➕ Algebra 1 · Equations
Order of Operations: 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.
Two equally close friends
Two friends who are equally close to you both text you. Neither one outranks the other, so you reply in the order the messages arrived: first come, first served.
How it connects: Multiplication and division are equals, so you go left to right: 30 ÷ 5 × 2 = 6 × 2 = 12, not 30 ÷ 10 = 3. The same goes for addition and subtraction. This tie-breaker works only inside one level: a higher level, such as exponents, still goes before multiplication and division.
Inception: dreams inside dreams
In the movie Inception, a team enters a person’s dream, then a dream inside that dream, and goes deeper still. The deepest dream is nested inside all the others.
How it connects: Parentheses can nest too. In 4 × (1 + (8 − 3)) you can spot the layers from the outside in, but you calculate from the innermost layer outward: 8 − 3 = 5, then 1 + 5 = 6, then 4 × 6 = 24.
Reading a box score
A basketball player makes 7 two-point baskets, 3 three-point baskets and 2 free throws worth 1 point each. Her points are 7 × 2 + 3 × 3 + 2 × 1.
How it connects: Multiplication ranks above addition: first find the points from each kind of shot, then add them: 14 + 9 + 2 = 25. Working strictly left to right through the symbols would give 53, a score no scoreboard would accept.
Three friends, one bill
Three friends each order a 7-dollar burger and a 2-dollar drink. The total is 3 × (7 + 2): find the cost of one friend’s meal first (7 + 2 = 9), then multiply by 3 to get 27 dollars.
How it connects: Parentheses are worked out first, as one package. Without them, 3 × 7 + 2 = 23 would mean three burgers but only one drink, so parentheses can change what an expression means.
Doubling with every octave
Each time a note goes up one octave, its frequency doubles. Starting from 110 hertz and going up three octaves gives 110 × 23 hertz.
How it connects: The exponent touches only the 2 (three doublings), and exponents rank above multiplication: 110 × 23 = 110 × 8 = 880 hertz. Multiplying first would give (110 × 2)3, which is over ten million: a very different expression.
A health bar
In a game your health is 20. You take 6 damage, then you heal 4. That story is 20 − 6 + 4.
How it connects: Subtraction and addition share a level, so go left to right: 20 − 6 = 14, then 14 + 4 = 18. Adding first would give 20 − (6 + 4) = 10, which is a different story (10 damage). The minus sign belongs to the 6, so you can also read it as 20 + (−6) + 4.
First things first
The proverb means you should deal with the most important or most necessary thing before anything else.
How it connects: On the ladder, “first” means highest rank: grouping, then exponents, then multiplication and division, then addition and subtraction. Position on the page only breaks ties between operations of the same rank.
Why exponents go first
Why do exponents get to go before multiplication? Because they have all the power.
How it connects: An exponent is repeated multiplication, so it ranks one level higher: 3 × 24 = 3 × 16 = 48, not 64 = 1296. (A power is the result of using an exponent, which is where the pun comes from.)
Memory help
PEMDAS, with a twist
Please Excuse My Dear Aunt Sally stands for Parentheses (grouping), Exponents, Multiplication, Division, Addition, Subtraction. The twist: M and D tie, and A and S tie. Think of it as PE(MD)(AS). Inside each pair, go left to right, not in the order of the letters.
The ladder rhyme
Groups go first, then powers climb. Times and divide share one line. Plus and minus come at the end. Ties go left to right, my friend.
The four-rung ladder
Picture a ladder with four rungs and step down from the top: grouping, exponents, then multiplication and division, then addition and subtraction. The bottom two rungs are wide enough for two operations each. When two operations stand on the same rung, they take turns from left to right.
Walk through your school
The front door is grouping: you have to go inside first, starting with the innermost room. The gym is exponents (power!). In the cafeteria, multiplication and division share one long table and are served left to right. At the library at the end of the hall, addition and subtraction share the last table, also served left to right.
Rewrite after every step
On paper, change only one thing per line. Underline the operation you are doing, solve it, and rewrite the whole expression with the result in its place. Doing three steps in your head and jumping to the answer is how slips sneak in.
The exponent hugs the number next to it
In −92 the exponent touches only the 9, so the value is −81. If you want the negative squared, add parentheses: (−9)2 = 81.