Topic 4: Calculus

Calculus is the one topic where a literal “same lesson, three programmes” plan isn’t honest: calculus is not IGCSE/GCSE content at all (confirmed explicitly in the Ecctis DP-vs-GCSE comparative report on file), and MYP deliberately treats it as pre-calculus rather than formal calculus. The three plans below are a genuine bridging sequence rather than three versions of the same lesson — each is flagged accordingly.

A. DP Mathematics: Analysis & Approaches — SL/HL, Year 1 (start) → Year 2 (completion)

Caveat: content below is drawn from the pre-publication draft syllabus (first assessment 2029, Topic E: Calculus). Verify against the final published guide once released.

Anchor concept: Limits and the derivative as a gradient function; introducing differentiation (syllabus ref. E1/E2). This lesson begins the Calculus strand in Year 1 per SOW_DP1_AAHL, which continues into Year 2 with integration and, at HL, differential equations and Maclaurin series — noted here as a forward link, not covered in this lesson.

Prior knowledge assumed: Functions (Lesson 2 above) — domain/range, function notation, and graph sketching; gradient of a straight line (MYP5/IGCSE).

Learning objectives

Command terms used

Find (derivative, gradient at a point); Determine; Evaluate (a limit, HL); Deduce (HL — the derivative rule from the first-principles pattern).

Assessment focus (2029 draft AOs)

Abstraction — recognising gradient-of-a-curve as a natural extension of gradient-of-a-line; Computation — accurate application of differentiation rules.

ATL / skills focus

Thinking (moving from an average rate of change, over a shrinking interval, to an instantaneous rate — a genuine conceptual leap); self-management (persisting through the abstraction before the mechanical rules make it feel easy).

Starter (≈10 min)

Show a curved graph (e.g. (y=x^2)) with two points marked, and ask students to calculate the gradient of the chord between them. Repeat with the two points closer together. Ask: “What do you notice happening to the gradient value?”

Main teaching sequence (≈35 min)

  1. Formalise the idea from the starter: as the second point approaches the first, the chord gradient approaches the gradient of the tangent at that point — this is the derivative. Connect to the notation (f’(x)) and (\frac{dy}{dx}).
  2. State (without full proof, for SL) the power rule: if (f(x)=x^n), (f’(x)=nx^{n-1}); extend to sums of terms. Several worked examples differentiating polynomials.
  3. Apply differentiation to find the gradient of a curve at a specific point, and to find where the gradient equals a given value (previews stationary points, covered in the next lesson of the unit).
  4. HL-only extension (last ~10 min): derive the power rule for (n=2) and (n=3) from the first-principles limit definition, making the abstraction from the starter fully rigorous. SL students instead complete extended practice differentiating a wider range of polynomial and simple root/reciprocal-power functions (e.g. (x^{1/2}), (x^{-1})) using the power rule.

Formative assessment / check for understanding

Mini whiteboard check: give a simple polynomial, ask for (f’(x)); check the class holds up matching answers before moving to point-evaluation questions.

Plenary / exit ticket (≈10 min)

“Find the gradient of the curve (y = x^3 - 4x) at the point where (x=2).” HL students additionally sketch what the tangent line at that point would look like relative to the curve.

Resources

GDC graphing to visualise the chord-to-tangent limiting process dynamically (see Use of GDC in 2026.pdf); Haese AA SL textbook chapter M14 (Integral/Differential Calculus) under IB DP Mathematics/PPT of SL Mathematics/ for further worked examples.

This lesson opens the Calculus strand; per SOW_DP2_AAHL, Year 2 completes it with integration techniques, kinematics/optimization applications, and — HL only — differential equations and Maclaurin series. Not part of this lesson; flagged here for continuity planning only.

This is the first formal calculus students encounter across the three programmes — MYP5 deliberately avoids formal calculus (see below) and IGCSE does not include it at all. The MYP optimization task below is explicitly designed by the MYP guide itself as this lesson’s bridge.

B. MYP Mathematics — Grade 10 (MYP Year 5)

Note: This is not a calculus lesson — MYP deliberately treats rate of change and optimization graphically/numerically via GDC, as an explicit pre-calculus bridge to DP. This substitution is a deliberate curricular choice, not an oversight.

Anchor concept: Rate of change and optimization using graphical/GDC methods — the can-optimization problem (minimizing surface area for a fixed volume), taken directly from the MYP mathematics guide’s own worked bridging example (Thinking with Models branch).

Prior knowledge assumed: Functions and graphing (Lesson 2 above); working with formulae for surface area and volume of a cylinder.

Learning objectives

Assessment criteria addressed

Criterion D (Applying mathematics in real-life contexts) — primary focus: modelling, selecting an appropriate technology-based strategy, and justifying the solution’s accuracy and sense in context; Criterion A (Knowing and understanding) — correct algebraic setup of the surface-area function.

