AMC 10 · 2021 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

order-of-operationsexponentsmulti-digit-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: order-of-operations

📏 Short solution 💡 1 insight

Problem

What is the value of $(2^2-2)-(3^2-3)+(4^2-4)$

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 12$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

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.

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

#7 Identify Subproblems 6.EE.A.1 Step 1

Evaluate the first piece.
$2^2 = 4$, so $2^2 - 2 = 4 - 2 = 2$.

$$2^2 - 2 = 4 - 2 = 2$$

💡 Squaring a small whole number and subtracting is the Grade 6 "evaluate expressions with whole-number exponents" idea.

#7 Identify Subproblems 6.EE.A.1 Step 2

Evaluate the second piece.
$3^2 = 9$, so $3^2 - 3 = 9 - 3 = 6$.

$$3^2 - 3 = 9 - 3 = 6$$

💡 Same recipe as the first piece — exponent first, then subtract.

#7 Identify Subproblems 6.EE.A.1 Step 3

Evaluate the third piece.
$4^2 = 16$, so $4^2 - 4 = 16 - 4 = 12$.

$$4^2 - 4 = 16 - 4 = 12$$

💡 Three matching subproblems, all closed in a single step each.

#7 Identify Subproblems 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 $-4 + 12 = 8$.

$$2 - 6 + 12 = -4 + 12 = 8 \;\Rightarrow\; \textbf{(D)}$$

💡 Going below zero and back up is Grade 6 "positive and negative numbers describe quantities" — the temperature dipping then rising.

[1] #7 6.EE.A.1 Evaluate the first piece. $2^2 = 4$, so $2^2 - 2 = 4 - 2 = 2$.

[2] #7 6.EE.A.1 Evaluate the second piece. $3^2 = 9$, so $3^2 - 3 = 9 - 3 = 6$.

[3] #7 6.EE.A.1 Evaluate the third piece. $4^2 = 16$, so $4^2 - 4 = 16 - 4 = 12$.

[4] #7 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.1 Write 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.5 Understand 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 $8$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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