A sealed machine sits between two windows. Numbers go in one side; numbers come out the other. You never see the works — only the record of what went in and what came out.
The task
Each group receives a stack of cards: the in-out records of several machines. For each machine, decide three things and defend them at the board. First: can you predict what the machine does to a number it has never been fed? Second: would the machine give the same answer if you fed it the same number twice — and how do you know? Third: one of the machines in the stack is broken, or at least untrustworthy. Find it, and say exactly what trust means for a machine like this. Then invent your own: one trustworthy machine and one broken one, as in-out records only, and trade with another group.
Facilitation notes — for the teacher
Seed one card set with a many-to-one machine (e.g., squaring) and one with a one-to-many record (the same input appearing twice with different outputs) — the second is the “broken” one. Groups usually say “it changed its mind” before anyone says function; hold the vocabulary until the boards agree on the idea. If a group finishes, ask them to draw the records as graphs and hunt for a visual test — the vertical line arrives on its own. Keep squaring’s two-inputs-one- output machine in play: it is legal, and the debate about why is the whole lesson.
What mathematics tends to surface
The word function is a promise: one input, exactly one output, every time. Groups discover the promise by watching it break — the machine that answers the same question two ways is not a function, while the machine that gives two questions the same answer is fine. Mapping the records as graphs surfaces the vertical-line test without anyone announcing it. What Is a Function names what the boards built, and Function Notation gives the machines their nameplates.
Where it leads
Every machine in this course — quadratic, exponential, sinusoidal — keeps the same promise, and Domain and Range asks the follow-up questions: what may go in, and what can come out? Later, running a machine backwards raises the year’s best question: is the reverse still trustworthy?
The answer is not on this page
No card sets and no verdicts are printed here. The broken machine is your group’s to catch and convict at the boards.
Curriculum connection
A1.1
explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., identifying a one-to-one or many-to-one mapping; using the vertical-line test) Sample problem: Investigate, using numeric and graphical representations, whether the relation is a function, and justify your reasoning.
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A1.2
represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions [e.g., evaluate , given ]
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