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Week 1 · Lecture outline

Week 1 — Lecture Outline · Real Numbers, Exponents & Algebraic Expressions

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

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objectives covered: Objective 1 — Simplify algebraic expressions using the properties of real numbers, the order of operations, and the rules of integer exponents.
SLOs touched: A (apply procedures accurately) · B (connect symbolic/numerical representations)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.

Week at a Glance
The week's big question "What are the rules that never change — the ones that let us rewrite any expression without changing its value?"
By the end of the week, students can… (1) classify a number into the real-number sets (natural, whole, integer, rational, irrational); (2) evaluate any expression using the order of operations, including the −4² trap; (3) name and apply the properties of real numbers (commutative, associative, distributive, identity, inverse); (4) apply the integer-exponent rules (product, quotient, power, zero, negative) and simplify algebraic expressions by distributing and combining like terms.
Key vocabulary natural / whole / integer / rational / irrational / real numbers, order of operations, base, exponent, coefficient, term, like terms, commutative, associative, distributive, identity, inverse, product rule, quotient rule, power rule, zero exponent, negative exponent
Materials slides (Deck 1), the week's readings + video links, Desmos (or GeoGebra) and a calculator, one approved chatbot (Gemini / Claude / ChatGPT) for the AI-critique moment and the tutorial
Timing note 8 segments, ~150 min total. Session 1 = Segments 1–4 (~73). Session 2 = Segments 5–8 (~77).

Segment 1 — Hook & the Promise (8 min) · Session 1 opens

Hook. "Raise your hand if you've split a restaurant bill, doubled a recipe, or figured out a tip in the last week." Wait — most hands go up.
- "Every one of those is algebra. You took a rule and applied it to unknown numbers — the bill, the batch, the meal — without knowing the exact values in advance. That's what algebra is: arithmetic with the numbers left as letters."
- "This whole course is about doing that correctly and confidently — and Week 1 is the bedrock: the rules that never change, no matter what the letters stand for."

The promise (write it on the board): "By the end of this week you can take any messy expression — like 5x − 2(3x − 4) — and simplify it without fear, and you'll never again get tricked by whether −4² is −16 or 16."

Why it matters line (memory hook): "Algebra isn't a new kind of math. It's the same arithmetic you trust, with the rules made explicit so they keep working when the numbers are unknown."

Segment 2 — The Real Numbers and Their Sets (18 min)

Plain language first. Before we manipulate numbers, let's name the families they come in. Every number you'll meet this term is a real number — a point on the number line — but it belongs to smaller clubs too.

  • Natural numbers: 1, 2, 3, … (the counting numbers).
  • Whole numbers: the naturals plus 0.
  • Integers: whole numbers and their negatives: …, −2, −1, 0, 1, 2, …
  • Rational numbers: anything you can write as a ratio of two integers — fractions, terminating decimals (0.25 = 1/4), and repeating decimals (0.333… = 1/3). Integers count (5 = 5/1).
  • Irrational numbers: real numbers that cannot be written as a ratio — their decimals go forever without repeating: √2, π, √5.
  • Real numbers: the rationals and irrationals together — the whole number line.

Memory hook (put it on a slide):

Each club is inside the next: ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ, and the irrationals fill the gaps the fractions miss.

One fully worked example (classify each number, every step out loud).

Classify: 7, −3, 0, 5/8, √16, √2, 0.45, π.
- 7 → natural, whole, integer, rational (a counting number).
- −3 → integer, rational (negative, so not whole/natural).
- 0 → whole, integer, rational (but not natural).
- 5/8 → rational (a ratio of integers).
- √16 = 4 → natural, whole, integer, rational. (The trap: √16 simplifies to a whole number.)
- √2 ≈ 1.414… → irrational (never repeats, never ends).
- 0.45 = 45/100 → rational (a terminating decimal is a fraction).
- π ≈ 3.14159… → irrational.

