IGCSE · Pearson Edexcel · 4MA1

Higher: plot sparse polynomial and reciprocal graphs

Count the six numerical coefficients, calculate mixed reciprocal values exactly and draw separate branches on the original domain.

Section 3.3 · Graphs (teaching linked; review pending)

Pearson 4MA1 · Additional Higher Graphs · Graph families · Additional Higher 3.3A

Exam scope and assessment notes

Additional Higher only. Higher includes the shared Foundation content. Use the nine shared Graphs lessons and the two additional Higher line-equation lessons before these four graph-family lessons. H:3.3A requires all four lessons with worked examples and guided, independent and application practice: integer polynomial/cubic graphs, the sparse six-term family, both degree sine/cosine graphs and degree tangent with domain breaks. Lower-degree members with zero leading coefficients remain valid; the shared linear/quadratic work supplies genuine prerequisite practice. All nine Foundation and seven additional Higher Graphs source rows have teaching and staged-practice links. These links do not establish exhaustive subskill approval, mastery or examination readiness; specialist review and formal assessment validation remain pending. These fixed examples and bounded evaluators do not replace the full written plotting constructions, a general expression parser or a formal assessment. Written work is not automatically marked; explained answers support self-checking, not exam marks or proof of mastery. Specialist review and formal assessment validation remain unestablished. Pearson’s chapter-level modular-map tags and uncorrected-proof navigation remain separate evidence; final full-book and Teaching Hub alignment has not been verified.

Before you begin

Evaluate signed powers and reciprocal fractions exactly.
Plot coordinate pairs and keep track of excluded denominator inputs.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which coefficient list (A,B,C,D,E,F) satisfies this sparse family’s condition?

What you will learn

Recognise the six-term family with at least three zero numerical coefficients.
Generate values for mixed, inverse and inverse-square terms, including fractional inputs.
Preserve original exclusions and draw each permitted branch without bridging zero.

Choose a Higher graph example

y = x² − 1 + 2/x

Horizontal axis: x. Vertical axis: y. Read each numbered scale independently.

Mathematical domain: All real x except 0. The selected-value control accepts half-unit inputs from −1080 to 1080; this control limit does not redefine the function’s domain.

Graph window

View: x from −4 to 4; y from −10 to 18. Window edges neither add missing domain values nor move valid off-screen points.

Dashed vertical guides mark undefined inputs at 0. They are not pieces of the curve.

Selected Higher graph input

At input 1/2, exact output = 13/4. The selected point is shown in red.

Only three of the six allowed coefficients are nonzero. At x=0 the reciprocal term is undefined, so there is a break rather than a plotted point.

Plot the positive and negative branches separately. A curve leaving the window is not an endpoint of the mathematical domain.

Show the Higher value table and graph reasoning

x	y
-4	29/2
-2	2
-1	-2
-1/2	-19/4
0	Undefined
1/2	13/4
1	2
2	4
4	31/2

Use the table to generate your own points on labelled axes, then draw the appropriate smooth trace. Undefined entries remain gaps. More points can be needed near changes of direction; straight chords between sparse samples are not a permanent graph construction.

Exact fractions and standard-angle labels come from fixed model data. Plotted coordinates use numerical samples; a reading made only from a drawn graph is approximate. A table or display alone does not establish every property of an arbitrary function.

Original StudyForA fixed Higher graph examples. The polynomial and sparse evaluators have bounded coefficients, and this tool is not a general expression parser. It does not mark a student’s graph or written reasoning, award exam marks or establish mastery. Complete the guided, independent and application constructions on paper. Transformations, drawn tangents and nonlinear intersections are separate requirements.

Count the coefficients before plotting

The family is y=Ax³+Bx²+Cx+D+E/x+F/x², with numerical coefficients and at least three of the six equal to zero. Thus no more than three coefficients are nonzero. For y=x²+2x+4/x the ordered list (A,B,C,D,E,F) is (0,1,2,0,4,0). There are three zero entries and three nonzero entries; all three displayed terms matter.

Numerical does not mean integer in this family. The coefficient 1/2 is permitted, provided the zero-count condition is met. Signed and fractional inputs are also allowed wherever the expression is defined. Count coefficients in their fixed six slots; a rewritten expression does not gain a fourth-power or inverse-cube term simply because it is another graph you already know.

