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Pearson Edexcel International GCSE Mathematics B (4MB1) · single Higher tier · Pearson Edexcel ·

Inverse cube proportion and signed changes

An inverse-cube law combines an odd power with a denominator restriction. Both the original nonzero condition and the sign of an input change matter.

Pearson4MB1 · original variation-family teaching and practice · single Higher tier · Mathematics B 4G
Exam scope and assessment notes

Original eight-family variation expansion: eight lessons,32 original cases and every Worked,Guided,Independent and application stage for each enumerated direct/inverse linear,square,cube and principal-root form. Together with the original general variation introduction,all nine complete same-record teaching/worked/practice bundles are required for one structural4Glink. Calibrate from original observations using actual reversible coefficient-equation operations;retain signed coefficients,the complete natural and stated domain,exact rational/simple-root predictions,all distinct admitted recoveries and their original-law returns,changed input and baseline values,ratio availability and coefficient dimensions. Both learner working and separately entered final answers begin blank;neither repairs the other. Edits,removed rows,withdrawal,reset and source/session/stage/revision changes revoke previous comparisons and references. A complete wrong attempt can study its explanation without passing. These construction cases use identified nonzero rational coefficients and nonzero rational calibration bases;origin ambiguity and zero-coefficient distinctions are taught explicitly,while more general symbolic forms need review. Software bounds are not syllabus exclusions. Written formulas,derivations,whole-domain arguments,unit prose and empirical model validity need human review. All14Functions source rows have reviewed structural teaching/practice links;other Mathematics B source gaps remain explicit in the current course map. The wider Pearson curriculum remains incomplete. Automated controls and initial/completed rendering are checked;fresh mounted-browser and hosted four-stage interaction remain unverified. ISBN9780435044107 is metadata only;no full book body,chapter or Hub mapping is verified. Single Higher-tier linear4MB1,entry context2026/27only;no MathematicsA,Foundation,modular or later-series inheritance. No official marks,mastery,specialist approval or formal assessment readiness.

Before you begin

  • The original general variation-law introduction
  • Signed rational arithmetic,integer powers and the required real root
  • Original domain,interval endpoints and dependent/input units

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

Quick readiness check

Has the proportional family been stipulated before its coefficient is calibrated?

What you will learn

  • Derive the stipulated reciprocal-cube model from an admitted original observation using explicit equation operations.
  • Use the original natural and stated domain for exact predictions and complete real recoveries.
  • Check each recovered input in the original proportional basis and keep the coefficient dimensions.
  • Distinguish a valid numerical change ratio,an excluded original input change and an unavailable zero baseline.

Inverse cube: y = k/x³: Worked

This is an abstract real-variable law on its natural domain; no additional practical restriction is imposed. The stipulated law is inverse cube proportion with unknown coefficient k: y=k*(1/x^3). The observation x=2 gives y=8. Prediction inputs: 4, 0, -2. Recover original inputs for outputs 1, 0, -8. Multiply the original input 2 by -2. Evaluate the proportional basis at the observed input, write the substituted coefficient equation and show each reversible operation. Then complete every exact prediction, full recovery list, original-law return check and change-ratio decision before entering separate answers.

Stipulated family: Inverse cube: y = k/x³. Original observations: input 2 gives output 8.

Stated restriction: (-inf,inf). Retain the original denominator and principal-root restrictions too.

Predict outputs at 4, 0, -2. Recover all allowed inputs for outputs 1, 0, -8. Multiply original input 2 by -2. Input unit: none; output unit: none.

How to enter my mathematical working

Use k for the coefficient and * for multiplication. Begin calibration with the original substituted equation, then add each equation operation. Enter exact fractions, sqrt(...) or cbrt(...) rather than rounded values. Write sets as (-inf,inf), [0,2), or intervals joined by U. Use none for an empty set, unavailable value or absent unit. Add one finite recovery row per candidate; zero rows means no candidates.

Coefficient dimensions use the output unit multiplied by the input unit to your entered exponent. A change ratio needs two admitted inputs and a nonzero initial output. The working and separate-answer panels stay independent.

My actual working

My chosen law,coefficient and original real domain
My coefficient dimensions
My calibration using input 2 and output 8
My actual coefficient equation operations

0 entered equation rows. Add each row you can justify; an empty finite list claims no recoveries.

My prediction at original input 4
My prediction at original input 0
My prediction at original input -2
My recovery for output 1
My recovery for output 0
My recovery for output -8
My change from original input 2 by factor -2
My original model derivation and limitations

Current entries need recording.

My separate final answer

My separate final answer

My chosen law,coefficient and original real domain
My coefficient dimensions
My separate answer at original input 4
My separate answer at original input 0
My separate answer at original input -2
My separate recovery answer for output 1
My separate recovery answer for output 0
My separate recovery answer for output -8
My separate change-ratio answer
My independently entered formula and explanation

Current entries need recording.

Record your current original calibration,domain,predictions,recoveries and change working.

Scope of this self-check

All mathematical fields and equation/candidate rows begin blank. The task supplies only original observations,requested inputs/outputs,units,restrictions and a stipulated family.

The student’s actual coefficient-equation steps,original domain,predictions,complete recovery candidates and original-rule returns,change working and coefficient dimensions are checked separately from independent answers.

Edits,row removal,withdrawal,reset,stage changes and original-source/session/generation/revision changes withhold earlier comparisons and explained references. Removed dependent row contents are discarded.

A complete incorrect attempt can study its explained reference without passing. These fixed comparisons do not approve written formulas,derivations,whole-domain arguments,unit prose,empirical validity,marks,mastery,specialist review or formal assessment readiness.

Use the original reciprocal cube

Write y=k/x^3 with x different from zero. Because a cube keeps its input sign, opposite nonzero inputs give opposite outputs for the same nonzero k. This differs from an inverse-square law.

A positive input is not part of the abstract definition unless stated. A practical calibration may impose a positive operating interval, which must then be retained at every prediction and recovery.

Calibrate without dropping signs

For observation a,b, write b=k/a^3 and multiply both sides by the signed nonzero a^3. The result is k=ba^3. A positive observed output at a negative input can therefore identify a negative coefficient.

A zero observation input is excluded by the original denominator. Algebraic rewriting to x^3y=k does not authorise using that excluded original input.

Take a single real cube-root recovery

For nonzero k, a zero requested output has no finite recovery. For nonzero output b, solve x=cbrt(k/b). Keep the sign of the quotient and retain the single real cube root rather than a pair of square-root signs.

Check the recovered value in the original reciprocal cube and apply its original interval. The invented dial calibration allows only positive settings up to six; negative real algebraic candidates are rejected by that model, not called non-real.

