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

Drawn tangents,right triangles and local gradients

Draw a local tangent and your own reading triangle,then estimate its signed coordinate gradient and interpret its units and context.

Section 4 · Functions (all14rows linked;self-check practice)

Pearson4MB1 · drawn tangents and reading triangles · single Higher tier · Mathematics B 4K

Exam scope and assessment notes

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.

Before you begin

Original sparse graph families and excluded inputs
Independent Cartesian scales
Straight-line gradients and signed fractions

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

Quick readiness check

Which points should supply the two main readings for a tangent-gradient triangle?

What you will learn

Draw a reasonable tangent at a named point on the supplied original graph.
Construct a visible right-angled reading triangle from two well-separated points on the actual tangent.
Calculate signed rise divided by run using the original independent axis scales.
Distinguish horizontal zero-rise controls,original curve domains and contextual limits.

Drawn tangents and local gradients: Worked

The original blue graph and named point are given. Place your own tangent anchors A and B and triangle corner C. Record your drawing and calculation,then a separate interpretation,before studying the explanation. Fixed comparisons do not award marks or mastery.

Draw at the named point and read your own triangle

Use the supplied original graph of y = x^2 − 1 and its named point P=(1,0). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision.

Horizontal: -4 to 4,step 1/2 dimensionless. Vertical: -12 to 12,step 3 dimensionless. A gradient uses dimensionless.

The blue curve and P are supplied givens. My amber line and dashed reading triangle come only from my actual entries. Tap the graph to place tangent anchor A,or use the keyboard controls below. Each grid division keeps its original stated step;some labels span several divisions. Cropping the drawing never moves an entered point onto its boundary.

A: not plottedB: not plottedC: not plotted

Coordinate labels are rounded to four decimal places;the original plotted positions are retained.

My held x-coordinateMy held y-coordinate

My coordinate readings and calculation

Choose two well-separated visible anchors on your tangent. The third vertex should join a horizontal and a vertical reading leg. Keep the sign and use coordinate units rather than screen lengths.

My horizontal coordinate stepMy vertical coordinate stepMy horizontal axis labelMy vertical axis labelMy triangle decisionMy signed coordinate run,B minus AMy signed coordinate rise,B minus AMy calculated rise divided by runMy gradient unitExplain my construction and its limits

Construction: Not recorded or changed

My separate gradient interpretation

Enter your final estimate independently. Decide whether the named mathematical point is admitted by the original model context. A tangent-line extension does not change that context.

My separate final gradient estimateMy gradient unitMy original context decisionExplain my independent interpretation

Interpretation: Not recorded or changed

Draw at the named point and read your own triangle

Use the supplied original graph of y = −x^2 + 2x + 3 and its named point P=(2,3). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision.

Horizontal: -4 to 4,step 1 dimensionless. Vertical: -12 to 12,step 2 dimensionless. A gradient uses dimensionless.

A: not plottedB: not plottedC: not plotted

Coordinate labels are rounded to four decimal places;the original plotted positions are retained.

My held x-coordinateMy held y-coordinate

My coordinate readings and calculation

Choose two well-separated visible anchors on your tangent. The third vertex should join a horizontal and a vertical reading leg. Keep the sign and use coordinate units rather than screen lengths.

Construction: Not recorded or changed

My separate gradient interpretation

Enter your final estimate independently. Decide whether the named mathematical point is admitted by the original model context. A tangent-line extension does not change that context.

My separate final gradient estimateMy gradient unitMy original context decisionExplain my independent interpretation

Interpretation: Not recorded or changed

Draw at the named point and read your own triangle

Use the supplied original graph of y = 3 and its named point P=(0,3). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. This horizontal-contact control has zero rise and nonzero run. Identify its degenerate zero-rise construction explicitly;do not call it a nondegenerate right triangle. A straight-line graph is its own tangent. A constant graph has zero gradient at every named input.

Horizontal: -4 to 4,step 1/2 dimensionless. Vertical: -12 to 12,step 4 dimensionless. A gradient uses dimensionless.

A: not plottedB: not plottedC: not plotted

Coordinate labels are rounded to four decimal places;the original plotted positions are retained.

