IGCSE · Pearson Edexcel · 4MA1

Use density, mass and volume

Find density, mass and volume with compatible compound units, interpret liquid displacement and compare materials before combining explicitly additive volumes.

Section 4.4 · Measures

Pearson 4MA1 · Shared Geometry · Measures · Foundation 4.4G · inherited by Higher

Exam scope and assessment notes

Foundation and Higher share the seven published F:4.4A–G Measures rows. The Higher table says See Foundation Tier; there are no additional H4.4 rows. The chosen eight-lesson split covers instrument scales, time intervals, estimates, nearest-degree angle measurement, three-figure bearings, average speed, density and pressure. Eight lessons and thirty-two original worked/staged cases are StudyForA’s design, not a syllabus-prescribed count. Each structural source link requires the corresponding complete teaching record carrying that source ID, a worked explanation and all Guided, Independent and Transfer stages. F4.4F requires the complete speed lesson; broad F4.4G requires all three speed, density and pressure lessons, not a speed-only partial link. Source binding and substantive structure must belong to the same record; a separate untagged duplicate cannot supply missing evidence. These are structural links, not exhaustive subskill or specialist approval. Preserve the actual calibrated instrument ticks, pointer/meniscus positions, ruler endpoints, units and protractor alignment action. Read labelled scales rather than assuming CSS pixels are millimetres. Use only diagrams explicitly supplied as accurate for measurement; ordinary examination diagrams need not be to scale and must not be measured without instructions. Numerical reference readings, protractor openings and the measured bearing remain hidden before a calibrated learner attempt; revealing a reference is not proof of instrument operation. Keep ideal model values separate from approximate human readings and nearest-degree rounding; no universal automatic tolerance or measurement competence is established. Bearings start at the named origin and run clockwise from declared north, with parallel north assumptions stated and leading zeros retained. Distinguish reverse rays, interior route angles and forward heading changes; no magnetic or globe-navigation correction is inferred. Clock intervals retain am/pm, noon/midnight and day order. Average speed uses total route distance over the full named elapsed interval, including stated stops; displacement and instantaneous speed are different. Calculating time by division requires positive speed: at constant zero speed a positive distance is impossible, while zero distance alone does not determine elapsed time. Keep density mass/volume units compatible and volume positive; liquid displacement is final minus initial level. Combined volumes are added only under the stated no-void/no-volume-change assumption. The pressure formula is supplied in every pressure stage; do not infer that the density or average-speed formula is also supplied. Use a provided force in newtons, positive contact area, correct squared-area conversion and an explicitly limited average-pressure model. No real-world safety or route-feasibility certification follows. The source assessment context uses SI units and may require mathematical instruments. These lessons do not complete4.5 nearest-mm line drawing, formal shape/compass constructions or scale-drawing problems,5.2 transformations,4.8 bearing trigonometry,4.9/4.10 full mensuration or4.11 map/similarity work. No blanket formal-proof waiver is imported. Written practice is self-checked, not automatically marked; acknowledgements and attempts do not award marks or establish mastery. Specialist approval and formal assessment validation remain unestablished. Publisher navigation is separate evidence: literal F4.4D appears in Book2 Shape and Space6, map physical10/pages128–160, and Hub36 Bearings and Scale Drawings, co-tagged F4.5C/F4.11B; those co-tags confer no construction/map credit here. Literal F4.4G appears in Book1 Number5, map physical5/pages255–268, and Hub13 Compound Measures. A/B/C/E/F have no literal4.4 chapter or Hub tag in that map; do not invent one from nearby time or scale-drawing topics. Book1 public uncorrected-proof contents start Number5/Compound measures at256 and end its summary at269, conflicting with map255–268; preserve both references. Book2 chapter start128 and next start161 agree with its map range, but no explicit bearings contents heading locates every skill. Neither relevant chapter body is present in the public samples. Full final-book/Teaching Hub content and pagination alignment remain unverified; chapter links are not full-book or modular teaching credit. The current2026/2027 manual lists linear4MA1 F/H in November2026 and June2027; FR/HR are June2027 only, based on centre eligibility. Vietnam is on the regional-centre list; learner residence alone does not choose an option. The table points to page2 although the list is on physical/printed3. Modular4XMAF/4XMAH cash-ins and4WM units remain separate entries for international centres. This entry context does not create a2027 revision of the Issue2 specification.

Before you begin

Multiply and divide decimals and rearrange a quotient.
Read supplied liquid-volume levels and subtract them.
Use the supplied equivalences 1 mL=1 cm³ and 1 g/cm³=1000 kg/m³ without claiming a complete volume-conversion programme.