Keep the original domain

A genuine term involving 1/x or 1/x² excludes x=0. At x=−1/2, use 1/x=−2 but 1/x²=4: squaring the input changes the sign behaviour. An undefined table entry is not a height of zero and must not be filled with a point on the axis. If a supplied expression has a denominator, record its exclusions before any rewriting or cancellation.

When E=F=0 and the function is defined directly by the remaining polynomial, zero is allowed unless a separate restriction is stated. Absent reciprocal terms do not create an artificial hole. Conversely, rewriting a function whose original denominator forbade zero does not automatically restore that input. The rule and its original domain must travel together.

Draw the permitted parts separately

Plot exact pairs on each side of the excluded input and draw each branch separately. For 4/x, negative inputs near zero give large negative values and positive inputs give large positive values. In x²+2x+4/x those reciprocal values determine the near-zero direction; never draw a solid bridge through x=0. A dashed vertical guide can mark the excluded input, but it is not part of the function.

An inverse-square term can have the same sign on both sides of zero, yet the gap still remains. Test symmetry using the whole expression: a cubic term can destroy the symmetry of an inverse-square contribution. Choose a suitable view and keep the exact table even when some heights are off-screen. A cropped branch does not end at the border, and its point must not be moved to that border.

Sources for this lesson

Pearson Mathematics A linear specification · Issue 2 · printed page 33

Pause and explain

A function is defined directly by y=x²+2x with no further domain restriction. What happens at x=0?

Put the idea to work

Worked example

For y=x²+2x+4/x, identify the three nonzero family coefficients, make values at x=−2,−1,−1/2,0,1/2,1,2, and draw the two permitted parts. Label the exclusion before joining any points.

Show the worked solution

The coefficient list is (0,1,2,0,4,0), so the rule meets the three-zero condition. Because 4/x is present, its domain is x≠0.
The exact table entries are (−2, −2); (−1, −5); (−1/2, −35/4); 0 → undefined (no graph point); (1/2, 37/4); (1, 7); (2, 10). For instance, at x=−1/2 the sum is 1/4−1−8=−35/4; at x=1/2 it is 1/4+1+8=37/4.
Plot the negative and positive inputs as separate parts. As x approaches zero the left branch falls without bound and the right rises without bound. Draw no point or connecting segment at zero; use additional permitted samples if a branch needs more detail.

Answer Nonzero coefficients B=1, C=2, E=4. Domain x≠0. The table is (−2, −2); (−1, −5); (−1/2, −35/4); 0 → undefined (no graph point); (1/2, 37/4); (1, 7); (2, 10); draw the two parts separately.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

For y=x³−2+3/x², complete the same signed/half-unit table and plot both sides of the excluded input. Explain why the point at x=0 cannot be filled in. Use inputs −2,−1,−1/2,0,1/2,1,2.

Give me a hint

Compute x³ and 3/x² separately. The inverse-square contribution is positive on both sides of zero.

Compare my reasoning

The list (A,B,C,D,E,F) is (1,0,0,−2,0,3). Its three zero coefficients satisfy the condition, and x=0 is excluded.
The exact table entries are (−2, −37/4); (−1, 0); (−1/2, 79/8); 0 → undefined (no graph point); (1/2, 81/8); (1, 2); (2, 27/4). At x=−1/2 the value is −1/8−2+12=79/8; at x=1/2 it is 1/8−2+12=81/8.
Plot and draw each side separately; both branches rise without bound near zero because 3/x² dominates the finite nearby contributions. The cubic term means the whole graph is not even, despite that shared near-zero direction.

Domain x≠0; the table is (−2, −37/4); (−1, 0); (−1/2, 79/8); 0 → undefined (no graph point); (1/2, 81/8); (1, 2); (2, 27/4). Both sides rise near the excluded input, with no connecting segment.

Look for these in your work

Show the correct coefficient count and zero exclusion.
Keep the signed cube and positive inverse-square calculation separate.
Draw both permitted branches without claiming false symmetry.

2 · Independent

The graph uses u,v and v=(2u²−5)/u. Rewrite it in the sparse family, preserve its original domain, make values at u=−2,−1,−1/2,0,1/2,1,2 and plot the separate parts.

