Linear Relations
Warm-up
Project the pizza-night deal: "Delivery costs \$5 flat, plus \$9 per pizza." Ask three quick questions: cost of 2 pizzas? (23) Of 5? (50) Of ZERO pizzas — just the delivery guy showing up? (\$5 — laughter, but it's the y-intercept being born.)
Build the table together and notice: every extra pizza adds the SAME \$9. Constant change is the signature of LINEAR relations — today they get tables, graphs, and equations, all saying the same thing.
Explore
Representation-quartet lab: each pair takes a scenario (pizza deal; savings starting at \$12 growing \$6/week; a candle 20 cm tall burning 2 cm/h) and produces all four representations — story, table, graph, equation — on one placemat.
Cross-checks with structure: the constant difference in the table IS the coefficient in the equation IS the steepness of the graph; the start value IS the zero-column entry IS where the line meets the vertical axis. The candle adds the decreasing case: negative rate, downhill line, and a meaningful end (height 0 at 10 h — where the model stops).
Formalize
Formalize the linear form and the two parameters' jobs:
Reading each representation for and : table — the per-step difference and the row; graph — steepness (rise per run) and the vertical-axis crossing; story — the "per" quantity and the flat/starting amount. Fluency is hopping between the four without recomputing from scratch.
Practice
Practice: complete a placemat for a new scenario; extract equations from two tables and one graph; two prediction questions ("when does the candle die?"); one comparison — two phone plans' lines crossing, with the crossover interpreted.
Exit ticket: a table shows . Equation? ( — and say which number was read from WHICH feature.)
Exit ticket
Practice: complete a placemat for a new scenario; extract equations from two tables and one graph; two prediction questions ("when does the candle die?"); one comparison — two phone plans' lines crossing, with the crossover interpreted.
Exit ticket: a table shows . Equation? ( — and say which number was read from WHICH feature.)
The table: week 0 → \$12, week 1 → \$18, week 2 → \$24, week 3 → \$30.
Step 1: Rate: each week adds 6 → dollars/week.
Step 2: Start: week 0 holds 12 → . Equation: .
Step 3: Predict week 10: dollars.
Step 4: Reverse-predict: when does the balance hit $90? → → weeks.
Step 5: Say the two directions' names: evaluating (plug in ) answers "how much by then?"; solving (find ) answers "when do we get there?" One line, both services.
The graph: a diver's depth over time — a straight line from rising to .
Step 1: Intercept: at , depth m — the dive starts 30 m down. (.)
Step 2: Rate from rise-over-run: climbs 30 m in 6 min → m/min.
Step 3: Equation: .
Step 4: Interrogate it: depth at ? m. When does the diver pass m? → min.
Step 5: Model boundaries, stated like a pro: past the equation predicts positive depth — a flying diver. The LINE continues; the MODEL stops at the surface. Every linear model wears a domain, and graduates of this unit say so unprompted.
The story: a print shop charges $3 setup plus $0.75 per poster. The table: . A student asks whether they describe the same deal.
Step 1: Extract the story's equation directly: .
Step 2: Extract the table's equation: differences — every posters adds dollars → rate ✓. Back up to : from , remove one step: → ✓.
Step 3: Verdict: same line, — the table just sampled it at multiples of 4.
Step 4: The subtle skill exposed: the table hid (no zero row) and hid (steps of 4, not 1) — and both were recoverable by structure. Representations disguise the parameters; they can't destroy them.