Reasoning · Grade 6-2 Top, Front, and Side Views of a Stack
Problem
Views of clear and colored cubes
18 small cubes form a block 3 across, 3 deep and 2 high. Only 6 are coloured: upper layer back-left, middle-right, front-middle; lower layer back-right, middle-right, front-left. The other 12 are clear and invisible. Draw the views from above, the front and the right.
Your answer
Watch outA near cube seems to block the view, but the rest are clear so it hides nothing.
How to solve
StrategyVisualize Spatial Relationships — The pictures of see-through cubes are hard to read, so I stop looking at the drawing and give every one of the 18 cubes a plain address instead: which column (left, middle, right), which row (back, middle, front), and which layer (lower, upper). Then the whole problem becomes a list of six addresses and a single yes-or-no question repeated once per square of each answer grid: is any cube on this line of sight colored? Changing focus to the complement makes it even faster, because a square is blank only if every cube on its line is clear, which is a rarer thing to check. Before trusting the list I test it against the front view the book has already filled in; if my addresses reproduce that picture exactly, they are right, and the top view and side view can be read off with confidence.
1STEP 1
The rule for see-through cubes: one colored cube anywhere on the line is enough
One coloured cube on the line is enough to show.
square colored ⇔ at least one colored cube on the line; square blank ⇔ all cubes on the line are clear
This is exactly like holding a stack of clear plastic sheets up to the window with a dot inked on some of them: the dot shows through no matter which sheet it is on. Turning the question into 'is this line all clear?' means each square of the answer needs only a quick scan of two or three cubes.
6.G.A.4Change Focus Count The Complement
With see-through cubes, one coloured cube anywhere along the line of sight is enough to colour that square.
Why?
A line of sight either contains a coloured cube or it does not, and those two cases cover every square with no overlap.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
Each line of sight gives exactly one square of the view, so testing the lines one by one fills the picture with none missed.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Three position words pin a cube down as surely as a seat number in a theatre pins down a seat: row, column, layer. Once the cubes have names, the spatial puzzle turns into bookkeeping that cannot be muddled by a confusing picture.
K.G.A.1Make A Systematic List
3STEP 3
Test the addresses against the front view the book already drew
The printed front view shows the addresses are right.
upper row : ■ ■ ■ lower row : ■ □ ■
Using the answer that is already given as a test of your own reading of the figure is the cheapest insurance in the whole problem. If the front view had not come out right, the mistake would be in the addresses, not in the views still to be drawn.
5.MD.C.4Draw A Diagram
4STEP 4
The top view: look straight down each of the nine little towers
From above 5 squares are coloured.
back: ■ □ ■ middle: □ □ ■ front: ■ ■ □
Looking down, the two layers stack into one square of the answer, so each of the nine questions involves only two cubes. That is why the top view is the easiest of the three to draw even though it has the most squares.
5.MD.C.4Visualize Spatial Relationships
5STEP 5
The right-side view: look along each of the six rows of three
From the right all 6 come out coloured.
upper row : ■ ■ ■ lower row : ■ ■ ■
It is worth checking a fully shaded answer twice, but it is genuinely possible here: six colored cubes spread over six different row-and-layer lines fill every square. Because the sideways lines are three cubes long, they catch color much more easily than the two-cube towers seen from above.
Counting the same six cubes three different ways gives three chances to catch an error, and each count is small enough to do on your fingers. The rule 'one square per colored cube, minus the ones that hide behind each other' explains why the three views look so different from one another.
5.MD.C.3Make A Systematic List
Answer
5, 5 and 6 shaded squares
6 − 1 = 5, 6 − 0 = 6
The shapes and sizes of the two answers are right: looking down at a block that is 3 by 3 on the floor must give a 3 by 3 picture, and looking sideways at a block that is 3 deep and 2 high must give a picture 3 wide and 2 tall, which is what the answer grids provide. The shaded counts are consistent as well. There are 6 colored cubes, so no view can shade more than 6 squares, and the top view shades 5, the front view 5 and the right-side view 6 — never more. Each of those numbers is explained exactly by how many pairs of colored cubes happen to line up in that direction: one pair vertically, one pair front-to-back, none sideways. A final sanity check is that no view can be emptier than the block's own colored count allows: since each layer holds 3 colored cubes and each row of the side view is a whole layer, no row of the right-side view could have been blank.
Takeaway
Clear cubes hide nothing, so a square gets colored if even one colored cube sits anywhere along the line you are looking down.
The rule for see-through cubes: one colored cube anywhere on the line is enough
Give all six colored cubes an address
Test the addresses against the front view the book already drew
The top view: look straight down each of the nine little towers
The right-side view: look along each of the six rows of three