Give me a hint

Divide each numerator term by u, but retain the original restriction u≠0.

Compare my reasoning

For u≠0, v=(2u²−5)/u=2u−5/u. The coefficient list is (0,0,2,0,−5,0), so four coefficients are zero.
The exact (u,v) table is (−2, −3/2); (−1, 3); (−1/2, 9); 0 → undefined (no graph point); (1/2, −9); (1, −3); (2, 3/2). For example, at u=−1/2 the value is −1+10=9; at u=1/2 it is 1−10=−9.
Label u horizontally and v vertically, plot both signed sets of pairs, and draw separate smooth branches. The original denominator still forbids u=0, so neither rewriting nor the table permits a bridge there.

v=2u−5/u for u≠0. The table is (−2, −3/2); (−1, 3); (−1/2, 9); 0 → undefined (no graph point); (1/2, −9); (1, −3); (2, 3/2); the two drawn parts remain separated at zero.

Look for these in your work

Rewrite without losing the original domain.
Calculate exact signed and half-unit values.
Use the renamed axes and preserve the branch break.

3 · Transfer

For r=(1/2)s²+1−3/s², generate values at s=−2,−1,−1/2,0,1/2,1,2 and draw the graph. Test the claims that numerical coefficients must all be integers and that equal values at ±s justify drawing across zero.

Give me a hint

Place 1/2 in the x² slot and −3 in the inverse-square slot; count zeros before checking symmetry.

Compare my reasoning

The list is (0,1/2,0,1,0,−3): three coefficients are zero and the noninteger coefficient 1/2 is permitted. The term −3/s² excludes s=0.
The exact (s,r) table is (−2, 9/4); (−1, −3/2); (−1/2, −87/8); 0 → undefined (no graph point); (1/2, −87/8); (1, −3/2); (2, 9/4). At s=1/2, 1/8+1−12=−87/8; the same value occurs at −1/2.
Because replacing s by −s leaves every squared term and the constant unchanged, r(−s)=r(s) throughout the original domain. Draw the symmetric branches separately: symmetry does not define a value at zero or authorise a joining segment.

The integer-only claim is false. Domain s≠0; values are 9/4, −3/2, −87/8, undefined, −87/8, −3/2, 9/4 in the stated order. The graph is even but remains split at zero.

Look for these in your work

Apply the numerical-coefficient and zero-count conditions.
Verify symmetry using the full expression.
Draw the two branches with the zero exclusion intact.

Common mistakes

Counting displayed terms without checking the six coefficient slots.
Replacing an undefined zero input by a zero output.
Mirroring or connecting branches without checking the full rule and its domain.

Recall without your notes

What can you explain now?

Can the missing value at x=0 in x³−2+3/x² be filled with zero?

Compare with the explanation

No; it is undefined.

The term 3/x² divides by zero there. A blank or undefined entry is a domain exclusion, not a plotted origin.

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Make it stick

Distinguish an inverse-square contribution from an inverse contribution. Explain why symmetry, cancellation or a cropped view cannot fill an original hole.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics A

