MathJewels.com
Problem 34 · Least Action LabPrint one blank worksheet now, download its PDF, or prepare a family or classroom packet.

Problem 34

Least Action Lab

Student worksheets
1 generic worksheet · no answer keyIf this browser cannot open its printer dialog, the ready-to-print PDF opens automatically.
MathJewels.com
Name
Date

Least Action Lab

Complete to collectTumbled JadeLevel 2 · 4 Glints

Compare whole motions

Physicists call a complete motion from start to finish a history. Each history has a number called action.

A nearly friction-free puck starts at and must finish at exactly 3 seconds later. Four candidate histories show where it could be after each second.

For each 1-second step

Total action is the sum of the three step actions.

Read one step before comparing whole histories

1 · Find the change

Subtract coordinates to get .

2 · Score the step

Square both changes and add them.

3 · Score the history

Add its three step actions only after each step is clear.

For example, moving from to gives , so the step action is .

Four coordinate-grid panels show puck histories from (0, 2) at 0 seconds to (6, 2) at 3 seconds. History A passes through (2, 2) and (4, 2). History B passes through (1, 2) and (4, 2). History C passes through (2, 3) and (4, 3). History D passes through (2, 1) and (4, 3).
The dots are the puck's positions at 0, 1, 2, and 3 seconds.

Page 2 · Test the histories

Complete the table here. Use separate paper for your explanations.

  1. Complete the table. Each ordered pair is (horizontal change, vertical change). Write each step action and each total, then circle the history with the least total action.

    History 0–1 s 1–2 s 2–3 s Total
    A
    B
    C
    D

  2. Two histories have the same total action even though one stays straight and the other bends. Name them. Use their three step actions to explain the tie.

  3. Describe the motion of the least-action history. Use its time marks and step actions as evidence. Then write one sentence describing the principle of least action in this puck model.

Physics note: In this free-puck model, the action is truly least. In more advanced systems, the precise idea is stationary action, which need not be the absolute smallest action.

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 7

Uses Middle Grade 6 math.

Thinking

Standard challenge

The solver chooses and coordinates familiar methods.

Time & response

About 8–16 minutes

Moderate amount of written work · Short Explanation response.

How to support: The unfamiliar idea is defined completely on the page. Ask what each time mark contributes to the total, but do not point out which history is straight and steady.

Worksheet arc · Introduction · 1 of 3 on Complete arc

Discover Action

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

How did it feel?

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

Keep exploring

Keep exploring

Technical reference

Rating, skills & curriculum

Designed for Grade 7 · Uses Middle Grade 6 math · About 8–16 minutes

Grade 7 · Algebraic structure and equations · Expressions and symbolic structure

MJ 6.5 · 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 6.5

Grades 6–8

Grade 6 progression · point 5

Likely range
MJ 6.4–6.6
Confidence
82%

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
8–16 minutes · typically 11
Response
Short Explanation
Known-method steps
19
  • 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.

  • Evaluate expressions for specified variable values

    The learner can evaluate expressions for specified variable values. This capability is distinct from mere exposure to or isolated use of substitution; operations.

    • Developing mastery
    • 100% focus
    • MJ 6.5 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Write and interpret numerical expressions without necessarily evaluating them

    The learner can write and interpret numerical expressions without necessarily evaluating them. This capability is distinct from mere exposure to or isolated use of operation language.

    • Fluent readiness
    • Required
    • MJ 5.4 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
Standard components1 location · Representative: Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).