Author: Admin

  • Reactivity 1.1 – Measuring enthalpy change

    R1.1.3 – Energetic stability and the direction of change

    📌 Direction of change

    • Chemical reactions change to decrease their enthalpy (ie the chemical potential energy)
    • The natural change in enthalpy is generally meant to reach a more stable state
    • The change in enthalpy is denoted using the positive (+) and negative (-) signs in relation to Δ𝐻
    • In an exothermic reaction , the reactants have greater energy than the products. This is because the system less energy to the surroundings in the form of heat during the reaction. This means that many exothermic reactions occur spontaneously in an attempt to reach a state of stability.
    • Conversely, in an endothermic reaction, the products have greater energy than the reactants. This is because the system is gaining energy in the form of heat from the surroundings. Endothermic reactions often require external effort as products may be less stable than reactants.
    • A positive change in enthalpy is associated with endothermic reactions and a negative enthalpy change is associated with exothermic reactions

    ⭐️ some exceptions to this rule do exist – certain endothermic reactions can occur spontaneously [eg. : SOCl(I) + FeCl3.6H,O(s) → FeCl(s) + 6SO,(g) + 12HCl(g) is endothermic but highly spontaneous ]

    ！the concept of spontaneous reactions is further discussed in R1.4 (HL)

    The following diagrams demonstrate the change in enthalpy in endothermic and exothermic reactions :

    [image from Chemistry for the IB Diploma Programme]

  • Reactivity 1.1 – Measuring enthalpy changes

    Reactivity 1.1.1 – Chemical reactions involve heat transfer

    📌 Heat versus temperature

    • Heat can be defined as a form of energy transfer occurring as a result of temperature difference. The transfer of heat to a system causes an increase in kinetic energy in molecules in the system.
    • Temperature is defined as the measure of the average kinetic energy of these molecules.

    📌 System and the surroundings

    • The system is the area of ‘interest’ of a reaction while the surroundings are theoretically, everything else in the universe.
    • There are 3 types of systems :
    1. Open systems : where energy and matter can be exchanged with the surroundings
    2. Closed systems : where energy can be exchanged with the surroundings but matter cannot
    3. Isolated systems : where neither energy nor matter can be exchanged with the surroundings
    • The first law of thermodynamics states that energy cannot be created or destroyed, therefore the total energy cannot change during a reaction. However, heat can be exchanged between a system and the surroundings. The total heat content of a reaction is known as its ‘enthalpy’

    📌 Enthalpy changes

    • When a system transfers heat to its surroundings, the total enthalpy of the system decreases
    • When a system gains heat from its surroundings, the total enthalpy of the system increases
    • Changes in enthalpy are denoted by the sign Δ𝐻 (where 𝐻 is enthalpy)

  • 🧠 Formation of Stereotypes and Their Effects on Behaviour

    📌 Key terms

    TermDefinition
    StereotypeA mental representation (schema) of a group and its members — a fixed, oversimplified belief often resistant to change.
    StereotypingThe process of assigning characteristics to someone based solely on group membership.
    Social CategorizationClassifying people into groups based on perceived similarities (e.g., gender, ethnicity, occupation).
    Illusory CorrelationPerceiving a relationship between two variables (e.g., group membership and behavior) even when none exists.
    Stereotype ThreatFear of confirming a negative stereotype about one’s group, which can negatively affect performance.
    Self-Fulfilling ProphecyWhen expectations about others lead them to behave in ways that confirm those expectations.
    Confirmation BiasTendency to interpret information in a way that confirms pre-existing beliefs.
    Grain of Truth HypothesisSuggests stereotypes may originate from small but real differences, later exaggerated or overgeneralized.

    📌 Notes

    Stereotypes are a key form of social cognition, shaping how individuals perceive and interact with others.
    They simplify complex social information, allowing people to categorize quickly — but often at the cost of accuracy and fairness.

    Stereotypes influence behavior, perception, and even memory, making them a central topic in both cognitive and sociocultural psychology.
    They can form automatically, through cognitive shortcuts (heuristics), or socially, via cultural norms and media.

    📌 Key Studies

    1️⃣ Social Categorization (Tajfel, 1971)

    • Categorization into in-groups and out-groups simplifies the world.
    • Once categorized, people accentuate similarities within groups and differences between groups.
    • Leads to overgeneralization — forming stereotypes.

    💡 Example: Seeing “all athletes” as extroverted or “all artists” as emotional.

    2️⃣ Illusory Correlation – Hamilton & Gifford (1976)

    Aim:
    To investigate how stereotypes form through illusory correlation between group membership and behavior.

    Procedure:

    • Participants read statements about two groups (A and B) performing positive or negative behaviors.
    • Group A (majority) performed more positive acts; Group B (minority) fewer acts overall, with the same positive-to-negative ratio.

    Findings:

    • Participants overestimated the number of negative behaviors by Group B.
    • They associated the minority group with negative traits, even though both groups behaved similarly.

    Conclusion:

    • Stereotypes can form through cognitive errors — humans notice distinctive events (minority + negative) and exaggerate correlations.

    Evaluation:
    ✅ High control → cause-and-effect link between cognition and stereotype formation.
    ⚠️ Artificial task → low ecological validity.
    ✅ Highly influential in cognitive origins of prejudice.

