Box counting: investigate a pattern

At a glance

Learning goals

Define a counting rule; make a labeled table and graph; explain how a measurement depends on a method.

Materials

Squared paper and a pencil. A spreadsheet or short script is optional for larger grids.

Suggested time

Two 45-minute sessions, with the teacher adjusting scope for readiness.

Deliverable

One page containing a question, method, results table, figure and a statement of limits.

Learner instructions

1. Make the pattern. Divide a square into nine equal squares and leave the center empty. Repeat within each remaining square. Draw the first two stages on squared paper.
2. Define a measurement. State the grid size, what you count as occupied and how you align your counting grid. Count the occupied squares at each stage.
3. Keep a record. Record stage, side length in cells, occupied squares and your rule. Keep corrections visible.
4. Show the result. Make a labeled table and figure. Explain the pattern in your own words.
5. Question the method. Compare your result with the worked example. Explain one reason a count could change without the underlying pattern changing.
6. Credit and reflect. Cite the example, identify help received and describe one thing you would check next.

For a larger analysis, use the six-scale example data and its figure. Label supplied data as supplied; do not describe them as your own observations.

Adapt the challenge

First investigation

Draw two stages, count carefully, compare ratios and explain the rule. Logarithms are not required.

High school

Reproduce the aligned box counts and log–log slope. Compare shifted-grid counts and explain the finite-grid difference.

College / independent study

Compare depths and offsets, justify scale selection and report residuals or uncertainty. A new analysis must be distinguished from the supplied example.

Suggested assessment

For each criterion, use 0 = absent, 1 = partly explained, 2 = clear and supported. Maximum 8 points; adapt before assigning.

Method

The rule, grid and alignment are sufficiently clear for another learner to repeat.

Evidence

Counts, calculations and any supplied data are traceable and labeled.

Communication

The table, figure and explanation agree; axes and units or scale are clear.

Interpretation

The conclusion stays within the evidence and identifies a limitation or useful next check.

Use a short feedback session before the final version.

Credit and reuse

This assignment’s original instructions and rubric are offered by SME under CC BY 4.0. Credit “SME, Box counting: investigate a pattern, version 1 (2026)” and identify changes. Linked materials retain their own source and license information.

Prepare your adapted teaching resource or learner’s project. Publishing a student’s work requires its own permissions.