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MathJewels worksheet arc · 3 problems · 3 stages

Discover Action

A three-problem arc from comparing a few timed histories to building a frame-independent action integral and locating stationary action in an infinite family.

About 38 minutes for the complete route6 printable pages1 routesWorked answers included
Start with Least Action LabView the arc map

The arc map

Ideas branch, deepen, and reconnect.

Arrows are represented by each card’s “builds on” relationship. Problems in the same stage can be completed in either order.

  1. Stage 1

    • A glowing air-hockey puck crosses a dark coordinate-grid lab beneath the words Least Action Lab.

      Introduction · Problem 34

      Least Action Lab

      Compare a small set of complete timed motions and notice that both route and timing affect action.

      Entry point: No earlier worksheet in this arc is required.

      Designed for
      Designed for Grade 7
      Difficulty
      MJ 6.5 · C2 · W2
      Active time
      About 8–16 minutes
      • Extends into: The equal-one-second squared-step score is rebuilt for arbitrary time slices, revealing the rate-times-duration structure hidden in the first lab.
      Start hereWorked answer

  2. Stage 2

    • Steady and accelerating position graphs glow across an equal-time grid beneath the title Action Between Frames.

      Guided Practice · Problem 41

      Action Between Frames

      Repair the discrete score so it is independent of camera resolution, then recognize its continuous integral.

      Builds on: Least Action Lab

      Designed for
      Designed for Grade 12
      Difficulty
      MJ HS:CALC.8 · C2 · W2
      Active time
      About 12–24 minutes
      • Prepares: The frame-independent integral action becomes S(a), a differentiable function that compares a continuous one-parameter family of histories.
      Open worksheetWorked answer

  3. Stage 3

    • A glowing knob fans a straight motion into several curved histories beneath the title One Knob, Infinitely Many Histories.

      Synthesis · Problem 50

      One Knob, Infinitely Many Histories

      Compress infinitely many endpoint-sharing histories into one action function and find its stationary member.

      Builds on: Action Between Frames

      Designed for
      Designed for Grade 12
      Difficulty
      MJ HS:CALC.8 · C2 · W2
      Active time
      About 13–27 minutes
      Open worksheetWorked answer

Ways through the graph

Choose a route

A route is a valid linear journey through part or all of the arc. Every listed dependency still comes first.