Ten teacher scripts · 25… | Year 11 Methods · T1 W1–2 · Lessons

Set-up: Use Cards A–J 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 and optional surd explorer only when they clarify a step. The two public checks are fresh relative to cards; collect first attempts before opening the staff key. 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. 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. 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 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, 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. 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. 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. 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 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. 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, then independently transfer to Check A. 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, 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 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. 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. 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 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. 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 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. 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 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. 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, then transfer independently to new Check B. 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 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 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.