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IGCSE · Pearson Edexcel · 4MA1

Higher: recover counts and missing histogram scales

Read frequencies from areas, recover a common density calibration or missing bar, and make explicitly conditional partial-class estimates.

Pearson 4MA1 · Graphical representation of data · Additional Higher 6.1A
Exam scope and assessment notes

The first six lessons are shared Foundation work inherited by Higher; the last four are additional Higher histograms and cumulative-frequency work. Construct actual labelled diagrams with their scales, units and keys. Unequal continuous intervals need frequency density and area; partial-class and straight-guide readings are estimates, and a flat inverse can be an interval. Numerical agreement does not award marks or establish mastery; written explanations need human review. Statistical measures and probability remain separate. Linear November 2026/June 2027 entries and regional-centre eligibility differ from modular awards. Publisher map/contents are navigation only: F6.1Aii is a modular subpart, not the whole linear F6.1A row. Book1 Handling Data3 map398–428 differs from proof397/summary426/next428; Book2 Handling Data6 map208–222 differs from proof207/summary221/next222. Relevant chapter and Hub bodies remain unavailable; specialist and formal-assessment approval remain pending.

Before you begin

  • Construct and interpret continuous unequal-width histograms.
  • Calculate rectangle areas and recover a missing part of a total.
  • Split a numerical interval at stated class boundaries.

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

Quick readiness check

An 8-unit-wide histogram bar is 5 grid divisions high and contains 20 observations. What does one vertical grid division represent?

What you will learn

  • Recover frequencies and a missing common density calibration from actual rectangle areas.
  • Use a complete total to recover an absent class count and density without pre-drawing its target bar.
  • Estimate partial-class counts under a stated uniform-density assumption and explain their limits.

A supplied continuous histogram has 0 < x ≤ 4: density 3; 4 < x ≤ 10: density 2; 10 < x ≤ 20: density 3/2. Recover the full frequency table and keep the density scale labelled in observations per unit of x. Estimate the count with 8 < x ≤ 14, explicitly assuming uniform density within each class. Show the split overlap areas. Draw or annotate the actual rectangles, check the total and compare at least two bar heights and frequencies without confusing them.

Choose your representation, enter its actual categories, counts, geometry and labels, and explicitly record it. No table margin, scale, missing sector, density, cumulative point or reading trace is filled for you.

Original supplied data

boundaries
0 ; 4 ; 10 ; 20
d
3 ; 2 ; 3/2
q
8 ; 14
interval Convention
lower < x ≤ upper
complete Sample
true

Only literal data are supplied. A missing entry is displayed as not supplied; requested totals, frequencies, angles, density calibration, cumulative constructions and read-off traces are not generated on this board.

All metric panels in this board share one camera and equal Cartesian drawing units. Equal full widths, heights and pie radii therefore stay visibly comparable; no individual panel is independently fitted. Histogram numerical frequency density is its entered vertical calibration. Tables and tally strokes are symbolic displays.

Supplied known histogram bars; a missing bar stays absent

Supplied data only. This is a display state, not a correctness or marking result.

Supplied known histogram bars; a missing bar stays absentOnly literal supplied data or the explicit actual learner representation is drawn here. No absent bar, sector, cumulative point or reading elbow is inserted. A common equal Cartesian drawing scale and camera preserve circles, bar widths and radii; the numerical vertical calibration is separately labelled where required.05101520actual corner (0, 3)actual corner (4, 3)actual corner (4, 2)actual corner (10, 2)actual corner (10, 3/2)actual corner (20, 3/2)measurement x (unit of x)Frequency density; whole-view turn 0°

Entered density per vertical drawing unit: 1. Tick positions and labels are retained exactly as entered; mismatched calibration is not repaired.

Blue is supplied geometry, amber is my actual work, purple is dirty retained work and green is a separately gated reference. Rendering never repairs a wrong count, overlap, radius, calibration, cumulative decrease or read-off elbow. Tool coordinates are bounded to±1000000; finite unsupported portions are not silently clamped. Up to1000 explicit pictogram pieces and100 tally bundles per displayed row are rendered. Screen size is not physical instrument calibration. Readiness and fixed numerical comparison do not mark reasoning, award marks, prove mastery or provide teacher/specialist/formal-assessment approval.

Observed and whole-class counts stay distinct from within-class histogram or cumulative-guide estimates. A zero-frequency plateau can give an interval of inverse readings. No median, quartile, midpoint-mean or probability teaching credit is supplied by this display.

My actual complete rectangles and calibrated density axis

No actual representation is recorded for this action.

The board places only the coordinates you explicitly choose into the selected slot. It generates no other corner, bar, sector or reading elbow. Interval-area bounds are two measurements, not a planar point.

My quantities, conclusions and explanation

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Enter an exact fraction or finite decimal, with the precision and quantity unit explicitly requested in the question.

Explain the actual construction and its conditions. Text presence does not establish a correct argument.