My held x-coordinateMy held y-coordinate

My coordinate readings and calculation

Choose two well-separated visible anchors on your tangent. The third vertex should join a horizontal and a vertical reading leg. Keep the sign and use coordinate units rather than screen lengths.

Construction: Not recorded or changed

My separate gradient interpretation

Enter your final estimate independently. Decide whether the named mathematical point is admitted by the original model context. A tangent-line extension does not change that context.

My separate final gradient estimateMy gradient unitMy original context decisionExplain my independent interpretation

Interpretation: Not recorded or changed

Record every current tangent,triangle and calculation.

These are approximate checks on original authored exercises. Written reasoning,tangent judgement,model validity and formal assessment require human review. The horizontal control is explicitly degenerate. Teaching and practice are mapped to4K;eleven Functions source rows remain unlinked. Specialist approval and formal assessment readiness remain pending.

A local direction,not any line through two curve points

A gradient at a named point describes the graph’s local direction. Draw a reasonable tangent there. A chord through two different curve points usually gives a secant gradient instead. A straight-line graph is its own tangent;for a constant graph the gradient is zero at every input. A tangent to a cubic can cross the curve locally or meet it again elsewhere;global nonintersection is not a definition of tangent.

Construct an appropriate reading triangle

Choose two well-separated visible points on your actual drawn tangent. Add a horizontal and a vertical side using a third vertex. Read the two coordinate changes in one endpoint order;the points need not be other points on the original curve. The triangle belongs to the tangent line.

Keep the signed coordinate quotient and independent scales

Calculate gradient as signed rise divided by signed run. Each numerical axis step must be applied independently;screen lengths or the displayed angle do not determine the quotient. Reversing both endpoint orders preserves the gradient. A horizontal tangent has zero rise;its zero-height control is degenerate rather than a nondegenerate right triangle. A vertical line has no finite gradient.

Distinguish the curve,tangent and supplied context

A reciprocal curve excludes zero,but an auxiliary tangent line may pass through that input without adding it to the original curve. An invented contextual model admits only its stated inputs. A mathematical tangent outside that context is not an admitted physical rate. Numerical drawing tolerances and finite checks do not certify visual judgement,written reasoning or formal assessment readiness.

Sources for this lesson

Pause and explain

The original reciprocal curve excludes zero. Does a tangent line passing through x=0 add that input to the original curve?

Put the idea to work

Worked example

Task 1: Use the supplied original graph of y = x^2 − 1 and its named point P=(1,0). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. Task 2: Use the supplied original graph of y = −x^2 + 2x + 3 and its named point P=(2,3). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. Task 3: Use the supplied original graph of y = 3 and its named point P=(0,3). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. This horizontal-contact control has zero rise and nonzero run. Identify its degenerate zero-rise construction explicitly;do not call it a nondegenerate right triangle. A straight-line graph is its own tangent. A constant graph has zero gradient at every named input.

Show the worked solution

Task 1: Locate the given named point P=(1,0) on the original graph;use the two original numerical axis scales.
Task 1: Draw a reasonable local tangent at P. These separate reference anchors are A=(-1,-4) and B=(3,4);other adequate anchors on a reasonable learner tangent can be valid.
Task 1: For this reference,choose the third vertex C=(3,-4). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 1: Keep one anchor order: coordinate run = 3 − (-1) = 4;coordinate rise = 4 − (-4) = 8.
Task 1: The reference quotient is rise/run = (8)/(4) = 2 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 1: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 2: Locate the given named point P=(2,3) on the original graph;use the two original numerical axis scales.
Task 2: Draw a reasonable local tangent at P. These separate reference anchors are A=(0,7) and B=(3,1);other adequate anchors on a reasonable learner tangent can be valid.
Task 2: For this reference,choose the third vertex C=(3,7). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 2: Keep one anchor order: coordinate run = 3 − (0) = 3;coordinate rise = 1 − (7) = -6.
Task 2: The reference quotient is rise/run = (-6)/(3) = -2 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 2: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 3: Locate the given named point P=(0,3) on the original graph;use the two original numerical axis scales.
Task 3: Draw a reasonable local tangent at P. These separate reference anchors are A=(-2,3) and B=(2,3);other adequate anchors on a reasonable learner tangent can be valid.
Task 3: For this reference,choose the third vertex C=(2,3). The rise is zero,so the control is explicitly degenerate;do not claim three distinct right-triangle vertices.
Task 3: Keep one anchor order: coordinate run = 2 − (-2) = 4;coordinate rise = 3 − (3) = 0.
Task 3: The reference quotient is rise/run = (0)/(4) = 0 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 3: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 3: This horizontal-contact control has zero rise and nonzero run. Identify its degenerate zero-rise construction explicitly;do not call it a nondegenerate right triangle.
Task 3: A straight-line graph is its own tangent. A constant graph has zero gradient at every named input.

