Math / Counting a floor
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Lesson 1 · How many meets Place
You know how many tiles across. You know how many down. The floor is too big to tap every tile. The count is the job (multiplikation: räkna en rektangel).
Later Place × How-many becomes a length you cannot pace: .
Scribes already counted rectangular fields. The Rhind and Moscow papyri treat area as length times width — land after the flood, not a chant. Bricks in a wall and soldiers in ranks are the same job: rows of a known length.
Greek classrooms later sold “multiplication” as repeated addition. That is a teaching trick. The original job was area: a floor, a field, a rank. Adding 4 + 4 + 4 is how you check a small one. It is not why the tool exists.
Place-value names made huge products speakable. You no longer needed a new word for every size of harvest.
Once a side can be unnamed, the same rectangle is a pile you solve for — the floor is the answer, not the given. School later says .
Honesty: nobody invented times. We have copies of jobs, not a single inventor.
Drag a side. Each black square is one tile. The number comes after the floor exists — not before.
Across = how many in a row. Down = how many rows. (gånger) means “that many of those rows.”
3 × 4 = 12 tiles
Cap is 8 across and 8 down on a phone. A real floor can be larger; the idea does not change.
3 down and 4 across is the same floor as 4 down and 3 across. You rotated it. You did not add tiles.
3 × 4 — three rows of four
4 × 3 — four rows of three
12 either way. That is why the chant “3 times 4 is 4 times 3” is true: it is one rectangle, two ways to name the sides.
Kommutativa lagen: a·b = b·a. Same floor, turned.
Start with 3 × 4 = 12. Add one tile to the long side: you glue on one extra column. That is +3, so 15.
Add one to both sides and you get an L of new tiles: a new column and a new row, plus the corner where they meet. 4 × 5 = 20. Not 12 + 1 + 1.
Grey = the old 3×4. Black = what you added. The corner tile is why “plus one each way” undercounts.
People already counted rows: plants in a bed, bricks in a wall, soldiers in a line. One row you can point along. Many rows, you stop pointing and you take “this many rows of that many.”
Egyptian problem-texts ask for the area of rectangular fields — length times width in cubits, then the count of setjat (field units). That is this lesson, on dirt instead of bathroom tile.
Honesty: we know this from copies, especially the Rhind Mathematical Papyrus (Ahmes, about 1650 BCE, from older material). The scribes were not inventing a cute story. They were measuring land after the flood.
2 children have 2 apples each. How many apples?
Nothing is at stake. Nobody is buying, sharing, or running out.
You are tiling a 2-by-2 patch by the door, and two more patches the same size along the hall. Do you have 12 tiles in the box, or do you go back to the shop?
Three patches, four tiles each: 3 × 4 = 12. The box is enough. That is a job.
School word in the margin: area (area) is just “how many unit squares cover it.” If the unit is a tile, the area is the tile count.
Cover the answer. Say a range out loud. Then open.
1. Bathroom floor. The room is 5 tiles across and 6 tiles down. Tiles come in boxes of 10. How many boxes do you buy? (No extra for cuts — assume they fit.)
Estimate: more than 2 boxes, less than 10? Pick a number before you open.
5 × 6 = 30 tiles. 30 ÷ 10 = 3 boxes. If you added 5 + 6 = 11, you counted a walk around part of the edge, not the floor.
2. Hall seats. 8 seats in a row, 7 rows. Fire rule: you may not exceed 60 people. Are you over?
Estimate: 8 × 7 is a bit under 8 × 8 = 64, so near 60. Over or under?
8 × 7 = 56. Under 60. You have four seats of slack. 8 × 8 would be 64 — that would fail. This hall is not a square.
3. Mural. A wall takes 4 tiles across and 3 down. You want the same mural twice as wide and twice as tall. Tiles needed?
Estimate: not “twice 12.” Twice both sides is four copies of the original rectangle.
Original: 4 × 3 = 12. New: 8 × 6 = 48. Four times the tiles, because both sides grew. Doubling one side only would be 24.
4. Feed a table. 6 plates in a row, 4 rows. Each plate gets 2 rolls. How many rolls from the bakery?
Estimate: about 25 plates? Then twice that in rolls.
Plates: 6 × 4 = 24. Rolls: 24 × 2 = 48. Two rectangles stacked: first the plates, then two rolls on each plate.
Each right gate item is 20 points. Hands-on is 20. Wrong is 0 until you retry and get it. Next unlocks only at 100 — and only if that page exists.
1. What is “times” doing?
2. A path is 7 tiles across and 3 tiles down. How many tiles to buy?
3. Why is 3 × 4 the same count as 4 × 3?
4. Which is a real question?