Lesson 7 — Algorithms: Step-by-Step Procedures

Summary

Learners describe an algorithm as a precise step-by-step procedure for a real problem — sorting a pile, splitting a bill fairly, finding the largest value — and analyze it for the two things that make an algorithm work: it must always finish (termination) and produce the right result (correctness). They meet the very word’s origin in the Persian scholar al-Khwarizmi, and see the procedure as something anyone can write and check.

Objectives

• Describe an algorithm as a step-by-step procedure for a real problem and analyze whether it works correctly. (D05.S2.12.02)

Connection

A recipe, a bus route, a rule for sharing a harvest fairly, the steps of long division — each is an algorithm: a precise list of steps that a person (or machine) can follow without judgment, and that ends with a result. The power and the risk are the same: because an algorithm runs exactly the steps you write, a missing step or a wrong order does real damage — so learning to read and test one is a survival skill in a world run by them.

Materials

• Algorithm sheet
• Math journal
• Low-tech: cards or stones to sort, a recipe or instruction list

Preparation

• Prepare two procedures (a sort and a fair split) and a deliberately flawed one to debug.
• Retrieval: from Lesson 6, row reduction was itself an algorithm (ordered row operations). Today we study procedures themselves.
• Prepare the worked example (trace a sort by hand) to model first (S-011).

Facilitator note

This lesson is written to the learner (“you”). The idea to land: an algorithm is a finite, ordered, unambiguous set of steps that turns an input into an output; we analyze it for termination (it always stops) and correctness (it stops on the right answer). Teach tracing by hand as the skill — read each step, write the state, repeat — with a worked example and guided practice (S-011). Debugging a flawed algorithm is the deepest learning; let learners find the flaw rather than only receiving it.

The ethics lens: an algorithm that is correct is a promise kept — but correctness is only one virtue, and a procedure can be correct yet cruel; naming that distinction prepares Lesson 8. The egalitarianism lens: because an algorithm is written in steps anyone can follow and check, it can be inspected and questioned by anyone — transparency is the egalitarian heart of procedure. The global lens: the word algorithm comes from the name of Muhammad ibn Musa al-Khwarizmi (c. 780–850, Baghdad), whose book on calculation by completion and balancing also gave us the word algebra (S-431, S-293) — a reminder that the idea of explicit procedure has deep, multi-rooted history. The technology lens: algorithms are what runs — every recommendation, every search, every payment. The environment lens: resource-sharing and allocation algorithms (who gets water, how a harvest is divided) decide real environmental outcomes, so their correctness is a matter of survival, not just arithmetic. Preview: Lesson 8 asks whether an algorithm is fair.

Procedure

1. Recall (5 min). From Lesson 6, row reduction was a set of ordered steps. Name other procedures you follow without thinking: a recipe, a route, a game’s rules.
2. Define it (8 min). An algorithm is a finite, ordered, unambiguous list of steps that takes an input and produces an output. It must terminate (always stop) and be correct (stop on the right answer).
3. Worked example — sort by hand (15 min). Trace this algorithm on the pile [3, 1, 2]: repeat — find the smallest remaining card and move it to the front — until the pile is empty. Show each pass: [1, 3, 2], then [1, 2, 3]. Check both properties: it always finishes (the pile shrinks each pass) and it is correct (each pass places the next smallest). This is selection sort.
4. Test it (8 min). Run the same algorithm on [2, 1] and on an empty pile. Does it still terminate? Is it correct? A good algorithm handles edge cases — the empty pile, the one-item pile.
5. Find the flaw (12 min). Debug a flawed fair-split algorithm: “give each person one item in turn, repeating until nothing is left, but never check who has had a turn.” Trace it on 5 items and 2 people. Where does it break? Rewrite the step so it is unambiguous and always finishes. With a partner, swap fixes and test each other’s.
6. Write your own (10 min). Choose a real problem (finding the tallest in a line, splitting a bill evenly, sharing water fairly) and write a precise algorithm. Trade with a partner and trace theirs by hand; report whether it terminates and is correct.
7. Close (2 min). Say — or write, sign, gesture, or use AAC to express — in one sentence each, what termination and correctness mean, and why a missing step matters.

Differentiation

• Support: Sort a physical pile following a given set of steps, then describe the steps you used in one sentence each.
• Extension: Write the algorithm so it also counts how many steps it takes, and compare the count on a sorted vs a reversed pile (a first taste of efficiency).
• Number access (dyscalculia): Provide a pre-printed trace table for step 3 with a “pile” box and a “next smallest” box, so the learner records each pass ([3, 1, 2] → [1, 3, 2] → [1, 2, 3]) by copying rather than comparing numerals; offload the counting (“find the smallest remaining”) to a partner; and take the physical route — sort real cards or stones by the rule and describe each pass aloud, checking termination and correctness in words without number work.
• Communication access: Every spoken step — saying, naming, reading aloud, discussing, or closing — can be done in writing, sign, gesture, or AAC instead. Learners who are non-speaking, d/Deaf, hard-of-hearing, or who use AAC complete every task in their preferred mode; no step requires producing or hearing sound.

Assessment

• Formative (peer + self): Can the learner describe an algorithm precisely (in speech, writing, sign, or AAC), trace it by hand, and judge termination and correctness — including finding a flaw?
• Portfolio artifact (unit): One written algorithm with a hand trace and a correctness check, in the math journal.

Home connection

Describe one procedure from home (cooking, packing, a game) as a numbered list, then test it by having someone follow it exactly. Write one sentence on what they did that you did not intend.

Resources

• On worked examples and guided practice: Kirschner, Sweller & Clark (2006) (S-011).
• On the origin of the word algorithm and of algebra in al-Khwarizmi’s work: Wikipedia — al-Khwarizmi (S-431); MacTutor — al-Khwarizmi (S-293).