Reasoning · Grade 6-2 Top, Front, and Side Views of a Stack
Problem
Views of a cube stack from each direction
A stacking table gives each square's cube count in the plan from above, 0 for none. Rows run back to front: (1) 1, 0, 3 / 2, 3, 1. (2) 0, 2, 1 / 3, 1, 2 / 1, 0, 0. (3) 2, 2, 1 / 1, 1, 3 / 2, 3, 0. Draw (1) front and back, (2) back and left, (3) left and right.
Your answer
Watch outAdding a line's counts is wrong: the height seen is the largest of them.
How to solve
StrategyVisualize Spatial Relationships — Guessing at views from a picture of the solid is exactly how mistakes happen, so I turn the whole job into two mechanical questions and answer them the same way every time. First, what does one direction of view do to the table? It collapses a whole line of the table into a single column of the picture, and the height of that column is the largest number in the line, because the tallest pile hides every shorter one behind it. Second, in what order do those columns appear from left to right? That is settled once and for all by asking which line of the table is on my left when I stand in that spot. With those two rules fixed I can list, line by line, the heights for each of the six pictures, and then check the whole answer against the fact that walking to the opposite side of the shape flips the picture left-to-right. Real cubes, or a hand held up like a wall, make both rules obvious to a beginner.
1STEP 1
The rule for a view: each column is as tall as the tallest pile in that line
A column is as tall as the tallest pile in its line.
height of a column = max(numbers in that line), max(3,2)=3, max(3,1)=3
This is the same reason a short child standing behind a tall one disappears from a class photo. Nothing is being added up, so no arithmetic beyond 'which of these small numbers is biggest' is needed, and the cubes themselves make it visible.
5.MD.C.4Visualize Spatial Relationships
2STEP 2
Decide which line of the table you are looking down, and in what order
Decide which line, and in what order.
front: L, M, R back: R, M, L left side: back, mid, front right side: front, mid, back
The only thing to be careful about is that 'left' belongs to the viewer, not to the paper. Walk around a chair in the classroom and watch what used to be on your left swing over to your right; that single experience is all the geometry this step needs.
K.G.A.1Draw A Diagram
3STEP 3
Notice the mirror: opposite directions give mirror-image pictures
The opposite view is left-right mirrored.
back = reverse(front), right side = reverse(left side)
Mirror images are already familiar from folding a paper heart in half: the two halves match when the paper is folded along the line. Here the fold line is the vertical line between the two drawings, and the matching parts are the two views of one shape from opposite sides.
4.G.A.3Visualize Spatial Relationships
Views from opposite directions are mirror images of each other, so one drawing hands over the other for free.
Why?
Walking round to the far side does not change the solid, so the second view is the first one seen the other way round.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Each line of the table gives exactly one column of the picture, so reversing the reading order reverses the columns and nothing else.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
4STEP 4
Part (1): the front view and the back view
In (1) the front is 2, 3, 3 and the back mirrors it.
max(1,2)=2, max(0,3)=3, max(3,1)=3 → front 2,3,3 → back 3,3,2
Writing the three maximums out in a row before shading anything keeps the drawing honest. Treating a blank square as the number 0 makes it behave like every other square, so no special case is needed for the empty spot.
6.G.A.4Make A Systematic List
5STEP 5
Part (2): the back view and the left-side view
In (2) the back is 3, 2, 2 and the left side 2, 2, 3.
The two views use the very same nine numbers, once grouped into columns and once grouped into rows. Re-sorting the table rather than re-imagining the solid is what makes a nine-square problem no harder than a four-square one.
6.G.A.4Make A Systematic List
6STEP 6
Part (3): the left-side view and the right-side view
In (3) the left is 2, 3, 3 and the right mirrors it.
max(2,2,1)=2, max(1,1,3)=3, max(2,3,0)=3 → left 2,3,3 → right 3,3,2
Notice that the tall pile of 3 in the middle row sits at the right-hand end, and the tall pile of 3 in the front row sits in the middle, yet neither position shows up in a side view. From the side only the height of a row survives, which is exactly why one shape can look the same from one direction and quite different from another.
6.G.A.4Make A Systematic List
7STEP 7
Check the six pictures against the cubes themselves
Every check here is a count of squares or a comparison of small numbers, which is why this is a good problem to verify with your hands. Building the pile and turning it round is slower than the rule, but it convinces you that the rule was right.
5.MD.C.3Create A Physical Representation
Answer
2 3 3 and 3 3 2, 2 2 3 and 2 2 3, 2 3 3 and 3 3 2
Every answer is a whole number of cubes between 1 and 3, which is right because the tallest pile written anywhere in the three tables is 3, and no view can show a column taller than the tallest pile it contains. All six pictures therefore fit inside the 3 × 3 answer grids the book supplies, and none of them is completely full, which matches the fact that none of the three shapes is a solid block. The mirror rule provides an independent check on four of the six pictures: the back view of (1) is the front view of (1) reversed, and the right-side view of (3) is the left-side view of (3) reversed, both of which came out that way. The shaded-square totals are also consistent with the cube counts: shape (1) uses 10 cubes and its front view shades only 8 squares, shape (2) uses 10 cubes and its left-side view shades 6, shape (3) uses 15 cubes and its side views shade 8, and a view can never shade more squares than there are cubes, since hiding only ever removes squares.
Takeaway
From any side you only ever see the tallest pile in each line, and walking round to the opposite side just flips the whole picture left to right.
The rule for a view: each column is as tall as the tallest pile in that line
Decide which line of the table you are looking down, and in what order
Notice the mirror: opposite directions give mirror-image pictures
Part (1): the front view and the back view
Part (2): the back view and the left-side view
Part (3): the left-side view and the right-side view
Check the six pictures against the cubes themselves