Use an inverse-cube ratio and its dimensions

For an allowed nonzero input multiplier a, the output multiplier is 1/a^3. A negative multiplier gives a negative output ratio on the unrestricted abstract domain. It may nevertheless be inadmissible in a positive practical interval.

The coefficient has output-unit times input-unit cubed dimensions. A zero input multiplier reaches an original denominator hole and supplies no model prediction or numerical ratio. These exercises do not establish an empirical device law.

Sources for this lesson

Pause and explain

Can an algebraic rearrangement or a correct separate answer replace the original-domain and actual-work checks?

Put the idea to work

Worked example

This is an abstract real-variable law on its natural domain; no additional practical restriction is imposed. The stipulated law is inverse cube proportion with unknown coefficient k: y=k*(1/x^3). The observation x=2 gives y=8. Prediction inputs: 4, 0, -2. Recover original inputs for outputs 1, 0, -8. Multiply the original input 2 by -2. Evaluate the proportional basis at the observed input, write the substituted coefficient equation and show each reversible operation. Then complete every exact prediction, full recovery list, original-law return check and change-ratio decision before entering separate answers.

Show the worked solution
  1. Use y=k*(1/x^3) on (-inf,0) U (0,inf). The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions.
  2. At x=2, the proportional basis is 1/8. The observation gives 8=k*(1/8). Divide both sides by the nonzero basis 1/8 to obtain k=64.
  3. Thus y=(64)*(1/x^3) on that original domain. Changing the original domain would change the task.
  4. Prediction at x=4: y=1.
  5. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it.
  6. Prediction at x=-2: y=-8.
  7. For y=1, real algebraic candidates are 4; the complete original-domain recovery set is 4. Check each retained candidate in the original rule.
  8. Original-law return at x=4: its proportional basis is 1/64, so y=(64)*(1/64)=1. This input satisfies every original natural and stated restriction.
  9. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule.
  10. For y=-8, real algebraic candidates are -2; the complete original-domain recovery set is -2. Check each retained candidate in the original rule.
  11. Original-law return at x=-2: its proportional basis is -1/8, so y=(64)*(-1/8)=-8. This input satisfies every original natural and stated restriction.
  12. Changing x from 2 to -4 changes the original outputs from 8 to -1. The exact numerical output ratio is -1/8.
  13. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Both variables are dimensionless in this abstract task.

Answer Use y=k*(1/x^3) on (-inf,0) U (0,inf). The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions. At x=2, the proportional basis is 1/8. The observation gives 8=k*(1/8). Divide both sides by the nonzero basis 1/8 to obtain k=64. Thus y=(64)*(1/x^3) on that original domain. Changing the original domain would change the task. Prediction at x=4: y=1. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it. Prediction at x=-2: y=-8. For y=1, real algebraic candidates are 4; the complete original-domain recovery set is 4. Check each retained candidate in the original rule. Original-law return at x=4: its proportional basis is 1/64, so y=(64)*(1/64)=1. This input satisfies every original natural and stated restriction. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule. For y=-8, real algebraic candidates are -2; the complete original-domain recovery set is -2. Check each retained candidate in the original rule. Original-law return at x=-2: its proportional basis is -1/8, so y=(64)*(-1/8)=-8. This input satisfies every original natural and stated restriction. Changing x from 2 to -4 changes the original outputs from 8 to -1. The exact numerical output ratio is -1/8. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Both variables are dimensionless in this abstract task. The law, coefficient, original domain, exact predictions, complete recoveries, input change and coefficient dimensions describe different parts of this same original model. Written derivation,whole-domain reasoning,unit prose and empirical validity require human review. No marks,mastery,specialist or formal assessment approval is inferred.

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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

This is an abstract real-variable law on its natural domain; no additional practical restriction is imposed. The stipulated law is inverse cube proportion with unknown coefficient k: y=k*(1/x^3). The observation x=-3 gives y=2. Prediction inputs: 3, 0, -3/2. Recover original inputs for outputs -2, 0, 16. Multiply the original input -3 by -1/2. First state the original real input set. Compute your observed basis and show the coefficient equation operations. For each recovery, check every sign or root against the original rule. Complete the change and coefficient-unit fields yourself; no numerical answer is supplied.

Give me a hint

Use the family’s stated proportional basis. Evaluate it at the original observation,show each reversible coefficient operation,retain natural and stated domains,then check every recovery and the numerical ratio conditions.

Compare my reasoning
  1. Use y=k*(1/x^3) on (-inf,0) U (0,inf). The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions.
  2. At x=-3, the proportional basis is -1/27. The observation gives 2=k*(-1/27). Divide both sides by the nonzero basis -1/27 to obtain k=-54.
  3. Thus y=(-54)*(1/x^3) on that original domain. Changing the original domain would change the task.
  4. Prediction at x=3: y=-2.
  5. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it.
  6. Prediction at x=-3/2: y=16.
  7. For y=-2, real algebraic candidates are 3; the complete original-domain recovery set is 3. Check each retained candidate in the original rule.
  8. Original-law return at x=3: its proportional basis is 1/27, so y=(-54)*(1/27)=-2. This input satisfies every original natural and stated restriction.
  9. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule.
  10. For y=16, real algebraic candidates are -3/2; the complete original-domain recovery set is -3/2. Check each retained candidate in the original rule.
  11. Original-law return at x=-3/2: its proportional basis is -8/27, so y=(-54)*(-8/27)=16. This input satisfies every original natural and stated restriction.
  12. Changing x from -3 to 3/2 changes the original outputs from 2 to -16. The exact numerical output ratio is -8.
  13. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Both variables are dimensionless in this abstract task.

Use y=k*(1/x^3) on (-inf,0) U (0,inf). The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions. At x=-3, the proportional basis is -1/27. The observation gives 2=k*(-1/27). Divide both sides by the nonzero basis -1/27 to obtain k=-54. Thus y=(-54)*(1/x^3) on that original domain. Changing the original domain would change the task. Prediction at x=3: y=-2. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it. Prediction at x=-3/2: y=16. For y=-2, real algebraic candidates are 3; the complete original-domain recovery set is 3. Check each retained candidate in the original rule. Original-law return at x=3: its proportional basis is 1/27, so y=(-54)*(1/27)=-2. This input satisfies every original natural and stated restriction. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule. For y=16, real algebraic candidates are -3/2; the complete original-domain recovery set is -3/2. Check each retained candidate in the original rule. Original-law return at x=-3/2: its proportional basis is -8/27, so y=(-54)*(-8/27)=16. This input satisfies every original natural and stated restriction. Changing x from -3 to 3/2 changes the original outputs from 2 to -16. The exact numerical output ratio is -8. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Both variables are dimensionless in this abstract task. The law, coefficient, original domain, exact predictions, complete recoveries, input change and coefficient dimensions describe different parts of this same original model. Written derivation,whole-domain reasoning,unit prose and empirical validity require human review. No marks,mastery,specialist or formal assessment approval is inferred.

