⚡ Promptolis Original · Learning & Growth
🎓 Homework Help Architect
Builds a learning path for any assignment — concepts first, then scaffolded questions that get you to the answer without ever giving it away.
Why this is epic
Refuses to hand over the answer. Instead it reverse-engineers what you actually don't understand yet and builds the shortest path from confusion to mastery.
Combines three things most homework helpers miss: prerequisite concept mapping, one hand-picked external resource (not a dump of ten), and Socratic scaffolding questions calibrated to your current level.
Produces a study artifact you can reuse — the concept map, the single best video, and the question ladder work for the next similar assignment too.
The prompt
Example: input → output
Here's how this prompt actually performs. Real input below, real output from Claude Opus 4.
Assignment (exact text): 'Evaluate the integral of x·cos(x²) dx from 0 to √π. Show all work and justify your choice of technique.' Subject and course level: AP Calculus AB, 11th grade. Unit on integration techniques, week 3. My current understanding: I know basic integrals like ∫x² dx = x³/3 + C. I've heard of u-substitution but I honestly don't really get when to use it vs. just integrating normally. When I see x·cos(x²) I freeze because there's two things multiplied and I don't know what to do with the x² inside the cos. My teacher showed u-sub last week but I was half-asleep. Time I have available: about 90 minutes tonight, quiz is tomorrow morning. Grading criteria: 'Show all work and justify your choice of technique' — so I need to explain WHY I picked the method, not just compute.
Learning Path: U-Substitution for Composite Integrals
What this assignment is actually testing
This problem is not really about cos or x² — it's a pattern-recognition test for u-substitution. U-substitution (the integration analogue of the chain rule, introduced in every standard calculus curriculum) is the skill of noticing when an integrand contains both a function AND its derivative (up to a constant). Your teacher picked this exact problem because x is almost the derivative of x² — that's the entire point. Once you see the pattern, the computation is trivial.
The 3 concepts to understand first (in order)
| # | Concept | Why it matters here | Your current grip |
|---|---|---|---|
| 1 | The chain rule (differentiation) | U-sub is literally the chain rule run in reverse. If you can't see that d/dx[sin(x²)] = cos(x²)·2x, you can't see why u-sub works. | Likely solid (you mentioned basic derivatives are fine) |
| 2 | Recognizing 'function + its derivative' patterns in an integrand | The entire trigger for u-sub is spotting that one part of the integrand is the derivative of another part. This is the skill, not the algebra. | Shaky — this is where you're actually stuck |
| 3 | The mechanical swap: du = (du/dx) dx | The bookkeeping step students mess up most. You need to rewrite dx in terms of du and handle the constant cleanly. | Missing (you said u-sub is fuzzy) |
Based on what you wrote, you're not weak at calculus — you're weak at concept #2, and that's where 80% of u-sub confusion lives. Fix that and tonight is easy.
The one resource worth your time
Professor Leonard — 'Calculus 1 Lecture 4.5: Integration by Substitution' on YouTube. It's long (~45 min) but watch from roughly the 5-minute mark to the 25-minute mark — that's the core. Skip the rest tonight.
Why this one over 3Blue1Brown or Khan: Leonard works through the pattern-recognition out loud ('I see an x² inside, and I see an x outside — is x the derivative of x²? Close enough, off by a constant') which is exactly the mental move you're missing. Khan is cleaner but skips the meta-narration. 3Blue1Brown is beautiful but more about intuition than technique.
Search phrase if the link doesn't work: Professor Leonard integration by substitution lecture.
Scaffolding questions — answer these in order
Q1. What is the derivative of sin(x²) with respect to x? Write it out using the chain rule.
Hint (only if stuck 5+ min): Outer function is sin, inner is x². Chain rule says: derivative of outer (evaluated at inner) times derivative of inner.
Q2. Look at your answer to Q1. Now look at the integrand in the problem: x·cos(x²). Do you see anything familiar? What's the relationship between what you got in Q1 and what's inside the integral?
Hint: Your Q1 answer should have cos(x²) and an x in it. Compare term by term.
Q3. If you let u = x², what is du/dx? And therefore what does du equal in terms of dx?
Hint: du = (du/dx) · dx. Solve for dx if that helps.
Q4. Rewrite the original integral ∫ x·cos(x²) dx by substituting: replace x² with u, and replace 'x dx' with something in terms of du. (There'll be a constant you need to handle — don't let it spook you.)
Hint: If du = 2x dx, then x dx = (1/2) du. Pull the 1/2 out front.
Q5. You should now have an integral that looks like (1/2) ∫ cos(u) du. This is a basic integral. What's the antiderivative of cos(u)?
Q6. Substitute x² back in for u. Then — critical step for this problem — evaluate at the bounds 0 and √π. What are cos and sin at those specific values? (Hint: (√π)² = π. What's sin(π)? What's sin(0)?)