    3️⃣ Grain of Truth Hypothesis

    • Suggests that stereotypes may stem from real experiences with individuals from a group, then overgeneralized.
    • Example: Meeting one rude tourist → believing “all tourists are rude.”
    • Criticized for assuming all stereotypes have factual bases.

    🧠 Effects of Stereotypes on Behaviour

    1️⃣ Stereotype Threat – Steele & Aronson (1995)

    Aim:
    To test whether stereotype threat affects African-American students’ academic performance.

    Procedure:

    • Participants: African-American and White college students.
    • Two conditions:
      1. Diagnostic — test described as measuring intelligence.
      2. Non-diagnostic — same test, described as unrelated to ability.

    Findings:

    • African-American students performed worse in the diagnostic condition.
    • When the stereotype of “African-Americans perform poorly” was activated, anxiety impaired performance.

    Conclusion:

    • Awareness of stereotypes can negatively impact performance — a self-fulfilling prophecy effect.

    Evaluation:
    ✅ Strong experimental design → causal relationship.
    ⚠️ Cultural bias (US context).
    ✅ Replicated in gender studies (e.g., Spencer et al., 1999).

    2️⃣ Gender Stereotype Threat – Spencer et al. (1999)

    Aim:
    To investigate if women’s math performance is affected by stereotype activation.

    Procedure:

    • Male and female university students completed math tests under two conditions:
      1. Stereotype-activated: Told men perform better in math.
      2. Control: Told no gender differences exist.

    Findings:

    • In the stereotype-activated condition, women performed worse than men.
    • In the control condition, performance was equal.

    Conclusion:

    • Awareness of gender stereotypes can impair cognitive performance.

    3️⃣ Self-Fulfilling Prophecy – Rosenthal & Jacobson (1968)

    Aim:
    To investigate whether teacher expectations affect student performance.

    Procedure:

    • Teachers were told certain students were likely “academic bloomers” (randomly chosen).
    • Over the year, those students showed greater IQ improvement.

    Conclusion:

    • Expectations influence behavior of both perceiver and target — supporting stereotype effects.

    Evaluation:
    ✅ Real-world application in education.
    ⚠️ Ethical concerns (deception).
    ✅ Strong field evidence of stereotype impact.

    🔍Tok link

    Knowledge Question: “To what extent do labels shape our perception of reality?”

    Stereotypes reveal the interaction between knowledge and bias — how mental shortcuts (heuristics) simplify reality but distort truth.

    TOK discussion: Are stereotypes ever useful for understanding? Can categorization exist without bias?

     🌐 Real-World Connection

    Used to understand racial profiling, gender inequality, and media representation.

    Research has informed diversity training, inclusive education, and implicit bias interventions.

    Understanding stereotype threat helps design fair testing environments.

    ❤️ CAS Link

    • Organize a bias-awareness campaign in school.
    • Conduct a survey on perceptions of different school groups, then design activities to reduce stereotypes.
    • Reflect on personal experiences of being stereotyped or stereotyping others.

    🧠  IA Guidance

    Design an experiment showing illusory correlation or stereotype activation.
    Example:
    Participants read behavior statements about groups with varied ratios and estimate trait frequency.

    Ensure no sensitive social categories (ethnicity, religion) are used — keep it ethical.

    🧠 Examiner Tips

    • Always distinguish formation (how stereotypes originate) vs. effects (how they influence behavior).
    • Use Hamilton & Gifford (1976) for formation and Steele & Aronson (1995) for effects.
    • Link to cognitive biases (heuristics, illusory correlation).
    • Evaluation: discuss ecological validity, ethics, cross-cultural replication.

  • 🧠 Formation of Stereotypes and Their Effects on Behaviour

    📌 Key terms

    TermDefinition
    StereotypeA mental representation (schema) of a group and its members — a fixed, oversimplified belief often resistant to change.
    StereotypingThe process of assigning characteristics to someone based solely on group membership.
    Social CategorizationClassifying people into groups based on perceived similarities (e.g., gender, ethnicity, occupation).
    Illusory CorrelationPerceiving a relationship between two variables (e.g., group membership and behavior) even when none exists.
    Stereotype ThreatFear of confirming a negative stereotype about one’s group, which can negatively affect performance.
    Self-Fulfilling ProphecyWhen expectations about others lead them to behave in ways that confirm those expectations.
    Confirmation BiasTendency to interpret information in a way that confirms pre-existing beliefs.
    Grain of Truth HypothesisSuggests stereotypes may originate from small but real differences, later exaggerated or overgeneralized.

    📌 Notes

    Stereotypes are a key form of social cognition, shaping how individuals perceive and interact with others.
    They simplify complex social information, allowing people to categorize quickly — but often at the cost of accuracy and fairness.

    Stereotypes influence behavior, perception, and even memory, making them a central topic in both cognitive and sociocultural psychology.
    They can form automatically, through cognitive shortcuts (heuristics), or socially, via cultural norms and media.

    📌 Key Studies

    1️⃣ Social Categorization (Tajfel, 1971)

    • Categorization into in-groups and out-groups simplifies the world.
    • Once categorized, people accentuate similarities within groups and differences between groups.
    • Leads to overgeneralization — forming stereotypes.

    💡 Example: Seeing “all athletes” as extroverted or “all artists” as emotional.

    2️⃣ Illusory Correlation – Hamilton & Gifford (1976)

    Aim:
    To investigate how stereotypes form through illusory correlation between group membership and behavior.