Look for these in your work

  • My original observation,basis and actual coefficient-equation operations are explicit.
  • I retain the complete original domain and every permitted exact recovery,with return checks.
  • My change uses both original inputs and distinguishes an excluded journey from a zero baseline.
  • My coefficient dimensions use the original input and output units.
  • My independent final values cannot repair incorrect own working;written arguments and model validity remain for human review.
2 · Independent

This is an abstract real-variable law on its natural domain; no additional practical restriction is imposed. The stipulated law is inverse cube proportion with unknown coefficient k: y=k*(1/x^3). The observation x=3 gives y=1/8. Prediction inputs: 3, 0, -3. Recover original inputs for outputs 27/16, 0, -1/8. Multiply the original input 3 by -2/3. Derive the model and original domain, then calculate every named exact prediction, complete recovery set and change ratio with its actual working. Enter the final formula, values, domains and coefficient dimensions independently.

Give me a hint

Use the family’s stated proportional basis. Evaluate it at the original observation,show each reversible coefficient operation,retain natural and stated domains,then check every recovery and the numerical ratio conditions.

Compare my reasoning
  1. Use y=k*(1/x^3) on (-inf,0) U (0,inf). The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions.
  2. At x=3, the proportional basis is 1/27. The observation gives 1/8=k*(1/27). Divide both sides by the nonzero basis 1/27 to obtain k=27/8.
  3. Thus y=(27/8)*(1/x^3) on that original domain. Changing the original domain would change the task.
  4. Prediction at x=3: y=1/8.
  5. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it.
  6. Prediction at x=-3: y=-1/8.
  7. For y=27/16, real algebraic candidates are (1)*cbrt(2); the complete original-domain recovery set is (1)*cbrt(2). Check each retained candidate in the original rule.
  8. Original-law return at x=(1)*cbrt(2): its proportional basis is 1/2, so y=(27/8)*(1/2)=27/16. This input satisfies every original natural and stated restriction.
  9. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule.
  10. For y=-1/8, real algebraic candidates are -3; the complete original-domain recovery set is -3. Check each retained candidate in the original rule.
  11. Original-law return at x=-3: its proportional basis is -1/27, so y=(27/8)*(-1/27)=-1/8. This input satisfies every original natural and stated restriction.
  12. Changing x from 3 to -2 changes the original outputs from 1/8 to -27/64. The exact numerical output ratio is -27/8.
  13. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Both variables are dimensionless in this abstract task.

Use y=k*(1/x^3) on (-inf,0) U (0,inf). The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions. At x=3, the proportional basis is 1/27. The observation gives 1/8=k*(1/27). Divide both sides by the nonzero basis 1/27 to obtain k=27/8. Thus y=(27/8)*(1/x^3) on that original domain. Changing the original domain would change the task. Prediction at x=3: y=1/8. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it. Prediction at x=-3: y=-1/8. For y=27/16, real algebraic candidates are (1)*cbrt(2); the complete original-domain recovery set is (1)*cbrt(2). Check each retained candidate in the original rule. Original-law return at x=(1)*cbrt(2): its proportional basis is 1/2, so y=(27/8)*(1/2)=27/16. This input satisfies every original natural and stated restriction. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule. For y=-1/8, real algebraic candidates are -3; the complete original-domain recovery set is -3. Check each retained candidate in the original rule. Original-law return at x=-3: its proportional basis is -1/27, so y=(27/8)*(-1/27)=-1/8. This input satisfies every original natural and stated restriction. Changing x from 3 to -2 changes the original outputs from 1/8 to -27/64. The exact numerical output ratio is -27/8. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Both variables are dimensionless in this abstract task. The law, coefficient, original domain, exact predictions, complete recoveries, input change and coefficient dimensions describe different parts of this same original model. Written derivation,whole-domain reasoning,unit prose and empirical validity require human review. No marks,mastery,specialist or formal assessment approval is inferred.

Look for these in your work

  • My original observation,basis and actual coefficient-equation operations are explicit.
  • I retain the complete original domain and every permitted exact recovery,with return checks.
  • My change uses both original inputs and distinguishes an excluded journey from a zero baseline.
  • My coefficient dimensions use the original input and output units.
  • My independent final values cannot repair incorrect own working;written arguments and model validity remain for human review.
3 · Transfer

An invented calibration stipulates an inverse-cube response at positive dial settings up to six units. Do not extrapolate it to negative or zero settings or claim it is a validated device law. The stipulated law is inverse cube proportion with unknown coefficient k: y=k*(1/x^3). The observation x=3 gives y=8. Prediction inputs: 2, 0, -3. Recover original inputs for outputs 27, 0, -1. Multiply the original input 3 by 2. Retain the stipulated operating interval as well as the natural reciprocal or principal-root conditions. Derive the coefficient with units and distinguish excluded model inputs from non-real expressions. Complete original predictions, all permitted recoveries and the stated change; explain the model limits.

Give me a hint

Use the family’s stated proportional basis. Evaluate it at the original observation,show each reversible coefficient operation,retain natural and stated domains,then check every recovery and the numerical ratio conditions.

Compare my reasoning
  1. Use y=k*(1/x^3) on (0,6]. The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions.
  2. At x=3, the proportional basis is 1/27. The observation gives 8=k*(1/27). Divide both sides by the nonzero basis 1/27 to obtain k=216.
  3. Thus y=(216)*(1/x^3) on that original domain. Changing the original domain would change the task.
  4. Prediction at x=2: y=27.
  5. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it.
  6. Prediction at x=-3: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it.
  7. For y=27, real algebraic candidates are 2; the complete original-domain recovery set is 2. Check each retained candidate in the original rule.
  8. Original-law return at x=2: its proportional basis is 1/8, so y=(216)*(1/8)=27. This input satisfies every original natural and stated restriction.
  9. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule.
  10. For y=-1, real algebraic candidates are -6; the complete original-domain recovery set is empty. Check each retained candidate in the original rule.
  11. Changing x from 3 to 6 changes the original outputs from 8 to 1. The exact numerical output ratio is 1/8.
  12. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Original units are response units and dial units.
  13. The operating interval is a model assumption. A correct calibration or numerical fit does not establish a real device,price,staffing or experimental law. Keep the physical interpretation and any continuous-versus-discrete assumption explicit.

