Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #2
Grade 6 arithmeticPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 ◆ 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.
Work inside the parentheses first
Split at the parentheses into two subproblems: first find 5 ◆ 3, then apply bigstar 6 to that result.
Reading the parentheses first is the Grade 5 "use parentheses in numerical expressions" rule, and it gives us our two subproblems.
5.OA.A.1Identify SubproblemsApply the first rule
Apply the ◆ rule with a = 5, b = 3: since a ◆ b = a² - b², substitute to write 5² - 3².
Plugging numbers into letters a and b of a defined formula is exactly Grade 6 "evaluate expressions where letters stand for numbers".
The inner expression 5 ◆ 3 equals 5² - 3².
▸ Why?
The symbol ◆ is a made-up operation, and its definition a ◆ b = a² - b² is simply a recipe: it tells you what to build out of the two numbers written on its left and right.
▸ Why?
In that recipe the letters a and b are placeholders for the two inputs, and here the left input is 5 and the right input is 3, so a stands for 5 and b stands for 3 (the order matters, since ◆ is not symmetric).
▸ Why?
Because each letter equals the number assigned to it, swapping the letter for its number leaves the value of a² - b² unchanged, turning it into 5² - 3².
Square and subtract
Square and subtract: 5² - 3² = 25 - 9 = 16.
Evaluating 5² and 3² uses Grade 6 "whole-number exponents"; the subtraction 25 - 9 is well-within-grade arithmetic.
6.EE.A.1Identify SubproblemsApply the second rule
Substitute back into a bigstar b = (a-b)²: 16 bigstar 6 = (16 - 6)² = 10² = 100.
We reuse the same Grade 6 substitution move on a different defined operation, and the exponent finishes the work.
6.EE.A.2Identify SubproblemsMatch against the choices
Among the choices -20, 4, 16, 100, 220 only 100 matches, so the answer is (D); 16 is the stop-halfway trap.
Lining the result up against the five choices is the AMC multiple-choice habit and catches the "stopped halfway" trap labelled (C).
5.OA.A.1Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 expression evaluation — plug numbers into the letters of a formula — that you already know!
- Work inside the parentheses first
- Apply the first rule
- Square and subtract
- Apply the second rule
- Match against the choices
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