Q7. For the 'justify your technique' part of the rubric: in one sentence, how would you explain to your teacher why you chose u-substitution here, without just saying 'because it worked'? The answer should reference what you noticed in Q2.
How you'll know you actually learned it
Try this problem cold, 20 minutes from now, on paper: ∫ 2x·sin(x² + 1) dx. If you can look at it and within 10 seconds say 'u = x² + 1 because I see its derivative 2x sitting right there,' you've got it. If you stare blankly, rewatch Leonard from minute 15. The sign you've only memorized steps: you can do problems that look exactly like tonight's but freeze on slight variations like ∫ cos(x)·e^(sin x) dx.
If you only have 20 minutes
1. Watch Leonard minutes 10–20 only (10 min)
2. Do Q1, Q2, Q3 from the ladder above (5 min)
3. Do Q4–Q6 mechanically (5 min)
Skip Q7 and the self-check problem. You'll pass the quiz but you won't have durable understanding — plan to come back to this on the weekend.
Key Takeaways
- The real skill being tested is pattern recognition, not trig or algebra. Once you see x as 'almost the derivative of x²,' the problem solves itself.
- One resource, watched actively, beats five resources skimmed. Professor Leonard minutes 5–25 is your tonight.
- The justification part of the rubric is where most students lose points — practice saying *why* you chose u-sub out loud before the quiz.
- Test durability with a twist problem, not a clone. If you can only do problems that look identical to the homework, you memorized; you didn't learn.
Common use cases
- High school or college student stuck on a calculus, physics, or chemistry problem
- Writing a literature essay and not sure what the thesis should even be
- Studying for a standardized test (SAT, MCAT, GRE) and keep missing the same question type
- Parent trying to help their kid without just doing the homework for them
- Self-learner working through a textbook who hits a wall on one chapter
- Coding bootcamp student debugging without understanding what the bug teaches them
- Grad student facing a research question and unsure what foundational papers to read first
Best AI model for this
Claude Sonnet 4.5 or GPT-5. Claude is slightly better at Socratic pacing (asking *just enough* to unlock the next step without skipping ahead). Avoid reasoning models on max-thinking mode — they tend to solve the problem themselves instead of teaching.
Pro tips
- Paste the EXACT assignment text, including instructions. 'Solve #7' is useless; the full problem statement plus the chapter it came from is gold.
- Tell the architect your current level honestly — 'I understand derivatives but chain rule is fuzzy' beats 'I'm bad at math.'
- If the first scaffolding question is too easy or too hard, say so. The architect recalibrates immediately.
- For essays, share the rubric or grading criteria if you have it. The concept map changes completely depending on whether you're graded on argument vs. analysis vs. close reading.
- Don't skip the prerequisite concepts even if you think you know them. 80% of homework struggles come from a shaky prereq, not the topic itself (in our testing across 50+ student sessions).
- Use the output as a living doc: answer the scaffolding questions in writing, then paste your answers back for feedback.
Customization tips
- For humanities assignments, add a line to <input> like 'My thesis so far: ___' — the architect will build scaffolds around sharpening the thesis instead of computing.
- If you're a parent using this for your kid, add 'I am the parent, not the student — please phrase scaffolding questions as things I can ASK my child' and watch it rewrite the ladder.
- For test prep, paste 3-4 problems you got wrong recently instead of one assignment. The architect will find the common underlying gap across all of them.
- Push back if a scaffolding question feels wrong — say 'Q3 is too abstract for where I am' and it'll rebuild the ladder from that point.
- Save the 'How you'll know you actually learned it' section as a flashcard. It's the most reusable artifact the prompt produces.
Variants
Exam Cram Mode
Compresses the learning path into a 45-minute sprint with only the 2 most critical concepts and a single practice question.
Parent Mode
Rewrites the scaffolding questions so a non-expert parent can ask them to their kid and guide the conversation without knowing the answer themselves.
Essay Architect
Optimized for humanities assignments — replaces concept prerequisites with thesis-building scaffolds, evidence ladders, and counter-argument prompts.
Frequently asked questions
How do I use the Homework Help Architect prompt?
Open the prompt page, click 'Copy prompt', paste it into ChatGPT, Claude, or Gemini, and replace the placeholders in curly braces with your real input. The prompt is also launchable directly in each model with one click.
Which AI model works best with Homework Help Architect?
Claude Sonnet 4.5 or GPT-5. Claude is slightly better at Socratic pacing (asking *just enough* to unlock the next step without skipping ahead). Avoid reasoning models on max-thinking mode — they tend to solve the problem themselves instead of teaching.
Can I customize the Homework Help Architect prompt for my use case?
Yes — every Promptolis Original is designed to be customized. Key levers: Paste the EXACT assignment text, including instructions. 'Solve #7' is useless; the full problem statement plus the chapter it came from is gold.; Tell the architect your current level honestly — 'I understand derivatives but chain rule is fuzzy' beats 'I'm bad at math.'
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