Use y=k*(1/x^3) on (0,6]. The original domain includes the stipulated restriction as well as natural reciprocal or principal-root conditions. At x=3, the proportional basis is 1/27. The observation gives 8=k*(1/27). Divide both sides by the nonzero basis 1/27 to obtain k=216. Thus y=(216)*(1/x^3) on that original domain. Changing the original domain would change the task. Prediction at x=2: y=27. Prediction at x=0: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it. Prediction at x=-3: no value is assigned by the original allowed model. This input is excluded by a natural reciprocal/principal-root condition or the stated original model restriction;an algebraic extension does not authorise it. For y=27, real algebraic candidates are 2; the complete original-domain recovery set is 2. Check each retained candidate in the original rule. Original-law return at x=2: its proportional basis is 1/8, so y=(216)*(1/8)=27. This input satisfies every original natural and stated restriction. For y=0, real algebraic candidates are none; the complete original-domain recovery set is empty. Check each retained candidate in the original rule. For y=-1, real algebraic candidates are -6; the complete original-domain recovery set is empty. Check each retained candidate in the original rule. Changing x from 3 to 6 changes the original outputs from 8 to 1. The exact numerical output ratio is 1/8. The coefficient dimension is the output unit multiplied by the input unit to exponent 3. Original units are response units and dial units. The operating interval is a model assumption. A correct calibration or numerical fit does not establish a real device,price,staffing or experimental law. Keep the physical interpretation and any continuous-versus-discrete assumption explicit. The law, coefficient, original domain, exact predictions, complete recoveries, input change and coefficient dimensions describe different parts of this same original model. Written derivation,whole-domain reasoning,unit prose and empirical validity require human review. No marks,mastery,specialist or formal assessment approval is inferred.

Look for these in your work

  • My original observation,basis and actual coefficient-equation operations are explicit.
  • I retain the complete original domain and every permitted exact recovery,with return checks.
  • My change uses both original inputs and distinguishes an excluded journey from a zero baseline.
  • My coefficient dimensions use the original input and output units.
  • My independent final values cannot repair incorrect own working;written arguments and model validity remain for human review.

Common mistakes

  • Using an always-positive inverse-square factor for an odd-power law.
  • Recovering input zero for requested output zero.
  • Dropping the sign of a cube root.
  • Assuming abstract negative settings are also admitted by a positive calibration interval.
Recall without your notes

What can you explain now?

Can an algebraic rearrangement or a correct separate answer replace the original-domain and actual-work checks?

Compare with the explanation

No.

Retain the original law and every stated restriction;independent answers do not supply the learner’s missing or incorrect construction.

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Make it stick

State this family’s original input domain,derive its calibration equation,explain its complete recovery rule and give its allowed change factor. Keep actual units and numerical-ratio conditions separate. Fixed checks do not establish marks,mastery,empirical validity,specialist review or formal assessment readiness.

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 B

Pearson2026/2027 International GCSE entry manual Pearson Mathematics B official qualification
Course scope and assessment

