Problem
Five views of stacked rectangular blocks
Three blocks of different sizes are stacked into one solid. A small cube and a long bar sit on a big cube. Different directions hide different parts. Draw the view from each of five directions.
Reasoning · Grade 6-2 Top, Front, and Side Views of a Stack
Your answer
Watch out Opposite views seem mirrored, but only the outline is — the colour lines can differ.
How to solve
Strategy Visualize Spatial Relationships — Unlike a pile of identical stacking cubes, these blocks have different edge lengths, so a view is not a grid of squares — it is a rectangle cut up by lines that have to land in the right places. I therefore start by ruling the green cube's 4 cm top face into four 1 cm strips and saying exactly which strips each upper block covers. That single plan of the top face answers the top view immediately and turns each of the other four views into a small separate subproblem: draw the 4 by 4 green square, then work out where the red cube and the yellow bar project onto it and, crucially, which of the two hides part of the other from that particular side. Handling the five views one at a time, and finishing with a symmetry check between the two opposite pairs, is what keeps the hidden parts from being missed.
1STEP 1
Rule the top face into 1 cm strips and say what each block covers
Rule the top into strips and note what covers each.
strips 1,2 (back 2 cm) : red, strip 3 (all 4 cm) : yellow, else : green
Cutting the square into equal 1 cm strips is the same partitioning of a rectangle into rows and columns that makes array pictures work, and it is what lets the 2 cm block and the 1 cm block be compared fairly. Laying the real blocks on a piece of centimeter paper does the whole step for you.
2.G.A.2Create A Physical Representation2STEP 2
The top view: at every spot you see whichever block is highest there
From above you see the highest block.
red 2 × 2 = 4 cm², yellow 1 × 4 = 4 cm², green 16 - 4 - 4 = 8 cm²
From above nothing sticks out beyond the green cube, so the outline is easy and only the inside lines need care. Checking that the three colored areas add back up to the 16 cm² of the whole face is a quick way to be sure no strip was drawn the wrong width.
6.G.A.4Draw A DiagramLooking straight down, at every spot you see whichever block reaches highest there.
Why?
A taller block hides everything beneath it at that spot, and the tallest hides all the rest, so only its top shows.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Each spot on the floor gives exactly one square of the top view, so the picture has one square per spot and no more.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
The front view: the red cube and the end of the bar sit side by side on the green square
From the front the cube and bar sit side by side.
outline : 6 cm tall over the left 2 cm, 5 cm over the next 1 cm, 4 cm over the right 1 cm
Splitting the picture into 'the green square' and 'what pokes up above it' turns one awkward outline into two easy rectangles. The little 1 cm yellow square is the part beginners forget, so it helps to remember that the bar has to show its end face somewhere.
6.G.A.4Identify Subproblems4STEP 4
The back view: the mirror image of the front view
The back view is the front's mirror.
back view = mirror image of the front view across a vertical line
Fold a sheet of tracing paper down the middle and the front drawing lands exactly on the back drawing: that is what a line of symmetry means. Walking round an object is the physical version of that fold, which is why opposite views come in mirror pairs.
4.G.A.3Visualize Spatial Relationships5STEP 5
The left-side view: the red cube hides the back half of the yellow bar
From the left the cube hides part of the bar.
bar length seen = 4 - 2 = 2 cm, outline area = 16 + 4 + 2 = 22 cm²
This is the first view where being careful pays: the bar really is 4 cm long, but you may only draw the part you can actually see. Because the red cube is exactly 2 cm deep, it swallows exactly half the bar, and the drawing must show the 2 cm to 2 cm split at true scale.
7.G.A.1Visualize Spatial Relationships6STEP 6
The right-side view: now the yellow bar hides the bottom half of the red cube
From the right the bar hides part of the cube.
red height seen = 2 - 1 = 1 cm, outline area = 16 + 4 + 2 = 22 cm²
Which block does the hiding depends entirely on which side you are standing, and that is the whole lesson of this page. From the left the tall red cube is in front; from the right the low yellow bar is in front, so it cuts the bottom 1 cm off the red cube instead.
7.G.A.1Visualize Spatial Relationships7STEP 7
Check the pairs: outlines mirror, but the color lines need not
Outlines pair up, but the colour lines differ.
left: 16 + 4 + 2 = 22 cm² = 16 + 4 + 2 : right, front = 16 + 4 + 1 = 21 cm² = back
The outline of a solid seen from one side is the shadow it casts, and a shadow from the opposite side is the same shadow flipped. The colors are a different matter, because they record which block is nearest, and that changes the moment you walk round.
4.G.A.3Make A Systematic ListAnswer
five pictures: top, front, back, left, right
4 + 4 + 8 = 16
The sizes are right. The solid is 4 cm wide, 4 cm deep and 6 cm tall at its highest, so the top view must fit exactly a 4 cm square and each of the four side views must fit inside a 4 cm by 6 cm rectangle, which they all do. Areas confirm the drawings: the three regions of the top view come to 4 + 4 + 8 = 16 cm², exactly the area of the green cube's face; the front and back outlines both come to 16 + 4 + 1 = 21 cm²; and the left and right outlines both come to 16 + 4 + 2 = 22 cm². Opposite views having equal areas is a necessary consequence of their being mirror images, so three separate equalities all check out. The hidden parts are also the right sizes: the red cube is 2 cm deep and the bar 4 cm long, so exactly half the bar disappears from the left view, and the bar is 1 cm high against the red cube's 2 cm, so exactly half the red cube's height disappears from the right view. Finally, a mistake worth ruling out: the side views are 22 cm² rather than 21 cm², because sideways the bar shows its long face and not just its 1 cm square end.
Takeaway
Which block hides which changes as you walk around, so opposite views always share an outline but need not share the lines inside.
- Rule the top face into 1 cm strips and say what each block covers
- The top view: at every spot you see whichever block is highest there
- The front view: the red cube and the end of the bar sit side by side on the green square
- The back view: the mirror image of the front view
- The left-side view: the red cube hides the back half of the yellow bar
- The right-side view: now the yellow bar hides the bottom half of the red cube
- Check the pairs: outlines mirror, but the color lines need not