AMC 10 · 2006 · #2
Grade 6 algebraPick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The expression h ⊗ (h ⊗ h) hides one operation inside another, so Tool #7 (Identify Subproblems) says: settle the parentheses first, then handle the outer ⊗. Tool #4 (Introduce a Variable) lets us hold the inner answer as one lump and drop it into the y slot of the rule, so we never lose track of which input gets cubed. Tool #3 (Eliminate Possibilities) reads the choices — the outcome must stay a simple expression in h, and choice (E) h³ is exactly the term that ends up cancelling, so it is a natural trap to watch for.
Do the inside operation first
Parentheses first: feed x = h and y = h into the rule, so h ⊗ h gives h³ - h.
A rule wrapped in parentheses has to be finished before anything outside it can use its result.
6.EE.A.2Identify SubproblemsFeed that result into the outer rule
Apply the rule again with x = h and y = h³ - h; the whole second input is subtracted as one block: h³ - (h³ - h).
The second input, however long, drops into the y slot as a single quantity to be subtracted whole.
6.EE.A.2Introduce A VariableDistribute the subtraction and simplify
Flip the signs inside: h³ - h³ + h. The two h³ terms cancel, leaving h — choice (C).
Subtracting a difference flips the sign of each piece inside, so the matching h³ terms wipe each other out.
Subtracting a difference flips the sign of each piece inside, so the matching terms wipe each other out.
▸ Why?
A factor spread across a sum reaches every term inside it, and a minus sign is such a factor.
▸ Why?
A quantity added to its own opposite leaves nothing behind, so the pair simply disappears.
Finish the parentheses first, then plug that whole result into the rule as the part you subtract — here the two h³ pieces cancel and only h is left.
- Do the inside operation first
- Feed that result into the outer rule
- Distribute the subtraction and simplify