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Problem 37

The Safe Zone

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The Safe Zone

Complete to collectEmerald CrystalLevel 2 · 4 Glints

Can one check control every step?

Start at and repeat the recurrence

An interval is invariant under a rule if every current value in that interval produces a next value in the same interval. In plain language: once the process is in the safe zone, the rule cannot send it out.

A value enters a band from zero to two, passes through the rule one plus y squared over four, and reaches an output gate marked with a question mark.
Test the whole zone, not just a few sample values.

What an invariant interval proof must do

Assume the current value is anywhere in apply the rule to the whole range prove the next value is still in
Examples are clues

A few calculated values can suggest a safe zone, but they do not cover every possible current value.

Bounds are a certificate

An inequality that starts with an arbitrary value in the interval proves one safe step for all of them.

The repeat engine: start inside + prove every step stays inside = every later value stays inside.

Page 2 · Certify the safe zone

Use the page for notes; write your complete explanation on separate paper.

  1. Calculate exactly. Do these examples suggest that may be a safe zone?

  2. Now test every possible current value in the interval. Assume and complete the chain. Explain why the final bounds keep the next value inside .

  3. The process starts at . Use that fact and Part 2 to explain why every later value stays in , even though you cannot list infinitely many values.

  4. Is the smaller interval invariant? Test the allowed current value and justify your answer.

At a glance

About this problem

These tags show what the problem is about and the math you’ll use. Choose one to find similar problems.

Skills

Helpful first

Grade

Math area

Topics

Is this a good fit?

Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 9 · Grade 10

Uses HS DISC math.

Thinking

Standard challenge

The solver chooses and coordinates familiar methods.

Time & response

About 11–28 minutes

Moderate amount of written work · Proof response.

How to support: The page defines invariant interval as a safe zone that the rule cannot leave. Ask why several successful examples are only evidence, and what an if-then statement about every possible current value can prove.

Worksheet arc · Guided Practice · 3 of 5 on Complete arc

Edge of Escape

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

How did it feel?

Pick the option that best matches this challenge. Recommendations use topic, math area, grade, story, and thinking challenge.

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Technical reference

Rating, skills & curriculum

Designed for Grade 9 · Grade 10 · Uses HS DISC math · About 11–28 minutes

Grade 9 · Grade 10 · Discrete mathematics and logic · Algorithms and recursion

MJ HS:DISC.4 · C2 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ HS:DISC.4

High school

HS DISC · point 4

Likely range
MJ HS:DISC.3–HS:DISC.5
Confidence
81%

Challenge

C2 · Standard

The solver chooses and coordinates familiar methods.

  • Method selection1
  • Reasoning depth2
  • Novelty2
  • Constraint management1
  • Proof or justification2

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
11–28 minutes · typically 18
Response
Proof
Known-method steps
16
  • Mechanical execution2
  • Written output2
  • Representation production1
  • Bookkeeping1

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Define and analyze simple recursive processes and recurrence relations

    The learner can define and analyze simple recursive processes and recurrence relations. This capability is distinct from mere exposure to or isolated use of sequences; functions.

    • Developing mastery
    • 100% focus
    • MJ HS:DISC.4 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Write and graph one-variable inequalities such as x>c

    The learner can write and graph one-variable inequalities such as x>c. This capability is distinct from mere exposure to or isolated use of number line; comparison.

    • Fluent readiness
    • Required
    • MJ 6.9 milestone

  • Multiply a fraction or whole number by a fraction

    The learner can multiply a fraction or whole number by a fraction. This capability is distinct from mere exposure to or isolated use of fraction of a quantity.

    • Fluent readiness
    • Required
    • MJ 5.6 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 1 location
Standards1 location · Representative: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★