Lesson 10: Calculus Differentiation Capstone Portfolio Synthesis
Nine lessons of technique, one idea underneath all of it. The capstone is synthesis: build a problem that needs three rules at once, solve it, and be able to say what a derivative is without using the word.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Comprehensive synthesis of differentiation methods: power rule, trig/exp/log derivatives, product/quotient/chain rules, curve sketching, optimisation, and related rates.
Synthesise your complete Level 3 Calculus differentiation logbook, present a real-world optimisation engineering solution, and complete final mastery revision check.
🎥 Media Anchor & Pedagogical Scaffold
The Paradox of the Derivative — 3Blue1Brown
Video (16 min 50 sec, 3Blue1Brown, ⏱ 17 min): The celebrated visual essay on what a derivative is — instantaneous rate of change as a paradox resolved by limits. The right capstone: after nine lessons of rules and technique, this reconnects every method in your logbook to the single idea underneath.
🧠 1. Before Viewing (Activate & Predict)
How does calculus serve as the foundation for modern physics, engineering, economics, and computer science modelling?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 16 min 50 sec essay as a final capstone review. In your logbook synthesis notes, record:
- Rule mastery checklist: For each of the nine lessons completed, write one line summarising the key formula or concept (first principles and the power rule; trig, exponential and log derivatives; the product and quotient rules; the chain rule; tangents, normals and rates of change; stationary points and concavity; optimisation; related rates; exam technique). Rate your confidence 1–5 for each.
- Synthesis task: Design ONE original problem that combines at least 3 different differentiation techniques (e.g., product rule + chain rule + optimisation). Solve it completely and write it into your logbook.
- Reflection: What was the biggest insight you gained about derivatives? How would you explain the derivative concept to a Year 10 student?
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Facilitate logbook review and celebrate student mastery of Level 3 Calculus Differentiation.
Immediate Task: Final review of your Level 3 Calculus differentiation logbook, and an exam-readiness check against the standard.
⚡ Whakaoho | Do Now: Choosing the Rule (5 mins)
For each of x²eˣ, sin(3x), (x + 1)/(x − 1) and x⁵: name the rule you would use and why. Naming only. Rule selection under pressure is what the external actually tests.
📖 Activity 1: Build the Problem, Then Solve It (20 mins)
Design ONE original problem of your own that cannot be solved without at least three of this unit's methods — for example a composite function (chain rule) inside a product, optimised on a restricted domain, or a related rate whose geometry needs the quotient rule first. Write the problem, then solve it completely, showing every line. A problem you can state but cannot solve is not finished; a problem that only needs one rule is not a capstone.
📝 Activity 2: Trade, Solve, and Explain It Back (15 mins)
Swap your problem with a partner and solve theirs cold, with no hints. Then hand both solutions back and do the part that matters: explain to your partner what their problem was actually testing, and whether it tested what they intended. Finish by writing, in your logbook, one sentence saying what a derivative is — without using the word "derivative", and without writing a formula. If you can do that, the nine lessons have landed.
🎫 Exit Ticket: Name the Gap (5 mins)
Two lines. (a) The one method from this unit you would most want a worked example of before the external. (b) How you will get that worked example before your next lesson.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 alignment: Direct preparation for NCEA Level 3 Achievement Standard AS 91578, Apply differentiation methods in solving problems (external, 6 credits), against The New Zealand Curriculum (2007) Mathematics and Statistics Level 8 — Calculus. Emphasise complete algebraic working and a conclusion written in the context of the question, which is what separates Merit from Achieved.
Materials: The complete calculus logbook (this lesson is unusable without it), graphics calculator, Desmos, and blank paper for the original problem and its worked solution.
Pacing (67 mins): Do Now 5 · media anchor 22 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5. The clip is 16 min 50 sec — the longest in the unit. Set it as pre-viewing homework. Watching it in class leaves no room for the synthesis work, which is the point of the lesson.
Formative assessment — what to look for: Part (a), aggregated across the class, is your revision timetable. If four or more ākonga name the same method, reteach it to everyone; if the answers scatter across nine different methods, run a clinic instead. Either way the class has told you what to do next, which is more reliable than a mark out of ten.
Differentiation. Entry: An original problem combining two rules is enough. The skill being assessed is construction, not difficulty. On level: Three rules combined, solved completely, written up as it would be marked. Extension: Write the assessment schedule for your own problem — Achieved, Merit and Excellence criteria — then trade problems with a partner and mark theirs against the schedule they wrote.