    Procedure:

    • Participants read statements about two groups (A and B) performing positive or negative behaviors.
    • Group A (majority) performed more positive acts; Group B (minority) fewer acts overall, with the same positive-to-negative ratio.

    Findings:

    • Participants overestimated the number of negative behaviors by Group B.
    • They associated the minority group with negative traits, even though both groups behaved similarly.

    Conclusion:

    • Stereotypes can form through cognitive errors — humans notice distinctive events (minority + negative) and exaggerate correlations.

    Evaluation:
    ✅ High control → cause-and-effect link between cognition and stereotype formation.
    ⚠️ Artificial task → low ecological validity.
    ✅ Highly influential in cognitive origins of prejudice.

    3️⃣ Grain of Truth Hypothesis

    • Suggests that stereotypes may stem from real experiences with individuals from a group, then overgeneralized.
    • Example: Meeting one rude tourist → believing “all tourists are rude.”
    • Criticized for assuming all stereotypes have factual bases.

    🧠 Effects of Stereotypes on Behaviour

    1️⃣ Stereotype Threat – Steele & Aronson (1995)

    Aim:
    To test whether stereotype threat affects African-American students’ academic performance.

    Procedure:

    • Participants: African-American and White college students.
    • Two conditions:
      1. Diagnostic — test described as measuring intelligence.
      2. Non-diagnostic — same test, described as unrelated to ability.

    Findings:

    • African-American students performed worse in the diagnostic condition.
    • When the stereotype of “African-Americans perform poorly” was activated, anxiety impaired performance.

    Conclusion:

    • Awareness of stereotypes can negatively impact performance — a self-fulfilling prophecy effect.

    Evaluation:
    ✅ Strong experimental design → causal relationship.
    ⚠️ Cultural bias (US context).
    ✅ Replicated in gender studies (e.g., Spencer et al., 1999).

    2️⃣ Gender Stereotype Threat – Spencer et al. (1999)

    Aim:
    To investigate if women’s math performance is affected by stereotype activation.

    Procedure:

    • Male and female university students completed math tests under two conditions:
      1. Stereotype-activated: Told men perform better in math.
      2. Control: Told no gender differences exist.

    Findings:

    • In the stereotype-activated condition, women performed worse than men.
    • In the control condition, performance was equal.

    Conclusion:

    • Awareness of gender stereotypes can impair cognitive performance.

    3️⃣ Self-Fulfilling Prophecy – Rosenthal & Jacobson (1968)

    Aim:
    To investigate whether teacher expectations affect student performance.

    Procedure:

    • Teachers were told certain students were likely “academic bloomers” (randomly chosen).
    • Over the year, those students showed greater IQ improvement.

    Conclusion:

    • Expectations influence behavior of both perceiver and target — supporting stereotype effects.

    Evaluation:
    ✅ Real-world application in education.
    ⚠️ Ethical concerns (deception).
    ✅ Strong field evidence of stereotype impact.

    🔍Tok link

    Knowledge Question: “To what extent do labels shape our perception of reality?”

    Stereotypes reveal the interaction between knowledge and bias — how mental shortcuts (heuristics) simplify reality but distort truth.

    TOK discussion: Are stereotypes ever useful for understanding? Can categorization exist without bias?

     🌐 Real-World Connection

    Used to understand racial profiling, gender inequality, and media representation.

    Research has informed diversity training, inclusive education, and implicit bias interventions.

    Understanding stereotype threat helps design fair testing environments.

    ❤️ CAS Link

    • Organize a bias-awareness campaign in school.
    • Conduct a survey on perceptions of different school groups, then design activities to reduce stereotypes.
    • Reflect on personal experiences of being stereotyped or stereotyping others.

    🧠  IA Guidance

    Design an experiment showing illusory correlation or stereotype activation.
    Example:
    Participants read behavior statements about groups with varied ratios and estimate trait frequency.

    Ensure no sensitive social categories (ethnicity, religion) are used — keep it ethical.

    🧠 Examiner Tips

    • Always distinguish formation (how stereotypes originate) vs. effects (how they influence behavior).
    • Use Hamilton & Gifford (1976) for formation and Steele & Aronson (1995) for effects.
    • Link to cognitive biases (heuristics, illusory correlation).
    • Evaluation: discuss ecological validity, ethics, cross-cultural replication.

  • 🧠 SOCIAL COGNITIVE THEORY (SCT)

    📌 Key terms

    TermDefinition
    Social Cognitive Theory (SCT)A learning theory proposed by Albert Bandura suggesting that people learn behaviors, attitudes, and emotional reactions through observation, imitation, and modeling.
    Observational LearningLearning by watching the actions of others and the consequences of those actions.
    ModelingDemonstrating or imitating a behavior shown by another person (model).
    Vicarious ReinforcementWhen an individual learns by observing the rewards or punishments of others.
    Self-EfficacyOne’s belief in their ability to perform a specific behavior successfully.
    Reciprocal DeterminismThe idea that behavior, personal factors (cognitive), and environment all influence each other.
    RetentionRemembering the observed behavior for later imitation.
    MotivationThe desire to reproduce observed behavior, often influenced by anticipated outcomes or identification with the model.

    📌 Notes

    Social Cognitive Theory (SCT) expands upon the earlier Social Learning Theory (SLT), emphasizing that learning occurs in a social context and involves reciprocal interactions between behavior, personal cognition, and environment.