Partial B-specific teaching: Number1A–K,Sets2A–I,Algebra3A–L,all14Functions source rows4A–N and Matrices5A–G have reviewed original teaching/worked/practice bundles. The27Functions lessons combine existing substantive theory and learner constructions with eight new variation-family lessons and the corrected composition counterexample. Twelve Matrices lessons supply seven further structural source links,with all six named transformation families required jointly for5F. Geometry 6F, 6G, 6H, 6I, 6J, 6K and 6L have reviewed joint teaching/practice bundles. Seven other Geometry rows and 31 rows in the four later topics remain unlinked. Those later topics remain unstarted. A structural source link does not establish exhaustive subskill,practical,specialist or formal assessment approval. No MathematicsA,Foundation or modular inheritance. Single Higher-tier linear4MB1; no Foundation or additional-Higher inheritance. These seven original Number lessons address1A–F; 1C needs both integer/root and fractional/negative-index strands. These seven lessons make no claim for later Number strands or other topics; consult their separate teaching map and actual lessons. Work exactly with original signed values and grouping; retain nonzero-denominator and real-root domains, principal even roots and signed odd roots. Surd rationalisation keeps the original denominator restriction even when a conjugate is used. Number-set membership uses the stated natural-number convention; irrationality proofs are outside1F. Model feedback is for self-checking, with human review of reasoning; no exam marks, mastery, specialist or formal-assessment approval. Both linear papers may draw on any topic. The2026/27 manual confirms standard November2026/June2027 and regional June2027 subject to centre location; the2016specification's January/June wording is historical. ISBN9780435044107 is verified metadata only; no Bbook chapter/body/Hub map or modular/onscreen route is verified. No4MA1 book or teaching credit is imported. Single Higher-tier linear4MB1. These eleven original Number1G–K lessons use exact given units, currency directions and proportional models; units and physical/count domains are explicit. Decimal-place/significant-figure answers preserve requested precision, and signed midpoint conventions and bound endpoint inclusion are stated. Source1G metric/SI and money needs all three measure/time/currency strands; 1H representations, ratios/proportion and percentages need all three strands. Bounds need both rounded-input and derived-quantity lessons, and standard form needs representation/comparison plus calculation/context lessons. This Number set does not complete any later Sets, Algebra, Functions, Matrices, Geometry, Mensuration, Vectors/transformations, Trigonometry or Statistics/probability topic. Structural links are separate from exhaustive subskill, human reasoning, mastery and formal-assessment approval. Original self-check practice only; no official marks are awarded. The2026/27 manual confirms standard November2026/June2027 and regional June2027 subject to centre location; no later examination-year availability is inferred. ISBN9780435044107 is metadata only: no verified B book-body/chapter/Hub mapping, no Mathematics A book inheritance and no B Foundation/modular/onscreen route. Single Higher-tier linear4MB1. These eight original Sets lessons cover membership and descriptions,universe-relative complement,union/intersection,inclusion,logical regions,two/three-set counts and feasibility. A source link requires its complete reviewed lesson bundle; number-set recognition1F is prerequisite context and gains no new link. Equality-allowing inclusion and proper subset are stated explicitly where the printed subset glyph is ambiguous. Missing universes,inconsistent counts and insufficient information are not replaced with invented diagrams or values. Two/three sets and the particular logical identities are chosen breadth,not a printed maximum or extra source row. Number1F’s irrationality-proof note is not a blanket waiver of Sets arguments and counterexamples. Your actual symbolic/Venn work and reasoning need human review; self-checks award no official marks or mastery/assessment approval. Standard November2026/June2027 and regional June2027 depend on the2026/27 manual and centre location; no later series is inferred. ISBN9780435044107 is B product/access metadata only; no verified B Sets chapter/body/Hub map or MathematicsA book inheritance. No Foundation,modular or onscreen B route is evidenced. Single Higher-tier linear4MB1. These eight original Algebra lessons address operations and integer/fractional indices with real domains,formula construction/substitution and subject changes,factorisation,cubic factor theorem including ax±b,cubic division and four operations with simple algebraic fractions. Each structural row needs its whole reviewed lesson bundle. Original denominator exclusions,principal square roots/branches and declared units remain part of the learner work. The finite symbolic checker accepts bounded explicit syntax,not every equivalent expression or explanation; unmatched alternative radical cancellation or general symbolic equivalence needs human review. A self-check or nonblank explanation does not award official marks or certify exhaustive subskills,reasoning,mastery,specialist or formal assessment readiness. Algebra3G–L and all later topics remain unmapped by this batch; Number and Sets links are separate. Standard November2026/June2027 and regional June2027 depend on the2026/27 manual and centre location; no later series is inferred. ISBN9780435044107 is B product/access metadata only; no verified Algebra body/chapter/Hub map or MathematicsA book inheritance. No Foundation,modular or onscreen B route is evidenced. Single Higher-tier linear4MB1. Five original equation lessons jointly link3G only when every teaching,worked,Guided,Independent and application strand remains intact: linear equations and contexts,quadratic factorisation,completing the square AND the quadratic formula,actual graphical quadratic construction,and cubic equations. Graph work requires learner-entered table values,labelled scales,actual points and curve strokes,zero-axis marks and readings. A generated ideal curve or algebra-only solution cannot substitute for that construction. Exact roots and approximate graphical readings are separate. Bounded symbolic and sampled geometry checks do not approve all equivalent methods,whole-curve smoothness,global reasoning,reading precision,mastery or official marks. Written explanations need human review; specialist and formal assessment approval remain pending. Algebra3H–L and all later topics remain unmapped by this batch; previous3A–F,Number and Sets are separate bundles. Standard November2026/June2027 and regional June2027 follow the2026/27 manual and centre location; no later entries are inferred. ISBN9780435044107 is metadata only; no verified Algebra body/chapter/Hub map or MathematicsA book inheritance. No Foundation,modular or onscreen B route is evidenced. Single Higher-tier linear4MB1. Both original3H lessons must remain complete: algebraic elimination/substitution with exact original-equation checks AND actual graphical simultaneous-equation construction on common labelled scales. Each graphical line needs the learner own calculated table,independently plotted points,two chosen retained anchors and an explicit straightedge-extension action. No generated ideal line or table substitutes for this construction. Vertical lines,parallel distinct lines,coincident lines,cropping,nonnegative context prices,exact check pairs and approximate readings remain separate. Finite rational-symbolic and geometric checks do not approve all alternative methods,written explanations,reading precision,physical instruments,mastery or official marks; specialist and formal assessment approval remain pending. Algebra3I–L and later topics remain unmapped by this batch. Entry context is2026/27 only,with regional options governed by centre location. ISBN9780435044107 is metadata only; no verified book-body/chapter/Hub map or MathematicsA inheritance. No Foundation,modular or onscreen B route is evidenced. Single Higher-tier linear4MB1. Original3I teaching includes substitution of the actual line into the quadratic,complete reduced equations,formula/factorisation or negative-discriminant reasoning,and separate partner recovery with checks in both originals. All four stages must remain available; Independent includes two distinct systems,no-real pairs and zero/fractional coordinates; Application explains one repeated contact. The finite exact rational polynomial checker requires a genuine quadratic after substitution and unrestricted real coordinates. Task-specific domains,lower-degree or line-component reductions and other methods require separate review. A supporting sketch does not replace algebra or certify graphical precision. Written reasoning,specialist and formal assessment approval remain pending; no official marks or mastery. Algebra3J–L and all later topics remain unmapped by this batch. Entry context2026/27 only. ISBN9780435044107 is metadata only; no book-body/chapter/Hub coverage or MathematicsA/Foundation/modular/onscreenB inheritance. Single Higher-tier linear4MB1,Algebra3J. Two original lessons jointly cover linear inequalities on number lines and simultaneous straight-line regions in two dimensions. Every Worked,Guided,Independent and Application construction is separate from its final answer. Negative scaling reverses order; open/closed endpoints and solid/dashed boundaries retain strictness. A finite integer list never replaces the real solution