AMC 8 · 2011 · #8
Grade 3 countingProblem
Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Bag A holds three chips marked $1$, $3$, $5$ and Bag B holds three chips marked $2$, $4$, $6$. One chip is drawn from each bag and the two numbers are added. How many different sums are possible?
Givens: Bag A $= \{1, 3, 5\}$ (all odd); Bag B $= \{2, 4, 6\}$ (all even); One chip is drawn from each bag and the values are added; Answer choices: (A) $4$, (B) $5$, (C) $6$, (D) $7$, (E) $9$
Unknowns: The number of different possible values of the sum $a + b$, where $a \in \{1,3,5\}$ and $b \in \{2,4,6\}$
Understand
Restated: Bag A holds three chips marked $1$, $3$, $5$ and Bag B holds three chips marked $2$, $4$, $6$. One chip is drawn from each bag and the two numbers are added. How many different sums are possible?
Givens: Bag A $= \{1, 3, 5\}$ (all odd); Bag B $= \{2, 4, 6\}$ (all even); One chip is drawn from each bag and the values are added; Answer choices: (A) $4$, (B) $5$, (C) $6$, (D) $7$, (E) $9$
Plan
Primary tool: #11 Make a Table
Secondary: #1 Find a Pattern
There are only $3 \times 3 = 9$ ways to pick one chip from each bag, so Tool #11 (Make a Table) is the cleanest way to lay every sum out in a $3 \times 3$ addition grid — nothing can be missed. Tool #1 (Find a Pattern) then explains the structure of the answer: odd $+$ even is always odd, and the sums are evenly spaced from the smallest ($1+2=3$) to the largest ($5+6=11$), so the distinct sums must be $3, 5, 7, 9, 11$.
Execute — Answer: B
3.OA.A.1 Step 1 - Set up a $3 \times 3$ addition table with Bag A values down the side and Bag B values across the top.
- Each cell holds the sum $a + b$ for that row and column.
💡 An addition table is the Grade 3 way to organize "each from this set combined with each from that set" — every possible pair gets exactly one cell.
3.OA.D.9 Step 2 - Read the nine sums off the table: $3, 5, 7, 5, 7, 9, 7, 9, 11$.
- Several values repeat — for example, $1+4 = 3+2 = 5$ and $1+6 = 3+4 = 5+2 = 7$.
- The problem asks for different sums, so collapse the duplicates.
💡 Spotting that sums repeat in a regular pattern — and that the set of different sums forms a Grade 3 arithmetic pattern — is the Tool #1 move.
3.OA.A.1 Step 3 - Count the distinct sums in $\{3, 5, 7, 9, 11\}$.
- There are five of them, so there are $5$ different possible values of the sum.
💡 Counting the distinct entries that appear in the addition table is the same equal-groups counting move from Grade 3 multiplication.
3.OA.A.1 Set up a $3 \times 3$ addition table with Bag A values down the side and Bag B v 3.OA.D.9 Read the nine sums off the table: $3, 5, 7, 5, 7, 9, 7, 9, 11$. Several values r 3.OA.A.1 Count the distinct sums in $\{3, 5, 7, 9, 11\}$. There are five of them, so ther Review
Reasonableness: Every chip in Bag A is odd and every chip in Bag B is even, so every sum is odd. The smallest sum is $1+2=3$ and the largest is $5+6=11$. The odd numbers from $3$ to $11$ are $3, 5, 7, 9, 11$ — exactly $5$ values. Each of these is reachable (for instance $3=1+2$, $5=1+4$, $7=1+6$, $9=3+6$, $11=5+6$), so the count of $5$ is tight: no gaps and no extras. Answer (B) is consistent.
Alternative: Tool #1 (Find a Pattern) directly: the smallest sum is $1+2=3$, the largest is $5+6=11$, and every sum is odd + even $=$ odd. The odd numbers from $3$ to $11$ form the arithmetic sequence $3, 5, 7, 9, 11$, and a quick check confirms each is actually attainable. That gives $5$ different sums without writing out the full table.
CCSS standards used (min grade 3)
3.OA.A.1Interpret products of whole numbers and the equal-groups model of multiplication (Recognizing that the $3 \times 3 = 9$ entries of the addition table list every possible (Bag A, Bag B) pair exactly once, and counting the $5$ distinct sums that appear in the table.)3.OA.D.9Identify arithmetic patterns and explain them using properties of operations (Observing the pattern that odd $+$ even is always odd and that the distinct sums $3, 5, 7, 9, 11$ form a regular arithmetic sequence, which justifies collapsing the duplicates.)
⭐ This AMC 8 problem only needs Grade 3 skills — make an addition table, spot the pattern, and count — that you already know!
⭐ This AMC 8 problem only needs Grade 3 skills — make an addition table, spot the pattern, and count — that you already know!