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Week 8 · Lecture outline

Week 8 — Lecture Outline · Midterm Review & Exam

College Algebra · MATH 120 Fall 2026 · Prof. Calloway Fictional sample

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objectives covered: cumulative — Objectives 1–6 (Weeks 1–7). Obj 1 — real numbers, exponents & simplifying; Obj 2 — linear equations & inequalities (incl. absolute value); Obj 3 — functions: notation, domain & composition; Obj 4 — linear functions, graphs & systems; Obj 5 — polynomials & factoring; Obj 6 — quadratic equations.
SLOs touched: A (apply procedures accurately) · B (connect symbolic representation to interpretation)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.

This is a review-and-exam week — no new content. Each segment briskly re-teaches one objective from Weeks 1–7 with one worked example and the single misconception most likely to cost points, then the final segment frames the midterm itself. Built to be taught from cold as a review: an instructor (or a substitute) can run it without having taught the first seven weeks, because every definition and cure travels with the segment. The midterm covers Objectives 1–6; it does not reach the quadratic functions/graphs that begin in Week 9.

Week at a Glance
The week's big question "Across the whole first half — simplify, solve, functions, lines, factor, quadratics — what is the one core move each topic asks of us, and where does everyone slip?"
By the end of the week, students can… (1) run each objective's core move on demand — simplify with order-of-ops/exponent rules (Obj 1); solve linear equations, inequalities, and absolute-value equations (Obj 2); evaluate, find a domain, and compose functions (Obj 3); find a slope, write a line, and solve a system (Obj 4); multiply and factor polynomials (Obj 5); solve quadratics and read the discriminant (Obj 6); (2) name and avoid the highest-cost misconception in each objective; (3) walk into the Midterm knowing its format, its weight (20%), and a concrete preparation plan built around the Study Guide, the Exam-Prep Tutorial, and the Practice Exam.
Key vocabulary (all review) order of operations, exponent rules, power of a product, distributing a negative; linear equation, inequality & interval notation, absolute-value equation; function notation, domain, composition; slope, slope-intercept & point-slope, parallel/perpendicular, system (substitution/elimination); FOIL, special products, difference of squares, factoring; standard form, zero-product property, square root property, completing the square, quadratic formula, discriminant
Materials slides (Deck 8 — the review deck), the Study Guide, the Exam-Prep Tutorial (AI), the Practice Exam, Desmos (or GeoGebra) and a calculator, one approved chatbot (Gemini / Claude / ChatGPT) for the audit-the-AI review moment
Timing note 8 segments, ~150 min total. Session 1 (Tue) = Segments 1–4 (~75): Objectives 1–3. Session 2 (Thu) = Segments 5–8 (~75): Objectives 4–6 + the midterm frame. Scale to your own pattern.

Segment 1 — Hook & the Map of the First Half (8 min) · Session 1 opens

Hook. Put one expression on the board with no instruction: −2³ + (−3)². Ask: "What's this equal to? Is it 1, 17, or −17?" Let the room split. "If even the first line of the first week can trip us up on a sign, then today's job is clear: walk the whole first half once, fast, and find the exact spot in each topic where points quietly disappear." (Answer: −8 + 9 = 1.)

The promise (write it on the board): "By Thursday you'll be able to take any of the six big skills — simplify, solve, functions, lines, factor, quadratics — and on demand run the one core move it requires and dodge the one mistake that sinks it. That's the midterm."

The map (one slide, say it out loud — this is the photograph slide of the week):

Obj 1 — SIMPLIFY (order of ops, exponent rules). Obj 2 — SOLVE (equations, inequalities, absolute value). Obj 3 — FUNCTIONS (evaluate, domain, compose). Obj 4 — LINES & SYSTEMS (slope, equations, systems). Obj 5 — POLYNOMIALS (multiply, factor). Obj 6 — QUADRATICS (four methods, discriminant).