Check the official syllabus Issue 2 · November 2026 / June 2027 entries checked

Course scope and assessment

Linked teaching: sections 1.1 to 1.11, 2.1 to 2.8, 3.1 to 3.4,4.1–4.11 and5.1–5.2 and6.1–6.3; 2.5, 3.2,3.4 and5.1 are Higher only. This includes Foundation content inherited by Higher and Higher extensions for recurring decimals, surds, fractional powers, algebraic sets, Venn counts, repeated percentage changes, calculations with measurement bounds and standard-form application problems. Algebra includes Foundation symbols, integer index laws, substitution, like terms, brackets and monic factorisation, plus Higher fractional indices, multiple brackets, non-monic factorisation, algebraic fraction operations with domain restrictions, completing the square and proofs. Formulae include derivation from words and diagrams, substitution, single-occurrence subject changes including squares and roots, and Higher repeated-subject and harder power/root rearrangements. Linear equations cover both sides, brackets, fractional coefficients and constructing models from data. Section 2.5 adds all eight direct/inverse x, square, cube and square-root proportion families with actual graph exploration and domain checks. Section 2.6 adds shared elimination and substitution, exact fractions, contextual feasibility and repeated or inconsistent conditions; the additional Higher graphical lesson interprets genuine-line intersections, vertical lines and off-screen crossings. Section 2.7 adds Foundation monic quadratic factorisation, with Higher general factorisation, quadratic-formula and completing-square alternatives, context derivation and line–parabola/line–circle systems including full branches and exact paired solutions. Section 2.8 adds required open/closed number-line intervals, linear and compound solving, construction and identification of Cartesian regions, and Higher quadratic sign intervals and harder linear regions. Its Cartesian boundary-drawing convention note does not remove number-line endpoint requirements. Section 3.1 adds term-to-term and position-to-term generation, integer continuation and linear nth terms, plus Higher arithmetic a,d recovery, the general nth-term formula and finite arithmetic sums. Section 3.2 adds Higher-only mappings, both function notations, actual domain/range reasoning, composition in Pearson order with original restrictions, and affine/rational/restricted-branch inverses. Section 3.3 has nine shared coordinate, straight-line, travel, speed, conversion and quadratic-plotting lessons, followed by two additional Higher line-equation lessons, four Higher graph-family lessons, nine transformation/equation-recovery lessons and three drawn-tangent/intersection lessons. All nine Foundation source rows have lesson links, with paired interpretation and linear/quadratic plotting breadth guards. The broad additional Higher H3.3A row requires integer polynomial/cubic, sparse mixed reciprocal, degree sine/cosine and degree tangent lessons with every practice stage. H3.3B requires all four transformation forms across linear, quadratic, degree sine and degree cosine functions in eight core lessons; H3.3C also requires their algebra recovery and the ninth domain/equivalence/order synthesis lesson. H3.3D requires drawn-tangent construction and signed rise/run using actual units; calculus is not a replacement. H3.3E requires both line/quadratic and genuine line/cubic intersections and their nonlinear-minus-linear difference graphs, with all practice stages. All sixteen Graphs source rows now have teaching and staged-practice links; exhaustive subskill approval, mastery and assessment readiness remain unestablished. Section 3.4 adds six Higher-only lessons for variable rates, integer-power differentiation, gradient/rate conditions, stationary coordinates classified by original graph shape, one-dimensional motion and other practical models. H3.4C requires gradient and stationary slices; H3.4E requires both motion and other-practical slices. Preserve original domains, negative-power exclusions, admissible counts and boundary candidates. All five source rows have structural links, not exhaustive subskill approval or specialist/assessment validation. Section4.1 adds six shared Foundation/inherited-Higher lessons for selected acute/right/obtuse/reflex arcs, point/straight/intersection and parallel-transversal angles, triangle sums/exteriors, isosceles/equilateral and right/mixed triangles. All four shared rows have structural links; B requires both local and parallel strands and D both special and right-triangle strands, each with every practice stage. No additional H4.1 rows are invented. Preserve labelled geometry and reasons, the auxiliary-parallel triangle-sum argument, actual straight extensions, inclusive equilateral/isosceles and right-isosceles categories, and impossible/insufficient-data checks. Section4.5 constructions,4.4 measurement/bearings and the full H4.7 reasoning scope remain separate, without a blanket formal-proof waiver. Section4.2 adds seven shared Foundation/inherited-Higher Polygons lessons for all nine named polygon examples, convex/concave