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MathJewels worksheet arc · 5 problems · 4 stages

Edge of Escape

A worksheet arc that branches from recurrence into complex-orbit and safe-zone reasoning, then reunites those ideas in two exact Mandelbrot arguments.

About 85 minutes for the complete route10 printable pages2 routesWorked answers included
Start with Rook’s Halfway SignalView 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 brass rook faces a luminous observatory dial labeled 0 and 8 beneath the title Rook’s Halfway Signal.

      Introduction · Problem 35

      Rook’s Halfway Signal

      Introduce recurrence as a next-step rule and turn it into an explicit gap pattern.

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

      Designed for
      Designed for Grade 9 · Grade 10
      Difficulty
      MJ HS:DISC.4 · C2 · W2
      Active time
      About 9–23 minutes
      • Prepares: The recurrence notation becomes the orbit rule used with complex numbers.
      • Extends into: The familiar next-step process is extended from calculation to a statement about every iterate.
      Start hereWorked answer

  2. Stage 2 · choose the order of these branches

    • A glowing complex-plane lens displays the symbol i beneath the title The Orbit of i.

      Guided Practice · Problem 36

      The Orbit of i

      Open the complex-orbit branch by introducing complex arithmetic, magnitude, and a bounded repeating orbit.

      Builds on: Rook’s Halfway Signal

      Designed for
      Designed for Grade 10 · Grade 11
      Difficulty
      MJ HS:A2.2 · C2 · W2
      Active time
      About 11–28 minutes
      • Prepares: Complex arithmetic, magnitude, and orbit notation are required by the full-plane capstone.
      Open worksheetWorked answer
    • A glowing interval from 0 to 2 passes through a rule toward a question-mark gate beneath the title The Safe Zone.

      Guided Practice · Problem 37

      The Safe Zone

      Open the proof branch by developing invariant intervals and distinguishing examples from a universal argument.

      Builds on: Rook’s Halfway Signal

      Designed for
      Designed for Grade 9 · Grade 10
      Difficulty
      MJ HS:DISC.4 · C2 · W2
      Active time
      About 11–28 minutes
      • Feeds synthesis: The invariant-interval proof pattern becomes one half of an exact endpoint argument.
      Open worksheetWorked answer

  3. Stage 3

    • A computed Mandelbrot-set image crossed by a gold real axis with a question mark at the positive real-axis endpoint; the full set visibly extends farther right above and below the axis.

      Synthesis · Problem 38

      The Last Safe Positive Real Number

      Synthesize recurrence and safe-zone reasoning to prove the exact positive real-axis endpoint.

      Builds on: The Safe Zone

      Designed for
      Designed for Grade 10 · Grade 11
      Difficulty
      MJ HS:A1.6 · C3 · W2
      Active time
      About 14–34 minutes
      • Extends into: The capstone extends a two-sided boundary proof from the real axis to the full complex set.
      Open worksheetWorked answer

  4. Stage 4

    • A computed Mandelbrot set with a gold point inside it and a dashed vertical gold wall to its east beneath the title How far east?

      Capstone · Problem 39

      How Far East Can the Mandelbrot Set Go?

      Reunite complex arithmetic and invariant-region proof to cage the full set’s easternmost point.

      Builds on: The Orbit of i · The Last Safe Positive Real Number

      Designed for
      Designed for Grade 11 · Grade 12
      Difficulty
      MJ HS:A2.3 · C3 · W3
      Active time
      About 20–44 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.