Why it matters line (memory hook): "Weeks 1–7 are one sentence: simplify honestly, solve carefully, read functions, command lines, factor cleanly, and crack quadratics."

Segment 2 — Objective 1 Review: Exponents & Simplifying (16 min)

Re-teach in plain language. Algebra starts by doing arithmetic in the right order with the right signs, and rewriting expressions without changing their value. Two things cost the most: the sign on a power and distributing a negative.

One worked example (do every step out loud):

Simplify and evaluate.
- −2³ + (−3)²: the minus in −2³ is outside the cube, so −2³ = −(8) = −8; (−3)² = 9 (even power → positive). Sum: −8 + 9 = 1.
- (2x²y)³: power of a product hits every factor — including the coefficient: 2³ = 8, (x²)³ = x⁶, y³ = y³ → 8x⁶y³.
- 3(2x − 4) − 2(x − 5) = 6x − 12 − 2x + 10 = 4x − 2 (−2 · −5 = +10).

Highest-cost misconception + cure:
- ❌ "(2x²y)³ = 2x⁶y³," and "−2(x − 5) = −2x − 10."
✅ Cure: raise every factor — cube the 2 to get 8. And distributing a negative flips every sign: −2 · −5 = +10. "The exponent and the minus both reach every term."

Segment 3 — Objective 2 Review: Solving Equations & Inequalities (20 min)

Re-teach in plain language. Solving means isolating x by undoing operations in reverse. Two wrinkles change the rules: inequalities flip on a negative multiply/divide, and absolute-value equations split into two cases.

One worked example (compute it live):

  • 4(x − 1) = 2x + 6 → 4x − 4 = 2x + 6 → 2x = 10 → x = 5. Check: 16 = 16 ✓.
  • −3x + 2 ≤ 11 → −3x ≤ 9 → divide by −3 and flip: x ≥ −3, i.e. [−3, ∞).
  • |2x − 3| = 7 → two cases: 2x − 3 = 7 → x = 5; 2x − 3 = −7 → x = −2 → x = 5 or x = −2. Both check.

Highest-cost misconception + cure:
- ❌ "−3x ≤ 9 means x ≤ −3," and "|2x − 3| = 7 has one answer, x = 5."
✅ Cure: dividing by a negative flips the inequality → x ≥ −3. And absolute value splits into two cases — never stop at one. "Negative divide flips; absolute value doubles."

Segment 4 — Objective 3 Review: Functions (12 min) · Session 1 closes (~75)

Re-teach in plain language. A function is a rule: one input, one output. Three moves — evaluate, find the domain, compose.

One worked example (do every step):

Let f(x) = x² − 2x, g(x) = x².
- f(−3) = (−3)² − 2(−3) = 9 + 6 = 15 (−2 · −3 = +6; put the input in parentheses).
- Domain of √(x − 4): the inside must be ≥ 0 → x − 4 ≥ 0 → x ≥ 4, [4, ∞) (x = 4 is allowed: √0 = 0).
- (f ∘ g)(2) = f(g(2)): do g first → g(2) = 4 → f(4) = 2(4) + 1… wait, use the stated f: here f(x) = x² − 2x, so f(4) = 16 − 8 = 8; if instead f(x) = 2x + 1, f(4) = 9. Emphasize: compose inside-out, and the order matters.

Quick interaction — rapid-fire "what's restricted?" (think-pair-share, ~5 min):
Put three functions on a slide; students name the domain restriction. √(x + 1) → x ≥ −1; 1/(x − 3) → x ≠ 3; x² − 5 → all reals. Vote, then debrief the difference between a radical (≥ 0 inside) and a denominator (≠ 0).

Highest-cost misconception + cure:
- ❌ "f(−3) = 9 − 6 = 3," and "(f ∘ g) means f first."
✅ Cure: −2(−3) = +6, so f(−3) = 15 — protect signs with parentheses. And (f ∘ g) does g first (read right-to-left). "Substitute in parentheses; compose inside-out."