set. Closure corners may be excluded; unbounded rays continue beyond the displayed window. Empty planar claims use nonnegative-multiplier contradictions; the inclusive nonnegative-sum control is a single point. The checker uses bounded exact rational first-degree systems; the SVG is numerical display only,not approved graphical precision. No optimization or linear-programming coverage. Reasoning,specialist review and formal assessment approval remain pending; no official marks or mastery. Algebra3K–L and all later topics remain unmapped by this batch. Entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book-body/chapter/Hub mapping or MathematicsA/Foundation/modular/onscreenB inheritance. Single Higher-tier linear4MB1,Algebra3K. One original lesson constructs quadratic-inequality sign charts and actual number-line unions across Worked,Guided,Independent and Application stages,with eleven separate constructions. Show the polynomial moved to zero,descending rational coefficients,discriminant,distinct exact roots,actual rational test substitutions,selected open sign intervals and zero membership. Include negative leading coefficients,repeated-root empty/singleton/punctured/all-real cases,exact irrational boundaries and no-real-root controls. Each final answer is entered separately. The bounded checker uses exact algebra for rational plus one quadratic radical; numeric SVG positions do not certify graphical precision. Nonperfect radical numerator times denominator is bounded by10^12; input rational coefficients by10^30; internal arithmetic by4096bits. Unsupported forms and unreadably close displayed points are rejected rather than rounded or merged. no official marks or mastery. Algebra3L and all later topics remain unmapped by this batch. Entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book-body/chapter/Hub mapping or MathematicsA/Foundation/modular/onscreenB inheritance. Single Higher-tier linear4MB1,Algebra3L. Two original lessons cover constant differences and familiar common integer sequences through all eight Worked,Guided,Independent and Application stage definitions:thirteen actual sequence tasks and two explicit ambiguity proofs. Enter successive differences,exact signed/fractional continuations,actual powers,triangular seed sums,updated predecessor pairs and candidate-by-candidate prime certificates where relevant. Keep a separate independent final answer. A familiar finite prefix does not force a unique continuation:consistent declared alternatives are reported for discussion rather than automatically marked wrong. Named families are original examples under the source note,not an exhaustive list prescribed by Pearson. No compulsory nth-term or arithmetic-series sum is inferred. Exact rational input bounds10^30,prefixes2–16terms,continuations1–16terms,prime candidates2–10000 with at most128literal certificates,triangular seed0–64,power exponents0–24. Unsupported work is explicitly rejected. no official marks or mastery. All seven topics after Algebra remain unmapped by this batch. Entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book-body/chapter/Hub mapping or MathematicsA/Foundation/modular/onscreenB inheritance. This first Functions lesson teaches finite supplied mappings through four stages and six original tasks. Draw your own arrows,record distinct images for every original input,and give a separate final classification and range. Missing and multiple images,valid many-to-one functions,unused targets and explicit domain limits are distinguished. This complete original finite-mapping record supports4B and contributes to the reviewed4A/4C/4Djoint bundles:All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. Variable evaluation,real domains/ranges,composition and inverses have separate reviewed original lessons; this finite record does not replace them. Exact rational sets are bounded to16distinct elements and64entered arrows; unsupported work is rejected. Written reasoning requires human review; specialist review and formal assessment remain pending; no official marks or mastery. Single Higher-tier linear4MB1,entry context2026/27 only. ISBN9780435044107 is metadata only; no book-body,chapter or Hub verification or MathematicsA/Foundation/modular inheritance. Original polynomial function notation and deliberate substitution, with four stages and nine actual term-table tasks. Enter the whole input, domain decision, coefficients, powers, products, sum and later output changes; record a separate final answer. A complete incorrect attempt may study the explained reference but is not accepted by fixed comparisons. Exact rational polynomials are bounded to degree three and numerical or linear-in-a inputs; original domain holes are not cancelled away. The practical finite input set is retained even when arithmetic elsewhere is possible. This is introductory4C/4Dteaching, not a complete real-domain/range,composition or inverse bundle:All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. Written reasoning needs human review. No official marks,mastery,specialist review approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27 only. ISBN9780435044107 is metadata only; no book-body,chapter or Hub verification or MathematicsA/Foundation/modular inheritance. Original selected real-domain/range teaching,with four stages,eight tasks and fourteen actual branch tables. Keep original denominator/radical conditions,stated restrictions,each branch image,attained endpoint input witnesses,units and separately entered final sets/input decisions. Seven bounded rule families and exact rational interval/point endpoints only; arbitrary expressions and irrational image endpoints are explicitly unsupported,not declared non-real. A complete incorrect attempt may study the explained reference but does not pass fixed comparisons. This is selected4Dteaching,not exhaustive Functions coverage:All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. Composition,inverses and wider Functions have separate reviewed original teaching; these bounded calculations do not establish formal approval. Written explanations need human review. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book body,chapter or Hub mapping and no MathematicsA/Foundation/modular inheritance. Continuity questions are outside the cited4Drow. Original selected composition teaching,with four stages,eight tasks and24two-step input traces. Keep the rightmost-first order,both original domains,the outer-domain preimage,actual intermediate and final images,units and separately entered final sets/input decisions. Seven bounded rule families and exact rational interval/point endpoints only; arbitrary expressions and irrational image endpoints are explicitly unsupported,not declared non-real. A complete incorrect attempt may study the explained reference but does not pass fixed comparisons. This is selected4Eteaching,not exhaustive Functions coverage:All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. General symbolic alternatives beyond this bounded self-check need human review; separate original inverse and other Functions lessons remain required. Written explanations need human review. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book body,chapter or Hub mapping and no MathematicsA/Foundation/modular inheritance. The4Estructural link requires this complete original teaching/worked/practice record and its corrected counterexample; it does not approve all symbolic alternatives. Original selected inverse teaching,with four stages,eight tasks,nine reversed branches and27 named inverse-input checks. Retain the original domain,actual image,stated target,one-to-one evidence,original holes,branch signs,return checks and swapped units. A larger target is not automatically the original image. An inverse undoes the mapping; it is not the reciprocal. Endpoint sets remain exact rational intervals. Selected simple square-root inverse values are compared exactly against the original branch and returned through the original equation; general symbolic forms and arbitrary inverse solving remain outside this bounded self-check. A squared return alone cannot choose an inverse side or change the original domain. A complete incorrect attempt may study the explained reference without passing fixed comparisons. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map; general symbolic inverse alternatives need human review and the remaining Pearson curriculum is incomplete. Written derivations,reversed equations and whole-domain proofs need human review. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book body,chapter or Hub mapping and no MathematicsA,Foundation or modular inheritance Original selected variation teaching with four stages,eight tasks,26named prediction traces and18recovery targets. Calibrate the eight listed families from retained original observations and actual equation operations. Keep natural and stated input restrictions,all real recoveries,original-law return checks,change factors and coefficient dimensions. Exact irrational values remain defined and must be entered symbolically; selected equivalent square/cube-root values are compared exactly. More general symbolic coefficients/forms need human review. A complete incorrect attempt may study the explained reference without passing fixed comparisons. Written formulas,derivations,whole-domain arguments,unit prose and empirical model validity remain unreviewed. Finite observations do not prove a physical law. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map; wider Functions and remaining Pearson teaching/practice are incomplete. