AMC 10 · 2010 · #4
Grade 8 algebraPick an answer.
The obvious move is to drop a number like x=-1 into all five expressions. That move is honest for throwing choices away, but it can never crown one: a single value says nothing about the infinitely many other negative numbers. So instead of picking a value of x, I rename it. Every negative x can be written as x=-t with t ＞ 0, and that one substitution rewrites all five expressions as questions about a positive number, where the sign rules are automatic. Before substituting I settle how the expressions are meant to be read, because a minus sign in front of a power is easy to misplace.
What "must" actually demands
Accepting needs it for every input; rejecting needs just one.
One bad example destroys a claim about everything, but a hundred good examples still prove nothing.
One bad example destroys a claim about everything, but a hundred good examples still prove nothing.
▸ Why?
A claim that something always holds fails the moment one case violates it.
▸ Why?
Checking a handful of values only tests a small part of the range, and a part cannot speak for the whole.
Read powers before minus signs
Reading powers before minus signs prevents a misreading.
The minus in front subtracts the finished power; it is not part of the base.
6.EE.A.2Organize Information In More WaysTrade x ＜ 0 for t ＞ 0
Substituting a positive letter makes every sign visible.
Renaming the variable moves the whole question to the side of the number line where signs are easy.
6.EE.B.6Introduce A Variable(A) is always exactly -1
The first candidate is always exactly negative one.
Dividing a number by its own size erases everything except its sign.
6.NS.C.7Eliminate Possibilities(B): the minus sits outside
The second has its minus outside the square.
The square erases the sign of x, and then the outside minus hands it a new one.
7.NS.A.2Eliminate Possibilities(C): base 2 stays positive
The third's base keeps it positive before the minus.
Positive numbers multiplied and divided stay positive, so the outside minus always wins.
8.EE.A.1Eliminate Possibilities(E): cube roots keep signs
Cube roots keep the sign of their input.
Cubing never hides a sign the way squaring does, so the cube root cannot invent one.
8.EE.A.2Eliminate PossibilitiesWhy (D) is always positive
Only the reciprocal one flips positive, choice (D).
A reciprocal keeps the sign of the number it came from, so flipping a negative and then negating it lands on the positive side.
7.NS.A.2Introduce A Variable"Must be positive" means positive for every negative x, so rename x as -t with t ＞ 0 one time and read all five signs at once.
- What "must" actually demands
- Read powers before minus signs
- Trade x ＜ 0 for t ＞ 0
- (A) is always exactly -1
- (B): the minus sits outside
- (C): base 2 stays positive
- (E): cube roots keep signs
- Why (D) is always positive