These original fixed cases support self-checking. They do not mark proofs or written explanations, award marks or mastery, certify instrument accuracy, or provide specialist or formal assessment approval. Within-class histogram and cumulative-guide readings are estimates, while supplied whole-class counts remain exact. A plateau may give an interval rather than one inverse reading.

Read an area with its calibration

For a histogram whose density axis is already calibrated, multiply each actual class width by its density height. Write the measurement intervals and recovered frequencies in a table, then check the total. A count is represented by area, not by the height label on its own. When no physical measurement unit is supplied, label the horizontal variable x and the density as observations per unit of x.

If the vertical calibration is missing but equal grid divisions are shown, first recover one density from a known class frequency divided by its width. Divide that density by the known bar’s number of grid divisions to find the density represented by one division. Apply that one common calibration to all heights; choosing a different calibration for each bar would destroy the area interpretation.

Recover an absent bar from a complete sample

A complete sample total can determine one missing class count. Calculate the frequencies of the known rectangles, add them, and subtract that sum from the stated total. Divide the remaining count by the missing class width to obtain its density, then draw the recovered rectangle at the actual boundaries on the same scale.

This recovery needs sufficient information. An unknown total and an entirely absent class do not generally determine a unique frequency. Negative remaining counts expose inconsistent data. Keep the missing rectangle and its numerical height as unknown work until they have been recovered; a supplied empty interval is not permission to invent an attractive height.

State what a partial-class reading assumes

For an interval cutting through classes, divide it at every class boundary. Multiply the overlap width within each class by that class’s density and add the resulting areas. This uses a uniform-within-class guide: it treats the class density as constant across that rectangle. The resulting partial count is an estimate of the grouped observations, even when its numerical value happens to be a whole number.

Whole-class areas recover the supplied grouped frequencies, but partial areas do not locate the raw individuals exactly. Label the requested overlap on the histogram and explain the assumption rather than announcing an exact observed count. For a complete supplied sample, intervals outside its class range contribute no additional observations; do not extend a bar beyond its actual class domain.

Sources for this lesson

Pause and explain

A partial-class area is 7.5 under an explicitly uniform within-class guide. Which statement is justified?

Put the idea to work

Worked example

A supplied continuous histogram has 0 < x ≤ 4: density 3; 4 < x ≤ 10: density 2; 10 < x ≤ 20: density 3/2. Recover the full frequency table and keep the density scale labelled in observations per unit of x. Estimate the count with 8 < x ≤ 14, explicitly assuming uniform density within each class. Show the split overlap areas. Draw or annotate the actual rectangles, check the total and compare at least two bar heights and frequencies without confusing them.

Show the worked solution
  1. The widths are 4, 6, 10. The supplied heights are already calibrated densities 3, 2, 3/2, so multiply each width by its stated density rather than treating a height as a frequency.
  2. Multiply widths by calibrated densities: 4×3=12; 6×2=12; 10×3/2=15. The frequencies are 12, 12, 15 and total 39.
  3. Mark 8 < x ≤ 14 on the histogram and split at its internal class boundaries. Overlap widths 0, 2, 4 give contributions 0, 4, 6; their sum is 10. This is an estimate under the stated uniform-within-class guide, not an exact raw-location count.
  4. Keep the original given bars and their actual boundaries. Compare area, not just height: the second and third bars have frequencies 12 and 15, even though their densities are 2 and 3/2. Whole-class areas match the grouped counts; the guide does not reveal individual positions within a class.

Answer Widths 4, 6, 10; densities 3, 2, 3/2; frequencies 12, 12, 15; total 39. The requested partial-interval count is estimated as 10, conditional on uniform within-class density. Retain actual class widths and use rectangle areas to compare counts.

Try a topic test

Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

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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

The supplied histogram has boundaries 10, 15, 25, 45, 60, and consecutive bar heights 6, 5, 4, 2 equal vertical grid divisions. Its first class contains 18 observations; the numerical density calibration is missing. Recover the density per grid division, all density heights, every class frequency and the total. Draw the calibrated density-axis labels on the supplied histogram and compare heights with class counts.

Give me a hint

Separate horizontal width, displayed grid height and calibrated density; use a known area or complete total before drawing an unknown target.

Compare my reasoning
  1. The first width is 5. Its density is 18÷5=18/5. Since that bar is 6 grid divisions high, each division represents (18/5)÷6=3/5 frequency-density units.
  2. Multiply widths by calibrated densities: 5×18/5=18; 10×3=30; 20×12/5=48; 15×6/5=18. The frequencies are 18, 30, 48, 18 and total 114.
  3. Label equal vertical grid divisions with increments of 3/5; the four actual heights correspond to densities 18/5, 3, 12/5, 6/5. Preserve the supplied widths 5, 10, 20, 15. The first bar is taller than the third but contains 18 rather than 48 observations, so compare their areas.
  4. Keep the original given bars and their actual boundaries. Compare area, not just height: the second and third bars have frequencies 30 and 48, even though their densities are 3 and 12/5. Whole-class areas match the grouped counts; the guide does not reveal individual positions within a class.

