Wednesday
Wednesday: Boathouse Lab, the Cardboard Oval
Comet pushes two pins into a cardboard sheet, 8 centimeters apart. "Lab day. The lawn, shrunk to a desk."
Wren ties a 10 centimeter string between the pins and pulls a pencil around inside it. A clean oval.
"Now the test," he says. "Pick any point on the curve. Measure to each pin. Add."
Comet measures the first: "1 and 9. Sum 10." Then another: "3.5 and 6.5. Sum 10."
Nova hovers over the sheet. "Would you like a hint? Try the very top and the very end."
"Top: 5 and 5. End: 9 and 1," Wren says. "Always 10. The string keeps its promise."
"Width 10, height 6," Comet says, measuring. "Just what b² = a² - c² predicted."
What you need
- A sheet of thick cardboard, two pushpins, string, a pencil, a ruler and your Boathouse Log.
- A grown-up pushes the pins in and takes them out. Scissors for cutting the string to length.
- Graph paper to copy the oval onto afterward, with the pins as the foci.
The grown-up handles the pins. Keep pins in a lid when not in use and off the floor.
Cut string with scissors only, sitting down. Nothing heavy is lifted alone, and a grown-up handles any ladder.
Run the lab
- Mark two focus points 8 centimeters apart on the cardboard. The grown-up sets a pin at each.
- Cut a string so that, tied to both pins, it measures 10 centimeters pin to pin. Pull a pencil around inside it.
- Pick five points on your oval. For each, measure the distance to each pin with the ruler.
- Add the two distances for each point. Record every pair and every sum in a table.
- Measure the full width and the full height of the oval. Halve them to get a and b.
- Compute a² - c², using c = 4, and compare with your b². Record the difference.
The crew's lab log
Here are the crew's own readings from the cardboard, to the nearest tenth of a centimeter. Made-up Boathouse data; yours will differ a little.
| Point | To pin 1 (cm) | To pin 2 (cm) | Sum (cm) |
|---|---|---|---|
| point 1 | 1 | 9 | 10 |
| point 2 | 3.5 | 6.5 | 10 |
| point 3 | 5 | 5 | 10 |
| point 4 | 7.2 | 2.8 | 10 |
| point 5 | 9 | 1 | 10 |
Every sum is 10, the string length, within the width of a pencil line. Width 10, height 6.
Prediction: b² = a² - c² = 25 - 16 = 9, so b = 3 and the height is 6. The cardboard agrees.
- Read the question.
- Tap your answer.
| What the lab shows | True or false? |
|---|---|
| Every measured sum came out close to the string length. | ? |
| The distance to pin 1 is the same at every point. | ? |
| The oval's measured height matched 2b from b² = a² - c². | ? |
| A point at the very end of the oval is farther from the near pin than from the far pin. | ? |
Careful lab work. Tomorrow the sum becomes a difference, and the curve breaks into two pieces.