➕ Algebra 1 · Equations
The Distributive Property: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Game night
You invite 5 friends over and ask each of them to bring some snacks and some games. Say each friend brings s snacks and g games. You can count the stuff two ways: 5 friends times “snacks and games,” or 5 groups of snacks plus 5 groups of games.
How it connects: 5(s + g) = 5s + 5g. The number of friends multiplies every item on the list, so nobody’s second item gets forgotten. The limit: it works because every friend brings the same amounts, s snacks and g games. If only one friend brought games, there would be g games, not 5g.
Honey, I Shrunk the Kids
In the movie Honey, I Shrunk the Kids, an inventor’s shrink machine accidentally shrinks some kids to a tiny size. Now imagine a much gentler ray that cuts everything it hits to half its size. Zap a 12-centimeter pencil and an 8-centimeter marker, and you get a 6-centimeter pencil and a 4-centimeter marker.
How it connects: A fraction outside the parentheses works like any other number: (1/2)(12 + 8) = (1/2)(12) + (1/2)(8) = 6 + 4 = 10. Halving the whole set gives the same result as halving each piece. The limit: every piece gets halved, not just the first one.
Measure twice, cut once
Carpenters say this to remind themselves to double-check before making a cut that cannot be undone.
How it connects: Before you trust an expansion, test it with a number. Suppose you expanded 7(x + 1) and got 7x + 1. Try x = 3: 7(3 + 1) = 28, but 7 × 3 + 1 = 22. They disagree, so something is wrong. Use a test number like 3 rather than 0, 1 or 2, because those can hide mistakes (at x = 2, x2 and 2x are both 4). One matching number is a good sign, not a proof, so if your answer has an x2, try a second number too.
Cover all your bases
To “cover all your bases” means to make sure you have handled every part of a situation so nothing catches you by surprise. The phrase is borrowed from baseball, where fielders have to cover the bases.
How it connects: In 3(x + 5) there are two bases: the x and the 5. The 3 has to reach both. Writing 3x + 5 leaves a base uncovered. The correct expansion is 3x + 15.
Points on the court
A basketball team has 5 players on the court. Say each player makes one 2-point shot and one 3-point shot. Each player scored 2 + 3 = 5 points, so the team scored 5 × 5 = 25. Or count shots instead: 5 twos and 5 threes make 10 + 15 = 25.
How it connects: 5(2 + 3) = 5 × 2 + 5 × 3, so 25 = 10 + 15. Multiplying the whole sum and multiplying each part separately give the same total.
Reversed controls
In some games, a power-up or curse reverses your controls: push the stick right and you go left, push up and you go down. Push right and up together, and you move left and down. Both parts of the move flip.
How it connects: A minus sign in front of parentheses flips every term inside: −(x + 4) = −x − 4, and −(x − 4) = −x + 4. A term that was negative becomes positive. The limit: a minus sign only flips signs. It does not change sizes, because it multiplies by −1 and nothing else.
Identical goodie bags
You have 12 cookies and 18 brownies, and you want to make identical goodie bags with nothing left over. The most bags you can make is 6, because 6 is the largest number that divides both 12 and 18. Each bag gets 2 cookies and 3 brownies.
How it connects: 12 + 18 = 6(2 + 3). Pulling out the greatest common factor, 6, is the distributive property run backward: the 6 is the number of bags and the parentheses hold one bag. (Making 3 bags of 4 cookies and 6 brownies also works, but 6 bags is the fully factored form.)
The left-out 2
Why did the 2 in 3(x + 2) feel left out? Somebody wrote 3x + 2, and the 3 never came over to say hi.
How it connects: In a(b + c) = ab + ac, the outside number multiplies every term inside. Skipping a term, like writing ab + c, is the classic mistake. For 3(x + 2) the correct answer is 3x + 6.
Memory help
DEAL with it
Draw an arrow from the outside number to Every term. Attend to the signs. Look back: test your answer with a number like 3. Say it as “DEAL with it.”
Outside in
Outside in, one by one, hit each term until you’re done. When you multiply, same signs make a plus, and different signs make a minus.
Draw the arrows
Draw an arc from the outside number to each term inside, like a rainbow with one stop for every term. Count the arcs. The number of arcs should match the number of terms. Then write each product.
Sketch the rectangle
Draw a rectangle as tall as the outside number. Split its width into the terms. Write the area of each small piece, then add them up. For 4(x + 6) the pieces are 4x and 24. (This sketch works best for positive terms, but the rule is true for all numbers.)
The skipped second term
9(y + 2) is not 9y + 2. The 9 multiplies the 2 as well, so the answer is 9y + 18. If an arrow does not land on every term, you are not finished.
A negative flips the sign inside
−8(x − 5): the second term is −5, and (−8) × (−5) = +40. So the answer is −8x + 40, not −8x − 40. Say it out loud: “negative times negative is positive.”
The matching jackets
Imagine the terms 12y and 30 are two players who both wear the same team jacket: the number 6. To factor, collect the jacket and hang it outside the parentheses. What is left of each player goes inside: 6(2y + 5). To check, hand the jacket back to everyone (distribute).