AMC 10 · 2021 · #1
Grade 6 arithmeticProblem
What is the value of (22−2)−(32−3)+(42−4)
Pick an answer.
AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Evaluate the arithmetic expression $(2^2 - 2) - (3^2 - 3) + (4^2 - 4)$ and match it to one of five choices.
Givens: Three grouped terms of the same shape: $n^2 - n$ for $n = 2, 3, 4$; Signs between groups: plus, minus, plus (the middle group is subtracted); Answer choices: (A) $1$, (B) $2$, (C) $5$, (D) $8$, (E) $12$
Unknowns: The numeric value of the expression
Understand
Restated: Evaluate the arithmetic expression $(2^2 - 2) - (3^2 - 3) + (4^2 - 4)$ and match it to one of five choices.
Givens: Three grouped terms of the same shape: $n^2 - n$ for $n = 2, 3, 4$; Signs between groups: plus, minus, plus (the middle group is subtracted); Answer choices: (A) $1$, (B) $2$, (C) $5$, (D) $8$, (E) $12$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #5 Look for a Pattern
The expression is three small pieces of the same shape $n^2 - n$ glued together with $+$, $-$, $+$. Tool #7 (Subproblems) says: compute each piece on its own, then combine. Tool #5 (Pattern) is a sanity helper — noticing $n^2 - n = n(n-1)$ gives the products $2 \cdot 1, 3 \cdot 2, 4 \cdot 3$, which makes each subproblem a one-step multiplication.
Execute — Answer: D
6.EE.A.1 Step 1 - Evaluate the first piece.
- $2^2 = 4$, so $2^2 - 2 = 4 - 2 = 2$.
💡 Squaring a small whole number and subtracting is the Grade 6 "evaluate expressions with whole-number exponents" idea.
6.EE.A.1 Step 2 - Evaluate the second piece.
- $3^2 = 9$, so $3^2 - 3 = 9 - 3 = 6$.
💡 Same recipe as the first piece — exponent first, then subtract.
6.EE.A.1 Step 3 - Evaluate the third piece.
- $4^2 = 16$, so $4^2 - 4 = 16 - 4 = 12$.
💡 Three matching subproblems, all closed in a single step each.
6.NS.C.5 Step 4 - Combine the three pieces with the original signs $+, -, +$: $2 - 6 + 12$.
- Going left to right, $2 - 6 = -4$, then $-4 + 12 = 8$.
💡 Going below zero and back up is Grade 6 "positive and negative numbers describe quantities" — the temperature dipping then rising.
6.EE.A.1 Evaluate the first piece. $2^2 = 4$, so $2^2 - 2 = 4 - 2 = 2$. 6.EE.A.1 Evaluate the second piece. $3^2 = 9$, so $3^2 - 3 = 9 - 3 = 6$. 6.EE.A.1 Evaluate the third piece. $4^2 = 16$, so $4^2 - 4 = 16 - 4 = 12$. 6.NS.C.5 Combine the three pieces with the original signs $+, -, +$: $2 - 6 + 12$. Going Review
Reasonableness: Each piece is tiny ($2, 6, 12$), and only the middle one is subtracted. Roughly: $12 + 2 = 14$ minus the $6$ in the middle leaves $8$. That matches choice (D), and it falls between the smallest and largest choices, which is reasonable.
Alternative: Tool #5 (Look for a Pattern). Use the factoring $n^2 - n = n(n-1)$: pieces become $2 \cdot 1 = 2$, $3 \cdot 2 = 6$, $4 \cdot 3 = 12$. Same combine: $2 - 6 + 12 = 8$. Same answer with one less squaring step.
CCSS standards used (min grade 6)
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Computing each $n^2 - n$ piece by first squaring the small whole number and then subtracting.)6.NS.C.5Understand that positive and negative numbers describe quantities (Combining $2 - 6 + 12$ left to right — the $-4$ intermediate result is a negative quantity that gets brought back up to $8$.)
⭐ 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 $8$.
⭐ 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 $8$.
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