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27 only. ISBN9780435044107 is metadata only; no verified book body,chapter or Hub mapping and no MathematicsA,Foundation or modular inheritance. Original eight-family variation expansion: eight lessons,32 original cases and every Worked,Guided,Independent and application stage for each enumerated direct/inverse linear,square,cube and principal-root form. Together with the original general variation introduction,all nine complete same-record teaching/worked/practice bundles are required for one structural4Glink. Calibrate from original observations using actual reversible coefficient-equation operations; retain signed coefficients,the complete natural and stated domain,exact rational/simple-root predictions,all distinct admitted recoveries and their original-law returns,changed input and baseline values,ratio availability and coefficient dimensions. Both learner working and separately entered final answers begin blank; neither repairs the other. Edits,removed rows,withdrawal,reset and source/session/stage/revision changes revoke previous comparisons and references. A complete wrong attempt can study its explanation without passing. These construction cases use identified nonzero rational coefficients and nonzero rational calibration bases; origin ambiguity and zero-coefficient distinctions are taught explicitly,while more general symbolic forms need review. Software bounds are not syllabus exclusions. Written formulas,derivations,whole-domain arguments,unit prose and empirical model validity need human review. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. The wider Pearson curriculum remains incomplete. Automated controls and initial/completed rendering are checked; fresh mounted-browser and hosted four-stage interaction remain unverified. ISBN9780435044107 is metadata only; no full book body,chapter or Hub mapping is verified. Single Higher-tier linear4MB1,entry context2026/27only; no MathematicsA,Foundation,modular or later-series inheritance. No official marks,mastery,specialist approval or formal assessment readiness. Original selected rational Cartesian coordinate teaching with four stages and twelve tasks. Read and construct ordered pairs using the original separate axis scales,signed division counts and exact multiplication. Keep fractional between-grid coordinates,all four quadrants,both axes and the origin. A plotting window is not the whole plane:an off-window point remains real and is not clamped to an edge. Supplied closed time/temperature ranges govern the hypothetical context separately. Actual construction does not fill the independent answer. A complete incorrect attempt may study the explained reference without passing fixed comparisons. Written reasoning,broader plotting judgement and practical model validity remain for human review. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map; the4Hstructural link requires this complete original teaching,actual point construction and all four explained stages. The remaining Pearson curriculum is incomplete; broader symbolic and practical judgements still need review. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27only. ISBN9780435044107is metadata only; no verified book body,chapter/Hub mapping or MathematicsA/Foundation/modular inheritance. Original selected rational straight-line teaching with four stages and eight tasks. Build lines from learner-chosen anchors using their original independent axis scales. Any distinct admitted rational anchors are accepted, including reversed orders and off-window points. Actual coordinate rise/run, not ratios of picture lengths, determines gradient. Retain positive, negative, fractional and zero gradients, a rearranged equation, and vertical/coincident contrasts. Actual construction never fills the separate gradient/intercept answer. Rate units and intercept units differ. Extending a mathematical line does not extend a context domain; a complete-line intercept outside the domain is not an admitted initial observation. A complete wrong attempt may study the explained reference without passing fixed comparisons. Written reasoning and model validity need human review. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map; the4Ilink requires this complete line-construction record and4Jalso requires all three original sparse-graph strands. The remaining Pearson curriculum is incomplete; broader symbolic and practical judgements still need review. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27only. ISBN9780435044107is metadata only; no verified book body,chapter/Hub mapping or MathematicsA/Foundation/modular inheritance. Original selected sparse numerical graph teaching with three lessons,twelve lesson/stage groups and sixteen cases. Every curve retains all six source-family coefficients,with at least three zero coefficients. Exact original-input tables and real plotted points and curve strokes precede independent interpretations. Real pointer and keyboard coordinates remain unsnapped; explicit bounded drawing-position tolerances do not soften exact table arithmetic or original zero exclusions. Both original curves remain overlaid for intersection readings; no solutions,repeated contacts,irrational readings,context filtering and cancelled reciprocal exclusions are distinguished. Complete wrong attempts may study a separate explained reference without passing fixed comparisons. Finite table,vertex and segment samples do not prove a globally correct smooth curve. Written reasoning,drawing judgement and model validity need human review. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map; the4Jlink requires these three original graph strands plus actual affine-line construction; 4Ktangent construction has a separate original lesson. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27only. ISBN9780435044107is metadata only; no verified book body,chapter/Hub mapping or MathematicsA/Foundation/modular inheritance. Original drawn-tangent teaching with four stages and twelve tasks. The original curve and named point are supplied givens; both tangent anchors and the third reading vertex come from actual learner input. Signed coordinate changes,calculated quotient and independent final gradient/unit/context are entered separately. Adequate alternative and reversed anchors and either right-corner orientation are supported. Pointer coordinates remain unsnapped. Original independent scales,negative/fractional gradients,cubic crossings and later meetings,reciprocal exclusions and context restrictions remain explicit. A horizontal zero-rise control is degenerate rather than a nondegenerate right triangle. A vertical line has no finite gradient. An auxiliary tangent crossing zero does not repair a reciprocal curve domain. Complete wrong attempts may study a separate explained reference without passing. Finite approximate comparisons do not approve written reasoning,tangent judgement,physical validity or formal assessment. Written reasoning,tangent judgement and practical validity require human review. This complete reviewed drawn-tangent teaching/practice bundle supplies the4Kstructural link; All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. Source mapping is separate from human drawing judgement and assessment approval. No4Ldifferentiation teaching is inferred from an internal exact reference. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27only. ISBN9780435044107is metadata only; no verified book body,chapter/Hub map or MathematicsA/Foundation/modular inheritance. Original integer-power differentiation teaching with four stages and twelve tasks. Each displayed original term has actual coefficient, old-power, coefficient-times-power and lowered-power working. Constants use an explicit zero derivative without a spurious reciprocal. Reciprocals, fractional and negative coefficients, powers beyond cubics, sums, cancellation and original zero exclusions are retained. The learner enters a combined derivative and named-input substitution, then independent final expression, notation, value, rate unit, mathematical-domain and model-context decisions. Exact rational comparison accepts reordered and equivalent supported expressions without repairing original or added domain holes. The general integer-power rule is taught; the finite checker bounds are tool limits rather than syllabus exclusions. Complete wrong attempts can study an explanation without passing. The complete checked4Lteaching/practice bundle has a structural source link; All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. Source mapping is separate from specialist and formal assessment approval. No4Mstationary/turning-point or4Nmotion completion is inherited. Written reasoning, model validity and specialist review remain pending. No official marks, mastery or formal assessment readiness. Single Higher-tier linear4MB1, entry context2026/27only. BookISBN9780435044107 is metadata only; no verified full book body, chapter/Hub map or MathematicsA/Foundation/modular inheritance. Original joint4Mteaching/practice bundle across three lessons and25cases,each with Worked,Guided,Independent and Application stages. Students enter their own graph observations,term-by-term differentiation,proposed derivative,original graph markers,complete cleared root lists,domain/context decisions,sign-chart witnesses and classifications. The global comparison retains stationary values,closed endpoints,open bounds,poles,infinities,attainment and all attaining inputs. Exact rational/surd and certified interval references support bounded self-checks; the finite sparse-curve and degree-five limits are tool bounds,not syllabus exclusions. Original punctures survive simplification and both graphs; constant and