# Ten teacher scripts · 25 minutes each

**Set-up:** Use [Cards A–J](LEARNER-CARDS.md) in order. Keep exact form and an explanation visible in every route. Ask learners to estimate or substitute to check; a decimal is a check, not a replacement for an exact answer. Use [A4 aids](print/TEXT-ALTERNATIVES.md) and optional [surd explorer](surd-explorer.html) only when they clarify a step. The two [public checks](STUDENT-CHECKS.md) are fresh relative to cards; collect first attempts before opening the [staff key](teacher/KEY-AND-NEXT.md). All contexts are constructed, not real measurements.

Routine days use **25 = 2 + 5 + 6 + 7 + 5 minutes**. Check days use **25 = 2 + 3 + 12 + 5 + 3 minutes**. A teacher can extend a task beyond 25 minutes; this pacing is a ready starting point, not a speed standard. Routes are switchable access choices with the same mathematical target.

## Day 1 · Surd or rational root?

**Goal:** classify roots and distinguish exact equality from approximation using [Card A](LEARNER-CARDS.md#card-a--which-roots-stay-irrational-day-1). **Say:** “The radical symbol alone does not make a value irrational.”

- **0–2:** Recall `3^2=9`, `4^2=16`; ask where 10 sits.
- **2–7:** Model `sqrt(9)=3` and `3<sqrt(10)<4`, since `9<10<16`. Say that `sqrt(10)≈3.162` is approximate.
- **7–13:** Partners classify the three sides; ask what “principal square root” means for a side length.
- **13–20:** Learners show exact/approximate notation and explain why `sqrt(10)` is a surd.
- **20–25:** Exit: can `sqrt(16)` be called a surd here? Collect the reason.

**Routes:** write a three-row exact/approximate table; tell the comparison while a peer captures `9<10<16`; arrange 9, 10, 16 square-number tiles and dictate the inequality. **Other domains:** [Day 1 swaps](PRACTICE-SWAPS.md#day-1--classification). **Optional/home:** find two invented square areas, one perfect and one not, then classify side expressions on scrap paper. **Move:** if a learner says all radicals are irrational, replace 10 with 16; if they write `=3.162`, ask what symbol marks rounding.

## Day 2 · Extract a square factor

**Goal:** simplify `sqrt(72)` from [Card B](LEARNER-CARDS.md#card-b--find-a-square-hiding-inside-day-2) and verify by squaring. **Say:** “We may split a *product* under a nonnegative square root, not a sum.”

- **0–2:** Recall that `sqrt(36)=6`.
- **2–7:** Model `72=36×2`, so `sqrt(72)=sqrt(36)sqrt(2)=6sqrt(2)`.
- **7–13:** Learners explain why `sqrt(36)+sqrt(2)` is not equivalent; compare squares or approximate values.
- **13–20:** Independent simplification with [square-factor mat](print/square-factor.pdf), then square the result: `(6sqrt(2))^2=72`.
- **20–25:** Ask why the answer lies between 8 and 9 despite its exact radical form.

**Routes:** symbolic factor-and-check; spoken factor tree with recorded equation; arrange a 36×2 rectangle card and label the equal roots. **Other domains:** [Day 2 swaps](PRACTICE-SWAPS.md#day-2--factor-transfer). **Optional/home:** make three factor pairs for 72 and find why the largest square factor is most efficient. **Move:** if a student writes `sqrt(a+b)=sqrt(a)+sqrt(b)`, test it at `a=b=1` and contrast `sqrt(ab)`.

## Day 3 · Compare exact expressions

**Goal:** simplify and compare [Card C](LEARNER-CARDS.md#card-c--equivalent-lengths-clearer-comparison-day-3). **Say:** “A common radical factor lets us compare without rounding.”

- **0–2:** Review `sqrt(72)=6sqrt(2)` from Day 2.
- **2–7:** Model `sqrt(50)=5sqrt(2)`; ask students to find the square in 98.
- **7–13:** Pairs get `sqrt(98)=7sqrt(2)` and explain why `7sqrt(2)>5sqrt(2)`.
- **13–20:** Independent exact difference; attach units to a length, not to a pure coefficient.
- **20–25:** Quick decimal check of order; discuss why rounding cannot prove exact equality.