Segment 5 — Objective 4 Review: Lines & Systems (20 min) · Session 2 opens

Hook back in: "Session 1 was simplify, solve, and functions. Now the workhorse of the course — lines and systems — and it's one of the two biggest slices of the midterm."

Re-teach in plain language. A slope and a point pin a line down. Perpendicular slopes are negative reciprocals. A system asks where two lines meet.

One worked example (compute it live):

  • Slope through (−1, 2) and (3, 10): (10 − 2)/(3 − (−1)) = 8/4 = 2 (rise over run — y's on top).
  • Line, slope −2 through (1, 5): y − 5 = −2(x − 1) → y = −2x + 7. Check (1, 5): −2 + 7 = 5 ✓.
  • Perpendicular to y = (1/4)x: negative reciprocal of 1/4 → −4.
  • System x + y = 7, x − y = 1: add → 2x = 8 → x = 4, then y = 3 → (4, 3).

Highest-cost misconception + cure:
- ❌ "Slope −2 through (1, 5), so y = −2x + 5," and "perpendicular slope is just the reciprocal, 4."
✅ Cure: 5 is the point's y, not the intercept — use point-slope → y = −2x + 7. And perpendicular is the negative reciprocal → −4. "Point-slope, then simplify; flip AND change the sign."

Segment 6 — Objective 5 Review: Polynomials & Factoring (18 min)

Re-teach in plain language. Multiplying is "every term times every term." Factoring runs it in reverse — and it's the engine for quadratics. The points-loser is the middle term of a squared binomial.

One worked example (do every step):

  • (x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10 (watch the middle sign).
  • (2x + 3)² = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9 (the middle term 2ab is required).
  • Factor: x² − 9 = (x − 3)(x + 3) [difference of squares]; x² + 6x + 9 = (x + 3)² [perfect square]; x² − x − 6 = (x − 3)(x + 2) [−3 and +2]; x² − 4x = x(x − 4) [GCF].

Highest-cost misconception + cure:
- ❌ "(2x + 3)² = 4x² + 9."
✅ Cure: the middle term 2·2x·3 = 12x is part of the square → 4x² + 12x + 9. "Square a binomial, keep the 2ab." And always pull the GCF first when factoring.

Segment 7 — Objective 6 Review: Quadratic Equations (22 min)

Re-teach in plain language. A quadratic can have two, one, or no real solutions. You have four methods, and the discriminant previews how many to expect.

One worked example (compute it live):

  • Factor & solve x² + 2x − 8 = 0 → (x + 4)(x − 2) = 0 → x = −4 or x = 2.
  • Repeated root x² − 6x + 9 = 0 → (x − 3)² = 0 → x = 3 (discriminant 36 − 36 = 0).
  • Discriminant of 2x² + 3x + 5 = 0: 9 − 40 = −31 < 0 → no real solutions.
  • Never divide by x: x² = 7x → x² − 7x = 0 → x(x − 7) = 0 → x = 0 or 7 (dividing by x loses x = 0).

Highest-cost misconception + cure:
- ❌ Solving x² = 7x by dividing by x, and "a negative discriminant means I made an error."
✅ Cure: never divide both sides by the variable — you delete a solution; move everything over and factor. And a negative discriminant is a valid conclusion: no real solutions. "Get to zero, then factor; the discriminant is a preview, not an error."

Segment 8 — The Midterm Frame: What's On It & How to Prepare (14 min) · Session 2 closes (~75)

Audit-the-AI review moment (the course's recurring habit, one last time before the exam — and a reminder that AI is not allowed on the exam itself):

Paste to an approved chatbot: "Solve |2x − 3| = 7, and find the domain of √(x − 4)."
Check it against what we taught. Chatbots often give only one case of an absolute-value equation (missing x = −2) and state a domain as x > 4 instead of x ≥ 4. The tool drafts; you judge. If you can catch the model here, you're ready.