Widths 5, 10, 20, 15; densities 18/5, 3, 12/5, 6/5; frequencies 18, 30, 48, 18; total 114. One vertical grid division represents density 3/5; label that calibration on the supplied graph. Retain actual class widths and use rectangle areas to compare counts.

Look for these in your work

  • Recover the common density calibration or missing class information from the supplied data.
  • Show actual bar areas, a labelled density scale and the complete frequency table.
  • Split requested partial overlaps at boundaries and state the uniform-density estimation limit.
2 · Independent

A supplied continuous histogram has 0 < x ≤ 3: density 2; 3 < x ≤ 9: density 5/3; 9 < x ≤ 15: final bar and height missing. The complete sample contains 28 observations. Recover the full frequency table and keep the density scale labelled in observations per unit of x. Estimate the count with 3/2 < x ≤ 12, explicitly assuming uniform density within each class. Show the split overlap areas. Draw or annotate the actual rectangles, check the total and compare at least two bar heights and frequencies without confusing them.

Give me a hint

Separate horizontal width, displayed grid height and calibrated density; use a known area or complete total before drawing an unknown target.

Compare my reasoning
  1. The class widths are 3, 6, 6. Known areas are 6, 10; their sum is 16. The complete total leaves 28−16=12 observations for the missing class, giving density 12/6=2.
  2. Multiply widths by calibrated densities: 3×2=6; 6×5/3=10; 6×2=12. The frequencies are 6, 10, 12 and total 28.
  3. Mark 3/2 < x ≤ 12 on the histogram and split at its internal class boundaries. Overlap widths 3/2, 6, 3 give contributions 3, 10, 6; their sum is 19. This is an estimate under the stated uniform-within-class guide, not an exact raw-location count.
  4. Keep the original given bars and their actual boundaries. Draw the recovered missing rectangle only after calculating its height. Compare area, not just height: the second and third bars have frequencies 10 and 12, even though their densities are 5/3 and 2. Whole-class areas match the grouped counts; the guide does not reveal individual positions within a class.

Widths 3, 6, 6; densities 2, 5/3, 2; frequencies 6, 10, 12; total 28. The requested partial-interval count is estimated as 19, conditional on uniform within-class density. The missing final bar has density 2. Retain actual class widths and use rectangle areas to compare counts.

Look for these in your work

  • Recover the common density calibration or missing class information from the supplied data.
  • Show actual bar areas, a labelled density scale and the complete frequency table.
  • Split requested partial overlaps at boundaries and state the uniform-density estimation limit.
3 · Transfer

A supplied continuous histogram has 0 < x ≤ 5: density 4; 5 < x ≤ 20: density 2; 20 < x ≤ 50: density 1. Recover the full frequency table and keep the density scale labelled in observations per unit of x. Estimate the count with 0 < x ≤ 25/2, explicitly assuming uniform density within each class. Show the split overlap areas. Draw or annotate the actual rectangles, check the total and compare at least two bar heights and frequencies without confusing them.

Give me a hint

Separate horizontal width, displayed grid height and calibrated density; use a known area or complete total before drawing an unknown target.

Compare my reasoning
  1. The widths are 5, 15, 30. The supplied heights are already calibrated densities 4, 2, 1, so multiply each width by its stated density rather than treating a height as a frequency.
  2. Multiply widths by calibrated densities: 5×4=20; 15×2=30; 30×1=30. The frequencies are 20, 30, 30 and total 80.
  3. Mark 0 < x ≤ 25/2 on the histogram and split at its internal class boundaries. Overlap widths 5, 15/2, 0 give contributions 20, 15, 0; their sum is 35. This is an estimate under the stated uniform-within-class guide, not an exact raw-location count.
  4. Keep the original given bars and their actual boundaries. Compare area, not just height: the second and third bars have frequencies 30 and 30, even though their densities are 2 and 1. Whole-class areas match the grouped counts; the guide does not reveal individual positions within a class.

Widths 5, 15, 30; densities 4, 2, 1; frequencies 20, 30, 30; total 80. The requested partial-interval count is estimated as 35, conditional on uniform within-class density. Retain actual class widths and use rectangle areas to compare counts.

Look for these in your work

  • Recover the common density calibration or missing class information from the supplied data.
  • Show actual bar areas, a labelled density scale and the complete frequency table.
  • Split requested partial overlaps at boundaries and state the uniform-density estimation limit.

Common mistakes

  • Reading an uncalibrated grid height directly as frequency density.
  • Pre-drawing a missing target bar instead of recovering it from the total.
  • Calling a partial-class estimate an exact raw count because its result is an integer.
Recall without your notes

What can you explain now?

Why is one known class frequency enough to calibrate all the equal vertical grid divisions of a consistent histogram?

Compare with the explanation

Its width gives the known class density; dividing that by its grid height gives density per division, which applies to every bar on the same axis.

The shared axis scale is common to the whole chart. A different scale for each bar would not be one consistent histogram.

After trying it yourself, choose your next review. This is your self-assessment.

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Make it stick

Describe recovery from a known bar and from a complete total, then explain how overlap areas yield a conditional partial-class estimate.

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.

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