    Albert Bandura (1986) proposed that humans learn indirectly — by observing others, mentally processing information, and then deciding whether to imitate.
    This model explains how behaviors like aggression, empathy, or gender roles are acquired without direct reinforcement, purely through social observation.

    Mechanisms of SCT

    1. Attention
      • To learn through observation, individuals must pay attention to the model.
      • Factors increasing attention: model’s attractiveness, authority, or similarity to observer.
      • Example: A child notices an admired teacher’s calm reaction under stress.
    2. Retention
      • Observed behaviors are stored in memory as mental representations.
      • Visualization and verbal coding (mental rehearsal) help with later reproduction.
    3. Reproduction
      • The ability to perform the behavior depends on physical and cognitive capabilities.
      • Example: A child may observe a gymnast but cannot yet reproduce the moves.
    4. Motivation
      • Individuals imitate behaviors they believe will lead to positive outcomes (rewards, approval).
      • Vicarious reinforcement (observing someone else being rewarded) increases motivation.
    5. Self-Efficacy
      • Belief in one’s own capability to perform the behavior successfully.
      • High self-efficacy leads to persistence; low self-efficacy reduces imitation likelihood.

    📌 Key Studies

    “Bobo Doll Experiment”

    Aim:
    To investigate whether children learn aggression through observation of adult models.

    Procedure:

    • 72 children (aged 3–6) divided into three groups:
      1. Observed an aggressive model
      2. Observed a non-aggressive model
      3. Control (no model)
    • After observation, children were allowed to play with toys, including a Bobo doll.

    Findings:

    • Children who observed the aggressive model showed more aggressive behaviors.
    • Boys imitated physical aggression more, girls imitated verbal aggression more.
    • Aggression was higher if the model was of the same sex.

    Conclusion:

    • Behavior can be learned through observation and imitation, even without direct reinforcement.
    • Identification with the model strengthens learning.

    Evaluation:
    ✅ High control → cause-and-effect inference.
    ⚠️ Low ecological validity (lab setting, toys).
    ⚠️ Ethical concerns — exposing children to aggression.
    ✅ Groundbreaking study → strong empirical support for SCT.

    🔬 Key Study 2: Perry, Perry & Rasmussen (1986)

    “Gender, Aggression, and Self-Efficacy”

    Aim:
    To investigate gender differences in aggression and perceptions of self-efficacy.

    Procedure:

    • Sample: Elementary school children.
    • Questionnaires measured beliefs about aggression, reinforcement, and self-efficacy in controlling aggression.

    Findings:

    • Boys were more likely to believe that aggression leads to positive outcomes.
    • Higher self-efficacy correlated with greater likelihood of imitating aggressive models.

    Conclusion:

    • Supports SCT — aggression is learned via observation, and self-efficacy influences whether the behavior is reproduced.

    Evaluation:
    ✅ Real-world application of SCT (gender role learning).
    ⚠️ Correlational → no causation.
    ✅ Good external validity; aligns with Bandura’s core principles.

    🔍Tok link

    Knowledge question: “To what extent do we learn who we are from observing others?”

    SCT explores how knowledge and behavior are socially constructed.

    Raises epistemological questions: Is imitation a form of understanding or replication?

    Can moral or cultural behavior be learned without personal experience?

     🌐 Real-World Connection

    Explains how children learn violence, gender norms, and social skills through media and family.

    Forms the basis for media regulation, anti-bullying programs, and positive role model campaigns.

    Used in health psychology: modeling healthy eating, safe sex practices, or quitting smoking.

    Education: Teachers model curiosity, empathy, and perseverance.

    ❤️ CAS Link

    • Design a peer-mentoring project where older students model good study habits.
    • Volunteer for a youth teaching initiative — reflect on how modeling affects learning.
    • Participate in theatre workshops exploring role modeling and social learning.

    🧠  IA Guidance

    An IA can test observational learning or vicarious reinforcement.
    Example: Participants observe one model rewarded vs. unrewarded for a behavior, then measure imitation.

    Keep ethical standards — avoid harm or deception and ensure debriefing.

    🧠 Examiner Tips

    • Always define SCT clearly: Learning via observation and imitation within a social context.
    • Mention Bandura’s 4 processes (attention, retention, reproduction, motivation).
    • For top marks, connect SCT to self-efficacy and reciprocal determinism.
    • Use Bandura (1961) as your primary study and one modern application (e.g., Perry et al. or Charlton, 2002).
    • Evaluate issues of ecological validity and ethics.

  • 🧠 SOCIAL IDENTITY THEORY (SIT)

    📌 Key terms

    TermDefinition
    Social Identity Theory (SIT)A theory developed by Tajfel & Turner (1979) proposing that an individual’s self-concept is derived from perceived membership in social groups.
    Social CategorizationThe process of classifying individuals (including oneself) into groups based on shared characteristics.
    Social IdentificationAdopting the identity, norms, and values of the group one belongs to.
    Social ComparisonComparing one’s group (in-group) to others (out-groups) to maintain positive self-esteem.
    Positive DistinctivenessThe motivation to show that the in-group is better or more valuable than the out-group, to enhance self-image.
    In-group BiasFavoring one’s own group over others, often leading to discrimination.
    Out-group DiscriminationNegative treatment or attitudes towards those not belonging to one’s group.
    Minimal Group ParadigmA method of studying intergroup bias where participants are divided into meaningless groups to test minimal conditions for discrimination.