Land the key idea: the only way to be irrational is to be a non-terminating, non-repeating decimal. A square root is irrational only when it doesn't simplify to a whole number — √16 is rational, √2 is not.

Segment 3 — The Order of Operations (25 min)

Plain language first. When an expression has several operations, we all have to agree on the order — otherwise "6 + 2 × 3" could be 24 or 12 depending on who's reading. The agreement is PEMDAS, but read it correctly:

  • P — Parentheses (and other grouping: brackets, fraction bars, the inside of a radical) first.
  • E — Exponents next.
  • MD — Multiply and Divide, left to right (they're a tie — whichever comes first).
  • AS — Add and Subtract, left to right (also a tie).

Memory hook: "PEMDAS is two pairs, not six steps: MD are equal partners read left-to-right, and so are AS."

One fully worked example (do every step):

Evaluate 6 + 2 × 3².
1. Exponent first: 3² = 9 → 6 + 2 × 9
2. Multiply: 2 × 9 = 18 → 6 + 18
3. Add: 6 + 18 = 24
Common wrong answer: adding 6 + 2 first to get 8 × 9 = 72. Multiplication outranks addition — 24 is correct.

The signature trap (give it real time):

What is −4²? And what is (−4)²?
- −4² means −(4²) = −(16) = −16. The exponent attaches to the 4 only; the negative sign is applied after.
- (−4)² means (−4)(−4) = +16. The parentheses put the negative inside the squaring.
The rule: an exponent binds tighter than a negative sign — the negative is only squared if it's in parentheses. Say it twice; it returns on the quiz, the assignment, and the midterm.

One more worked example (grouping + a fraction bar):

Evaluate (8 − 2·3)² ÷ (1 + 1).
1. Inside the first parentheses, multiply before subtract: 2·3 = 6 → (8 − 6)² = (2)²
2. Exponent: (2)² = 4
3. The other group: (1 + 1) = 2
4. Divide: 4 ÷ 2 = 2

Segment 4 — Properties of Real Numbers + Misconceptions (22 min) · Session 1 closes (~73)

Plain language first — the rules that let us rewrite without changing value.
- Commutative (order): a + b = b + a and ab = ba. You can reorder.
- Associative (grouping): (a + b) + c = a + (b + c) and (ab)c = a(bc). You can regroup.
- Distributive (the workhorse): a(b + c) = ab + ac. Multiplication spreads over a sum — this is the one we use constantly.
- Identity: a + 0 = a (additive), a · 1 = a (multiplicative). The value that leaves a number unchanged.
- Inverse: a + (−a) = 0 (additive), a · (1/a) = 1 for a ≠ 0 (multiplicative). The value that "undoes" a number.

Memory hook: Commutative moves the order; associative moves the parentheses; distributive breaks the parentheses open.

Name the misconceptions out loud, then cure each:
- ❌ "Subtraction is commutative — 7 − 3 is the same as 3 − 7."
✅ Cure: 7 − 3 = 4 but 3 − 7 = −4. Only addition and multiplication are commutative. (Rewrite subtraction as adding a negative if you want to reorder: 7 + (−3).)
- ❌ "a(b + c) = ab + c." (distributing to only the first term)
✅ Cure: distribute to every term inside: a(b + c) = ab + ac. Picture the a touching both.
- ❌ "−(x − 5) = −x − 5."
✅ Cure: the negative distributes to both terms: −(x − 5) = −x + 5. A negative sign in front is "times −1."

Interaction — Think-Pair-Share (rapid-fire, ~8 min):
Put 5 statements on a slide; students decide true property or imposter solo (30 sec), compare with a neighbor (1 min), then class votes thumbs up/down. Suggested items: 5 + 3 = 3 + 5 (commutative ✓) · 10 − 4 = 4 − 10 (✗) · (2·3)·4 = 2·(3·4) (associative ✓) · 3(x + 2) = 3x + 2 (✗, missing ·2) · x + 0 = x (identity ✓).
Debrief the two imposters — they're exactly the errors that cost points all term.