quadrilateral sums, all six named quadrilateral property families, general/regular polygon angles and congruence correspondence. Its seven rows A–G have structural links, with no additional Higher rows; C requires both family lessons and every practice stage. F/G remain two distinct published IDs supported by the same congruence lesson. Rhombus/square and kite endpoint-angle bisection and isosceles-trapezium equal diagonals are conservative standard-property breadth, not invented verbatim source bullets. Preserve reflex interiors, valid internal triangulations, local trapezium/kite conventions, exact integer side counts and impossible-data guards. Section4.3 symmetry,4.5 formal constructions,4.11 similarity and the full H4.7A reasoning programme remain separate. Literal book tags omit A; G has no Hub tag, and the Book2 map/proof congruence location and chapter7 page-range conflict remain unresolved. Section4.3 adds three shared Foundation/inherited-Higher Symmetry lessons for the one F4.3A source row, requiring every lesson and practice stage for the joint line/rotation capabilities. Draw all mirror lines, test a fixed original against a separate tracing about a chosen centre, count identity once and distinguish order1 from a blank circle’s non-finite rotations. Preserve indentations and true physical disk/colour features; the chosen missing-vertex parallelogram completion is original breadth, not an extra row or4.5construction credit. No additional H4.3 row is invented. The sole literal book link is Book2 Shape and Space6; no Book1 tag or explicit4.3Hub tag is invented, and its mapped chapter body/full final-book alignment remains unverified. Section4.5construction,5.2transformations,4.11similarity and full Higher reasoning remain separate. Section4.4 adds eight shared Foundation/inherited-Higher Measures lessons for seven A–G source rows and no additional Higher rows. Use complete instrument-scale, time-interval, sensible-estimate, nearest-degree protractor and three-figure-bearing teaching. F4.4F requires the full speed lesson; broad F4.4G requires speed, density and pressure, each with its worked explanation and every practice stage, with source binding on the same complete record. Preserve calibrated ticks and actual instrument action, compatible units, midnight/day order, approximate readings and north conventions. Keep numerical references hidden until calibrated attempts. Time recovery requires positive speed; at zero speed positive distance is impossible and zero distance alone is underdetermined. Density uses positive volume, displacement differences and explicit additive-volume assumptions; pressure uses the given formula, supplied forces and positive contact areas. Literal book navigation names only D and G; no A/B/C/E/F tags are invented, and Book1 map255–268/proof256–269 remains unresolved. These structural links do not complete4.5 instrument/compass constructions or scale drawings,5.2 transformations or formal assessment. Section4.5 adds six shared Foundation/inherited-Higher Construction lessons for four A–D rows and no additional Higher rows. A requires both nearest-millimetre measurement and actual line drawing. BroadB requires complete triangles and other two-dimensional shapes; broadD requires both straight-edge-and-compasses bisectors. Source binding and complete teaching/worked/all three practice stages must belong to the same required record. Use real ruler, protractor and compass action with retained construction evidence, physical calibration and honest approximate readings. Keep private endpoint readings, pose coordinates, solved vertices/crossings and reference directions hidden before instrument attempts. Respect valid mirror branches and selected minor/reflex angles; reject impossible triangles, insufficient data and degenerate crossings. Scale drawings require construction before measuring the unknown, compatible units and a labelled positive scale; drawing placement offsets remain givens. Literal book navigation names B/C/D only in Book2 Shape and Space6 with Hubs37/36/38. A has no literal chapter/Hub tag and no Book1 tag is invented. Public proof contents agree with map128–160 and show generic Constructions139, but the relevant body, full-book/Hub content and exact subtype pages remain unavailable. These structural links do not grant4.6 circle properties,4.7 whole geometrical reasoning,4.8 trigonometry,4.9 mensuration,4.11 maps/similarity,5.2 transformations or loci credit. The nearby circle formal-proof note is not imported. Section4.6 adds nine original Circle properties lessons for two shared Foundation and three additional Higher rows. The first three shared lessons cover all nine named parts, both chord/centre implications and both tangent facts. F4.6B requires both chord and tangent lessons. Additional Higher H4.6A requires both internal-chord and external-secant products with original named lengths, positive lengths and correct internal/near-far point order. H4.6B requires actual concyclic boundary vertices. H4.6C requires all five theorem parts across centre/circumference, diameter/same-segment, cyclic-opposite-angle and