Problem
Match a solid to its three views
Six cube solids are numbered. A box holds three flat grid figures. Which figure is which view is not stated. Pick the solid that matches all three.
Reasoning · Grade 6-2 Counting Cubes and Stackings
Your answer
Watch out Reading them in order as top, front, side fails: the order is not given.
How to solve
Strategy Visualize Spatial Relationships — Comparing six oblique drawings with three flat figures by eye is where this problem goes wrong, so the first move is to re-record every solid in a form that can be compared arithmetically: a small table holding the number of cubes in each stack. Once each solid is a table of heights, every view is a one-line calculation on that table, and the six candidates form a genuinely finite set that can be knocked out a direction at a time. Working out which flat figure has to be the top view first is what makes the elimination cheap, because it settles three of the six solids in a single sweep.
1STEP 1
Turn each drawing into a table of stack heights
Turn each drawing into a height table.
2 + 3 + 2 + 2 + 2 + 1 = 12
Counting how many cubes are stacked on one square is something you can do with your finger; doing it square by square converts a picture you have to imagine into numbers you can check.
5.MD.C.4Organize Information In More Ways2STEP 2
Only the right-hand figure can be the top view
Only one figure can be the top view.
Deciding whether a picture is taken from above or from the side is the everyday above/in-front-of thinking a young child already does, and the give-away here is simply that cubes cannot hang in the air.
K.G.A.1Visualize Spatial Relationships3STEP 3
Match the footprint and knock out three solids
Matching footprints knocks out three solids.
The footprint is the easiest of the three views to read off a table -- you just ask which squares carry any cubes at all -- so it is the cheapest test to apply first.
K.G.A.1Eliminate Possibilities4STEP 4
List the front view and the right-side view of the three survivors
List the front and side views of the survivors.
max(2, 2, 1) = 2, max(3, 2) = 3, max(2) = 2
Looking at a row of stacks from the side, the only one you actually see the top of is the tallest, so each bar is just the winner of a short height comparison.
1.MD.A.1Make A Systematic List5STEP 5
Eliminate down to one solid
Two more drop out.
Two near-misses are the useful part of this step: ③ matches one figure and ⑥ matches the other, which is exactly why a solid has to be tested from every direction before it is accepted.
1.MD.A.1Eliminate PossibilitiesOnce every other solid clashes with one of the views, the one still standing has to be the answer.
Why?
A single view that a solid cannot produce is enough to rule that solid out, whatever its other views look like.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
The candidates cover every possibility, so eliminating all but one leaves the answer with no guessing involved.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
6STEP 6
Read the three views back off solid ⑤ as a check
The remaining solid 5 matches all three.
2 + 3 + 2 = 7 and 1 + 2 + 3 = 6
Counting the squares in each printed figure and comparing with the bar totals is a check a young solver can do without imagining anything in three dimensions.
5.MD.C.4Visualize Spatial RelationshipsAnswer
5
The answer is a number from 1 to 6, which is the only kind of answer the question can take, and exactly one solid survived rather than none or several. The counts line up too: ⑤ uses 12 cubes on 6 squares with a tallest stack of 3, and the three printed figures contain 6, 7 and 6 squares -- the 6-square figure is the footprint, and 7 and 6 are the totals of the bar heights 2+3+2 and 1+2+3. The three near-misses are informative rather than worrying: ① fails on the footprint, ③ matches the left figure but not the middle one, and ⑥ matches the left figure but has a front view of 3, 2, 2. A single wrong direction is enough to rule a solid out, and only ⑤ has none.
Takeaway
Write each solid as a little table of stack heights, spot the one flat picture whose squares would have to float in the air (that one is the top view), and then knock out solids one direction at a time.
- Turn each drawing into a table of stack heights
- Only the right-hand figure can be the top view
- Match the footprint and knock out three solids
- List the front view and the right-side view of the three survivors
- Eliminate down to one solid
- Read the three views back off solid ⑤ as a check