    📌 Notes

    Social Identity Theory (SIT) explains how belonging to groups shapes our thinking, emotions, and behaviour.
    According to Tajfel & Turner (1979), humans are motivated to achieve a positive self-concept, and much of that self-concept comes from group identity.
    To maintain self-esteem, people tend to view their own groups (in-groups) as superior to others (out-groups), even when group membership is arbitrary or meaningless.

    This phenomenon can explain prejudice, discrimination, nationalism, sports rivalry, and social cohesion.

    Mechanisms of SIT

    1. Social Categorization
      • We categorize people (and ourselves) into groups based on perceived similarities.
      • Examples: nationality, school house, gender, religion, ethnicity.
    2. Social Identification
      • We adopt the identity of the group we belong to, including shared values and norms.
      • This strengthens belonging and emotional connection to the group.
    3. Social Comparison
      • We compare our group (in-group) to others (out-groups).
      • If our group is perceived as better, our self-esteem rises.
      • If not, we may derogate other groups or change group membership to maintain self-esteem.
    4. Positive Distinctiveness
      • We strive to make our group positively distinct — unique, successful, or superior — to protect social identity.

    📌 Key Studies

    🔬 Key Study 1: Tajfel et al. (1971)

    “Minimal Group Paradigm”

    Aim:
    To investigate whether minimal conditions are sufficient to cause in-group favoritism and out-group discrimination.

    Procedure:

    • 64 British schoolboys (aged 14–15) were randomly divided into two groups (Klee vs Kandinsky) based on preference for abstract paintings.
    • Participants allocated money to anonymous in-group and out-group members using matrices that represented reward trade-offs.

    Findings:

    • Boys consistently gave more rewards to members of their own group.
    • They even sacrificed absolute gains to ensure their group was better off (maximizing difference).

    Conclusion:

    • In-group bias occurs even with arbitrary group distinctions.
    • Supports SIT: group categorization alone is enough to produce discrimination.

    Evaluation:
    ✅ High control → strong internal validity.
    ⚠️ Lacks ecological validity (artificial task).
    ⚠️ All-male, Western sample → limited generalizability.
    ✅ Groundbreaking paradigm → influenced decades of intergroup research.

    🔬 Key Study 2: Cialdini et al. (1976)

    “Basking in Reflected Glory (BIRG)”

    Aim:
    To examine how group success influences self-esteem and identity.

    Procedure:

    • Observed U.S. university students after football matches.
    • Recorded frequency of wearing university apparel and pronoun use (“we won” vs “they lost”).

    Findings:

    • After victories, students used “we” more often and wore university symbols more.
    • After losses, they distanced themselves (“they lost”).

    Conclusion:

    • People associate with successful groups to enhance social identity and self-esteem (BIRGing).
    • Supports SIT’s idea of maintaining positive distinctiveness.

    Evaluation:
    ✅ High ecological validity.
    ✅ Demonstrates SIT in natural contexts.
    ⚠️ Correlational → cannot prove causation.
    ⚠️ Limited to Western sports culture.

    🔍Tok link

    SIT raises questions about how knowledge of identity is constructed:

    • Is group identity a social construct or a biological predisposition?
    • How do language and shared narratives shape “us vs them” distinctions?

    TOK discussion: To what extent do we need groups to define ourselves?

    SIT also links to Ethics in knowledge: when does social categorization become stereotyping or discrimination?

     🌐 Real-World Connection

    Explains nationalism, sports rivalry, workplace hierarchies, and ethnic conflict.

    Used in programs promoting diversity and inclusion, anti-bullying campaigns, and peace education.

    Also informs marketing (brand loyalty) and politics (group polarization).

    ❤️ CAS Link

    • Design a school event encouraging intergroup cooperation (e.g., mixed-group sports tournaments).
    • Volunteer in community integration projects promoting empathy and cultural awareness.
    • Reflect on group belonging in CAS journals to connect SIT to personal identity formation.

    🧠  IA Guidance

    SIT can inspire IAs testing ingroup bias or conformity.

    Example: experiment testing whether people allocate more points to members of their own class group vs. another class.

    Use ethical procedures—ensure anonymity, avoid real conflict, and debrief participants.

    🧠 Examiner Tips

    • Always define social identity theory clearly and mention three mechanisms (categorization, identification, comparison).
    • In essays, refer to at least two studies (Tajfel & Cialdini recommended).
    • For top marks, evaluate cultural bias, ecological validity, and SIT’s reductionism.
    • Connect SIT to real-world discrimination or group behaviour examples in conclusions.

  • AHL 2.2 FUNCTIONS, DOMAIN & INVERSE FUNCTIONS

    📌 Purpose: Understand what a function is, how to express it, determine domain and range, interpret graphs, and find/check inverse functions. Clear definitions, short examples, and practical tips for exams and calculators.