alternate-segment lessons. Every required slice needs substantive teaching, explained worked work and all three practice stages on one source-bound record; no half-row or untagged duplicate closes the bundle. Keep given marks separate from derived marks and retain selected major/minor arcs, tangent rays and physical algebraic-root domains. The no-formal-proof note belongs to H4.6C, not a blanket H4.6A/H4.7 exemption. Literal Book2 chapter5 tags F4.6B/H4.6B/H4.6C and chapter8 tags H4.6A/H4.6C are navigation only; no F4.6A or Book1 tag is invented. Chapter8 map326–349 versus proof chapter-start325 and next chapter349 remains unresolved; relevant final bodies and Hub content are unverified. These structural links do not complete construction, whole geometrical reasoning, trigonometry or mensuration, or establish specialist approval, mastery or formal assessment. Section4.7 adds five original Geometrical reasoning lessons for one shared F4.7A and one additional H4.7A row. The shared row jointly requires line/triangle, polygon and chord/tangent numerical reasoning, including lengths; the Higher row jointly requires both circle and mixed-angle standard-statement chains. Every required record needs substantive teaching, its explained worked example and all three fresh practice stages, with the source ID on that same complete record. Preserve actual diagram hypotheses, common arcs, contact radii, selected tangent rays, cyclic/straight point order, valid internal triangulations and the unknown regular-polygon outline after deduction. Each geometric equation needs a named reason; a correct number or sufficient text length does not validate an arbitrary written argument. The H4.6C formal-theorem-proof note remains local, not a blanket4.7 waiver. Only literal H4.7A Book1 Shape and Space1/Hub89,137,138,139 navigation is used; no Foundation4.7A or Book2 tag is invented. Public Book1 chapter60–99 is available but does not establish full Higher circle-angle or Hub body scope. These links do not award4.5construction skill, specialist approval, mastery or formal assessment. Section4.8 adds eleven original Trigonometry and Pythagoras’ theorem lessons for three shared F4.8A–C and six additional H4.8A–F rows. The first four are shared; the last seven are additional Higher. F4.8B jointly requires lengths with all acute ratios and inverse-angle work; H4.8C jointly requires sine-rule and cosine-rule teaching. Every row needs all its named source-bound records, substantive teaching, explained worked work and every Guided/Independent/Transfer stage. Preserve both missing-leg and missing-hypotenuse Pythagoras, signed bearing components with local north, all three obtuse functions, actual elevation/depression horizontals, admissible SSA branches and included-angle area restrictions. Three-dimensional work requires perpendicular sections and the actual foot/projection for a line-plane angle; plane-plane angles are excluded. Positive physical lengths, inverse domains, triangle inequalities, degree mode, exact forms and final-only rounding remain explicit. Chosen SSA two/one/no and inverse-area comparisons are safeguards, not extra syllabus rows. Literal navigation uses Book1 SS1–4 only: no Book2 tag or dedicated H4.8A Hub is invented. Hub145 is tagged F4.8A in SS1 and H4.8D in SS3; the former title does not extend Foundation into3D. Public Pythagoras body74–76 is available, but SS2/3/4 bodies and Hub content are unverified. SS2 map176–197 versus summary198, and SS3 map243–254 versus proof-start244/summary255 remain unresolved. No4.5construction,4.9mensuration, full-book, specialist, mastery or formal-assessment credit follows from these structural links. Section4.9 adds eight original Mensuration lessons for five shared F4.9A–E and one additional H4.9A row. The first six are shared; the last two are Higher. F4.9E jointly requires complete circle and semicircle/compound-boundary lessons; H4.9A jointly requires degree arcs/perimeters and sector-area/recovery lessons. Every required source-bound record needs substantive teaching, its explained worked example, checks and all Guided/Independent/Transfer stages. Retain both-direction linear/area conversions, exposed polygon boundary traces and internal joins, triangle/rectangle decomposition and contained disjoint cutouts, outside perpendicular heights, both parallelogram/trapezium methods, radius/diameter recovery and semicircle arc-versus-perimeter distinctions. Preserve major/remaining sector sweeps, positive physical roots, inverse dimensions, annular inner arcs and radial gaps, compatible square/linear units and question-specific precision; no universal3sf rule is invented. Radians are excluded; ordinary sector arc-plus-two-radii boundaries require0<θ<360, while a full disk has no radial cuts. Literal Book1 SS1 tags B/C/D/E and HigherA with Hubs47/48/118, plus Number4A/Hub14, are navigation only; no Book2 tag or verified Hub body is inferred. Public SS1contents summary98 versus actual99 and Number4map198–206 versus proof199–207/next208 stay explicit. Chosen compound/inverse/annular safeguards do not add source rows or grant construction, volume, similarity, full-book, specialist, mastery