Problem
Count ways to tile with one piece
An 8 m by 4 m wall is covered by four sheets, each 4 m by 2 m, with no gaps. A sheet may go flat or upright. The areas match exactly, so four sheets fit. Count the different ways to paper it.
Reasoning · Grade 3-1 Filling and Joining Shapes
Your answer
Watch out Matching areas suggest many layouts, but the wall is shallow — only five actually fit.
How to solve
Strategy Make a Systematic List — If we measure the wall in 2 m units it is 4 units wide and 2 units tall, and each sheet is a 2-by-1 block (a domino). Covering the wall column by column from the left and listing every choice without missing or repeating any guarantees a complete count, and a quick drawing of each arrangement keeps the list honest.
1STEP 1
Shrink to an easier grid
Halving the lengths leaves a 4-by-2 grid.
8 ÷ 2 = 4, 4 ÷ 2 = 2
Cutting the picture into equal 2 m squares turns hard measurements into a tidy grid a third grader can count.
2.G.A.2Solve An Easier Related Problem2STEP 2
Work column by column
Each sheet becomes a two-cell domino.
Always filling the first open spot first means no arrangement can be skipped or double-counted.
2.G.A.2Make A Systematic ListAlways filling the first still-open square before anything else means no arrangement can be skipped or written down twice.
Why?
The first open square is decided by the picture, not by choice, so every arrangement is reached along exactly one path of decisions.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
At each step only a few tiles can cover that square at all, and every other option is ruled out on the spot.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
List every arrangement
Split on whether the leftmost cell takes an upright or a flat sheet — two cases.
vertical-vertical-vertical-vertical, 2 horiz + 2 horiz, VV+HH, HH+VV, V+HH+V
Sketching each filling makes it obvious they are all different and that no sixth one is possible.
2.G.A.2Draw A Diagram4STEP 4
Count the list
Following each through gives 5 distinct layouts.
1 + 1 + 1 + 1 + 1 = 5
Counting the finished pictures is just simple addition once the list is complete.
1.OA.A.2Make A Systematic ListAnswer
5 ways
Each arrangement uses four 8 m² sheets covering 32 m², exactly the wall's area, so every listed covering is gapless and overlap-free. Five is a small, believable count for such a tiny 4-by-2 grid, and trying to add a sixth always repeats one already drawn.
Takeaway
Cut the wall into equal squares, fill from the left, and list every way in order: only Grade 2 picture-counting is needed to find all 5!
- Shrink to an easier grid
- Work column by column
- List every arrangement
- Count the list