    Term Definition / Note
    Function A rule f that assigns to each input x in the domain exactly one output f(x). Notation: f(x), v(t), C(n) etc.
    Domain Set of permitted inputs (x-values) for which f(x) is defined. Example: for f(x)=√(2 − x), domain x ≤ 2.
    Range Set of possible outputs f(x). Example above: range f(x) ≥ 0.
    Graph of a function Visual representation: points (x, f(x)). Useful to see continuity, domain/range, intercepts and inverse reflection across y = x.
    One-to-one (injective) Each y in range is image of exactly one x. Horizontal line test: any horizontal line intersects graph at ≤ 1 point.
    Inverse function f−1 Function that reverses f: f−1(y) = x iff f(x) = y. Exists only for one-to-one functions; domain of f−1 = range of f.

    📌 Determining domain & range (quick rules)

    • Polynomials: domain all real numbers (ℝ). Range depends on degree and turning points.
    • Rational functions: exclude x where denominator = 0 (vertical asymptotes).
    • Roots: for even roots (√), require expression inside ≥ 0 (or solve inequality to find domain).
    • Logarithms: argument must be > 0.
    • Graphical approach: domain = projection of graph on x-axis; range = projection on y-axis.

    📌 Inverse functions: meaning & how to find

    1. Meaning: f−1 undoes f: if y = f(x) then x = f−1(y). Graphically, reflect graph of f across line y = x.
    2. Algebraic method (standard):
      1. Write y = f(x).
      2. Swap x and y: x = f(y).
      3. Solve for y in terms of x; result is f−1(x).
    3. Existence: Only invertible if one-to-one. If not one-to-one, restrict domain to a region where it is one-to-one (common for sqrt, sin, etc.).
    4. Check: Verify f(f−1(x)) = x and f−1(f(x)) = x on valid domains.

    📌 Short examples

    Example 1 (domain & range): f(x) = √(2 − x). Domain: x ≤ 2. Range: f(x) ≥ 0.

    Example 2 (finding inverse): f(x) = 2x + 3. Let y = 2x + 3. Swap: x = 2y + 3 ⇒ y = (x − 3)/2. So f−1(x) = (x − 3)/2. Check: f(f−1(x)) = x.

    Example 3 (non-invertible without restriction): f(x) = x2 is not one-to-one on ℝ (fails horizontal line test); restrict to x ≥ 0 to get inverse f−1(x) = √x.

    🧮 GDC Tips

    • Plot f(x) to inspect domain/range visually and to test one-to-one property (horizontal line test).
    • Use solver or algebra tools to swap x & y and solve for inverse; if GDC gives numeric inverse, also show algebraic steps in exam work.
    • For inverse check numerically: compute f(f−1(a)) and f−1(f(a)) for sample a in domain to validate.

    🧠 Examiner Tip

    When asked for domain/range show brief reasoning (e.g., set inside √ ≥ 0 or denominator ≠ 0). For inverses: explicitly state domain restriction if needed and demonstrate both compositions f(f−1(x)) and f−1(f(x)).

    📝 Paper tips

    • State domain explicitly and show algebraic steps (not just final interval).
    • In context questions (temperature/currency), explain units and meaning of inverse (e.g., converting back).

    ⚠️ Common pitfalls

    • Forgetting to restrict domain for inverse when function is not one-to-one.
    • Confusing domain and range — remember domain = input x-values, range = outputs f(x).
    • Not checking compositions to confirm inverse.

    📌 Quick checklist

    • Identify expression type (polynomial, rational, root, log) → apply domain rules.
    • Find range by algebra or by analyzing graph/transformations.
    • To find inverse: swap x & y, solve for y, then restrict domain if necessary.
    • Always check inverses via composition f(f−1(x)) and f−1(f(x)).

  • AHL 2.1 Straight Lines and Gradients

    📌 Purpose: Learn the common forms of a straight line, compute gradients, work with intercepts, test parallelism and perpendicularity, and apply these ideas to real-world inclines and simple modelling.

    Term Definition / Note
    Gradient (m) Slope of the line: m = (y2 − y1) ÷ (x2 − x1). Rise over run; positive → uphill to the right, negative → downhill to the right.
    y = mx + c Gradient-intercept form: m = gradient, c = y-intercept (value where x = 0).
    ax + by + d = 0 General form. Convert to y = mx + c by isolating y: y = (−a/b)x − d/b (if b ≠ 0).
    Point-gradient form y − y1 = m(x − x1). Use when a point and gradient are known.
    Parallel lines Two lines are parallel ⇔ their gradients are equal: m1 = m2.
    Perpendicular lines Two lines are perpendicular ⇔ m1 × m2 = −1 (i.e., m2 = −1/m1, assuming neither is vertical).
    Vertical / horizontal lines Vertical: x = k (undefined gradient). Horizontal: y = k (m = 0).

    📌 How to compute gradient (3 quick ways)

    • From two points (x1, y1) and (x2, y2): m = (y2 − y1)/(x2 − x1). Always subtract in the same order.
    • From y = mx + c: read off m directly.
    • From ax + by + d = 0 (b ≠ 0): rearrange: y = (−a/b)x − d/b → m = −a/b.

    📌 Common forms & quick conversions

    • Gradient–intercept: y = mx + c — easiest to interpret (m slope, c y-intercept).
    • Point–gradient: y − y1 = m(x − x1) — ideal when you know one point and slope.
    • General: ax + by + d = 0 — convert to y = mx + c by isolating y (if possible).
    • Two-point to equation: Given two points, compute m then use point-gradient to get equation.