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

Quick readiness check

For a sample of mass m and volume V, which relationship gives density?

What you will learn

Use density=mass/volume and rearrange to find either mass or volume.
Convert between g/cm³ and kg/m³ and interpret a displaced-volume diagram.
Compare materials in common units and state the assumptions needed for a combined-volume calculation.

Choose a measures case

A uniform solid sample has mass 540 g and volume 200 cm³. Find its density in g/cm³ and kg/m³. Then find the mass of 350 cm³ of the same uniform material. Use 1 g/cm³=1000 kg/m³ and explain the unit conversion.

Read, draw and calculate before comparing. Every requested part and explanation needs your own attempt. The teal protractor and orange rays show your actual placements, including wrong ones. Comparisons do not award marks or establish practical instrument competence.

Whole-worksheet orientation Show explanatory worksheet grid

Coordinates use the original worksheet grid. The entire drawing and your tools rotate together; clockwise bearings retain their direction. Case or orientation changes clear all responses, placements and references. Named given points can be placed exactly; other drawing clicks use the nearest half-unit grid point and never snap to a solution.

Given panel 1

Mass (g): 540
Volume (cm³): 200
New sample volume (cm³): 350

Material density (g/cm³)

Material density (kg/m³)

Mass for the stated new volume (g)

Explain the reading, relationship, units and relevant limit

The board retains the last submitted valid placement. Editing a draft invalidates its submission without moving the last drawing; submit again before comparison.

Original StudyForA Measures exploration · Pearson4MA1 Foundation4.4A–G, inherited by Higher. Screen drawings are calibrated by labelled marks, not physical screen size. Pressure uses the given P=F/A model for average pressure on stated full contact faces. Tool bounds and fixed comparisons do not establish full practical construction4.5, wider mensuration, final-book/Teaching Hub alignment, specialist approval or formal assessment.

Density compares mass with volume

Density is mass divided by volume. If a uniform sample has mass m and volume V, write density=m/V. A larger sample of the same uniform material has a larger mass in the same ratio, while its density stays the same. Attach the compound unit: grams divided by cubic centimetres gives g/cm³, not simply grams.

In these ordinary material models, mass, volume and density are positive. The density calculated from an object’s total mass and external volume is its average or bulk density; it need not identify the density of a constituent material if the object contains cavities. Do not infer a substance’s identity from a single quotient. A zero volume gives an undefined division and is not a valid input to this model.

Find the missing mass or volume

Rearrange the relationship before substituting: mass=density×volume and volume=mass/density. Choose the form that has the requested quantity on its own. If a sample displaces liquid, find the displaced volume as final level minus initial level, using the supplied readings. The final liquid reading alone includes the original liquid and therefore is not the object’s volume.

For a displacement task, the assumptions matter: the object is fully submerged, liquid is not lost, no air is trapped and the object does not dissolve. In these tasks use 1 mL=1 cm³. The before-and-after cylinder levels are given measurements to interpret, not extra unknown quantities. Keep the initial and final levels visible in your working so the subtraction can be checked.

Convert before comparing or combining

To convert a density, change both mass and volume units. One gram is 0.001 kg, while 1 cm³ is 0.000001 m³, so 1 g/cm³=1000 kg/m³. Consequently a number in kg/m³ is divided by 1000 to express it in g/cm³. Comparing the numerical parts before conversion can reverse the intended conclusion.

For combined samples, add their masses and their stated volumes only when the question explicitly says that the volumes combine without change or voids. Then divide total mass by total volume. The simple average of two densities works for equal volumes under that additive-volume assumption, but not as a universal mixing rule. The relationship is taught here; no promise that a density formula will be supplied in an assessment is inferred from the separate pressure-formula note.

Sources for this lesson

Pause and explain

Which is equivalent to 2.3 g/cm³?

Put the idea to work

Worked example

Show the worked solution

Use density=mass/volume: 540÷200=2.7 g/cm³. This is a mass-per-volume value, not a new mass for the sample.
Convert both parts of the compound unit. Since 1 g/cm³=1000 kg/m³, 2.7 g/cm³=2700 kg/m³.
For the new volume use mass=density×volume with compatible g and cm³ units: 2.7×350=945 g.
The same uniform-material assumption keeps density fixed as volume changes. The larger 350 cm³ sample has a larger mass than the original 200 cm³ sample, which checks the direction.

Answer Density: 2.7 g/cm³=2700 kg/m³. Mass of 350 cm³: 945 g.

