# Year 9 exponent laws · clean learner routes · Days 11–20

Use the day's [original model card](MODEL-CARDS.md) and an optional [print/text/tactile aid](print/TEXT-ALTERNATIVES.md). Choose **A, B or C** today; you can choose differently tomorrow. Routes offer different ways to think and communicate, not fixed “learning styles”. You may use speech, AAC, typed work, a calculator, a directed scribe, a factor strip or enlarged/tactile symbols. Tell the teacher what support you used and keep first work beside any revision. An exact adult reading supports comprehension but does not show independent print reading. Every setting is invented; no purchase, real measurement or personal information is needed.

## Day 11

Use Card A. Explain `2^3×2^4=2^7` as seven independent binary choices and check the count. Tell why adding the two result counts is a different question.

- **A · Build pairs.** Set three two-choice cards beside four two-choice cards; count total slots, write both product and single-power forms, and check with 8×16.
- **B · Factor explanation.** Write the repeated factors for `3^2×3^3` and compare with `3^2+3^3`. Explain which operation permits exponent addition and which does not.
- **C · Misleading post repair.** An invented post says “8 first-stage options and 16 second-stage options make 24 combined strings.” Write a source-bound correction using pairs and the matching law.

**Exit:** The product law needs ___; plus signs ___ follow this rule.

## Day 12

Use Card B. Explain quotient law with the unit **groups of cells**, then show what a non-zero base to exponent zero means.

- **A · Grouping diagram.** Represent 243 invented paper cells as groups of 9 with counters/labels; show 27 groups and the `3^5/3^2` exponent subtraction.
- **B · Ratio ladder.** Build `2^6/2^2`, cancel equal factors and verify the quotient; finish with `5^3/5^3=1` and say why the base matters.
- **C · Error hearing.** A fictional editor says `3^5/3^2=3^(5+2)`. Quote the operation, repair the exponent and state what the answer counts.

**Exit:** Equal non-zero numerator and denominator powers give ___, so a^0 is ___ for a≠0.

## Day 13

Use Card C. Show how a negative exponent means a reciprocal and retain the drawn length unit.

- **A · Fraction tiles.** Cancel factors in `10^2/10^4`, write `10^−2=1/100`, then label the paper strip's `0.01 m` result.
- **B · Scale note.** Explain to a fictional map editor why the factor is unitless but the scaled drawn strip has metres/centimetres; name the unmeasured real object.
- **C · False-claim test.** Refute “`10^−2=−100`” with a positive quotient and one different example `2^−3=1/8`. State that the base cannot be zero for a negative exponent.

**Exit:** `4^−2=___`, with base condition ___.

## Day 14

Use Card D. Explain why two eight-choice paper cues give `(2^3)^2`, then contrast a different eight-by-four model.

- **A · Bracket tree.** Draw two 8-choice branches, each labelled `2^3`, then show `2^3×2^3=2^6` and 64 model pairs.
- **B · Two-source comparison.** Match `(2^3)^2` and `2^3×2^2` to the eight-by-eight and eight-by-four descriptions; state 64 versus 32.
- **C · Caption editor.** Repair “Raise a power to a power by adding exponents” with one expanded-factor check and a plain-language caption for the invented cue notebook.

**Exit:** The inner power repeats ___ times, so the exponents ___.

## Day 15

Your teacher gives you a **new numerical source** today. Mark each operation and base before deciding a law; do not use remembered numbers from practice.

- **A · Annotated calculations.** Show product, quotient and reciprocal steps with one expanded-factor or ordinary-arithmetic check; add a sentence about the invented claim.
- **B · Law sorting.** Place the new source's product, quotient and reciprocal on three cards; direct a partner or scribe to record your reasons and a final correction.
- **C · Quiet oral/AAC explanation.** Explain the same results and overclaim privately to a teacher; approve an exact local transcript of your mathematics. No recording is published.

**Exit:** Point to a result and why its operation permits the chosen exponent rule.

## Day 16

Use Card E. Extend the product and quotient laws to variable base `x`; distinguish where `x=0` is allowed.

- **A · Slot strip.** Make three x-choice slots and two more; show `x^3×x^2=x^5`, check x2, then separate the `x^5/x^2` quotient and its denominator.
- **B · Symbol audit.** Write the product and quotient as two lines with domains. Test x2 and x0 in the **original** expressions; explain why only the quotient fails at zero.
- **C · Fictional designer memo.** Explain to an editor why five independent x-choice slots match `x^5`, but division by `x^2` requires x≠0. Do not claim real game usability.

**Exit:** Product form ___; quotient form ___ only if ___.

## Day 17

Use Card F. Turn `u^2/u^5` into a reciprocal and explain signs at two non-zero inputs.

- **A · Cancellation ladder.** Cross out equal u factors, write `u^−3=1/u^3`, then evaluate u2 and u−2 with brackets.
- **B · Error investigation.** An invented solver says `u^−3` is negative for every u. Give one positive and one negative-base substitution and state u≠0.
- **C · Domain interview.** Ask a teacher/partner “What happens in the original quotient at u0?” Direct a written or AAC response, then make a two-row table for u2 and u−2.

**Exit:** `v^−2=___` for v≠0; at v0 ___.

## Day 18

Use Card G. Keep the entire `n+2` exponent grouped when combining equal-base powers.

- **A · Stage table.** Fill n0 and n2 first-stage/second-stage counts, then match them to `2^(2n+2)`.
- **B · Algebra line repair.** Correct `2^n×2^(n+2)=2^(n²+2)` by showing exponent addition, not multiplication of n by itself; test n3 if useful.
- **C · Tactile/spoken paths.** Build n first-stage cards and n+2 second-stage cards. Explain the total number of binary slots, write `2^(2n+2)` and name n's whole-number context.

**Exit:** `3^m×3^(m+1)=___`; brackets around the exponent matter because ___.

## Day 19

Use Card H. Match the actual poster design to a power-of-power expression and compare a different two-row design.

- **A · Two grids.** Sketch `2^r` rows by `2^r` icons and two rows by `2^r` icons. At r3, count 64 and 16 respectively; label the changed assumption.
- **B · Make a model.** Invent a safe paper-only square array with `3^s` per side for non-negative whole s. Write `(3^s)^2=3^(2s)`, check s1 and name a real-world limit of the sketch.
- **C · Poster caption audit.** Repair “Our 64 icons prove the poster will be legible to everyone” by citing the r3 model and one missing reader or print check.

**Exit:** My expression matches ___; the count does not establish ___.

## Day 20

Your teacher gives you a **fresh variable source** today. Work from its own conditions, show a substitution check and refuse an invalid zero-base or real-world promise.

- **A · Algebra and check.** Simplify the new variable product and quotient, test the given values and state each domain.
- **B · Counterexample board.** Put SAME BASE, OPERATION, DOMAIN and CLAIM LIMIT under separate headings; move source details before dictating a correction.
- **C · Quiet oral/AAC route.** Explain the same algebra, substituted value and source boundary to a teacher; keep an exact private transcript, not a public recording.

**Exit:** Show the rule, one valid input check and one condition your answer needs.

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