    📌 Short worked examples

    Example 1 (gradient from points): Points (1,2) and (4,8): m = (8 − 2)/(4 − 1) = 6/3 = 2. Equation using (1,2): y − 2 = 2(x − 1) ⇒ y = 2x (c = 0).

    Example 2 (perpendicular): Line L has m = 3. A perpendicular line has m = −1/3. Use point-gradient to form equation if a point is given.

    🧮 GDC Tips

    • Use two-point input or line-fit routines (if available) to get equation quickly from data points.
    • Use matrix or function tools to convert between forms (solve for y to get m and c).
    • Check gradient numerically by computing (y2 − y1)/(x2 − x1) on the calculator to avoid arithmetic mistakes.

    🧠 Examiner Tip

    Write clean steps: show how you computed m, which form you used, and how you rearranged to the requested form. For perpendicular questions explicitly show the negative reciprocal step.

    📝 Paper tips

    • Paper 1: quick calculations — label points and show subtraction order for slope.
    • Paper 2: interpret gradient in context (e.g., incline steepness, rate of change).

    ⚠️ Common pitfalls

    • Mixing up the order of subtraction in slope formula — always (y2 − y1)/(x2 − x1).
    • For vertical lines (x = k), do not attempt to compute m (it is undefined).
    • When testing perpendicularity ensure neither line is vertical before using negative reciprocal; if one is vertical, the other must be horizontal (m = 0).

    📌 Quick checklist

    • Identify knowns: two points / point & gradient / equation coefficients.
    • Compute gradient using correct formula and order.
    • Write equation in requested form (y = mx + c / ax + by + d = 0 / point-gradient).
    • For parallel/perpendicular: compare gradients (equal / negative reciprocal).

  • 1.15 Eigenvalues and Eigenvectors

    AHL 1.15: EIGENVALUES & EIGENVECTORS

    📌 Key terms & quick notes

    Term Definition / Notes
    Eigenvalue (λ) Scalar λ for which a non-zero vector v exists with A v = λ v. Solve via characteristic polynomial det(A − λ I) = 0.
    Eigenvector (v) Non-zero vector direction preserved by A (only scaled). Eigenspace = nullspace of (A − λ I).
    Characteristic polynomial p(λ) = det(A − λ I). For 2×2: p(λ) = λ2 − (tr A) λ + det A.
    Trace / Determinant tr A = sum diag entries = λ1 + λ2. det A = product λ1 × λ2.
    Algebraic / Geometric multiplicity Algebraic = root multiplicity in p(λ). Geometric = dimension of eigenspace (≤ algebraic).
    Diagonalizable A = P D P−1 with independent eigenvectors as columns of P. Distinct eigenvalues ⇒ diagonalizable (2×2).

    🔢 GDC Tips

    • Use matrix → eigen routines to compute eigenvalues and eigenvectors quickly.
    • For powers An, diagonalize (A = P D P−1) then Dn just raises diagonal entries.
    • Verify results by computing A v − λ v; numerical residual should be ≈ 0. Use rational/exact mode when available.

    📌 Geometric intuition

    • Direction preservation: eigenvectors are directions that A maps onto the same line (scaled by λ). They are the “axes” along which the linear transformation purely stretches/compresses (or flips).

    • Magnitude effect: |λ| < 1 ⇒ decay along that mode; |λ| > 1 ⇒ growth. λ negative ⇒ flip sign (mirror) plus scaling. Complex λ give rotation+scaling.

    • Modes: decomposing an initial state into eigen-components isolates independent behaviours: each component evolves as λt (discrete) or eλ t (continuous).

    📌 Step-by-step computations & meaning

    1. Form A − λI and compute det(A − λI): detects λ that make matrix singular → non-trivial solutions for v.
    2. Solve p(λ)=0: yields the scaling factors (modes) — algebraic roots that indicate possible behaviours.
    3. Solve (A − λI) v = 0: finds eigendirections v (geometric meaning: preserved lines).
    4. Diagonalize: A = P D P−1 means change to eigenbasis where A acts as scaling (D). This simplifies repeated application and interpretation.
    5. Use decomposition to compute An: Dn = diag(λin) so long-term behaviour dominated by eigenvalues with largest |λ|.

    📌 Holistic example — two-town transition

    Model: xt = [At, Bt]T, At+1/Bt+1 = A × xt with

    A = [[0.90, 0.20], [0.10, 0.80]]

    1. Characteristic: p(λ) = λ2 − (tr A) λ + det A. Here tr A = 1.70; det A = 0.70 → p(λ) = λ2 − 1.70λ + 0.70.
    2. Solve → λ1=1, λ2=0.70. Interpretation: λ=1 gives steady-state (persistent mode); 0.70 decays (transient).
    3. Eigenvector for λ=1 (solve (A−I)v=0) gives long-term population fractions; initial conditions different only affect transient coefficients.

    Conclusion: eigen-analysis separates steady distribution (end behaviour) from decaying components (how fast equilibrium is reached).

    🧮 IA Tips

    Use a simple transition dataset (migration, market share) so you can compute eigenpairs and perform sensitivity (±δ changes). Explain model assumptions and validate with GDC and hand-calculations. Discuss limitations and measurement error impact on eigenvalues/eigenvectors.

    📌 Stability & continuous analogues

    • Discrete: xt+1 = A xt stable if all |λ| < 1; dominated by largest |λ| otherwise.
    • Continuous: x'(t) = B x(t) solved via eBt; eigenvalues of B (real parts) govern stability (Re(λ) < 0 stable).