**Routes:** symbolic steps and inequality; verbal comparison with a recorded common-factor statement; draw two bar lengths of 5 and 7 identical `sqrt(2)` tiles with labels. **Other domains:** [Day 3 swaps](PRACTICE-SWAPS.md#day-3--comparison). **Optional/home:** compare `sqrt(27)` and `sqrt(75)` exactly. **Move:** if learners compare 50 and 98 correctly but cannot state the exact difference, bring out the matching `sqrt(2)` tiles.

## Day 4 · Add only like surds

**Goal:** add [Card D](LEARNER-CARDS.md#card-d--add-like-radicals-day-4) after simplification. **Say:** “The root parts have to match, just as like variables do.”

- **0–2:** Write `2x+3x=5x`; ask for a radical analogue.
- **2–7:** Model `sqrt(12)=2sqrt(3)`; invite `sqrt(27)=3sqrt(3)`.
- **7–13:** Students reject `sqrt(39)` by a numerical bound: the true sum is above 8, while `sqrt(39)` is between 6 and 7.
- **13–20:** Independent total and a sentence about like terms.
- **20–25:** Ask if `sqrt(12)+sqrt(8)` can combine in one surd; hear why not after simplifying.

**Routes:** equations plus bound; talk through matching root tiles with recorded exact sum; arrange coefficient cards beside shared `sqrt(3)` strips and caption the result. **Other domains:** [Day 4 swaps](PRACTICE-SWAPS.md#day-4--addition). **Optional/home:** invent two non-identical radicals that simplify to like terms. **Move:** when students add radicands, square or estimate both sides to expose the difference.

## Day 5 · Subtract and transfer (public Check A)

**Goal:** consolidate [Card E](LEARNER-CARDS.md#card-e--subtract-carefully-day-5-practice), then independently transfer to [Check A](STUDENT-CHECKS.md#check-a--day-5-fresh-surd-transfer). **Say:** “Simplify first, combine matching root parts, and check the sign.”

- **0–2:** Model only `sqrt(32)=4sqrt(2)` from the practice card.
- **2–5:** Invite `sqrt(8)=2sqrt(2)` and the exact difference; do not preview check values.
- **5–17:** Students complete fresh Check A. Collect a first response in any route.
- **17–22:** Share [public worked key](teacher/KEY-AND-NEXT.md#check-a--day-5), then ask for a revised reason or calculation in a second colour/voice note.
- **22–25:** Record whether the next need is factor extraction, like-term combination or approximation notation.

**Routes:** write exact steps; explain aloud with exact expressions captured by a scribe; use square-factor and matching-root strips then dictate the equation. **Other domains:** [Day 5 swaps](PRACTICE-SWAPS.md#day-5--subtraction) only after check. **Optional/home:** devise a subtraction that produces `3sqrt(2)` and explain it. **Move:** if a response is negative where longer minus shorter was requested, compare the two original radicands first. This is not a secure examination.

## Day 6 · Multiply then simplify

**Goal:** multiply square roots and whole-number coefficients using [Card F](LEARNER-CARDS.md#card-f--multiply-surds-and-units-day-6). **Say:** “Multiply lengths to get area; the unit changes even when the radical stays.”

- **0–2:** Ask what units follow `cm × cm`.
- **2–7:** Model `sqrt(6)sqrt(15)=sqrt(90)` for nonnegative factors, then ask for its square factor.
- **7–13:** Pairs separate coefficients and radicals in `(2sqrt(3))(3sqrt(6))`.
- **13–20:** Independent exact answers and a reasonableness estimate for the area.
- **20–25:** Exit: check that squaring `3sqrt(10)` gives 90; label it `cm²` in the rectangle case.

**Routes:** symbolic calculation with units; oral decomposition into coefficient and radical products recorded in notes; arrange factor tiles and caption the exact result. **Other domains:** [Day 6 swaps](PRACTICE-SWAPS.md#day-6--products). **Optional/home:** create one product of unlike roots that simplifies to a whole number. **Move:** if `sqrt(6)sqrt(15)=sqrt(21)`, remind students the operation between factors is multiplication and estimate: area is much larger than `sqrt(21)`.

## Day 7 · Rationalise a denominator

**Goal:** preserve value while writing [Card G](LEARNER-CARDS.md#card-g--move-the-radical-out-of-the-denominator-day-7) with a rational denominator. **Say:** “Multiply by one, not by a new value.”

