AMC 8 · 2022 · #2

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

function-evaluationorder-of-operationsformula-substitution identify-subproblems ↑ Prerequisites: order-of-operationsexponents

📏 Short solution 💡 1 insight

Problem

Consider these two operations:
\begin{align*} a , \blacklozenge , b &= a^2 - b^2\ a , \bigstar , b &= (a - b)^2 \end{align*}
What is the output of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$

$\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220$

Pick an answer.

(A)

${-}20$

(B)

(C)

(D)

100

(E)

220

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Toolkit + CCSS Solution

Understand

Restated: Two brand-new operations are defined: $a \, \blacklozenge \, b = a^2 - b^2$ and $a \, \bigstar \, b = (a-b)^2$. They behave like functions that take two numbers as inputs. The task is to evaluate $(5 \, \blacklozenge \, 3) \, \bigstar \, 6$, doing the inner operation first because of the parentheses.

Givens: Rule 1: $a \, \blacklozenge \, b = a^2 - b^2$; Rule 2: $a \, \bigstar \, b = (a-b)^2$; Expression to evaluate: $(5 \, \blacklozenge \, 3) \, \bigstar \, 6$; Answer choices: (A) $-20$, (B) $4$, (C) $16$, (D) $100$, (E) $220$

Unknowns: The numerical value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6$

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

The expression has a clear inside-then-outside structure thanks to the parentheses, so Tool #7 (Identify Subproblems) is the natural fit: first compute the inner $5 \, \blacklozenge \, 3$, then plug that result into the outer $\bigstar \, 6$. We treat each unfamiliar symbol like a tiny recipe — read the letters $a$ and $b$, line them up with the actual numbers around the symbol, and substitute. Tool #3 (Eliminate Possibilities) is the AMC multiple-choice safety net: after computing, we check that the value matches one of the five offered choices and that the others can be ruled out, guarding against an arithmetic slip.

Execute — Answer: D

#7 Identify Subproblems 5.OA.A.1 Step 1

Split the expression at the parentheses.
The whole problem becomes two smaller problems: (i) find $5 \, \blacklozenge \, 3$, (ii) take that number and apply $\bigstar \, 6$ to it.
Doing them in this order respects the parentheses.

$$(5 \, \blacklozenge \, 3) \, \bigstar \, 6 \;=\; \big[\,5 \, \blacklozenge \, 3\,\big] \, \bigstar \, 6$$

💡 Reading the parentheses first is the Grade 5 "use parentheses in numerical expressions" rule, and it gives us our two subproblems.

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

Apply the $\blacklozenge$ recipe with $a = 5$ and $b = 3$.
The rule says $a \, \blacklozenge \, b = a^2 - b^2$, so substitute the numbers into the letters.

$$5 \, \blacklozenge \, 3 \;=\; 5^2 - 3^2$$

💡 Plugging numbers into letters $a$ and $b$ of a defined formula is exactly Grade 6 "evaluate expressions where letters stand for numbers".

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

Carry out the arithmetic: square each number, then subtract.

$$5^2 - 3^2 \;=\; 25 - 9 \;=\; 16$$

💡 Evaluating $5^2$ and $3^2$ uses Grade 6 "whole-number exponents"; the subtraction $25 - 9$ is well-within-grade arithmetic.

#7 Identify Subproblems 6.EE.A.2 Step 4

Now substitute the inner result back: the expression has become $16 \, \bigstar \, 6$.
Apply the $\bigstar$ recipe with $a = 16$ and $b = 6$, where the rule is $a \, \bigstar \, b = (a-b)^2$.

$$16 \, \bigstar \, 6 \;=\; (16 - 6)^2 \;=\; 10^2 \;=\; 100$$

💡 We reuse the same Grade 6 substitution move on a different defined operation, and the exponent finishes the work.

#3 Eliminate Possibilities 5.OA.A.1 Step 5

Match the computed value $100$ against the five answer choices: $-20$, $4$, $16$, $100$, $220$.
Only choice (D) hits it; the others can be eliminated.
(Notice the trap: $16$ in choice (C) is exactly the intermediate result $5 \, \blacklozenge \, 3$, designed to catch students who stop after the first subproblem.)

$$100 \;\Rightarrow\; \textbf{(D)}$$

💡 Lining the result up against the five choices is the AMC multiple-choice habit and catches the "stopped halfway" trap labelled (C).

[1] #7 5.OA.A.1 Split the expression at the parentheses. The whole problem becomes two smaller p

[2] #7 6.EE.A.2 Apply the $\blacklozenge$ recipe with $a = 5$ and $b = 3$. The rule says $a \, \

[3] #7 6.EE.A.1 Carry out the arithmetic: square each number, then subtract.

[4] #7 6.EE.A.2 Now substitute the inner result back: the expression has become $16 \, \bigstar

[5] #3 5.OA.A.1 Match the computed value $100$ against the five answer choices: $-20$, $4$, $16$

Review

Reasonableness: The two operations both involve squaring, so the final answer should be a non-negative number on the order of $10^2$. We got $100$, which is exactly $10^2$ and lands in the middle of the answer range — perfectly reasonable. We can also sanity-check the trap: choice (C) $16$ is the inner result alone, choice (A) $-20$ would only appear if we accidentally did $a - b^2$ instead of $(a-b)^2$, and choice (E) $220$ is roughly $16^2 - 6^2 = 220$, the answer if someone mistakenly applied $\blacklozenge$ on the outside instead of $\bigstar$. Each distractor maps to a specific misstep, and our path avoids all of them.

Alternative: Tool #9 (Solve an Easier Related Problem): replace the unfamiliar symbols with familiar function names — let $f(a,b) = a^2 - b^2$ and $g(a,b) = (a-b)^2$. The problem becomes $g(f(5,3),\,6)$, a plain composition of two functions. Computing $f(5,3) = 16$ and then $g(16,6) = 100$ gives the same answer (D). This reframing makes the structure feel like the everyday "plug-the-output-of-one-machine-into-another" idea, which is the whole point of defined operations.

CCSS standards used (min grade 6)

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate (Respecting the parentheses to evaluate the inner $5 \, \blacklozenge \, 3$ before the outer $\bigstar \, 6$, and then matching the final value to the multiple-choice list.)
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Computing $5^2 = 25$, $3^2 = 9$, and $10^2 = 100$ inside the two defined operations.)
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Substituting $a = 5,\, b = 3$ into the $\blacklozenge$ rule $a^2 - b^2$, and $a = 16,\, b = 6$ into the $\bigstar$ rule $(a-b)^2$, just like evaluating an algebraic expression with given variable values.)

⭐ This AMC 8 problem only needs Grade 6 expression evaluation — plug numbers into the letters of a formula — that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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