Segment 5 — Integer Exponents: The Rules (25 min) · Session 2 opens

Hook back in: "Last session: the properties let us rewrite sums and products. Today: the shortcuts for repeated multiplication — exponents — which are really just the properties applied over and over."

Plain language first. An exponent counts repeated factors: x³ = x·x·x. The base is x; the exponent is 3. Every rule below comes from just counting factors.

  • Product rule: xᵃ · xᵇ = x⁽ᵃ⁺ᵇ⁾. Same base, multiplying → add exponents. (x²·x³ = x·x · x·x·x = x⁵.)
  • Quotient rule: xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾ (x ≠ 0). Same base, dividing → subtract exponents.
  • Power rule: (xᵃ)ᵇ = x⁽ᵃᵇ⁾. A power of a power → multiply exponents.
  • Power of a product: (xy)ᵃ = xᵃyᵃ. The exponent reaches every factor.
  • Power of a quotient: (x/y)ᵃ = xᵃ/yᵃ (y ≠ 0).
  • Zero exponent: x⁰ = 1 for x ≠ 0. (Because xᵃ/xᵃ = x⁰ and anything over itself is 1.)
  • Negative exponent: x⁻ⁿ = 1/xⁿ (x ≠ 0). A negative exponent means reciprocal, not "negative number."

Memory hook: Multiplying adds, dividing subtracts, a power-of-a-power multiplies. Negative exponent = flip it.

One fully worked example (every step):

Simplify (2x³)⁴.
1. Power of a product — the exponent 4 reaches both factors: (2)⁴ · (x³)⁴
2. Evaluate the number: 2⁴ = 16
3. Power rule on the variable: (x³)⁴ = x⁽³·⁴⁾ = x¹²
4. Combine: 16x¹²
Two classic slips: writing 8x¹² (using 2·4 instead of 2⁴) or 16x⁷ (adding 3 + 4 instead of multiplying).

Segment 6 — Applying the Exponent Rules + the Famous Trap (20 min)

Worked example (combine several rules, positive exponents only):

Simplify (12x⁵) / (3x⁸).
1. Numbers: 12/3 = 4
2. Quotient rule on x: x⁵⁻⁸ = x⁻³
3. Make the exponent positive: x⁻³ = 1/x³
4. Result: 4/x³

Worked example (negative exponent on a coefficient — watch what it touches):

Simplify 3x⁻² and (3x)⁻².
- 3x⁻²: the −2 is only on the x → 3 · (1/x²) = 3/x². The 3 stays put.
- (3x)⁻²: the −2 is on the whole product → 1/(3x)² = 1/(9x²) = 1/(9x²).
The lesson: an exponent attaches only to what it touches — to spread it over a product, you need parentheses.

The famous trap (name it):

❌ "x² · x³ = x⁶." — multiplying the exponents when you should add.
✅ Count factors: x² · x³ = (x·x)(x·x·x) = x⁵. Multiplying bases adds exponents; a power of a power multiplies them. The two rules look alike and get swapped — keep them straight: "x²·x³ adds to x⁵; (x²)³ multiplies to x⁶."

Callback: this is the cure to the "add vs. multiply" confusion — point back to the product rule from Segment 5 explicitly.

Segment 7 — Simplifying Algebraic Expressions (20 min)

Plain language first. To simplify is to write the same expression with as few terms as possible. Two tools do almost all the work:
- Distribute to clear parentheses (the distributive property from Segment 4).
- Combine like terms — terms with the identical variable part (3x and 5x are like; 3x and 3x² are not). Add their coefficients.

One fully worked example (every step):

Simplify 5x − 2(3x − 4).
1. Distribute the −2 to both terms: −2 · 3x = −6x and −2 · (−4) = +8
→ 5x − 6x + 8
2. Combine like terms: 5x − 6x = −x
3. Result: −x + 8
The #1 error: writing −2(3x − 4) = −6x − 8 (forgetting that a negative times a negative is positive). Negative-times-negative = positive — every time.

