How to run this lesson. Send the pre-work a day ahead. Open with goals (5 min), then work quadratics (16), functions (12), and exponentials (15), the Vocabulary Lab (7), and a wrap that assigns the next-two-days work. Math cards open a “Solve it another way” panel. The dark navy notes are for the instructor only.
The frame for today
One line to open: “Factoring is just distributing in reverse — you already know the hard part.” Today connects last week’s linear fluency to curves. Keep the “two routes” habit: factor or test the answers; substitute or reason.
00 Before you begin
Pre-work handout
Five minutes of warm-up makes the lesson land faster.
Advanced Math — Pre-work
Complete on paper before our session · about 10 minutes
Warm-up
1. Multiply out: (x + 2)(x + 3) = ______
2. Solve: x² = 36, so x = ______
3. If f(x) = x + 5, what is f(4)? ______
4. What is 2³? ______
Answer key (instructor)
1 · x² + 5x + 6 2 · x = ±6 3 · 9 4 · 8
02 Quadratics
Factor, then solve
A quadratic equals zero when one of its factors equals zero. Factor first; if it won’t factor cleanly, the quadratic formula always works.
Three ways to crack a quadratic
Factor
Two numbers that multiply to c and add to b. Then each factor = 0.
Test the answers
Plug each answer choice in until the equation gives 0. Fast when choices are given.
Quadratic formula
Always works when factoring won’t come quickly — slower, but bulletproof.
Factoring1
Solve: x² − 5x + 6 = 0
Why A
Find two numbers that multiply to +6 and add to −5 → −2 and −3. So (x − 2)(x − 3) = 0 → x = 2 or 3.Solve it another way
Test the answers: x = 2 gives 4 − 10 + 6 = 0 ✓, x = 3 gives 9 − 15 + 6 = 0 ✓.
True shortcut Product positive, sum negative → both numbers are negative.
Difference of squares2
Solve: x² − 9 = 0
Why A
x² = 9, so x = +3 or −3. (It also factors as (x − 3)(x + 3) = 0.)Solve it another way
True shortcut a² − b² always factors to (a − b)(a + b). Spot it and skip the work.
03 Functions
f(x) just means “plug in”
Function notation looks scary and isn’t. f(3) means “put 3 wherever x appears, then compute.”
Evaluate3
If f(x) = 2x² − 3, what is f(3)?
Why A
Put 3 in for x: 2(3)² − 3 = 2(9) − 3 = 18 − 3 = 15.Solve for the input4
If f(x) = x² + 1, for what value(s) of x does f(x) = 10?
Why A
Set it equal: x² + 1 = 10 → x² = 9 → x = ±3. Don’t forget the negative root.04 Exponentials
Repeated multiplying
Linear change adds the same amount each step; exponential change multiplies by the same factor. Growth multiplies by more than 1; decay multiplies by less than 1.
Growth5
A population starts at 200 and doubles every year. Which expression gives the population after t years?
Why B
Doubling means multiply by 2 each year → 200 · 2ᵗ. Adding (200 + 2t) would be linear, not doubling.Solve it another way
True shortcut “Doubles / triples / halves each period” → start · (factor)ᵗ.
Decay6
A car worth $20,000 loses 10% of its value each year. Which expression gives its value after t years?
Why A
Losing 10% means it keeps 90% → multiply by 0.90 each year → 20000(0.90)ᵗ.Solve it another way
True shortcut Down r% → multiply by (1 − r); up r% → multiply by (1 + r).
Build the two-route habit
Have the student name their method first, then show one alternative. By August they should factor and be able to test answers, so no quadratic stalls them. Reinforce the linear-vs-exponential distinction with the words “add vs multiply.”
05 Vocabulary Lab
This week’s words — think in synonyms and antonyms
Tap a card to flip it. Learn each word next to its opposite — that’s how the test frames them.
Quick check
Synonym7
Which word is closest in meaning to feasible?
Why B
Feasible = able to be done; workable matches. “Impossible” is the opposite.Antonym8
Which word is most nearly opposite to systematic?
Why C
Systematic = orderly and methodical; its opposite is haphazard.Word in context9
Without ______ evidence from controlled experiments, the theory remained speculation.
Why A
Empirical = based on observation or experiment — exactly what experiments provide.06 Wrap-up
Three things to carry out the door
Say them out loud — that’s how they stick.
Factor first
Two numbers that multiply to c and add to b — then each factor equals zero.
f(x) = plug in
Substitute the input wherever x appears, then compute.
Multiply, don’t add
Exponential change multiplies by a fixed factor each step.
Close the loop
Have the student set one target for the mid-week mini-lesson. Confirm the two lesson days, point them to the workbook, and preview that Lesson 4 is Geometry & Trigonometry.
07 Keep the fire lit
Next-two-days handout
Short, daily, cumulative.
Advanced Math — Work for the Next Two Days
About 30 minutes a day · name your method on every problem
Day 1 · Quadratics & functions
Factor and evaluate
1. Solve x² − 7x + 12 = 0. x = ______
2. Solve x² − 16 = 0. x = ______
3. If f(x) = 3x² − 2, find f(2). ______
4. If g(x) = x² − 5, for what x is g(x) = 20? ______
5. Factor x² + 6x + 8. ______
Day 2 · Exponentials & timed
Multiply through
1. A colony of 50 bacteria triples each hour. Count after t hours? ______
2. $5,000 grows 8% per year. Value after t years? ______
3. Solve x² − 2x − 8 = 0. x = ______
4. If f(x) = 2ˣ, find f(4). ______
Answer key (instructor)
Day 1: 1 · x = 3, 4 2 · x = ±4 3 · 10 4 · x = ±5 5 · (x + 2)(x + 4)
Day 2: 1 · 50 · 3ᵗ 2 · 5000(1.08)ᵗ 3 · x = 4, −2 4 · 16
Day 2: 1 · 50 · 3ᵗ 2 · 5000(1.08)ᵗ 3 · x = 4, −2 4 · 16
This sets up Lesson 4. Geometry leans on solving equations — the algebra you just sharpened is the engine behind every area, angle, and triangle problem.