- **0–2:** Recall `sqrt(5)sqrt(5)=5`.
- **2–7:** Model multiplying `7/sqrt(5)` by `sqrt(5)/sqrt(5)`.
- **7–13:** Students explain why the denominator becomes 5 and why the numerator changes too.
- **13–20:** Independently rationalise `4/sqrt(2)` and compare equivalent decimal approximations.
- **20–25:** Ask what went wrong if someone writes only `7sqrt(5)`.

**Routes:** fraction steps; spoken “multiply top and bottom” with recorded exact equality; paired numerator/denominator strips labelled with the same factor. **Other domains:** [Day 7 swaps](PRACTICE-SWAPS.md#day-7--rationalising). **Optional/home:** write a ratio of your own with a single square-root denominator and verify both forms numerically. **Move:** if a student cancels `sqrt(5)` without multiplying numerator, have them test both values at three decimal places.

## Day 8 · Audit a multi-step surd expression

**Goal:** solve [Card H](LEARNER-CARDS.md#card-h--audit-a-multi-step-result-day-8) by two exact routes and reject an invalid sum rule. **Say:** “Different valid routes should meet at the same value.”

- **0–2:** Recall the false rule from Day 4.
- **2–7:** Model only `sqrt(18)=3sqrt(2)` and `sqrt(8)=2sqrt(2)`.
- **7–13:** Pairs finish numerator-first and term-by-term division routes.
- **13–20:** Independently identify the invalid step in `sqrt(26)/sqrt(2)` and approximate-check the correct result.
- **20–25:** Compare routes; note that exact simplification reaches a whole number here by design.

**Routes:** write both routes; orally compare routes with a teacher capturing both equalities; arrange radical tiles above and below a fraction bar then label the cancellation. **Other domains:** [Day 8 swaps](PRACTICE-SWAPS.md#day-8--multi-step-audit). **Optional/home:** invent a numerator of two like surds over one matching surd that simplifies to 6. **Move:** if a student says the incorrect draft is “close enough”, estimate `sqrt(26)/sqrt(2)=sqrt(13)`, which is below 4, while the true answer is 5.

## Day 9 · Read a quadratic from vertex form

**Goal:** identify graph features of [Card I](LEARNER-CARDS.md#card-i--first-quadratic-graph-day-9) without claiming mastery of the 7-hour sub-topic. **Say:** “A squared term cannot be negative for real x.”

- **0–2:** Recall a square table for `x²` from prior learning.
- **2–7:** Model substituting x=2 and x=1 into `(x-2)²+1`; connect symmetry.
- **7–13:** Students fill the five-row table and plot or arrange points.
- **13–20:** Explain turning point, axis, y-intercept and why there are no real x-intercepts.
- **20–25:** Ask what changes if the final `+1` becomes `-1`; note as preview, not required proof today.

**Routes:** table and hand sketch; verbal walkthrough of mirrored x-values with captured coordinates; place five labelled point tiles on a tactile grid and dictate features. **Other domains:** [Day 9 swaps](PRACTICE-SWAPS.md#day-9--vertex-form). **Optional/home:** draw two different curves of the form `(x-h)²+k` and compare their turning points. **Move:** if a learner calls `(2,1)` an x-intercept, point to the definition `y=0` for x-intercepts.

## Day 10 · Factor form and second transfer check

**Goal:** use zeros and symmetry in [Card J](LEARNER-CARDS.md#card-j--factor-form-shows-zeros-day-10-practice), then transfer independently to new [Check B](STUDENT-CHECKS.md#check-b--day-10-fresh-graph-and-surd-transfer). **Say:** “A factor gives a zero; the graph still needs a check of direction and scale.”

- **0–2:** Model solving `(x+1)(x-3)=0` for the two roots only.
- **2–5:** Learners locate the midpoint axis and leave remaining features to explain.
- **5–17:** Give Check B, which uses a different downward parabola and new surd. Collect first responses.
- **17–22:** Share the [public key](teacher/KEY-AND-NEXT.md#check-b--day-10) and ask students to revise one reason, not just a number.
- **22–25:** Record which next block is needed: more surd operations, more graph features or both.

**Routes:** write exact steps and labelled sketch; speak a graph-feature explanation while a partner records coordinates and reasons; place factor/point tiles on a tactile axis then dictate exact results. **Other domains:** [Day 10 swaps](PRACTICE-SWAPS.md#day-10--factor-form) are after-check practice. **Optional/home:** make an invented factored quadratic with roots -2 and 4, then test its midpoint. **Move:** if a learner uses the midpoint x-value as the turning *height*, substitute it into the function. Public checks are not secure exams or school assessments.