Command terms used

Construct (the algebraic model); Justify (why the graphical minimum is the practical answer, and its accuracy); Interpret.

ATL / skills focus

Thinking skills (translating a physical constraint — fixed volume — into an algebraic relationship between two variables); research/self-management (using GDC systematically to narrow in on a minimum).

Starter (≈10 min)

Show a cylindrical can and ask: “If we need this can to hold exactly 330ml, is there a radius/height combination that uses the least metal to make it?” Students discuss intuitively (taller and thin vs short and wide) without calculating.

Main teaching sequence (≈35 min)

  1. Set up the model: volume constraint (V=\pi r^2h=330) gives (h) in terms of (r); surface area (S=2\pi r^2+2\pi rh); substitute to get (S) as a function of (r) alone.
  2. Use GDC to graph (S(r)) and trace/use the minimum-finding feature to locate the minimum point.
  3. Interpret: read off the radius that minimizes surface area, calculate the corresponding height, and discuss whether the “ideal” can matches real cans on shelves (it usually doesn’t — good discussion point about real-world constraints beyond pure optimization).
  4. Guided/independent second context (e.g. maximizing the area of a rectangular pen with fixed fencing) for students to model and solve graphically themselves, with a written justification of their answer’s reasonableness.

Formative assessment / check for understanding

Check each student’s algebraic model (before they graph) to catch substitution errors early — this is the step most likely to go wrong and undermines the rest of the task if uncaught.

Plenary / exit ticket (≈10 min)

“Explain, in your own words, what the minimum point on your graph represents in terms of the original can problem, and how confident you are that it’s exactly right using only a graph.” (Deliberately invites the observation that a graph only approximates the true minimum — sets up the DP calculus lesson’s promise of an exact method.)

Resources

Physical cylindrical can(s) for the starter; GDC with graphing/minimum-finding functionality.

This is the MYP guide’s own explicit bridge task: the identical can-optimization problem reappears in DP Year 1–2 (SOW_DP1/2_AAHL), solved exactly using calculus rather than graphically — worth referencing directly with students (“you’ll solve this exact problem again next year, a different way”).

C. IGCSE Mathematics — Grade 10

Note: Calculus is not IGCSE/GCSE content (confirmed in the Ecctis DP-vs-GCSE comparative report — Calculus is listed as a DP-only topic with no GCSE equivalent). The closest genuine IGCSE content is gradient/rate of change from distance-time and speed-time graphs, framed below explicitly as pre-calculus foundation rather than a calculus lesson.

Anchor concept: Gradient of a curve and rate of change from distance-time and speed-time graphs.

Prior knowledge assumed: Gradient of a straight line; reading and interpreting travel graphs at a basic level.

Learning objectives

Assessment objectives addressed

AO2 (reasoning and interpretation — reading meaning from a graph’s gradient and area); AO3 (solve problems in context — real travel scenarios).

Command terms used

Estimate (a tangent gradient by eye); Calculate (average speed from a chord); Describe (what a graph’s shape indicates about motion, e.g. accelerating vs constant speed).

Starter (≈10 min)

Show a distance-time graph with a curved section. Ask: “How would you estimate the speed at this exact instant, not just the average speed over the whole journey?”

Main teaching sequence (≈35 min)

  1. Recap gradient of a straight-line section of a distance-time graph as constant speed; contrast with a curved section where speed is changing.
  2. Introduce estimating instantaneous speed by drawing a tangent by eye at a point on the curve and calculating its gradient — explicitly named as an estimate, not an exact method.
  3. Move to speed-time graphs: gradient as acceleration (including negative gradient as deceleration), and area under the graph (including trapezium-rule style estimation for curved sections) as distance travelled.
  4. Combined practice: a real journey scenario (e.g. a train accelerating, cruising, decelerating) with both graph types, requiring students to describe the motion and calculate/estimate speed, acceleration, and distance at various stages.

Formative assessment / check for understanding

Peer-check tangent lines drawn on a shared curved graph — do different students’ tangents (and resulting gradient estimates) roughly agree? Use disagreement productively to reinforce that this is an estimate.

Plenary / exit ticket (≈10 min)

“On the distance-time graph provided, estimate the speed at t = 3 seconds by drawing a tangent. Explain why this is only an estimate and not an exact value.” (Directly plants the seed for exact methods to come.)

Resources

Printed distance-time and speed-time graphs (including at least one curved section); ruler for tangent-drawing.

This lesson’s honest limitation — “only an estimate, not exact” — is precisely the gap that DP Year 1 calculus (above) closes with the formal limit definition of the derivative; MYP5’s GDC-based optimization work sits at an intermediate point between this graphical estimation and DP’s exact algebraic method.