Answer Task 1: Estimated local gradient about 2 dimensionless;original context allowed. Written reasoning and drawing judgement require human review. Task 2: Estimated local gradient about -2 dimensionless;original context allowed. Written reasoning and drawing judgement require human review. Task 3: Estimated local gradient about 0 dimensionless;original context allowed. Written reasoning and drawing judgement require human review.

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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

Task 1: Use the supplied original graph of y = (1/2)x^3 − 2x + 1 and its named point P=(1,-1/2). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. Task 2: Use the supplied original graph of y = 2/x and its named point P=(-1,-2). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve. Task 3: Use the supplied original graph of y = (1/2)x + 1 and its named point P=(1,3/2). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. A straight-line graph is its own tangent. A constant graph has zero gradient at every named input.

Give me a hint

Draw the local direction at the named point,choose separated anchors on your own tangent,and form the actual horizontal/vertical reading legs. Use the original coordinate units and enter the final interpretation independently.

Compare my reasoning

Task 1: Locate the given named point P=(1,-1/2) on the original graph;use the two original numerical axis scales.
Task 1: Draw a reasonable local tangent at P. These separate reference anchors are A=(-2,1) and B=(2,-1);other adequate anchors on a reasonable learner tangent can be valid.
Task 1: For this reference,choose the third vertex C=(2,1). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 1: Keep one anchor order: coordinate run = 2 − (-2) = 4;coordinate rise = -1 − (1) = -2.
Task 1: The reference quotient is rise/run = (-2)/(4) = -1/2 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 1: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 2: Locate the given named point P=(-1,-2) on the original graph;use the two original numerical axis scales.
Task 2: Draw a reasonable local tangent at P. These separate reference anchors are A=(-3,2) and B=(1,-6);other adequate anchors on a reasonable learner tangent can be valid.
Task 2: For this reference,choose the third vertex C=(1,2). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 2: Keep one anchor order: coordinate run = 1 − (-3) = 4;coordinate rise = -6 − (2) = -8.
Task 2: The reference quotient is rise/run = (-8)/(4) = -2 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 2: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 2: The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve.
Task 3: Locate the given named point P=(1,3/2) on the original graph;use the two original numerical axis scales.
Task 3: Draw a reasonable local tangent at P. These separate reference anchors are A=(-2,0) and B=(2,2);other adequate anchors on a reasonable learner tangent can be valid.
Task 3: For this reference,choose the third vertex C=(2,0). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 3: Keep one anchor order: coordinate run = 2 − (-2) = 4;coordinate rise = 2 − (0) = 2.
Task 3: The reference quotient is rise/run = (2)/(4) = 1/2 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 3: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 3: A straight-line graph is its own tangent. A constant graph has zero gradient at every named input.

Task 1: Estimated local gradient about -1/2 dimensionless;original context allowed. Written reasoning and drawing judgement require human review. Task 2: Estimated local gradient about -2 dimensionless;original context allowed. Written reasoning and drawing judgement require human review. Task 3: Estimated local gradient about 1/2 dimensionless;original context allowed. Written reasoning and drawing judgement require human review.

Look for these in your work

My actual tangent is reasonable at the named original point.
My two chosen tangent anchors and third reading vertex are retained.
My signed rise and run use consistent endpoint order and independent coordinate scales.
My final gradient,unit and context decision are entered separately.
I explain the horizontal/domain/model limits and retain the need for human drawing review.