or formal-assessment credit. Section4.10 adds nine original 3D shapes and volume lessons for six shared F4.10A–F rows and one additional H4.10A row. The first seven are shared; the final sphere and right-cone lessons are additional Higher. Teach all seven named solid families, complete face/edge/vertex incidence with hidden features, planar triangle/rectangle surface inventories and cylinder surfaces with actual open/closed conditions. F4.10E requires both complete prism and cylinder-volume slices; H4.10A requires both complete sphere and right-cone surface-and-volume slices. Every required source-bound record needs substantive teaching, worked methods, checks and all Guided/Independent/Transfer stages. Preserve actual face altitudes, positive inverse dimensions, full-length groove conditions, internal capacity and fill levels, contact-patch removal, cubic unit factors and1L=1000cm³. Curved-solid tasks state their flat-face/rim/apex convention without inventing a universal mark-scheme count. Nets, joins, inverse applications and solid hemispheres are chosen breadth, not extra official rows or instrument, frustum, similarity, pyramid-volume or hollow-capacity credit. Keep exact values/π and question-specific precision, including explicit significant trailing zeroes or scientific form. Literal Book2 Shape and Space9 tags C/D/E/F and HigherA are navigation only; no A/B or Book1 tags are inferred. Hubs56/57/55/84/131 do not establish body coverage, and no F4.10F Hub tag is invented. Map359–385 versus proof358–384/next385 remains unresolved; relevant chapter body and full-book/Hub content are unavailable. Specialist, formal-assessment and mastery approval remain pending. Section4.11 adds seven original Similarity lessons for exactly two shared F4.11A/B rows and three additional H4.11A/B/C rows. The first four are shared correspondence, forward/inverse lengths, maps and plans; the final three cover area factors, volume factors and mixed problems. F4.11A requires both complete correspondence and length slices, and F4.11B requires both map and plan slices, with source binding, substantive teaching, worked methods, checks and every explained Guided/Independent/Transfer stage in the same actual record. Retain matching angles and every corresponding side, concave reflex corners, positive factors, inverse recovery, parallel/shared-vertex hypotheses, whole/part order, compatible units, route versus direct distance, and uniform print-resizing scale recovery. Higher work distinguishes k, k² and k³, positive roots, similarity hypotheses, unit powers, actual closed-cuboid total surfaces, physical dimensions, exact money before rounding, and parallel-section cone subtraction as a chosen application. No extra official row, transformations, instrument calibration, formal construction or general frustum-formula credit is claimed. Literal Book2 Shape and Space6 F4.11B/Hub36, Shape and Space7 F4.11A/Hub85, and Shape and Space9 H4.11A/B/C with Hubs143/144 are navigation only, with no Book1 tag invented. Shape and Space6 map128–160/proof128/summary159/next161, Shape and Space7 map223–248/proof222/summary247/next248, and Shape and Space9 map359–385/proof358/summary384/next385 remain explicit. The actual proof places Similar shapes at147 under Shape and Space6 while the literal F4.11A map tag remains under Shape and Space7; this topic-location conflict is unresolved. Relevant chapter and Hub bodies are unavailable; no full-book, specialist, formal-assessment or mastery approval follows. Section5.1 adds nine original additional Higher Vectors lessons for exactly H5.1A–G and no shared Foundation rows. All required source-bound notation, scalar, arithmetic, magnitude, route, position/division, parallel/collinear, simple-proof and intersection strands need substantive original teaching, an explained worked example and exactly one complete Guided/Independent/Transfer task in the same record. Preserve directed endpoints, signed/scalar and zero-vector exceptions, exact fractions and radicals, final-only rounding, current own head-to-tail joins, one-origin positions, t=0/t=1 endpoints and external points without clamping. Geometric conclusions require actual nonzero, nonparallel, noncollinear or simple-boundary hypotheses; formal two-to-four-symbol planar identities do not assert an independent spatial basis. H5.1G requires genuine explained geometric proof, not numerical resemblance or unchecked text. The H4.6C local theorem-proof note is not a blanket vector-proof waiver. Literal Book2 Shape and Space10 H5.1A–G map427–450/proof426/summary449/next450 remains unresolved; Hub167 covers A/B/C/D/F,168 A/E and169 G. Navigation does not establish modular equivalence. No Book1 tag or unseen Vectors chapter/Hub body is invented: the public uncorrected2025proof exposes contents and Number8 body, not Vectors. November2026/June2027 linear entry and regional-centre limits remain separate. Full-book, specialist, mastery and formal-assessment approval remain pending. Section5.2 adds twelve original shared Transformation geometry lessons across exactly F5.2A–M and no additional Higher