problems.
Interactive Tasks
Quiz: Test Your Knowledge
Which expression is equivalent to 4(x + 3)? (4x + 12) (!4x + 3) (!x + 12) (!7x)
Which expression is equivalent to 5(2x − 7)? (10x − 35) (!10x − 7) (!7x − 35) (!10x + 35)
What is the result of distributing −3 through x + 4? (−3x − 12) (!−3x + 12) (!3x − 12) (!−3x + 4)
Which expression is equivalent to 3(2x + 5) + 4x? (10x + 15) (!6x + 19) (!10x + 5) (!7x + 15)
Which factored form is equivalent to 12x + 18? (6 times the quantity 2x + 3) (!3 times the quantity 4x + 3) (!6 times the quantity 2x + 18) (!12 times the quantity x + 18)
What is −2(5x − 7) after distribution? (−10x + 14) (!−10x − 14) (!10x + 14) (!−10x + 7)
What should you do first to simplify 4(x + 2) + 3x? (Distribute 4 to both terms inside) (!Add 4 and 3) (!Combine x and 2) (!Delete the parentheses)
Which equation results after distributing in 3(x + 4) = 24? (3x + 12 = 24) (!3x + 4 = 24) (!x + 12 = 24) (!7x = 24)
Which multiplication is easiest to rewrite as 50 times 8 minus 1 times 8? (49 times 8) (!51 times 8) (!48 times 8) (!50 times 9)
Why can an area model explain the distributive property? (The total area equals the sum of the areas of its parts) (!Every rectangle must be a square) (!Area can only be found by addition) (!Parentheses always represent length)
Memory Game
| Distributive property | Multiplying an outside factor by every term inside parentheses |
| Equivalent expressions | Different expressions that have the same value for all allowed variable values |
| Like terms | Terms with the same variable part |
| Factoring | Rewriting a sum or difference as a product |
| Coefficient | Numerical factor multiplying a variable |
| Area model | Rectangle representation that shows a product split into smaller products |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Multiply the outside factor by every term | Distribute |
| Add or subtract matching variable terms | Combine like terms |
| Rewrite a sum as a product | Factor |
| Replace the variable with a chosen value | Substitute |
| Keep both sides of an equation equal | Preserve equality |
...
Crossword Puzzle
| Distribute | What verb means to multiply an outside factor by every term inside parentheses? |
| Factor | What do you call a number or expression being multiplied? |
| Coefficient | What is the numerical multiplier of a variable called? |
| Equivalent | What word describes expressions that always have the same value? |
| Parentheses | What symbols group terms before distribution? |
| Rectangle | What shape is commonly used in an area model of the distributive property? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Area model: Draw a rectangle that shows 5(x + 3) and label the two smaller areas; then write the matching expanded expression.
- Mental calculation: Create three multiplication problems that become easier when you split one factor around 10, 50, or 100, and explain each rewrite.
- Algebra tiles: Build or sketch a tile model for 2(x + 4) and explain how the model shows the expanded form.
- Equivalent expression: Write four pairs of equivalent expressions using the distributive property and check each pair with one chosen value.
Standard
- Distributive property: Create a one-page mini-poster that explains distribution over addition and subtraction, including one positive-factor and one negative-factor example.
- Like terms: Write and solve five expressions that require both distribution and combining like terms, then annotate where each step occurs.
- Linear equation: Interview a classmate about how they solve 3(x + 5) = 27, compare your methods, and write a short reflection on any differences.
- Mathematical modeling: Invent a shopping, distance, or geometry situation represented by a(b + c), choose realistic values, and explain both the grouped and expanded forms.
Advanced
- Error analysis: Create a gallery of four believable distributive-property mistakes, explain why each is wrong, and provide a corrected solution.
- Mathematical proof: Use an area argument and symbolic reasoning to justify why a(b + c) = ab + ac, then compare the strengths of the two explanations.
- Instructional video: Produce a two-minute teaching video that demonstrates a problem involving a negative outside factor and a second problem involving factoring.
- Mathematics education: Design a short investigation that compares how students understand the distributive property through symbols, area models, and algebra tiles, then summarize what your evidence suggests.
Learning Assessment
- Equivalent expressions: Decide whether 4(2x − 3) + x and 9x − 12 are equivalent, justify your answer symbolically, and confirm it with substitution.
- Error analysis: A student writes −3(2x − 5) = −6x − 15; identify the exact reasoning error and explain how the sign should be handled.
- Factoring: Rewrite 18x + 30 in factored form using the greatest common factor and explain how distribution verifies your answer.
- Linear equations: Solve 4(2x + 1) − 3 = 21 and explain why distribution is a useful first step.
- Area model: Create and label an area model for 7(x + 4), then use it to explain why 7x + 4 is not equivalent to the original expression.
- Transfer of learning: Show how the distributive property can simplify 98 × 17, and explain how the same structure appears in algebraic expansion.
Evidence of Learning
Knowledge: You can state the distributive property, explain why it works, and distinguish expanding from factoring.
Skills: You can distribute positive and negative factors, combine like terms, factor common factors, solve equations with parentheses, and verify equivalent expressions.
Products: Strong evidence includes accurate area models, worked solutions, error analyses, posters, explanations, or short instructional videos that show reasoning rather than answers alone.
Transfer: You can recognize distributive structure in mental arithmetic, geometry, shopping situations, and unfamiliar algebraic expressions, then choose whether expanding or factoring is more useful.
OERs on the Topic
The English Wikipedia article on the distributive property provides broader mathematical context and examples.
Linked Learning Areas
The distributive property connects arithmetic, geometry, and algebra. It supports mental computation, the construction of equivalent expressions, factoring, equation solving, and later work with polynomials.