stationary-without-turn cases are explicit. Correct separate answers cannot replace incorrect actual working. Complete wrong attempts may study explained references without passing. The complete source and learner-route checks support a structural4Mlink only when all three original teaching/worked/practice records remain intact. All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. The wider Pearson curriculum remains incomplete. Supplied graph stimuli do not award separate4Jcurve-drawing,4Kdrawn-tangent or4Nmotion coverage. Written reasoning,graph judgment,model validity and specialist review remain pending. No official marks,mastery or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27only. BookISBN9780435044107 is official metadata only; no verified full book body,chapter/Hub map or MathematicsA/Foundation/modular inheritance. Original joint 4N teaching and practice across three lessons,25 cases and all twelve lesson/stage groups. Students enter term-by-term first and second derivatives,notation,units,all mathematical velocity candidates,original-domain/piece/context filtering,rest intervals and isolated/endpoint rest,reversal signs,every monotone leg,total distance,displacement and both averages. The two construction lessons require32 original graph constructions across their cases,with independently entered exact tables and axis quantities/units/scales,actual unsnapped points and stroke vertices,exact open-boundary decisions,separate branches at idealised speed jumps,and retained physical branches and ends beyond a cropped frame. Positive acceleration need not increase speed; speed cusps and mathematical two-sided versus physical one-sided gradients remain distinct. Practical rates retain their actual dependent/input units,positive magnitude,signed instantaneous value,original finite differences/averages,open physical ends and cancelled original holes. Correct independent answers cannot replace incorrect actual working. All three complete original teaching/worked/practice records are required for one structural 4N link; All14Functions source rows have reviewed structural teaching/practice links; other Mathematics B source gaps remain explicit in the current course map. The wider Pearson curriculum remains incomplete. This finite reference supports at most six continuous position pieces,eight terms per expression,powers -2 through6,complete cleared velocity degree at most five and exact rational/quadratic-field roots. These are software bounds,not syllabus exclusions; unresolved cases require review. Actual graph comparisons use finite position tolerances and65 probes per physical branch,including off-window portions; they do not prove a globally correct smooth curve. Automated calculation and drawing controls are checked; fresh browser interaction remains unverified. Original invented models and metadata-only bookISBN9780435044107 do not establish real physical/device/price validity,a full book body/chapter/Hub map,practical competence or specialist approval. No marks,mastery or formal assessment readiness. Single Higher-tier linear4MB1,exam entry context2026/27only; no MathematicsA/Foundation/modular inheritance. Original Mathematics B Matrices5A–G candidate: twelve teaching lessons,twelve worked examples,36 explained guided/independent/application practice tasks,and48 bound four-stage cases; all four specified reflection lines,rotations and enlargements have dedicated complete four-stage bundles. Matrix data,addition,multiplication,scalars,identity/zero,nonsingular2×2inverse,and plane transformations retain their own original givens. The learner enters blank working operands,intermediate results,constructions and separate final answers; edits,withdrawal,clearing,stage/reset and source/session/revision changes withhold old comparisons and explained references. Exact supported real square-root entries cover30°,45° and60° rotations in addition to quarter turns; the general cosine/sine formula is taught,while more general numerical trigonometry and equivalent forms outside the finite algebra engine require review. Written meanings,dimensions,units,geometric descriptions,drawn shapes and empirical validity need human review. All7Matrices source rows have reviewed structural teaching/practice bundles only while every original complete record remains intact. Row5Fjointly requires all six named transformation-family records,each with all four stages. Other Mathematics B source rows remain unlinked; see the current course map; the wider Pearson curriculum remains incomplete. ISBN9780435044107 is metadata only; no full book body,chapter sequence or Hub mapping is verified. Single Higher-tier linear4MB1; entry context November2026/June2027only. No MathematicsA,Foundation,modular or later-series inheritance. No fresh mounted-browser,hosted four-stage interaction,saved progress,official marks,mastery,specialist approval or formal assessment readiness is established. These two Mathematics B lessons reuse complete reviewed original StudyForA 2D and 3D Pythagoras teaching and eight stage cases. The Guided 2D case keeps both its missing-leg and exact/rounded missing-hypotenuse tasks; the 3D coordinate case still requires an originally unlocated horizontal projection endpoint. All actual triangle, hypotenuse, right-angle, section, foot and projection entries begin blank. Original numerical work, exact forms, requested precision, explicit units, equations and reasons are recorded separately from final answers. Correct answers never repair wrong geometry, and canonical figures appear separately only after a complete compared attempt. The original geometry engine and renderer are shared code; no other qualification completion or source metadata is inherited. The whole Geometry 6F source row also requires the complete acute-triangle altitude unit and all three jointly reviewed teaching and practice bundles, with actual learner routes. These two lessons alone earn no completed 6F link. Written reasoning, physical instruments and drawings, specialist approval and formal assessment remain separate. Single Higher-tier linear 4MB1; entry context November 2026/June 2027 only. No Foundation, modular or later-series inheritance. ISBN 9780435044107 is public metadata only; no verified full book body, chapter sequence or Teaching Hub mapping is claimed. Original content and local comparison controls are checked; each source link requires the complete joint bundle in the course map. Fresh mounted-browser events and saved account-progress isolation are unverified. This original acute-triangle lesson fills the specific 4MB1 Geometry 6F altitude subskill. It retains one worked example, guided practice, two separately required Independent examples and an application task across five bound cases. The source row also requires the complete 2D and 3D Pythagoras programme; this lesson alone earns no completed 6F link. Students construct an actual foot D, explain the perpendicular, retain both right-triangle calculations and record separate final answers. Incorrect geometry is not repaired by correct numbers or by studying a reference. Exact fractions and a positive irrational-height example are included; illustrative drawings are not measurement scales. Written equations, theorem reasons, assumptions, physical drawing/instrument accuracy and contextual decisions need human review. No marks, mastery, specialist approval or formal assessment readiness is established. Single Higher-tier linear 4MB1; entry context November 2026/June 2027 only. No Mathematics A, Foundation, modular or later-series inheritance. ISBN 9780435044107 is public metadata only; no full book body, chapter sequence or Teaching Hub content is verified. This original content and local comparison activity is part of the joint course bundle; a title or tag alone does not earn a source link. Fresh mounted-browser events and persisted account progress remain unverified. Nine Mathematics B circle lessons reuse reviewed original StudyForA teaching and geometry cases, with B-specific source, outcome and learner bindings. Geometry 6K jointly requires vocabulary, centre/chord bisection and converse, equal tangents and contact radii, centre/circumference angles including the reflex choice, diameter and same/opposite segment angles, the alternate-segment theorem with the selected tangent ray, and internal/external intersecting-chord products with positive ordered lengths and rejected algebraic roots. Geometry 6L requires the complete cyclic-quadrilateral lesson, including membership, opposite pairs, exterior-angle reasoning and the off-circle counterexample. Each lesson retains original teaching, an explained worked example and Guided, Independent and application practice. The interactive geometry begins with blank selections and responses; complete attempts can be compared and studied even when wrong. Written reasons, theorem use, application decisions, instrument competence and drawings still need human review. These links do not establish expert approval, marks, mastery or formal assessment readiness. Scope is single Higher-tier linear 4MB1, with November 2026/June 2027 entry context only. No Mathematics A, Foundation, modular, later-series or full-book completion is inherited. ISBN 9780435044107 is public metadata only: book bodies, chapter sequence and Teaching Hub material remain unverified. The other eleven Geometry rows and the wider Pearson curriculum remain separate gaps. Original content and local comparison controls are checked; source links require the complete bundles in the course map. Fresh mounted-browser events and saved-progress isolation are unverified.

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