Skill 0.05 · CoreStandard: pre 6.EE.3About 10 minutes
A multiplier outside parentheses has business with every term inside, not just the one it happens to be standing next to.
Six team kits, each holding a jersey and a set of gear. Nobody would buy six jerseys and one set of gear. Six of a bundle means six of everything in the bundle.
That is the distributive property. The number outside the parentheses multiplies each term inside, and the answer keeps the operation that joined them.
1. What the property says
Write 3(x + 4). The 3 counts how many bundles you have, and each bundle holds an x and a 4. Three bundles hold 3x and 12 altogether, so 3(x + 4) equals 3x + 12. An area picture makes the split visible.
Figure 1. One rectangle, measured two ways. Its width is x + 4 as a whole, yet its area splits cleanly into two smaller rectangles.
2. Every term gets a turn
In practice, draw an arrow from the outside number to each term inside. Two terms inside means two arrows and two products. The arrows are worth drawing until the habit is automatic.
Figure 2. The plus sign between the terms survives the trip, which is why the answer is a sum of two products.
Key idea
Multiply the outside term by every term inside, one at a time, and join the results with the same operation that was inside the parentheses.
3. The mistake: stopping after the first term
The most common error is arriving at 3x, feeling finished, and copying the 4 down untouched. The result, 3x + 4, is a different expression. Substituting a value settles it, exactly as in Skill 0.04.
Watch out: the second term is not a passenger
Test all three forms at x = 2. Only two of them agree.
Sense check: count your products before you write the answer. Two terms inside the parentheses must produce two products outside them.
4. Negative multipliers and subtraction inside
When the outside number is negative, or the inside operation is subtraction, the sign rules from Skill 0.02 take over. Rewrite the inside subtraction as a negative term first, then every product is a plain multiplication and the flip rule does the rest.
Figure 3. The second product turns positive because both of its factors are negative. That is the rule from Skill 0.02 doing its job here.
5. All four sign cases
The multiplier can be positive or negative, and the inside operation can be addition or subtraction. That gives four combinations, and the signs of the two products follow directly from counting negative factors.
Figure 4. Only the last row ends with a plus, because it is the only row whose second product has two negative factors.
6. Worked examples
Example A
Expand 5(2x + 3).
Two terms inside, so two products. First 5 × 2x = 10x, then 5 × 3 = 15, joined by the plus that was already there.
5(2x + 3) = 10x + 15. Sense check with x = 1: the original gives 5(5) = 25, and 10 + 15 = 25.
Example B
Expand −3(4 − y).
Read the inside as 4 and −y. First −3 × 4 = −12. Then −3 × (−y) has two negative factors, so it is +3y.
−3(4 − y) = −12 + 3y. Sense check with y = 4: the original gives −3(0) = 0, and −12 + 12 = 0.
Example C
A coach in Moreno Valley orders 6 team kits. Each kit holds a jersey costing j dollars plus $15 of gear. Write the total cost.
One kit costs j + 15, so six kits cost 6(j + 15). Distribute: 6 × j = 6j and 6 × 15 = 90.
The total is 6j + 90 dollars. Sense check: $90 is six lots of $15 in gear, which is what six kits must include.
Words to know
Distribute
To multiply an outside term by every term inside the parentheses.
Expand
To rewrite a product as a sum of terms with no parentheses left.
Factor (outside)
The single term standing in front of the parentheses.
Product
One result of a single multiplication inside the process.
Equivalent
Two expressions that give the same value for every value of the variable.
7. A look ahead
Coming next: Skill 0.06, evaluating an expression by substitution
Substitution has already been your checking tool twice. Skill 0.06 makes it the task itself. The trap there arrives with negative values: students drop the sign, or drop the parentheses around it, and turn a substituted −3 into something else entirely.
Check your understanding
Explain, using the rectangle picture, why 3(x + 4) is 3x + 12.
A classmate writes 3(x + 4) = 3x + 4. Use x = 2 to show this is wrong.
Explain why −2(x − 5) ends with a positive 10.
How many products should 7(a + b) produce, and how do you know?
Explain why the operation inside the parentheses stays the same after expanding.
What to carry away
The outside term multiplies every term inside the parentheses, without exception.
Two terms inside means two products; count them before writing the answer.
The area of a rectangle split into two parts is why the property is true.
The operation joining the inside terms also joins the products.
Rewrite inside subtraction as a negative term before distributing.
A negative multiplier changes the sign of every term it reaches.
Two negative factors in a product give a positive result, as in Skill 0.02.
Substituting one value into both forms catches a stopped-too-early answer instantly.