Question: Can a different constant factor match the fictional data better without proving a mechanism? Kit: offline model explorer or model table. 35 = 3 + 7 + 12 + 8 + 5.
- 0–3: Recall
P(n)=160×r^n, with0<r≤1; ifr=0.5, it is the poster's half-model. This extension complements the mathematics exponent-law lessons without reusing their practice values. - 3–10: Teacher runs or traces
r=0.50andr=0.55. The explorer displays predicted rows and mean absolute gap against invented B. Define the gap as an arithmetic comparison, not an uncertainty estimate or proof of cause. - 10–22: Learners run two candidate factors offline or fill a paper table. Compare which fits these five values better, then name at least two reasons fit might not transfer (method, calibration, other paper). A model can match data while describing the wrong mechanism.
- 22–30: Explain
r^(a+b)=r^a r^bfor repeated identical factors and whyr=0cannot be used with negative exponents. Routes: local keyboard or paired navigator with recorded action; tactile repeated-factor cards and teacher-read output; written calculator table with exact-word explanation. - 30–35: “If 0.55 gives a smaller gap, does it prove all paper passes 55% each layer?” Key: no. Response move: mark the model as a fitted conditional description and add a new test requirement.
Another domain: Fit a fictional stage-filter card set, keeping data labelled invented. Home: compare two exponent expressions on paper; no software needed.