Before you begin
You now hold all four operations. This quiz settles a question that only exists because of that: when several of them crowd into one expression like 3 + 4 × 5, who goes first? The answer is not a matter of manners. It is a matter of meaning: each order gives a different number. As in every quiz of this series: no tricks, no calculator (this time especially — you'll see why at question 10), and whatever this shows is useful news.
Why there must be an order at all
Read 3 + 4 × 5 two ways and you get two different numbers: left to right gives 35; multiplication first gives 23. An expression that could mean two things means nothing — so the world agreed on one reading. And the agreed reading follows sense: Quiz 4 taught that 4 × 5 is bulk addition — four fives, already glued together. So 3 + 4 × 5 says "three, plus four fives": 3 + 20 = 23. Multiplication goes first because it is already a package. The rules just write down what the sentence means.
The pyramid of precedence
Four courses — four rows of stones — worked from the top down:
You may know the summary word PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It is honest only if you read its letters in pairs: M and D are one team, A and S are one team. The single most common error on this entire topic is treating PEMDAS as six separate steps and doing all multiplication before any division. The pyramid keeps the teams visible: two of its courses have two names on one stone.
Grouping: whatever is fenced together happens together
Parentheses outrank everything because they are not an operation — they are a fence. (5 + 2) × 6 says: this sum is one thing; multiply the thing. And one fence hides in plain sight: the fraction bar. Written as 30 + 64, the bar fences the whole top together before anything is divided — it means (30 + 6) ÷ 4, never 30 + 6 ÷ 4.
The team rules, left to right
On a shared course, read like a sentence: 24 ÷ 4 × 2 works left to right, so 24 ÷ 4 = 6, then 6 × 2 = 12 (not 3). Likewise 10 − 3 + 2: take 3, then add 2, giving 9 (not 5). If you got 3 and 5, you played M before D and A before S; the stones share a course.
A first look at exponents
A small power may appear in expressions: 3² means 3 × 3 = 9 — multiplication so repeated it earned its own perch, one course above ×. That is all you need here; exponents get a whole quiz of their own later in the series.
Three to study before you start
3 + 4 × 5. The multiplication is a package — four fives, already glued: 20. Then the addition assembles: 3 + 20 = 23. Reading left to right (7 × 5 = 35) breaks the package open and changes what the sentence says.
24 ÷ 4 × 2: same course, so left to right: 6, then 12. And 10 − 3 + 2 goes left to right: 7, then 9. The wrong answers (3 and 5) both come from the same misread: giving M rank over D, and A rank over S. They share a stone.
2 × (9 − 4) + 3². Top course: the fence gives (9 − 4) = 5. Next course: the power gives 3² = 9. Then × : 2 × 5 = 10. Then + : 10 + 9 = 19. Four courses, one pass, no argument.
Now you
Work without a calculator, on paper. Write your answers on the lines. Questions 4 and 8 are multiple choice — choose the one best answer.
- Evaluate: 5 + 2 × 6
- Evaluate: (5 + 2) × 6
- Evaluate: 18 − 6 ÷ 3
- Evaluate: 24 ÷ 4 × 2
- A) 3
- B) 6
- C) 12
- D) 48
- Evaluate: 10 − 3 + 2
- Evaluate: 5 × (8 − 3) − 12 ÷ 4
▶ VideoA video rest stop, for anyone who has kept going this far.
Take two minutes with the leafy and weedy seadragons, filmed by Wired.
- Evaluate: 30 + 64
- Evaluate: 2 + 3² × 2
- A) 20
- B) 14
- C) 50
- D) 100
- An electrician charges a $45 service call plus $65 an hour. Compute the bill for a 3-hour job: 45 + 65 × 3
- Marisol's phone calculator says 8 + 4 × 5 = 60. Her scientific calculator says 28. Which machine is right — and why do they disagree?
Check your work
Open the key — after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 3 | Packages before assembly | × and ÷ are glued bundles; + and − assemble them afterward. |
| 2, 7 | Grouping — fences and the bar | Whatever is fenced together happens together; the fraction bar is a fence in disguise. |
| 4, 5 | The team rules | M·D share a course and A·S share a course — take turns, left to right. |
| 8 | The exponent's perch | A power is the base times itself, and it goes before ×. |
| 6 | The whole pyramid | Top course down, one pass — rework Example 3 aloud. |
| 9, 10 | Order in the world | Formulas are sentences; some machines can't read them. You can. |
Eight or more right: the pyramid is yours — on to Quiz 7. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news — we've found the right course to stand on. Read expressions aloud as sentences with your teacher — "three, plus four fives" — then come back to these same ten.
Next in the Seahorse Series: Quiz 7 — Factors, Multiples, Primes, and Divisibility. Numbers have family trees: which numbers divide which, which numbers cannot be broken down at all, and the quick tests that tell you at a glance.
Fellow travelers
The masthead trio holds formation above. Today's guests arrive in numbers — first a small delegation, then the whole membership.
