Competition · AMC preparation · step 4 of 4
AMC 10 · 2021A · #1
Grade 6 arithmeticPick an answer.
AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The expression is three small pieces of the same shape n² - n glued together with +, -, +. Tool #7 (Subproblems) says: compute each piece on its own, then combine. Tool #5 (Pattern) is a sanity helper — noticing n² - n = n(n-1) gives the products 2 · 1, 3 · 2, 4 · 3, which makes each subproblem a one-step multiplication.
Evaluate the first piece
Exponent first: 2² = 4, so the first piece is 4 - 2 = 2.
Squaring a small whole number and subtracting is the Grade 6 "evaluate expressions with whole-number exponents" idea.
6.EE.A.1Identify SubproblemsEvaluate the second piece
Same recipe: 3² = 9, so the middle piece is 9 - 3 = 6.
Same recipe as the first piece — exponent first, then subtract.
6.EE.A.1Identify SubproblemsEvaluate the third piece
Again: 4² = 16, so the third piece is 16 - 4 = 12.
Three matching subproblems, all closed in a single step each.
6.EE.A.1Identify SubproblemsCombine with the original signs
Combine with +, -, +: 2 - 6 + 12 = -4 + 12 = 8, choice (D).
Going below zero and back up is Grade 6 "positive and negative numbers describe quantities" — the temperature dipping then rising.
Dipping below zero and climbing back up is just adding signed amounts in turn.
▸ Why?
A quantity added to its own opposite leaves nothing behind, so the dips and rises settle against each other.
▸ Why?
The signs alternate down the expression, so the pieces pair off in a regular rhythm.
This AMC 10 problem only needs Grade 6 "square the small number, subtract, then add them up" — three quick pieces (2, 6, 12) glued with +, -, + give 8.
- Evaluate the first piece
- Evaluate the second piece
- Evaluate the third piece
- Combine with the original signs
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