From Foundation to Extension·Concept Bridges
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Connect familiar A-Level mathematics to the deeper ideas used in demanding TMUA problems.

Chapter 1. Understanding Function Relationships

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$Follow the information contained in an indirect function relationship without unnecessarily finding...$
Topic 1. Following Function Relationships
6 Practices
PUBLIC
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$Decide when information about an output can be traced backwards to determine the input.$
Topic 2. Working Backwards Through a Function
6 Practices
MEMBERLogin required
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$Recognise when several inputs can produce the same output and determine what information remains whe...$
Topic 3. When a Function Loses Information
6 Practices
MEMBERLogin required
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$Recognise the different ways unfamiliar functions and mathematical rules can be defined in TMUA prob...$
Topic 4. Functions Defined by Rules
16 Practices
MEMBERLogin required

Chapter 2. Functional Equations and Recurrence

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$Use relationships between function values to create new equations and extract the information you ne...$
Topic 1. Functional Equations
8 Practices
PUBLIC
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$Follow a relationship through several steps to reveal cycles, fixed behaviour and repeating structur...$
Topic 2. Following Repeated Relationships
6 Practices
MEMBERLogin required
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$Use relationships between successive terms to uncover patterns, cycles and structures without calcul...$
Topic 3. Recurrence Relations
6 Practices
MEMBERLogin required

Chapter 3. Modulus

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$You may have first learned modulus as a rule: if x is positive, |x| = x, and if x is negative, |x| =...$
Topic 1. As a Distance (1)
6 Practices
PUBLIC
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$Modulus as Distance You may have first learned modulus as a rule: if x is positive, |x| = x, and...$
Topic 2. As a Distance (2) - TMUA Extension
6 Practices
MEMBERLogin required
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$The Shape of a Sum of Two Distances When two modulus terms appear in the same expression, the i...$
Topic 3. Functions with Two Modulus Terms
6 Practices
MEMBERLogin required
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$Understand the difference between applying modulus to a function's output and applying it to its inp...$
Topic 4. Modulus and Graph Transformations
6 Practices
MEMBERLogin required

Chapter 4. The Number of Solutions

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$Beyond the Discriminant For a quadratic equation, the discriminant gives a familiar way to deter...$
Topic 1. Beyond the Discriminant
6 Practices
PUBLIC
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$Beyond Polynomial Functions So far, we have used turning points to understand how the number of...$
Topic 2. Critical Points Beyond Polynomial
6 Practices
MEMBERLogin required
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$Finding the Edge Cases So far, we have used turning points and extreme values to find the values...$
Topic 3. From Critical Points to Edge Cases
5 Practices
MEMBERLogin required
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$Use periodicity and interval restrictions to count repeated solutions efficiently.$
Topic 4. Counting Solutions in Repeating Patterns
6 Practices
MEMBERLogin required
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$Understand how intervals and domain restrictions can change the number of valid solutions.$
Topic 5. When the Domain Changes the Answer
5 Practices
MEMBERLogin required

Chapter 1. Understanding Function Relationships

iFollow the information contained in an indirect function relationship without unnecessarily finding...Topic 1. Following Function Relationships

iDecide when information about an output can be traced backwards to determine the input.Topic 2. Working Backwards Through a Function

iRecognise when several inputs can produce the same output and determine what information remains whe...Topic 3. When a Function Loses Information

iRecognise the different ways unfamiliar functions and mathematical rules can be defined in TMUA prob...Topic 4. Functions Defined by Rules

Chapter 2. Functional Equations and Recurrence

iUse relationships between function values to create new equations and extract the information you ne...Topic 1. Functional Equations

iFollow a relationship through several steps to reveal cycles, fixed behaviour and repeating structur...Topic 2. Following Repeated Relationships

iUse relationships between successive terms to uncover patterns, cycles and structures without calcul...Topic 3. Recurrence Relations

Chapter 3. Modulus

iYou may have first learned modulus as a rule: if x is positive, |x| = x, and if x is negative, |x| =...Topic 1. As a Distance (1)

iModulus as Distance You may have first learned modulus as a rule: if x is positive, |x| = x, and...Topic 2. As a Distance (2) - TMUA Extension

Modulus as Distance

iThe Shape of a Sum of Two Distances When two modulus terms appear in the same expression, the i...Topic 3. Functions with Two Modulus Terms

The Shape of a Sum of Two Distances

iUnderstand the difference between applying modulus to a function's output and applying it to its inp...Topic 4. Modulus and Graph Transformations

Chapter 4. The Number of Solutions

iBeyond the Discriminant For a quadratic equation, the discriminant gives a familiar way to deter...Topic 1. Beyond the Discriminant

Beyond the Discriminant

iBeyond Polynomial Functions So far, we have used turning points to understand how the number of...Topic 2. Critical Points Beyond Polynomial

Beyond Polynomial Functions

iFinding the Edge Cases So far, we have used turning points and extreme values to find the values...Topic 3. From Critical Points to Edge Cases

Finding the Edge Cases

iUse periodicity and interval restrictions to count repeated solutions efficiently.Topic 4. Counting Solutions in Repeating Patterns

iUnderstand how intervals and domain restrictions can change the number of valid solutions.Topic 5. When the Domain Changes the Answer

i
$Follow the information contained in an indirect function relationship without unnecessarily finding...$
Topic 1. Following Function Relationships

i
$Decide when information about an output can be traced backwards to determine the input.$
Topic 2. Working Backwards Through a Function

i
$Recognise when several inputs can produce the same output and determine what information remains whe...$
Topic 3. When a Function Loses Information

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$Recognise the different ways unfamiliar functions and mathematical rules can be defined in TMUA prob...$
Topic 4. Functions Defined by Rules

i
$Use relationships between function values to create new equations and extract the information you ne...$
Topic 1. Functional Equations

i
$Follow a relationship through several steps to reveal cycles, fixed behaviour and repeating structur...$
Topic 2. Following Repeated Relationships

i
$Use relationships between successive terms to uncover patterns, cycles and structures without calcul...$
Topic 3. Recurrence Relations

i
$You may have first learned modulus as a rule: if x is positive, |x| = x, and if x is negative, |x| =...$
Topic 1. As a Distance (1)

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$Modulus as Distance You may have first learned modulus as a rule: if x is positive, |x| = x, and...$
Topic 2. As a Distance (2) - TMUA Extension

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$The Shape of a Sum of Two Distances When two modulus terms appear in the same expression, the i...$
Topic 3. Functions with Two Modulus Terms

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$Understand the difference between applying modulus to a function's output and applying it to its inp...$
Topic 4. Modulus and Graph Transformations

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$Beyond the Discriminant For a quadratic equation, the discriminant gives a familiar way to deter...$
Topic 1. Beyond the Discriminant

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$Beyond Polynomial Functions So far, we have used turning points to understand how the number of...$
Topic 2. Critical Points Beyond Polynomial

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$Finding the Edge Cases So far, we have used turning points and extreme values to find the values...$
Topic 3. From Critical Points to Edge Cases

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$Use periodicity and interval restrictions to count repeated solutions efficiently.$
Topic 4. Counting Solutions in Repeating Patterns

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$Understand how intervals and domain restrictions can change the number of valid solutions.$
Topic 5. When the Domain Changes the Answer