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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

A uniform sample has mass 448 g and volume 160 cm³. Find its density. For the same material, find the mass of 75 cm³ and the volume occupied by 140 g. Show a suitable form of the relationship for each requested quantity.

Give me a hint

First calculate density. Then multiply for a mass and divide for a volume.

Compare my reasoning

Calculate density=448÷160=2.8 g/cm³. Keep this common density for the same uniform material.
For a 75 cm³ sample, use mass=density×volume: 2.8×75=210 g. The cm³ units cancel, leaving the requested mass in grams.
For a 140 g sample, use volume=mass/density: 140÷2.8=50 cm³. Dividing by grams per cm³ recovers a volume, not another mass.
Check the last result by multiplying 2.8 g/cm³×50 cm³=140 g. The units and the inverse operations agree.

Density: 2.8 g/cm³. Mass of 75 cm³: 210 g. Volume of 140 g: 50 cm³.

Look for these in your work

Find and label the 2.8 g/cm³ density first.
Use multiplication for the new mass and division for the new volume.
Check the 50 cm³ result by recovering 140 g with compatible units.

2 · Independent

An object of mass 561.6 g is fully submerged without dissolving, trapped air or lost liquid. The labelled measuring-cylinder levels rise from 350 mL to 422 mL. Using 1 mL=1 cm³, independently find the object’s volume and density. Then find the volume occupied by 156 g of the same uniform material. Interpret both levels in the supplied before-and-after diagram.

Give me a hint

Subtract the two level readings before dividing mass by volume.

Compare my reasoning

The displaced liquid volume is 422−350=72 mL. Using the supplied equivalence, the object volume is 72 cm³.
Calculate density=561.6÷72=7.8 g/cm³. Do not divide by 422 cm³, because that includes the original liquid.
For 156 g of the same uniform material use volume=156÷7.8=20 cm³. Keep the density at 7.8 g/cm³ because the material model is unchanged.
Check by multiplying 7.8×20=156 g and retain the stated submersion assumptions. They justify treating the liquid increase as the object volume.

Object volume: 72 cm³. Density: 7.8 g/cm³. Volume for 156 g: 20 cm³.

Look for these in your work

Use the actual before-and-after level difference from the diagram.
Find density with the 72 cm³ volume and the g/cm³ unit.
Rearrange to obtain 20 cm³ and retain the physical assumptions.

3 · Transfer

Material A has density 2.5 g/cm³ and material B has density 2700 kg/m³. Compare the masses of 100 cm³ samples of each. If these two stated volumes then combine without voids or any volume change, find the combined mass, volume and average density in g/cm³. Explain the condition that makes the volume addition legitimate.

Give me a hint

Convert B to g/cm³ before comparing; add masses and explicitly additive volumes before finding the combined density.

Compare my reasoning

Convert 2700 kg/m³ to 2700÷1000=2.7 g/cm³. Thus B has the larger density in the same unit, rather than merely the larger original numerical label.
For 100 cm³ samples, A has mass 2.5×100=250 g and B has mass 2.7×100=270 g.
The combined mass is 250+270=520 g. The task expressly allows additive volumes, so the combined volume is 100+100=200 cm³.
Average density is 520÷200=2.6 g/cm³. It happens to be the mean of 2.5 and 2.7 here because the volumes are equal and additive; do not generalise that as a rule for arbitrary mixtures.

A: 250 g; B: 270 g. Combined mass: 520 g; volume: 200 cm³; average density: 2.6 g/cm³ under the stated additive-volume assumption.

Look for these in your work

Convert B to 2.7 g/cm³ and compare like units.
Find both sample masses before adding them.
Use the explicit no-void/no-volume-change condition and explain why this is not a universal averaging rule.

Common mistakes

Dividing by the final cylinder level instead of the change in liquid volume.
Changing only the numerator or only the denominator when converting a compound density unit.
Averaging two densities without the necessary volume assumptions, or claiming that a quotient identifies a material.

Recall without your notes

What can you explain now?

A solid is fully submerged and the liquid level changes from 80 mL to 105 mL. What volume enters its density calculation, using 1 mL=1 cm³?

Compare with the explanation

25 cm³.

Use the displaced volume, 105−80=25 mL=25 cm³. The final 105 mL includes the original liquid, so it is not the object’s volume.

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

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Make it stick

Recall density=mass/volume, mass=density×volume and volume=mass/density. State the displaced-volume subtraction and the factor between g/cm³ and kg/m³, including why both unit parts matter.

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.