AMC 10 · 2007 · #2
Grade 6 arithmeticDefine the operation ⋆ by a⋆b=(a+b)b. What is (3⋆5)−(5⋆3)?
Pick an answer.
AMC 10 2007 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: A new operation $\star$ is defined by the rule $a \star b = (a+b)b$. Using this rule, find the value of $(3 \star 5) - (5 \star 3)$.
Givens: The operation is defined by $a \star b = (a+b)b$ — add the two numbers, then multiply that sum by the second number; The expression to evaluate is $(3 \star 5) - (5 \star 3)$; Answer choices: (A) $-16$, (B) $-8$, (C) $0$, (D) $8$, (E) $16$
Unknowns: The single number that $(3 \star 5) - (5 \star 3)$ equals
Understand
Restated: A new operation $\star$ is defined by the rule $a \star b = (a+b)b$. Using this rule, find the value of $(3 \star 5) - (5 \star 3)$.
Givens: The operation is defined by $a \star b = (a+b)b$ — add the two numbers, then multiply that sum by the second number; The expression to evaluate is $(3 \star 5) - (5 \star 3)$; Answer choices: (A) $-16$, (B) $-8$, (C) $0$, (D) $8$, (E) $16$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #4 Introduce a Variable, #3 Eliminate Possibilities
The expression is a difference of two separate $\star$ computations, so Tool #7 (Identify Subproblems) says: evaluate $3 \star 5$ on its own, evaluate $5 \star 3$ on its own, then subtract. Tool #4 (Introduce a Variable) keeps the rule $a \star b = (a+b)b$ in view so the first and second inputs land in the right slots — the trap here is assuming $\star$ behaves like ordinary multiplication and commutes. Tool #3 (Eliminate Possibilities) uses that trap: choice (C) $0$ is exactly what you get if you wrongly believe $3 \star 5 = 5 \star 3$.
Execute — Answer: E
6.EE.A.2 Step 1 Read the rule and its two slots
- The definition $a \star b = (a+b)b$ takes two inputs.
- The first input is $a$ and the second is $b$.
- You add them to get $a+b$, then multiply that sum by the second input $b$.
- Notice the sum $a+b$ is symmetric, but the outside factor is $b$ alone — so which number sits in the $b$ slot changes the answer.
💡 A custom symbol is only a set of instructions; label each slot before you plug in numbers.
5.OA.A.1 Step 2 Evaluate the first term
- For $3 \star 5$, put $a=3$ and $b=5$.
- The sum inside is $3+5=8$, and the outside multiplier is the second input $b=5$.
- So $3 \star 5 = 8\cdot 5 = 40$.
💡 Finish the parentheses first, then multiply by the second number.
5.NBT.B.5 Step 3 Evaluate the second term
- For $5 \star 3$, put $a=5$ and $b=3$.
- The sum inside is $5+3=8$ again, but now the outside multiplier is the second input $b=3$.
- So $5 \star 3 = 8\cdot 3 = 24$.
- The sum stayed $8$, yet the smaller multiplier makes this term smaller than the first.
💡 Swapping the inputs keeps the sum but hands the outside factor to a different number.
5.OA.A.1 Step 4 Subtract the two results
- Now subtract: $(3 \star 5) - (5 \star 3) = 40 - 24 = 16$.
- That is choice (E).
- The answer is not $0$: it would be $0$ only if the two terms were equal, but $\star$ does not commute here because the outside factor changed from $5$ to $3$.
- This rules out the trap choice (C).
💡 The gap between the two terms comes entirely from the different second factors.
6.EE.A.2 The definition $a \star b = (a+b)b$ takes two inputs. The first input is $a$ and 5.OA.A.1 For $3 \star 5$, put $a=3$ and $b=5$. The sum inside is $3+5=8$, and the outside 5.NBT.B.5 For $5 \star 3$, put $a=5$ and $b=3$. The sum inside is $5+3=8$ again, but now t 5.OA.A.1 Now subtract: $(3 \star 5) - (5 \star 3) = 40 - 24 = 16$. That is choice (E). Th Review
Reasonableness: Both terms share the same sum $3+5=5+3=8$, so the difference is $8\cdot 5 - 8\cdot 3 = 8(5-3) = 8\cdot 2 = 16$ — matching the step-by-step result. Because the first term uses the larger outside factor ($5$ vs $3$), the difference must be positive, which immediately kills the negative choices (A) $-16$ and (B) $-8$. A difference of $16$ is small and sensible for numbers this size, and the shared sum of $8$ makes the clean factor-of-$8$ answer believable.
Alternative: Skip computing each term separately. Factor out the common sum $8$: since $3 \star 5 = 8\cdot 5$ and $5 \star 3 = 8\cdot 3$, the difference is $8\cdot 5 - 8\cdot 3 = 8(5-3) = 16$. Spotting that $a+b$ is the same both ways turns the whole problem into one multiplication.
CCSS standards used (min grade 6)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Reading the definition $a \star b = (a+b)b$ and identifying which input is $a$ and which is $b$ before substituting.)5.OA.A.1Use parentheses, brackets, or braces in numerical expressions and evaluate (Evaluating $(3+5)\cdot 5$ and combining the two results as $40-24$ in the correct order.)5.NBT.B.5Fluently multiply multi-digit whole numbers (Computing the products $8\cdot 5 = 40$ and $8\cdot 3 = 24$.)
⭐ A made-up symbol is just a recipe: read which slot each number goes into, because swapping the inputs can change the result.
⭐ A made-up symbol is just a recipe: read which slot each number goes into, because swapping the inputs can change the result.
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