One more (two distributions):

Simplify 4(2x − 3) − (x + 5).
1. Distribute: 4·2x = 8x, 4·(−3) = −12; and −(x + 5) = −x − 5
→ 8x − 12 − x − 5
2. Combine like terms: (8x − x) + (−12 − 5) = 7x − 17
3. Result: 7x − 17

Misconception + cure:
- ❌ "3x + 2x = 5x², so 3x + 2x must be 5x²."
✅ Cure: adding like terms adds the coefficients, not the variables — 3x + 2x = 5x. You only get x² when you multiply (3x · 2x = 6x²). Adding keeps the exponent; multiplying grows it.

Segment 8 — Technology Workflow + AI-Critique, Callback & Hand-off (12 min) · Session 2 closes (~77)

Technology workflow — check a simplification in Desmos (exact steps):
1. Open desmos.com/calculator (free, no login).
2. Type the original expression on line 1: 5x - 2(3x - 4).
3. Type your simplified expression on line 2: -x + 8.
4. Both graph as the same line → your simplification is correct. (If the lines differ, you made an error — Desmos won't tell you where, but it tells you that.)
- Same trick checks an exponent simplification: graph (2x^3)^4 and 16x^12 — identical curves confirm it.

AI-critique moment (students verify, not consume):

Paste this to an approved chatbot: "Simplify −3² + (−2)³ and explain each step."
Then check its work by hand. Chatbots frequently mishandle −3² (treating it as (−3)² = 9 instead of −9) and sign errors on (−2)³ = −8. The correct value is −9 + (−8) = −17. Your job all semester: the tool drafts, you judge. This is exactly how the weekly Lecture Tutorial works — you'll catch the model, not trust it.

Callback + tease:
- Callback: "Everything this term — solving, graphing, every function — rides on this week: the order of operations, the properties, and the exponent rules."
- Tease next week: "Now that we can simplify an expression, Week 2 is the first thing we do with one: set it equal to something and solve — linear equations, inequalities, and the absolute-value twist."

Hand-off (the week's graded work):
- Lecture Tutorial 1 (AI tutor, share-link submission) — real-number sets, order of operations, properties, exponent rules.
- Quiz 1 (end of week, no AI) and Discussion 1 ("Find the Flaw" — fix a botched simplification).
- Assignment 1 ("The Rules That Never Change") — AI-coached, self-scored.

Instructor FAQ — Common Stumbles

Student says / does Quick cure
"Is √16 irrational? It has a radical." Simplify first — √16 = 4, a whole number, so it's rational. A root is irrational only when it doesn't simplify to a fraction/whole number (√2 does not).
Writes −4² = 16. The exponent binds tighter than the negative: −4² = −(4²) = −16. Only (−4)² = 16. Parentheses decide.
Writes x² · x³ = x⁶. Multiplying the bases adds the exponents → x⁵. You multiply exponents only for a power of a power, (x²)³ = x⁶.
Writes x⁰ = 0. Any nonzero base to the 0 power is 1 (because xᵃ/xᵃ = x⁰ = 1).
Writes 3x⁻² = 1/(3x²). The −2 touches only the x. 3x⁻² = 3 · (1/x²) = 3/x². To pull the 3 down you'd need (3x)⁻².
Writes −(x − 5) = −x − 5. The negative distributes to both terms: −(x − 5) = −x + 5. Negative times negative is positive.
"3x + 2x = 5x²." Adding like terms adds coefficients, not exponents → 5x. You get x² only by multiplying.
Thinks subtraction is commutative. 7 − 3 ≠ 3 − 7. Only addition and multiplication commute; rewrite a − b as a + (−b) if you must reorder.

Scope flag

This outline stays within Objective 1. The full set-inclusion chain (ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ) and the (3x)⁻² vs 3x⁻² contrast are added depth (not strictly required by the objective) — kept because they prevent the term's most common errors; trim them for a leaner 60-minute version.

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