What's on the Midterm (state it plainly — put it on the closing slide):
- Coverage: cumulative over Weeks 1–7, Objectives 1–6 — simplifying & exponents, equations & inequalities, functions, lines & systems, polynomials & factoring, and quadratic equations. It does not include quadratic functions/graphs (Week 9) or anything after.
- Weight & logistics: the Midterm is 20% of the course grade. The window opens Mon Oct 19 and the exam is due Sun Oct 25, 11:59 p.m. No AI on the exam. (There is no Quiz 8 or Assignment 8 — the midterm replaces them.)
- Format: 20 items, 100 points (5 each) — auto-gradable, application-skewed (simplify/solve/evaluate/factor in the spirit of the worked examples above), weighted toward Objectives 4 and 6.

The preparation plan (point at each artifact by name):
1. Study Guide — work it first; it's the checklist of every move in the six objectives.
2. Exam-Prep Tutorial — run it with an approved chatbot (Gemini / Claude / ChatGPT) and submit the share link; it drills the weak spots adaptively.
3. Practice Exam — sit it timed and AI-free, like the real thing, then review what you missed against the Study Guide.
4. Discussion 8 (the debrief) — after the exam, reflect on the one mistake that cost you the most and your fix going forward.

Callback + tease:
- Callback: "Every item on the exam is a move you already made in Weeks 1–7 — today we just named it and found where it slips."
- Tease next: "After the midterm, Week 9 turns the quadratic from something you solve into something you graph — the parabola, its vertex, its max or min. Completing the square comes back as the vertex tool."

Hand-off (the week's work): review the Study Guide, run the Exam-Prep Tutorial (share link), take the Practice Exam, sit the Midterm (due Sun Oct 25, no AI), and post Discussion 8 (the debrief, replies due Sun Oct 25).

Instructor FAQ — Common Stumbles (Review Week)

Student says / does Quick cure
"Is −2³ equal to 8 or −8?" −8 — the minus is outside the cube. Only (−2)³ cubes the negative, and that's also −8; but +8 is wrong here.
Forgets to cube the coefficient in (2x²y)³. Power of a product hits every factor: 2³ = 8 → 8x⁶y³.
Doesn't flip the inequality on a negative divide. Dividing/multiplying by a negative flips the sign: −3x ≤ 9 → x ≥ −3.
Solves 2x − 3
f(−3) = 9 − 6 = 3. −2(−3) = +6, so f(−3) = 9 + 6 = 15. Put the input in parentheses.
States a √ domain with > instead of ≥. The radicand can be 0 (√0 = 0), so use ≥: domain of √(x − 4) is [4, ∞).
Inverts the slope formula or uses a point's y as the intercept. Slope = rise/run (y's on top). Use point-slope for the equation, then simplify.
"Perpendicular slope is the reciprocal." It's the negative reciprocal — flip and change the sign.
(2x + 3)² = 4x² + 9. Keep the middle term: 2·2x·3 = 12x → 4x² + 12x + 9.
Solves x² = 7x by dividing by x. Never divide by the variable — you lose x = 0. Factor: x(x − 7) = 0 → x = 0 or 7.
"Negative discriminant means I made a mistake." No — it's a valid result: no real solutions. Compute b² − 4ac first as a preview.
Panics that the exam is "everything." It's Objectives 1–6 only (Weeks 1–7). Quadratic functions/graphs (W9+) are not on the midterm. Bound the studying.

Scope flag

This outline is pure review of Objectives 1–6 — no new material. The midterm covers Objective 6's quadratic-equations portion; quadratic functions and graphs (also Objective 6) are taught in Week 9 and appear on the final, not this midterm. The midterm and its bundle (Study Guide, Exam-Prep Tutorial, Practice Exam) are built separately and only referenced here by name. No readings are required this review week; revisit any Week 1–7 reading link from those modules if you want a refresher.

~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com