2 · Independent

Task 1: Use the supplied original graph of y = x^3 and its named point P=(0,0). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. This horizontal-contact control has zero rise and nonzero run. Identify its degenerate zero-rise construction explicitly;do not call it a nondegenerate right triangle. This tangent crosses the original cubic locally;local crossing does not prevent tangency. Task 2: Use the supplied original graph of y = 1/x^2 and its named point P=(-1,1). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve. Task 3: Use the supplied original graph of y = x^2 + 1/x + 1/x^2 and its named point P=(1,3). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve.

Give me a hint

Compare my reasoning

Task 1: Locate the given named point P=(0,0) on the original graph;use the two original numerical axis scales.
Task 1: Draw a reasonable local tangent at P. These separate reference anchors are A=(-2,0) and B=(2,0);other adequate anchors on a reasonable learner tangent can be valid.
Task 1: For this reference,choose the third vertex C=(2,0). The rise is zero,so the control is explicitly degenerate;do not claim three distinct right-triangle vertices.
Task 1: Keep one anchor order: coordinate run = 2 − (-2) = 4;coordinate rise = 0 − (0) = 0.
Task 1: The reference quotient is rise/run = (0)/(4) = 0 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 1: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 1: This horizontal-contact control has zero rise and nonzero run. Identify its degenerate zero-rise construction explicitly;do not call it a nondegenerate right triangle.
Task 1: This tangent crosses the original cubic locally;local crossing does not prevent tangency.
Task 2: Locate the given named point P=(-1,1) on the original graph;use the two original numerical axis scales.
Task 2: Draw a reasonable local tangent at P. These separate reference anchors are A=(-3,-3) and B=(1,5);other adequate anchors on a reasonable learner tangent can be valid.
Task 2: For this reference,choose the third vertex C=(1,-3). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 2: Keep one anchor order: coordinate run = 1 − (-3) = 4;coordinate rise = 5 − (-3) = 8.
Task 2: The reference quotient is rise/run = (8)/(4) = 2 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 2: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 2: The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve.
Task 3: Locate the given named point P=(1,3) on the original graph;use the two original numerical axis scales.
Task 3: Draw a reasonable local tangent at P. These separate reference anchors are A=(-2,6) and B=(2,2);other adequate anchors on a reasonable learner tangent can be valid.
Task 3: For this reference,choose the third vertex C=(2,6). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 3: Keep one anchor order: coordinate run = 2 − (-2) = 4;coordinate rise = 2 − (6) = -4.
Task 3: The reference quotient is rise/run = (-4)/(4) = -1 dimensionless. Reversing both endpoints reverses both changes and preserves the quotient.
Task 3: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 3: The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve.

Task 1: Estimated local gradient about 0 dimensionless;original context allowed. Written reasoning and drawing judgement require human review. Task 2: Estimated local gradient about 2 dimensionless;original context allowed. Written reasoning and drawing judgement require human review. Task 3: Estimated local gradient about -1 dimensionless;original context allowed. Written reasoning and drawing judgement require human review.

Look for these in your work

My actual tangent is reasonable at the named original point.
My two chosen tangent anchors and third reading vertex are retained.
My signed rise and run use consistent endpoint order and independent coordinate scales.
My final gradient,unit and context decision are entered separately.
I explain the horizontal/domain/model limits and retain the need for human drawing review.

3 · Transfer

Task 1: Use the supplied original graph of T = −t^2 + 4t + 2 and its named point P=(3,5). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. This is an invented temperature formula admitted only from0to4minutes;the tangent estimate is not a measured physical law. Task 2: Use the supplied original graph of q = x^2 + 1/x + 1/x^2 and its named point P=(-1,1). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. This hypothetical score model is admitted only for positive inputs from1/2to2. The named negative mathematical point is outside that context;state that its tangent is not an admitted model rate. The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve. Task 3: Use the supplied original graph of s = (1/2)t^3 and its named point P=(2,4). Draw your own local tangent with two chosen anchors. Add your own third triangle vertex,then enter signed coordinate run,rise and their quotient. Enter a separate final gradient,unit and original-context decision. This invented distance formula is admitted only from0to3seconds. An auxiliary tangent may extend to negative reading heights without asserting a negative observed distance or validating a physical model.

