Wednesday
Wednesday: Loft Lab, Build a Function Machine
"Lab day," Comet says, cutting a slot in a shoebox. "Our engineer builds a crank box at home."
Wren shuffles a stack of index cards. "Inputs on cards. A secret rule card inside. Outputs come out the other slot."
"Who is the box?" Comet asks. "Someone has to sit behind it and do the rule."
"A partner," Wren says. "Or you, with the rule hidden from whoever feeds the cards."
Nova hovers over the stack, her light counting cards. "Would you like a hint? Keep a table of every pair."
"Input column, output column," Comet says. "Then we check: did any input ever give two outputs?"
"What do you notice when it does?" Wren asks. "The box is broken, or the rule changed."
What you need
- A shoebox or cereal box with a slot cut in two opposite sides, done by a grown-up.
- Twenty index cards or slips of paper and a pencil.
- A partner to be the box, or a hidden rule card you follow yourself.
- Your Loft Log with a two-column table ready: input and output.
Keep the cards and the box on a table, not on the floor where someone could slip.
If the crank is a pencil through the lid, tape the point so nothing is sharp.
Run the lab
- The box keeper writes a secret rule on a card, such as "× 2 + 3", and hides it inside.
- Feed in an input card. The keeper writes the output on a new card and slides it out.
- Record the pair in your table. Feed in at least six inputs, and repeat one input on purpose.
- Check the table: did the repeated input give the same output? Then it is a function.
- Guess the rule from the table. Test your guess with two new inputs.
- Write the rule in function notation, such as m(x) = 2x + 3, and switch roles.
The crew's lab table
The crew's box ran the rule × 3 + 2, written f(x) = 3x + 2. These are their own cards, made up for the story.
| Input card | Output card, f(x) = 3x + 2 |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 5 | ? |
| 8 | ? |
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| Feeding the same input twice should give the same output twice. | ? |
| 3 × 8 + 2 = 26 | ? |
| For the crew's box, f(5) = 17. | ? |
| If one input gave two different outputs, the box is still a function. | ? |
| The input cards make up the domain of the box. | ? |
Then run a rule like "the number of letters in the word" with word cards. Is it a function?
Great lab work. Tomorrow the crew reads four Loft tables and decides what domain makes sense.