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Geometry 9-12 / Week 12 / Tuesday
2/6
Week 12 · Modeling the Set and Opening Night

Tuesday

Sheets, cups and seats
// Every prop a shape, every plan a number
⏱ about 20 min

Tuesday: Sheets, Cups and Seats

Wren counts the plywood row. "12 sheets covered 24 square meters of backdrop frame. What do you notice?"

"Half a sheet for every square meter," Comet says. "So a bigger frame is easy. Multiply."

"And the jug," Wren says, lifting it. "2 cubic decimeters, and it filled 8 of our cups."

"Four cups a cubic decimeter," Comet says. "What can we make of the seats? The floor is a rectangle."

Nova projects the floor plan with 96 chairs on 48 square meters. "Would you like a hint? Amount over space, every time."

"Two seats per square meter," Wren says. "That is density. Our designer, same idea three ways: by area, by volume, by area again."

Density: an amount divided by the space it fills

Density by area is an amount per square meter: sheets of plywood, tiles or seats per square meter.

Density by volume is an amount per cubic unit: cups of water per cubic decimeter, foam beads per cubic centimeter.

Once you know a density, multiply by a new area or volume to predict a new amount. Divide to go the other way.

  1. Measure the space: an area in square meters or a volume in cubic units.
  2. Count the amount that fills it: sheets, cups, tiles or seats.
  3. Divide the amount by the space. Write the unit: sheets per square meter.
  4. To predict: multiply the density by a new space. To check: divide a new amount by its space and compare.

The crew's densities

Every number here is the crew's own count from the Scene Shop and the theater floor.

WhatAmountSpaceDensity
plywood on the backdrop frame12 sheets24 square meters0.5 sheets per square meter
water in the jug8 cups2 cubic decimeters4 cups per cubic decimeter
seats on the floor96 seats48 square meters2 seats per square meter
tiles on the honeycomb panel114 tiles3 square meters38 tiles per square meter

Predict with it: a second frame of 30 square meters needs 0.5 × 30 = 15 sheets.

A 5 cubic decimeter jug would fill 4 × 5 = 20 cups.

FIND THE DENSITY
  • Read the question.
  • Tap your answer.
12 sheets of plywood cover 24 square meters of frame. What is the density in sheets per square meter?
The jug holds 2 cubic decimeters and fills 8 cups. What is the density in cups per cubic decimeter?
96 seats fill 48 square meters of floor. What is the density in seats per square meter?
USE THE DENSITY
  • Read the question.
  • Tap your answer.
At 0.5 sheets per square meter, how many sheets cover a 30 square meter frame?
At 4 cups per cubic decimeter, how many cups does a 5 cubic decimeter jug fill?
114 tiles cover the 3 square meter honeycomb panel. What is the density in tiles per square meter?
96 seats on 48 square meters. How many seats per square meter? Type the number.
At 2 seats per square meter, how many seats fit on 70 square meters? Type the number.
StatementTrue or false?
Density is an amount divided by the space it fills.?
Cups per cubic decimeter is a density by area.?
To predict an amount, multiply the density by the new space.?
The seat density is greater than the plywood density.?
WHY THIS EXERCISEKeeping area and volume densities apart is what makes each prediction come out in the right unit.
Try it
Count the tiles or boards on a floor at home and measure the floor. Find the tiles per square meter.
Fill a measuring jug with cups of water and find your own cups per liter.
Draw the jug and the floor plan side by side. Label the amount and the space on each, and write the two densities.

Clear thinking, designer. Tomorrow is Shop Lab: cardboard hexagons, and why they fit with no gaps.

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