## Chapter 1 — Mathematical reasoning ### Lesson map Lesson 01 published. Map, example, EX001. ```mermaid flowchart TD N0["Learning outcomes"] N1["1.1 The case behind the course"] N2["1.2 Proportions"] N3["1.3 Inequalities as targets"] N4["1.4 Checking claims"] N5["1.5 Orders of magnitude"] N6["Synthesis"] N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> N6 N6 --> EX["Worked example"] EX --> Q["Exercise EX001"] ``` ### Logical drawing ```mermaid flowchart LR N0["Learning outcomes"] N1["1.1 The case behind the course"] N2["1.2 Proportions"] N3["1.3 Inequalities as targets"] N4["1.4 Checking claims"] N5["1.5 Orders of magnitude"] N6["Synthesis"] N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> N6 --> Q["EX001"] ``` ```text [Learning outcomes] | v [1.1 The case behind the course] | v [1.2 Proportions] | v [1.3 Inequalities as targets] | v [1.4 Checking claims] | v [1.5 Orders of magnitude] | v [Synthesis] | v [Worked example] --> [Exercise EX001] ``` ### Learning outcomes You will (1) reason with counts, proportions and simple inequalities about devices; (2) check claims with arithmetic; (3) connect mathematics to the course case. ### 1.1 The case behind the course Aisha Rahman is a bachelor's student at the fictional Thornbury University. Her tutor, Dr Lewis Grant, links her to the fictional Bramley Housing Association, a charity with 45 staff laptops and four servers looked after by one IT officer, Gareth Lloyd. Last winter a laptop was stolen from a staff member's car. It was not encrypted and held tenant records, so Bramley had to report the loss and write to affected tenants. Gareth wants to know how many other laptops are in the same state and what a small charity can realistically do. Aisha's degree projects will follow this problem. ### 1.2 Proportions If 13 of 45 laptops lack encryption, the proportion is 13 divided by 45, about 29%. Proportions let Bramley compare itself over time and with guidance, even as the number of laptops changes. The same 13 laptops would be 13% of a fleet of 100 but nearly 29% of Bramley's 45, so the proportion carries the meaning that the raw count cannot. Aisha also writes the complement: 32 of 45 encrypted is about 71%, and the two figures must add to 100%. Checking that the parts add up is the simplest test that no device has been counted twice or left out of the total. ### 1.3 Inequalities as targets Targets are often inequalities: "at least 95% of laptops encrypted" or "no critical update older than 14 days". Writing them this way makes them checkable: either the measured value satisfies the inequality or it does not. Aisha notes that the direction of the sign matters as much as the number in a target. "At least 95% encrypted" means 43 of 45 laptops would pass, because 43 divided by 45 is 95.6%, while 42 laptops gives only 93.3% and fails. Writing the smallest passing count beside the percentage lets Gareth check the target without a calculator. ### 1.4 Checking claims A supplier claims its tool "fixes 90% of problems". Aisha asks: 90% of what, counted how, over what period? Mathematical reasoning starts with defining the quantity before believing the number. Applied to Bramley, the supplier's claim raises three concrete questions that Gareth can put in writing. Does a "problem" mean a missing update, a failed setting or a support ticket? Is the 90% measured on laptops like Bramley's or on large corporate fleets? And over what period was the figure taken, a month or a year? Without those answers the claim cannot be compared with Bramley's own counts. ### 1.5 Orders of magnitude With 45 laptops, one stolen laptop is about 2% of the fleet, but if it holds 3,000 tenant records, its impact is far larger than 2% suggests. Counting devices and counting affected people are different measures. Aisha therefore keeps two columns in her first summary table: devices affected and records exposed. The stolen laptop scores one in the first column and about 3,000 in the second. Managers who see only the device column could think the loss was minor, while the second column explains why letters went to tenants and the loss had to be reported. ### Worked example Gareth says "most laptops are fine". Aisha asks for the count. From his notes, 32 of 45 are encrypted: 71%. She rewrites his statement as "29% of laptops, 13 devices, are not encrypted", which makes the size of the problem clear to Bramley's managers. > **Synthetic case.** Aisha, Dr Grant, Gareth and Bramley are synthetic. ### Errors and limits Percentages of small numbers can mislead; always show the count too. ### Exercises - **EX001 — Case trigger.** What incident started Bramley's concern? *Rubric:* 3 = unencrypted stolen laptop with tenant records; 2 = partial; 1 = vague; 0 = none. - **EX002 — Proportion.** What share of 45 laptops is 13? *Rubric:* 3 = about 29%; 2 = partial; 1 = vague; 0 = none. - **EX003 — Two measures.** Why do device count and people affected differ? *Rubric:* 3 = one device can hold many records; 2 = partial; 1 = vague; 0 = none. ### Synthesis Define quantities clearly and show counts with proportions. ### Continue studying — chapter 01 of 40 [← Course home](../../README.md) · [Course chapter index](../SUBJECTS.md) · [Chapter 02 (draft) →](#ch-02) ## Worked answers **EX001.** A stolen, unencrypted laptop holding tenant records. **EX002.** About 29%. **EX003.** One device may hold thousands of people's records.