AMC 8 · 2003 · #1
Easy mode Grade 2Problem
Picture a cube sitting on a table.
Jamie counted all the edges of the cube. Jimmy counted all the corners. Judy counted all the faces. Then they added their three numbers together.
What was the total?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Three kids look at a single cube. Jamie counts its edges, Jimmy counts its corners (vertices), and Judy counts its faces. Add the three counts together. What is the total?
Givens: The shape is a cube; Jamie counts edges; Jimmy counts corners (vertices); Judy counts faces; Answer choices: (A) $12$, (B) $16$, (C) $20$, (D) $22$, (E) $26$
Unknowns: The sum: edges $+$ corners $+$ faces
Understand
Restated: Three kids look at a single cube. Jamie counts its edges, Jimmy counts its corners (vertices), and Judy counts its faces. Add the three counts together. What is the total?
Givens: The shape is a cube; Jamie counts edges; Jimmy counts corners (vertices); Judy counts faces; Answer choices: (A) $12$, (B) $16$, (C) $20$, (D) $22$, (E) $26$
Plan
Primary tool: #7 Break into Subproblems
Secondary: #2 Make an Organized List
The question asks for one number, but that number is built from three independent counts. Tool #7 (Break into Subproblems) splits the task into three small jobs — count edges, count corners, count faces — and the final step just adds them. Tool #2 (Make an Organized List) helps each sub-count: list the faces by direction (top, bottom, and four sides), the corners by floor and ceiling, and the edges by which group of parallel lines they belong to. Listing in groups makes sure no part of the cube is missed or double-counted.
Execute — Answer: E
2.G.A.1 Step 1 - Count the faces.
- A cube has a top, a bottom, and four side walls.
- Group them by direction so nothing is missed.
💡 Grade 2 identifies shapes by counting their flat sides; a cube has $6$ square faces.
2.G.A.1 Step 2 - Count the corners.
- Every corner sits at one of the four floor corners or one of the four ceiling corners.
💡 Each corner of the bottom square has a matching corner directly above it on the top square, giving $4 + 4 = 8$.
2.G.A.1 Step 3 - Count the edges by grouping them by direction.
- The cube's edges come in three families of parallel segments: $4$ around the bottom, $4$ around the top, and $4$ standing up between them.
💡 Sorting edges into three groups of $4$ guarantees each edge is counted exactly once.
2.NBT.B.5 Step 4 Add the three subtotals to finish.
💡 Grade 2 within-$100$ addition: $12 + 8 = 20$, then $20 + 6 = 26$.
2.G.A.1 Count the faces. A cube has a top, a bottom, and four side walls. Group them by 2.G.A.1 Count the corners. Every corner sits at one of the four floor corners or one of 2.G.A.1 Count the edges by grouping them by direction. The cube's edges come in three fa 2.NBT.B.5 Add the three subtotals to finish. Review
Reasonableness: Quick check with Euler's relation for any convex polyhedron: $V - E + F = 2$. Plugging in $V = 8$, $E = 12$, $F = 6$ gives $8 - 12 + 6 = 2$, which is correct, so the three counts are consistent. The sum $V + E + F = 8 + 12 + 6 = 26$ matches answer (E). Also, $26$ is the largest choice, which fits because we are adding three positive whole numbers, not subtracting.
Alternative: Tool #15 (Visualize): picture a die. Looking at it, you immediately see $6$ square faces, $8$ pointy corners where you could prick your finger, and $12$ straight edges where two faces meet. Add: $6 + 8 + 12 = 26$, so the answer is (E).
CCSS standards used (min grade 2)
2.G.A.1Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces (Identifying the cube's parts and counting its $6$ faces, $8$ corners, and $12$ edges by grouping them in organized lists.)2.NBT.B.5Fluently add and subtract within $100$ using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (Adding the three subtotals $12 + 8 + 6$ to reach the final total $26$.)
⭐ Big questions about a 3D shape become easy when you split them up: count faces, count corners, count edges, then add. A cube always gives $6$, $8$, $12$ — and they add to $26$.
⭐ Big questions about a 3D shape become easy when you split them up: count faces, count corners, count edges, then add. A cube always gives $6$, $8$, $12$ — and they add to $26$.
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