Depth · one concept from first principles · step 2 of 4
Count how many fit along each edge
A box is to be packed full of identical small cubes with no gaps. We need how many cubes that takes.
GeometryMeasurement & data
Your answer
How to solve
Strategy Analyze the Units — Dividing volume by volume would give the right number here, but counting along each edge shows why it works and would catch a box the cubes cannot actually fill. Do it edge by edge -- after putting both in the same unit.
1STEP 1
Put the box in centimeters
The cube is measured in centimeters, so the box has to be too.
6 × 100 = 600, 3.6 × 100 = 360, 4.8 × 100 = 480
The box is 600 by 360 by 480 centimeters.
5.MD.A.1Analyze The Units2STEP 2
Count the cubes along each edge
Divide each edge by 6, the length of one cube. All three divide exactly, so the cubes really do fill the box.
600 ÷ 6 = 100, 360 ÷ 6 = 60, 480 ÷ 6 = 80
100 along, 60 deep, 80 up.
6.G.A.2Visualize Spatial RelationshipsThe cubes fill the box exactly, with none sticking out.
Why?
Each edge of the box is a whole number of cube edges laid end to end, because , and each divide by with nothing left over.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Lengths that measure in whole cubes let the cubes meet face to face with no gap anywhere, so the packed cubes and the box are the same solid.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Multiply the three counts
One layer holds 100 × 60 cubes, and there are 80 such layers.
100 × 60 = 6000, 6000 × 80 = 480000
480000 cubes are needed.
6.G.A.2Identify SubproblemsAnswer
480,000 cubes
Check by volume: the box is 600 × 360 × 480 = 103680000 cm³ and each cube is 216 cm³, and 103680000 ÷ 216 = 480000.
Takeaway
Count how many cubes fit along each edge -- that also shows whether they fit at all.
- Put the box in centimeters
- Count the cubes along each edge
- Multiply the three counts
Where next?
Another one like thissuggested
Step back and solidifysuggested
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