Give me a hint

Compare my reasoning

Task 1: Locate the given named point P=(3,5) on the original graph;use the two original numerical axis scales.
Task 1: Draw a reasonable local tangent at P. These separate reference anchors are A=(0,11) and B=(4,3);other adequate anchors on a reasonable learner tangent can be valid.
Task 1: For this reference,choose the third vertex C=(4,11). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 1: Keep one anchor order: coordinate run = 4 − (0) = 4;coordinate rise = 3 − (11) = -8.
Task 1: The reference quotient is rise/run = (-8)/(4) = -2 degrees Celsius/minute. Reversing both endpoints reverses both changes and preserves the quotient.
Task 1: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 1: This is an invented temperature formula admitted only from0to4minutes;the tangent estimate is not a measured physical law.
Task 2: Locate the given named point P=(-1,1) on the original graph;use the two original numerical axis scales.
Task 2: Draw a reasonable local tangent at P. These separate reference anchors are A=(-3,3) and B=(1,-1);other adequate anchors on a reasonable learner tangent can be valid.
Task 2: For this reference,choose the third vertex C=(1,3). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 2: Keep one anchor order: coordinate run = 1 − (-3) = 4;coordinate rise = -1 − (3) = -4.
Task 2: The reference quotient is rise/run = (-4)/(4) = -1 score units/input unit. Reversing both endpoints reverses both changes and preserves the quotient.
Task 2: The named point is disallowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 2: This hypothetical score model is admitted only for positive inputs from1/2to2. The named negative mathematical point is outside that context;state that its tangent is not an admitted model rate.
Task 2: The original curve excludes zero. Its auxiliary tangent line may cross that input without adding a point to the original curve.
Task 3: Locate the given named point P=(2,4) on the original graph;use the two original numerical axis scales.
Task 3: Draw a reasonable local tangent at P. These separate reference anchors are A=(1,-2) and B=(3,10);other adequate anchors on a reasonable learner tangent can be valid.
Task 3: For this reference,choose the third vertex C=(3,-2). AC is horizontal and CB is vertical. Use a sizeable visible triangle,not the curve chord or a ratio of screen lengths.
Task 3: Keep one anchor order: coordinate run = 3 − (1) = 2;coordinate rise = 10 − (-2) = 12.
Task 3: The reference quotient is rise/run = (12)/(2) = 6 metres/second. Reversing both endpoints reverses both changes and preserves the quotient.
Task 3: The named point is allowed by the supplied context. A mathematical tangent or its extension does not establish a physical model rate outside the original context.
Task 3: This invented distance formula is admitted only from0to3seconds. An auxiliary tangent may extend to negative reading heights without asserting a negative observed distance or validating a physical model.

Task 1: Estimated local gradient about -2 degrees Celsius/minute;original context allowed. Written reasoning and drawing judgement require human review. Task 2: Estimated local gradient about -1 score units/input unit;original context disallowed. Written reasoning and drawing judgement require human review. Task 3: Estimated local gradient about 6 metres/second;original context allowed. Written reasoning and drawing judgement require human review.

Look for these in your work

My actual tangent is reasonable at the named original point.
My two chosen tangent anchors and third reading vertex are retained.
My signed rise and run use consistent endpoint order and independent coordinate scales.
My final gradient,unit and context decision are entered separately.
I explain the horizontal/domain/model limits and retain the need for human drawing review.

Common mistakes

Using a chord through two different curve points as if every secant were a tangent.
Reading screen lengths without the two original coordinate scales.
Taking the two coordinate differences in inconsistent endpoint orders.
Supplying a correct final gradient while the actual line misses the named point.
Treating a zero-rise horizontal control as a nondegenerate right triangle.
Applying a reciprocal curve’s zero exclusion to its auxiliary tangent line.
Treating a mathematical tangent outside a supplied context as an observed physical rate.

Recall without your notes

What can you explain now?

What happens to the estimated gradient when both endpoints of a reading triangle are reversed?

Compare with the explanation

The signed rise and run both reverse,so their quotient stays the same.

Keep one consistent endpoint order for both coordinate differences;the geometric line has not changed.

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Make it stick

Close the reference. Draw a tangent and your own appropriate reading triangle. Calculate signed coordinate rise divided by run,explain why reversing both endpoint orders preserves the quotient,and distinguish the original curve,tangent extension and supplied context.

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.