All domains, settings, quantities and quotes below are fictional classroom models. The cards deliberately move from the opening fortnight's coordinate/linear models to Week 3 numerical exponent laws and Week 4 variables in exponents. A count model uses non-negative whole numbers unless a card says otherwise; a reciprocal or quotient requires a non-zero base. A calculator may verify an arithmetic result, but the learner must show why the exponent rule applies. No card proves a real technology specification, measurement, safety property or price. An exact adult reading is listening-supported access, not independent print reading.
Card A Nested message choices
A fictional game editor gives a player three independent yes/no choices in one stage and four independent yes/no choices in a second stage. The first stage has 2^3=8 possible choice strings; the second has 2^4=16. Each first string can pair with each second string, so there are 2^3×2^4=2^(3+4)=2^7=128 combined strings. This counts model strings, not whether players want or can use all of them. Adding 8+16 would count the two lists separately rather than all pairs. Counting assumes each yes/no option is available independently.
Card B Grouping a paper-cell array
An invented square-paper array has 3^5=243 equal cells. The editor divides it into groups of 3^2=9 cells without leftovers: 3^5÷3^2=3^(5−2)=3^3=27 groups. The quotient says groups of cells, not a new physical sheet size. Division by a base-zero power is not permitted. A separate dimensionless ratio of equal non-zero counts, 3^2÷3^2=3^0=1, explains why this non-zero base to exponent zero equals 1.
Card C A small paper scale
For a drawn length model, a scale factor 10^−2=1/10^2=1/100 multiplies an invented 1 m strip to give 0.01 m = 1 cm. The factor is dimensionless; the result is a length. The BIPM SI-prefix table identifies centi- as 10^−2. No real object or site was measured. A negative exponent does not make the length negative, and 10^−2 is not −100.
Card D Cue combinations
A fictional stage notebook has 2^3=8 possible paper cue cards. An ordered pair of cards has (2^3)^2=2^(3×2)=2^6=64 model pairs. This is different from 2^3×2^2=2^5=32: the second expression gives one eight-choice stage paired with one four-choice stage. The bracket changes the repeated factor. No actual lighting or stage device is being operated.
Card E Variable choices in independent slots
Let x be a positive whole number of abstract choices at each independent slot. Three slots have x^3 choice strings; two more slots have x^2. Together, x^3×x^2=x^5. If x=2, the model has 2^5=32 strings. If x=0, the count context has no available choice, but the quotient x^5÷x^2=x^3 would be undefined because it divides by zero; that identity requires x≠0. The symbolic product and quotient have different domain requirements.
Card F A non-zero scale ratio
Let u be a non-zero real, unitless scale factor. The ratio u^2÷u^5=u^(2−5)=u^−3=1/u^3. At u=2, the result is 1/8. At u=−2, it is −1/8 because an odd power preserves the negative sign. At u=0, the original quotient and reciprocal are undefined. This is a ratio, not a count of objects or a negative physical length.
Card G A variable exponent in a game rule
For a non-negative whole stage number n, a fictional puzzle maker uses 2^n first-stage paths and 2^(n+2) second-stage paths, independently. Total model path pairs are 2^n×2^(n+2)=2^(2n+2). At n=2, this is 2^2×2^4=4×16=64=2^6. The exponents add as algebraic expressions. This does not assert a game has been built or that all paths are usable.
Card H Poster icon grid
For a non-negative whole r, a fictional design sketch has 2^r paper icons in each row and the same number of rows. The model total is (2^r)^2=2^(2r) icons, not 2^(r+2). At r=3, eight rows of eight make 64 icons. The sketch says nothing about print clarity or available wall space. If a different design has only two rows of 2^r icons, its total is 2×2^r=2^(r+1), a different condition.
Shared rule boundary
For non-zero base a and integers m,n: a^m a^n=a^(m+n), a^m/a^n=a^(m−n), (a^m)^n=a^(mn), a^0=1, a^(−n)=1/a^n. These are same-base product/quotient and power-of-power rules, not rules for sums: a^m+a^n cannot generally be simplified to a^(m+n). For negative integer exponents or a quotient, the base must be non-zero. Parentheses control whether a negative sign is part of the base: (-2)^4=16, whereas -2^4=−16 under the usual order of operations. The teacher key holds feedback examples; fresh check numbers are separate.
Original resource rights: © NeuroForgeIO Pty Ltd 2026, SubjectNest, CC BY 4.0. Credit author, source, licence and changes. SI-prefix source credited above; no external text/artwork has been copied into these cards.