row. Full source-bound bundles require substantive teaching, explained worked and exactly one complete Guided/Independent/Transfer task in each record. Retain every labelled corner and corresponding trace, signed/general rotations, complete equivalent mirror lines, common translation columns, rigid distances/angles, positive and fractional factors, supplied/free centres and factor1 identity exceptions. General-degree written3dp is separate from unrounded geometry. Ordered compositions are chosen breadth rather than an extra source row; compare both orders, including commuting perpendicular reflections, and distinguish reverse order from the inverse. Source/book/year and proof/physical-construction limits remain explicit. Graphical representation6.1 adds ten original lessons: six shared Foundation tabulation/two-way/pictogram/bar/pie/interpretation slices and four additional Higher histogram construction/reading and cumulative construction/use slices. Each complete source-bound record and every practice stage are required; tables or diagram names alone do not establish actual construction. Keep scales/units/keys, zero categories, unequal continuous widths and density/area, hidden missing calibration and pie labels, uniform partial-class estimates, cumulative boundaries and nonunique flat inverses. No Statistical measures6.2 or Probability6.3 credit follows. Literal book/Hub metadata is navigation only: Book2 Handling Data4 F6.1Aii is a modular subpart; Book1 Handling Data3 map398 starts differently from proof397 and Book2 Handling Data6 map208 from proof207. Relevant chapter/Hub bodies remain unavailable. Statistical measures6.2 adds eight original lessons: four shared Foundation list/frequency averages, simple mean problems and grouped midpoint estimates/modal classes; four additional Higher spread, discrete quartiles, median and IQR from cumulative diagrams. Every source-bound record needs substantive teaching, its explained worked example and all three unique practice stages. Preserve repeated positions, weighted counts, observed zero-frequency extremes, compatible units, grouped estimation limits, the explicitly chosen median-of-halves convention, actual graph guides, each curve’s own total and nonunique plateau intervals. Literal Book2 Handling Data4/5/6 tags and Hubs are navigation metadata only; no Book1 tag, complete chapter body or Hub access is inferred. Handling Data6 map208–222 versus proof207/summary221/next222 stays explicit. Statistical measures alone does not replace Probability6.3 teaching. Probability6.3 adds nine original lessons for ten shared F6.3A–J and four additional H6.3A–D rows. The first five are shared language/scales/models, complete single/two-event sample spaces, Venn probabilities, experimental estimates/expected frequencies and complement/mutually-exclusive addition; the final four are Higher independent trees, actual without-replacement trees, simple conditioning and contextual inverse/mixed models. Each required source-bound strand needs substantive teaching, an explained worked example and exactly one complete Guided/Independent/Transfer task in the same actual record. F6.3C requires both model and sample-space slices; H6.3A requires independent, dependent and application trees; H6.3B requires independent and application slices; H6.3C requires without-replacement, simple-conditioning and application slices; H6.3D requires conditioning and application slices. Retain physical identities, nonuniform weights, original ordered histories, complete branches/terminal events, unchanged versus changing contents, restricted positive conditional denominators and undefined empty conditioning. Experimental variability and fractional expected counts are model predictions rather than guaranteed observations. Inverse physical counts keep original nonnegative integer/positive-total domains. Chosen three-set, three-stage and inverse safeguards do not add source rows or a named general conditional theorem. Literal Book1 HandlingData1/2/3 tags and Hubs are navigation only; no Book2 tag or unseen probability chapter/Hub body is invented. Preserve map228–242/proof229–243/next244, map303/proof304–328/next329 and map398–428/proof397/summary426/next428; missing explicit Hub A/B/F/J tags remain absent. The common two-coin elementary model is not claimed as novel. All39 source sections and242 requirements now have structural teaching/practice links; that is not exhaustive subskill, teacher, full-book, mastery, specialist or formal-assessment approval. The independent source inventory contains 162 Foundation requirements and 80 additional Higher requirements; some lettered requirements contain multiple skills. A lesson link alone does not verify every skill or examination readiness. Written practice is self-checked, not automatically marked. Specialist approval and cumulative assessment have not been established. This course follows the linked Issue 2 linear 4MA1 specification. The 2026/2027 entry manual lists F/H in November 2026 and June 2027; regional FR/HR entries are June 2027 only and depend on centre eligibility. Modular 4XMAF and 4XMAH coverage is separate.