# Chapter 1 notebook: Algebra: See the Structure Before You Solve **Goal:** Choose a useful representation and decide whether a first-order estimate is accurate enough. **Start with:** Rules 1.1.1, 1.1.3, 1.3.7. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data. **Source:** [Finished Chapter 1](../skills/math-thumb-algebra/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions. ## 1. Frame your decision Write a question in which an answer would change something you do. Gather: Starting and ending values; whether change is additive or multiplicative; domain exclusions; required accuracy. - My question and intended decision: - Known inputs and units: - Required accuracy or threshold: - What I expect before calculating: - What I still need to find out: Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step. ## 2. Choose a route - **Is the question about change?** Use a ratio when the baseline matters. If the model is exponential, translate rate into doubling time. - **Is there a small, dimensionless quantity?** Match the surrounding function: binomial power, exponential, logarithm, or nearby radical. Estimate the first omitted correction. - **Is there a long sum or tail?** Check for constant spacing or a constant ratio before touching individual terms. - **Is magnitude enormous or multiplicative?** Move to a logarithmic representation. - **Is the object a polynomial?** Use the leading term for distant behavior. For a quadratic, normalize, inspect for factors, and compute the discriminant before choosing a full root procedure. For a higher-degree integer polynomial, use the rational-root theorem only as a finite screen. - **Is the object rational?** Factor the denominator. If decomposing a function, let the factors build the partial-fraction template. If solving an inequality, let the factor zeros build the sign chart. **Chapter-specific stop check:** A small-looking percentage is insufficient: compare the actual discarded effect with the requested tolerance. Preserve denominator exclusions and inequality signs. ## 3. Work the lab A component's response is proportional to the fifth power of its input. The input rises 2%. Estimate the response factor and check whether an absolute error of 0.005 in that factor is acceptable. First-order gives 1 + 5(0.02) = 1.10; exact evaluation gives 1.1040808032. The error is about 0.004081, so the estimate meets this particular tolerance. The quadratic correction, 10(0.02)^2 = 0.004, is an error scale; the exact comparison supplies the check here. Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime. ```python x, exponent, tolerance = 0.02, 5, 0.005 assert 1 + x > 0 and tolerance > 0 estimate = 1 + exponent*x exact = (1+x)**exponent correction = exponent*(exponent-1)*x*x/2 error = abs(exact-estimate) print(f"Factor estimate: {estimate:.6f}; exact: {exact:.9f}") print(f"Quadratic correction: {correction:.6f}; absolute error: {error:.6f}") print("Tolerance met:", error <= tolerance) assert exact > estimate ``` **My prediction, observed result, and explanation:** _Record your work here._ ## 4. Practise without the answers ### Exercise 1 Two groups each gain 60 members, one from 120 and one from 1,200. Compare the changes using Rule 1.1.1. **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 2 Repeat the lab at a 10% input increase with the same tolerance. Is the approximation still sufficient? **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 3 Solve (x+1)/(x-3) <= 0. Explain the failed shortcut of multiplying by x-3 without splitting signs. **My approach, assumptions, calculation, and check:** _Write your attempt here._ **Coaching prompt:** “Use the Chapter 1 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.” ## 5. Answer key and reasoning Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear. ### Answer 1 60/120 = 0.50, or 50%; 60/1,200 = 0.05, or 5%. The equal absolute gains answer a different question from proportional growth. ### Answer 2 The estimate is 1.5; exact value is 1.1^5 = 1.61051. Error 0.11051 exceeds 0.005, so compute exactly or keep more terms and bound the remainder. ### Answer 3 The numerator changes sign at -1 and the denominator at 3. The valid interval is [-1,3); x=3 is excluded. Multiplication reverses the inequality on x<3, so a single unsigned multiplication loses cases. ## 6. Build the complete chapter toolkit The new lab samples the chapter; the following checklist covers all 17 rules. Study one thematic group at a time. A large group can take several sessions. - **Change, Growth, and Local Approximation:** work with rules 1.1.1, 1.1.2, 1.1.3, 1.1.4, 1.1.5. - **Compress the Calculation and Expose Dominant Structure:** work with rules 1.2.1, 1.2.2, 1.2.3, 1.2.4, 1.2.5. - **Diagnose Structure Before Choosing a Procedure:** work with rules 1.3.1, 1.3.2, 1.3.3, 1.3.4, 1.3.5, 1.3.6, 1.3.7. For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source. Read each complete numbered profile in the [chapter reference](../skills/math-thumb-algebra/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below. | Rule | Book role | First assumptions to inspect | Practice status | |---|---|---|---| | 1.1.1: Compare proportional change with ratios, not differences | Independent | common reference quantity; nonzero denominator | new | | 1.1.2: Estimate doubling time from the exponential rate | Independent | constant exponential rate; positive growth rate | new | | 1.1.3: Linearize a small binomial perturbation | Independent | small parameter; dimensionless perturbation | new | | 1.1.4: Replace a small exponential by one plus its exponent | Independent | small argument; dimensionless argument | new | | 1.1.5: Replace log one plus x by x for small x | Independent | small parameter; argument greater than minus one | new | | 1.2.1: Average the endpoints of an arithmetic list | Independent | constant spacing; finite sequence | new | | 1.2.2: Estimate a geometric tail from its first omitted term | Independent | geometric sequence; ratio magnitude below one | new | | 1.2.3: Let the leading term predict polynomial end behavior | Independent | polynomial function; large argument | new | | 1.2.4: Rationalize a difference of nearby square roots | Independent | real square roots or consistent complex branch | new | | 1.2.5: Use logarithms to tame products and powers | Workflow | positive real factors or consistent complex branch | new | | 1.3.1: Make a quadratic monic before reading its structure | Workflow | nonzero leading coefficient | new | | 1.3.2: Complete the square to read a quadratic's geometry | Workflow | quadratic function | new | | 1.3.3: Factor before reaching for the quadratic formula | Workflow | quadratic polynomial; visible simple factors | new | | 1.3.4: Use the discriminant to triage quadratic roots | Workflow | quadratic polynomial | new | | 1.3.5: Use the rational-root theorem as a shortlist | Workflow | integer coefficients; nonzero leading coefficient | new | | 1.3.6: Let denominator factors dictate partial fractions | Workflow | proper rational function; factored denominator | new | | 1.3.7: Cross-multiply inequalities only after checking signs | Workflow | known denominator signs; nonzero denominators | new | For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key. ## 7. Apply it to your own problem Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic). - Selected rule number(s) and reason: - Assumptions that hold, fail, or remain uncertain: - Substitution with units: - Result and error, uncertainty, or bound: - Independent check or limiting case: - Decision this supports: - Stop here, do a named next calculation, or gather missing information: ## 8. Transfer and continue Useful nearby chapters: Chapter 2: geometry; Chapter 4: calculus; Chapter 20: numerical methods. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need. **Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.