AMC 10 · 2016 · #9
Grade 6 geometry-2dPick an answer.
The post counts on the two sides are linked by a 'twice as many' relationship and a fixed total, so naming the shorter-side count with a variable turns the words into one equation. A quick sketch keeps the shared corner posts from being counted twice, and once a side's post count is known the length is a small multiplication subproblem.
Name the posts per side
One letter names the posts on both kinds of side.
Giving the unknown count a name lets the 'twice as many' rule become arithmetic instead of guesswork.
6.EE.B.6Introduce A VariableCount posts around the rectangle
The corners must be corrected for twice.
A corner post is shared by two sides, so counting it once for each side overcounts it by exactly one.
A corner post is shared by two sides, so counting it once for each side overcounts it by exactly one.
▸ Why?
Each corner belongs to exactly two sides, so summing over sides visits it twice.
▸ Why?
Every other post belongs to exactly one side, so only the corners need correcting.
Solve for n
Solving gives 4 posts on a short side.
Undoing the subtraction and then the multiplication peels the equation back to the single value of n.
6.EE.B.7Introduce A VariableTurn posts into side lengths
Gaps, not posts, give the lengths.
Posts are fenceposts, not fence pieces, so a row of posts has one fewer gap than it has posts.
4.OA.A.3Draw A DiagramCompute the area
Multiplying gives 336, choice (B).
Area of a rectangle is just the two side lengths multiplied together.
4.MD.A.3Identify SubproblemsCount the gaps, not the posts: a row of posts always has one fewer space than posts, and corners get shared by two sides.
- Name the posts per side
- Count posts around the rectangle
- Solve for n
- Turn posts into side lengths
- Compute the area