    📌 Numerical cautions & exam checks

    • Close eigenvalues: ill-conditioning — small A changes alter eigenvectors strongly.
    • Repeated eigenvalue: check geometric multiplicity; if less than algebraic, not diagonalizable — show dimension of nullspace.
    • Floating point: prefer rational/exact results when possible; state rounding explicitly where needed.
    • Verification: always compute A v − λ v (or residual) to validate eigenpairs numerically.

    🧠 Examiner Tip

    Show determinant computation (det(A − λI)), factor the characteristic polynomial, and solve (A − λI)v = 0. If eigenvalues repeat, compute nullspace dimension and explain diagonalizability. When using the GDC, include a short algebraic justification to secure method marks.

    🔍 TOK Perspective

    How do idealizations (linearity) shape what we consider knowledge? Discuss assumptions, model domains and ethical consequences of applying simplified linear models to policy decisions.

    🌍 Real-World Connection

    Markov chains, population transitions, Google’s PageRank, vibration modes in engineering — all use eigen-analysis to separate persistent modes from transients and to identify dominant behaviour.

    📌 Glossary & symbols

    Symbol Meaning
    λ Eigenvalue (scalar)
    v Eigenvector (column)
    P Matrix of eigenvectors (columns)
    D Diagonal matrix of eigenvalues
    An Matrix power — use diagonalisation for efficient computation

    📝 Exam Strategy

    Practice forming det(A − λI), solving polynomials, and computing eigenspaces. If using GDC, display the algebraic steps too (polynomial factorisation, nullspace dimension). When eigenvalues repeat, explicitly show geometric multiplicity check.

  • AHL 1.14 MATRICES

    AHL 1.14: MATRICES

    📌 Key Definitions

    Term Definition Example
    Matrix A rectangular array of numbers arranged in rows and columns. Example: [1 2]
    [3 4]
    Element The number at the intersection of a row and column in a matrix (notation: Aij). 2 is element (1,2) of [1 2]
    [3 4]
    Order The matrix’s size: number of rows × number of columns (m × n). Above matrix is order 2 × 2
    Identity Matrix Square matrix with 1s on diagonal and 0s elsewhere; acts as the multiplicative identity. I = [1 0]
    [0 1]
    Zero Matrix Matrix with all entries equal to zero; additive identity. 0 = [0 0]
    [0 0]

    🔢 GDC & Technology Integration

    Use the calculator’s matrix editor to input matrices (confirm rows × columns). Use built-in determinant, inverse and linear-solve functions. Store matrices in memory to avoid re-entry; verify inverses by computing A × A−1 = I. For larger problems, spreadsheets or CAS systems allow symbolic manipulation and numerical stability checks.

    📌 Algebra of Matrices

    • Equality: A = B if orders match and Aij = Bij for all i,j.
    • Addition: (A + B)ij = Aij + Bij (defined only for same order).
    • Subtraction: (A − B)ij = Aij − Bij.
    • Scalar multiplication: (kA)ij = k × Aij.

    🔍 TOK Perspective

    Matrices compress relational information into compact structures. Discuss whether the abstraction hides assumptions (e.g. linearity) and how that affects trust in model outputs. Does mathematical compactness enhance or reduce explanatory power?

    📌 Multiplication of Matrices

    • To multiply A (m × n) by B (n × p): (AB)ij = Σk=1..n Aik × Bkj.
    • Multiplication is associative: (AB)C = A(BC), and distributive over addition, but generally not commutative (AB ≠ BA in general).
    Example: A = [1 2]
    [3 4], B = [2 0]
    [1 3] → AB computed entrywise:

    AB = [ (1×2 + 2×1) , (1×0 + 2×3) ]
    [ (3×2 + 4×1) , (3×0 + 4×3) ] = [4 , 6] [10 , 12].

    🌐 EE Focus

    An EE might investigate the stability of repeated matrix multiplication (e.g., Markov chains) or the effect of rounding error in iterative matrix methods — both rich topics linking linear algebra theory to numerical analysis.

    📌 Identity & Zero Matrices

    • Identity I: A × I = I × A = A for conformable sizes.
    • Zero 0: A + 0 = 0 + A = A.

    📌 Determinants & Inverses

    • 2 × 2 determinant: |A| = a d − b c for A = [a b]
      [c d].
    • Inverse (if |A| ≠ 0): A−1 = (1 ÷ |A|) × [d −b]
      [−c a].
    Example: A = [4 7]
    [2 6] → |A| = 24 − 14 = 10 → A−1 = (1/10)[6 −7]
    [−2 4].

    🧮 IA Tips & Guidance

    Use real datasets to model systems (economic input-output, population transitions) and demonstrate how singular matrices (|A|=0) reveal dependencies or lack of invertibility. Discuss limitations and numerical stability when computing inverses with technology.

    📌 Solving Linear Systems

    • Matrix form: A x = b, where A is coefficient matrix, x vector of unknowns, b constants.
    • If A−1 exists: x = A−1 b. If A singular, investigate consistency (no solution or infinitely many).

    🧠 Examiner Tip

    Practice both by-hand methods (2×2 inverses, determinant checks) and GDC workflows (matrix entry, inverse, solve). In exam answers, indicate the order and clearly state intermediate checks (e.g., A×A−1=I).