Problem
Find the cubes removed twice over
Three straight lines of cubes are taken out of a large cube, one through each pair of opposite faces. We need how many cubes are left.
Geometry
Your answer
How to solve
Strategy Change Focus / Count the Complement — Three lines of five is fifteen cubes only if they are separate, and they are not -- all three run through the middle. Count the lines, then correct for the cube that got counted more than once.
1STEP 1
Count the lines as if they were separate
Each line runs the full width of the cube, so it holds 5 cubes, and there are three lines.
5 × 3 = 15
15 -- but not all of them are different cubes.
5.MD.C.3Visualize Spatial Relationships2STEP 2
Find what was counted twice
All three lines pass through the cube at the very centre, so that one cube was counted three times when it should be counted once.
3 - 1 = 2
The count is 2 too many.
5.MD.C.3Change Focus Count The ComplementThe centre cube was counted three times but can only be removed once.
Why?
All three lines run through the middle of the cube, so that one cube belongs to every one of them.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Counting the removed cubes means counting each cube once, so a cube belonging to three lines still contributes one to the count.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Correct the count of removed cubes
Take those two extra countings off.
15 - 2 = 13
13 different cubes come out.
5.MD.C.3Identify Subproblems4STEP 4
Take them off the whole cube
Subtract the removed cubes from the 125 the cube started with.
125 - 13 = 112
112 cubes are left.
6.G.A.2Identify SubproblemsAnswer
112 cubes
Count the removed cubes another way: the centre cube, plus 4 from each of the three lines once the centre is set aside, is 1 + 4 × 3 = 13.
Takeaway
When things you are counting overlap, count them all and then take the overlaps back off.
- Count the lines as if they were separate
- Find what was counted twice
- Correct the count of removed cubes
- Take them off the whole cube